Identification in Stochastic Choice

Iden tiﬁcation in Sto c hastic Choice ∗ P eter Caradonna † and Christopher T uransic k ‡ F ebruary 24, 2026 Abstract W e c haracterize the identiﬁed s ets of a wide range of sto c hastic c hoice mo dels, including random utilit y , v arious models of b oundedly-rational b eha vior, and dynamic discrete c hoice. In each of these settings, w e sho w tw o distributions ov er choice rules are observ ationally equiv alen t if and only if they can b e obtained from one another via a ﬁnite sequence of simple sw apping transforms. W e leverage this to obtain complete descriptions of b oth the deﬁning inequalities and extreme points of these iden tiﬁed sets. In cases where choice frequencies v ary smo othly with some parameters, w e provide a no vel global-inv erse result for practically testing identiﬁcation. 1 In tro duction A fundamen tal question for an y mo del of b eha vior is to what degree its underlying primi- tiv es can b e uniquely determined from individual or aggregate-lev el data. Ev en when exact iden tiﬁcation fails, knowledge of the iden tiﬁed set can provide crucial information for the design of p olicy , counterfactual sim ulations, and w elfare analysis. 1 ∗ W e are grateful to Ro y Allen, Jose Apesteguia, Pierpaolo Battigalli, Andr´ es Carv a jal, Simone Cerreia- Vioglio, Christopher Cham b ers, Emel Filiz-Ozba y , Satoshi F ukuda, Christopher Kops, SangMok Lee, Ja y Lu, Y usufcan Masatlioglu, P aulo Natenzon, Axel Niemey er, Luciano Pomatto, Justus Preusser, Collin Raymond, Kota Saito, F edor Sandomirskiy , Bumin Y enmez, Kemal Yildiz, and Xinhan Zhang as well as seminar participan ts at Bilken t, BRIC 11, BSE Summer F orum W orkshop on Choice and Decision, EC’25, Italian Junior W orkshop on Economic Theory 2025, Liverpo ol, R UD 2025, UC Davis, and WUSTL for discussions and helpful commen ts. This pap er subsumes “Iden tifying Restrictions on the Random Utilit y Mo del”. † Division of the Humanities and So cial Sciences, Caltech. Email: ppc@caltech.edu . ‡ Departmen t of Decision Sciences and IGIER, Bo cconi Univ ersit y . Email: christopher.turansick@ unibocconi.it . 1 E.g. Manski ( 2007 ); Agarwal and Somaini ( 2018 ); Kalouptsidi et al. ( 2021 ). See also Marschak ( 1953 ) and Allen and Rehbeck ( 2019 ). 1 Despite their practical imp ortance, the identiﬁcation prop erties of many w orkhorse mo d- els of sto c hastic choice, including the random utility mo del, ha ve remained p o orly under- sto o d. 2 In resp onse, an extensiv e literature has emerged, studying additional restrictions under whic h analysts can uniquely reco ver an underlying distribution of parameters go vern- ing behavior from choice share data. 3 In this pap er, we pro vide complete characterizations of the identiﬁed sets of a wide range of sto c hastic choice theories, under arbitrary supplemen tal restrictions. Our primary obser- v ation is that, in eac h of these settings, eac h set of empirically indistinguishable distributions (e.g. ov er preferences) can b e straigh tforw ardly described in terms of a simple family of trans- formations. W e term these Ryser swaps , in ligh t of a connection to the discrete tomograph y literature. 4 In particular, we show t wo distributions generate identical choice shares ov er an y menu if, and only if, they are related by a ﬁnite sequence of such sw aps. F or exp ositional clarit y , w e fo cus our discussion on the random utility mo del. Ho wev er our ﬁndings apply straigh tforw ardly to many other settings, including to mo dels of incomplete or b oundedly-rational b ehavior, correlated decision-making, and dynamic sto c hastic choice. 5 Example 1. A p opulation of voters are faced with four p otential urban planning p olicies. P olicy a calls for regional high-sp eed rail expansion, and b for large-scale inv estmen t in lo cal bus and sub wa y systems, whereas c for mo derate congestion-pricing, and d an expansion of bik e lanes and p edestrian zones. The p opulation is comprised of four types of v oters. The ﬁrst are ‘big picture’ activists, who prefer large-scale in v estment in public transit and a cen tralized policies ( a ≻ b ≻ c ≻ d ), whereas incremen talists prefer more funding to less, but prize more immediate, practical transit options, and a ground-up approach to p olicy ( b ≻ a ≻ d ≻ c ). Similarly , suburban comm uters prefer rail to buses and dislike taxes on their vehicles ( a ≻ b ≻ d ≻ c ), whereas urban comm uters primarily use public transit and prioritize less congested streets ( b ≻ a ≻ c ≻ d ). W e summarize these four preferences b elow. 6 Supp ose stra w-p oll data o v er v arious subsets of these p olicies is consistent with the so ci- etal distribution of preferences b eing µ =  1 4 , 1 4 , 3 8 , 1 8  , and consider now a new p olicy , lik ely 2 See, e.g., Fishburn ( 1998 ); McClellon ( 2015 ); Strzalecki ( 2025 ). 3 E.g. Luce ( 1959 ); McF adden ( 1972 ); Gul and Pesendorfer ( 2006 ); Ap esteguia et al. ( 2017 ); Manzini and Mariotti ( 2018 ); Filiz-Ozbay and Masatlioglu ( 2023 ); Y ang and Kop ylo v ( 2023 ); Suleymanov ( 2024 ). Similar questions arise in the context of demand inv ertibility , see Berry and Haile ( 2024 ). 4 E.g. Ryser ( 1957 ); Fishburn et al. ( 1991 ); Kong and Herman ( 1999 ). These to ols hav e found recen t application in economic theory , see He et al. ( F orthcoming ). 5 W e discuss these extensions in Section 7 . See also Online App endix B for further details. 6 The observ ation that a distribution with full support on these four preferences is unidentiﬁed from c hoice frequency data is due to Fishburn ( 1998 ). Moreo ver, such examples are minimal: the random utility mo del is known to b e identiﬁed whenev er there are three or few er alternatives, e.g. Blo ck and Marschak ( 1959 ). 2 ≻ 1 ≻ 2 ≻ 3 ≻ 4 a b a b b a b a c d d c d c c d to b eneﬁt suburban comm uters. 7 What can b e said ab out the size of this share of the p opulation? Our results sho w suburban commuters constitute anywhere b etw een 1 4 and 5 8 of the p op- ulation. Moreov er, these b ounds are tigh t: for any fraction 1 4 ≤ α ≤ 5 8 , there is a distribution o ver these four preferences where the p opulation share of suburban commuters is precisely α , and whic h yields iden tical vote shares to µ ov er any men u of p olicies. The k ey feature in this example is that all four preferences rank a and b ab o v e c and d , but disagree ab out the relativ e rankings of the alternatives within these groupings. In particular, ≻ 3 and ≻ 4 can b e obtained by taking ≻ 1 and ≻ 2 , and exchanging their respective rankings betw een c and d , while preserving all other comparisons. Giv en any distribution ov er these four preferences with full supp ort, b y subtracting some small, but equal amount of probability mass from either ≻ 1 and ≻ 2 , or ≻ 3 and ≻ 4 , and re- assigning it equally to the opp osing pair (here, obtained by exchanging only the rankings of c and d ) one obtains a mo diﬁed distribution with iden tical choice probabilities on all menus. In tuitively , such transformations aﬀect only the c orr elation b etw een preferring a to b and c to d , but not the lik eliho o d of these even ts. In this setting, it turns out these are the only transformations which preserve all choice probabilities. As such, the minimal and maximal p ossible p opulation frequencies for ≻ 3 can b e obtained by simply maximizing the amount of mass uniformly transferred from the pair {≻ 3 , ≻ 4 } to {≻ 1 , ≻ 2 } or vice-v ersa, yielding the extremal, empirically indistinguishable distributions µ =  3 8 , 3 8 , 1 4 , 0  and ¯ µ =  0 , 0 , 5 8 , 3 8  . ■ As illustrated by Example 1 , failures of iden tiﬁcation arise for a straightforw ard reason: c hoice probabilities do not, in general, uniquely determine the c orr elation b et ween c hoices. 8 Indeed, the identiﬁed sets of the random utilit y mo del are spanned precisely by the p ossible correlation patterns that cannot b e ruled out by an empiricist with access only to aggregate, rather than individual, level data. 7 That is, voters who hold ≻ 3 . 8 F or instance, in Example 1 , the correlation b et ween choices on menus { a, b } and { c, d } . 3 Our ﬁrst main result characterizes the full collection of preference distributions consistent with a giv en dataset. W e sho w any transformation of a distribution whic h preserv es all c hoice probabilities is equiv alen t to applying a ﬁnite sequence of simple swaps of the form considered in Example 1 . Using this represen tation, we pro vide a straigh tforw ard, geometric description of the random utilit y mo del’s identiﬁed sets, b oth in terms of their deﬁning inequalities, and their extreme p oints. As consumer welfare measures t ypically rely on preference information ( Deaton 1980 ; McKenzie 1983 ), this can b e used, e.g., to obtain tigh t b ounds on the estimated aggregate or distributional eﬀects of p olicy in terven tions. 9 W e then inv estigate ho w our results sp ecialize under common, structured classes of sup- plemen tal restrictions. F or example, in man y instances an analyst may wish to a priori restrict the set of preferences assumed held by a p opulation, but not their relative frequen- cies. 10 W e characterize identiﬁed sets sub ject to arbitrary supp ort restrictions, as w ell as those families of preferences S with the prop ert y that any data is consisten t with at most a single measure whose supp ort b elongs to S . In practice, S often consists of restrictions of some parametric family of utilities. 11 In this case, our results pro vide an exact test for prac- titioners to determine when choice probabilities uniquely identify a p opulation distribution o ver parameters. W e also consider cases in which choice probabilities are sp eciﬁed directly , as smooth functions of some vector of parameters. Using top ological techniques, w e pro vide a pair of necessary and suﬃcien t conditions for the global inv ertibility of general, non-linear systems. In con trast to many existing results, we require no monotonicity prop erties. 12 Th us, for example, applied to the problem of determining the inv ertibilit y of a smo oth demand, our results are applicable ev en when go o ds ma y fail to be substitutes, and in settings where the la w of demand ma y fail. W e demonstrate the practical applicability of our conditions by pro ving a no vel result, that the dynamic, logit habit formation mo del of T uransic k ( 2025 ) is not iden tiﬁed from unconditional choice data. 9 See, e.g., Deb et al. ( 2023 ). T ebaldi et al. ( 2023 ) and Kamat and Norris ( 2025 ) consider similar questions in the con text of attribute v ariation. Our results ma y be seen as providing complemen tary to ols for settings with menu v ariation, e.g. McF adden ( 2005 ); Kitam ura and Sto ye ( 2018 ), as well as to dynamic, or imperfectly rational, mo dels of b ehavior; see Section 7 . 10 F or example, exp ected utility preferences on domains of lotteries ( Gul and Pesendorfer 2006 , Ap esteguia and Ballester 2018 ), exp onentially discoun ted or quasi-hyperb olic preferences ov er streams ( Lu and Saito 2018 ), sub jective exp ected utilit y ov er acts ( Lu 2016 ; Dura j 2018 ), or single-p eaked preferences on ordered domains ( Ap esteguia et al. 2017 ). Such restrictions are also implicit in more general, dynamic settings, e.g. Rust ( 1987 ); Lu et al. ( 2024 ); see also F rick et al. ( 2019 ). 11 E.g. Ap esteguia and Ballester ( 2018 ) consider the case of random CARA or CRRA exp ected utility . 12 C.f. Gale and Nik aido ( 1965 ); Bec k ert and Blundell ( 2004 ); Simsek et al. ( 2005 ); Berry et al. ( 2013 ); Lindenlaub ( 2017 ); W ang ( 2021 ); Allen ( 2022 ). See also ‘dominant diagonal’ conditions, e.g. McKenzie ( 1960 ); Rosen ( 1965 ). 4 Finally , when the set of alternatives is ordered, an elegant approach due to Ap esteguia et al. ( 2017 ) sho ws that requiring single-crossing structure on the supp ort of a rational- ization yields iden tiﬁcation. 13 Ho wev er, such rationalizations need not alw ays exist, even for data compatible with the random utility mo del. Using our notion of a Ryser swap, w e pro vide a principled generalization we term swap pr o gr essivity . Whenev er the data admit a single-crossing represen tation, this will alw ays b e the unique swap-progressiv e rational- ization. Ho wev er, we sho w a unique sw ap-progressive represen tation exists for any data compatible with the random utility mo del. This pro vides a no vel, practical identiﬁcation strategy that incorp orates the natural ordering o ver alternativ es. The pap er pro ceeds as follows. In Section 2 , we formally introduce the random utility mo del. In Section 3 , w e formalize our core notion of a Ryser sw ap, and present our main geometric characterization. Section 4 pro vides a dual c haracterization of iden tiﬁed sets in terms of their extreme p oints, and characterizes identifying support restrictions. In Section 5 , w e introduce our notion of sw ap progressivity and establish its prop erties, and Section 6 considers the identiﬁcation of general, parametric sto chastic c hoice mo dels. Finally , Section 7 discusses extensions to other environmen ts. T o more concretely relate our contributions to existing work, we defer our discussion of related literature to Section 8 . Section 9 concludes. 2 The Random Utilit y Mo del Let X denote a ﬁxed, ﬁnite set of alternatives ov er which an individual chooses. A pref- erence is a linear order on X ; we denote the set of all preferences by L . 14 T o conserve on notation, w e will often write preferences as strings of alternatives, arranged in descend- ing order, i.e. ‘ abcd ’ denotes the ranking a ≻ b ≻ c ≻ d . F or any preference ≻ and any 0 ≤ k ≤ | X | , the k - initial and k - terminal segmen ts of ≻ are the ordered strings consisting of the k most-preferred alternativ es and the | X | − k least-preferred. 15 W e will denote these b y s ↑ k ( ≻ ) and s k ↓ ( ≻ ) resp ectiv ely . F or example, if ≻ corresp onds to the ordering abcde , then: s ↑ 2 ( ≻ ) = ab and s 2 ↓ ( ≻ ) = cde. A function ρ : X × 2 X \ { ∅ } → [0 , 1] deﬁnes a random choice rule if, for all non-empt y subsets A ⊆ X , X x ∈ A ρ ( x, A ) = 1 . 13 See also Filiz-Ozbay and Masatlioglu ( 2023 ); Ap esteguia and Ballester ( 2023 ). 14 A linear order is a complete, transitive, and antisymmetric binary relation on X . 15 F or k = 0 (resp. k = | X | ) w e deﬁne s ↑ k ( ≻ ) (resp. s k ↓ ( ≻ )) as the empty string. 5 Random c hoice rules are assumed observ able; they constitute the basic data in our identiﬁ- cation problem. F or any ﬁnite set A , w e use ∆( A ) to denote the set of probabilit y measures o ver A . A restriction of the random utility mo del is any subset M ⊆ ∆( L ). W e say a ran- dom c hoice rule is rationalizable sub ject to a restriction M if there exists some probability distribution µ ∈ M suc h that, for all x ∈ A ⊆ X , we ha v e: ρ ( x, A ) = µ {≻ ∈ L | x is maximal in A } = X ≻∈L µ ( ≻ ) 1 { x ≻ A \ x } . Under the interpretation of a restriction M as describing a set of p ossible makeups of a heterogeneous so ciet y , a collection of observed c hoice frequencies ρ are M -rationalizable if and only if they arise as the distribution of constrained-optimal outcomes according to some p opulation comp osition µ ∈ M . W e sa y t w o measures µ, ν ∈ ∆( L ) are observ ationally equiv alent if, for all x ∈ A ⊆ X , µ {≻ ∈ L | x is maximal in A } = ν {≻ ∈ L | x is maximal in A } , (1) i.e. they generate identical choice frequencies on every choice set ∅ ⊊ A ⊆ X . Finally , a restriction M is iden tifying if it contains no pair of distinct, observ ationally equiv alen t measures. 3 The Geometry of Iden tiﬁcation The unrestricted random utility mo del (i.e. M = ∆( L )) has long been kno wn to b e uniden ti- ﬁed. 16 Th us, to each distribution µ ∈ ∆( L ), there corresp onds some generally non-singleton collection of observ ationally equiv alent distributions. Let P denote the set of random c hoice rules on X , and deﬁne the mapping Φ : ∆( L ) → P via: Φ( µ ) ( x,A ) =    µ {≻ ∈ L | x is maximal in A } if x ∈ A 0 if x ∈ A. (2) By ( 1 ) tw o distributions µ, ν ∈ ∆( L ) are observ ationally equiv alent if and only if Φ( µ ) = Φ( ν ). How ever, as Φ is linear in µ , there exists some subspace R ⊆ R L suc h that, whenever Φ( µ ) = ρ , the iden tiﬁed set Φ − 1 ( ρ ) is precisely  µ + R  ∩ ∆( L ). 17 In particular, ev ery iden tiﬁed set is fully characterized by this subspace. 16 See Fishburn ( 1998 ). 17 Here we identify R L with the space of all signed measures ov er linear orders on X . 6 3.1 Compatible Pairs and the Ryser Subspace W e say that a pair of preferences are k - compatible , 0 ≤ k ≤ | X | , if b oth preferences agree on the set of k -b est alternatives. 18 In other words, ≻ 1 and ≻ 2 are k -compatible if s ↑ k ( ≻ 1 ) can b e obtained b y p ermuting the terms of the sequence s ↑ k ( ≻ 2 ), and analogously s k ↓ ( ≻ 1 ) from s k ↓ ( ≻ 2 ). This ensures the concatenated sequences: ≻ ′ 1 = s ↑ k ( ≻ 1 ) s k ↓ ( ≻ 2 ) and ≻ ′ 2 = s ↑ k ( ≻ 2 ) s k ↓ ( ≻ 1 ) (3) deﬁne v alid preferences. W e refer to the pair of preferences obtained in this manner as the k - conjugates of ≻ 1 and ≻ 2 . If ≻ 1 and ≻ 2 disagree on b oth (i) the ranking of the k -b est alternativ es and (ii) the ranking of the ( | X | − k )-w orst, ≻ ′ 1 and ≻ ′ 2 will b e distinct from either ≻ 1 or ≻ 2 ; in this case w e refer to ≻ 1 and ≻ 2 as non-trivially k -compatible. Our interest in compatible pairs of preferences is motiv ated by the following prop osition, whic h says that any uniform distribution supp orted on a compatible pair of preferences is b eha viorally equiv alent to the uniform distribution ov er the pair obtained via ( 3 ). Prop osition 1. Let ≻ 1 , ≻ 2 ∈ L b e k -compatible, with k -conjugates ≻ ′ 1 and ≻ ′ 2 . Then the uniform distributions on {≻ 1 , ≻ 2 } and {≻ ′ 1 , ≻ ′ 2 } are observ ationally equiv alent. As the follo wing example illustrates, the observ ational indeterminacy highligh ted by Prop osition 1 is a consequence of the fact that a join t distribution is not, in general, uniquely determined b y the requiremen t that it p ossess uniform marginals. Example 2. Consider again Example 1 , where X = { a, b, c, d } , and there are four pref- erences: ≻ 1 : abcd , ≻ 2 : badc , ≻ 3 : abdc , and ≻ 4 : bacd . Note that the pair ≻ 1 and ≻ 2 are 2-compatible, with 2-conjugates ≻ 3 and ≻ 4 . 19 W e ma y view these four preferences as el- emen ts of a pro duct set { ab, ba } × { cd, dc } , b y asso ciating an ordered pair of strings with their concatenation, e.g.: ( ab, cd ) 7→ abcd. A join t distribution ov er { ab, ba } × { cd, dc } generates the same c hoice probabilities on every men u as the uniform distribution on {≻ 1 , ≻ 2 } if and only if b oth of its marginals are them- selv es uniform. 20 Th us, in this setting, Prop osition 1 asserts the observ ational equiv alence of the tw o extremal joint measures with uniform marginals: µ 12 , the uniform on {≻ 1 , ≻ 2 } , whic h p erfectly correlates the choice of a from { a, b } with c from { c, d } , and µ 34 , the uniform on {≻ 3 , ≻ 4 } whic h p erfectly an ti-correlates these choices. See Figure 1 . ■ 18 In particular, any pair of preferences is v acuously 0-compatible. 19 That is, ≻ 3 = s ↑ 2 ( ≻ 1 ) s 2 ↓ ( ≻ 2 ), and ≻ 4 = s ↑ 2 ( ≻ 2 ) s 2 ↓ ( ≻ 1 ). 20 This can b e veriﬁed by direct computation or, e.g., follows from Theorem 5 of F almagne ( 1978 ). 7 ba ab cd dc ≻ 1 ≻ 4 ≻ 3 ≻ 2 (a) The preferences of Example 1 , viewed as a pro duct set. The uniforms µ 12 and µ 34 diﬀer only in their correlation structure. ≻ 4 ≻ 3 ≻ 2 ≻ 1 µ 34 µ 12 (b) Prop osition 1 asserts the observ ational equiv alence of the uniform distributions µ 12 and µ 34 . Figure 1: By Prop osition 1 , µ 12 and µ 34 are observ ationally equiv alent. By the linearity of ( 2 ), b oth are also equiv alent to any distribution in their conv ex hull, and any pair of distributions b elonging to some common translation of this set (e.g. the dashed segments) are observ ationally equiv alent. W e say a signed measure R ∈ R L deﬁnes a Ryser swap if: R = 1 {≻ ′ 1 , ≻ ′ 2 } − 1 {≻ 1 , ≻ 2 } , where ≻ 1 and ≻ 2 are k -compatible, and ≻ ′ 1 and ≻ ′ 2 are k -conjugate for some k . 21 A signed measure is a w eigh ted Ryser sw ap if it is proportional to a Ryser sw ap. Adding a weigh ted Ryser swap to some initial distribution µ ma y b e regarded as a comp ound transformation, whic h ﬁrst separates out from µ a product measure, applies to it a marginal-preserving transformation, then recom bines the result with the remaining mass from µ to obtain a mo diﬁed measure µ ′ . As in Example 1 , whenever µ ′ is a probability distribution, it diﬀers from µ only in the c orr elation b etw een c hoices, not their frequencies. Finally , we deﬁne the Ryser subspace R ⊆ R L as the span of the Ryser swaps: R = span  R ∈ R L | R is a Ryser sw ap  . Ev ery signed measure in R corresp onds to a transformation whic h applies a ﬁnite sequence of w eighted sw aps. By linearity of Φ, giv en ¯ R ∈ R and µ ∈ ∆( L ), if µ + ¯ R deﬁnes a probability distribution, Prop osition 1 implies it is observ ationally equiv alen t to µ . 22 Our next theorem 21 Note ≻ 1 and ≻ 2 are trivially k -compatible if and only if R = 0. 22 Prop osition 1 asserts that, given compatible ≻ 1 , ≻ 2 with conjugates ≻ ′ 1 , ≻ ′ 2 , the uniforms µ 12 and µ 1 ′ 2 ′ are observ ationally equiv alent, and hence Φ( µ 12 ) = Φ( µ 1 ′ 2 ′ ). By linearity: Φ  µ + ε ( µ 1 ′ 2 ′ − µ 12 )  = Φ( µ ) + ε (0) = Φ( µ ) . 8 sho ws that, in fact, every pair of observ ationally equiv alent distributions are related in this manner. Theorem 1. Tw o distributions µ, µ ′ ∈ ∆( L ) are observ ationally equiv alent if and only if µ − µ ′ ∈ R . In particular, for an y µ ∈ M ⊆ ∆( L ), the set of observ ationally equiv alent distributions in M is precisely  µ + R  ∩ M . Theorem 1 shows that not only do Ryser swaps preserv e choice probabilities, they in fact generate the set of al l transformations which do so. In other words, tw o distributions ov er preferences are observ ationally equiv alen t if and only if the ﬁrst can b e generated from the second (or vice-versa) by a ﬁnite sequence of suc h swaps. Geometrically , this implies the set of distributions observ ationally equiv alent to some µ ∈ ∆( L ) is simply the intersection of the aﬃne subspace µ + R with ∆( L ). This provides a practical means of computing features of the iden tiﬁed set. Since R do es not dep end on the data, once a c hoice of X is sp eciﬁed, a basis R 1 , . . . , R N for R can b e obtained from the set of Ryser sw aps. Then, given any rationalization µ ∈ ∆( L ), optimal b ounds on, e.g., the sizes of diﬀerent shares of the p opulation across all rationalizations, can b e computed b y ev aluating a straigh tforward linear program ov er the weigh ts α 1 , . . . , α N assigned to these basis elemen ts, sub ject only to the constrain t µ + P i α i R i ≥ 0 . 23 T o illustrate, consider again Example 1 . As there are only four alternativ es, an y non- trivially compatible pair must agree on the b est (and w orst) tw o alternativ es, but not their ordering. As a consequence, if the measure µ + R is a probability distribution for some R ∈ R , R m ust subtract mass only from ≻ 1 , . . . , ≻ 4 , and reassign it to preferences formed by matc hing terminal segments s 2 ↓ ∈ { cd, dc } to initial segments s ↑ 2 ∈ { ab, ba } . But ≻ 1 , . . . , ≻ 4 are the only preferences formed this wa y . It follo ws R must b e prop ortional to the sole (up to sign) non-trivial Ryser sw ap supp orted on these preferences, ¯ R = 1 {≻ 1 , ≻ 2 } − 1 {≻ 3 , ≻ 4 } . In particular, this implies the orange line segmen ts in Figure 1 are precisely the identiﬁed sets. 24 Example 3. In more complex environmen ts, elements of R ma y hav e richer form. Supp ose X = { a, b, c, d, e, f } , and consider the follo wing six preferences. Since every ¯ R ∈ R is a weigh ted sum of Ryser swaps, µ and µ + ¯ R are observ ationally equiv alent. 23 One could ev aluate a similar linear program ov er ∆( L ) using Φ( · ) = ρ as a constrain t, without app ealing to Theorem 1 (e.g. Deb et al. 2023 ). How ever, doing so requires requires | X | ! − 1 v ariables and hence may b ecome computationally diﬃcult in larger en vironments. In contrast, iden tiﬁed sets are generally far low er dimensional; in Example 1 , ∆( L ) is 23-dimensional whereas the identiﬁed set w as 1-dimensional. Thus the explicit description provided by Theorem 1 may allow for signiﬁcant computational savings in practice. 24 Indeed, given some rationalization µ , tight bounds on any prop erty of the identiﬁed set can b e computed simply by optimizing µ + α ¯ R o v er the scalar α , sub ject to the constraint this vector b elong to the simplex. 9 ≻ 1 ≻ 2 ≻ 3 ≻ ′ 1 ≻ ′ 2 ≻ ′ 3 a b c a b c b a d b a d c e b e c b d f a f d a e c f c f e f d e d e f Here, µ 123 , the uniform distribution on {≻ 1 , ≻ 2 , ≻ 3 } , is observ ationally equiv alen t to µ ′ 123 , the uniform on {≻ ′ 1 , ≻ ′ 2 , ≻ ′ 3 } . How ev er, unlike in Example 1 , these distributions do not diﬀer b y an y single weigh ted Ryser sw ap. 25 Nev ertheless, they do so b y a se quenc e of such swaps. ≻ 1 ≻ 2 ≻ 3 a b c b a d c e b d f a e c f f d e − → ≻ ′ 1 ≻ ≻ 3 a b c b a d e c b f d a c e f d f e − → ≻ ′ 1 ≻ ′ 2 ≻ ′ 3 a b c b a d e c b f d a c f e d e f T o construct such a sequence, ﬁrst transfer all mass from ≻ 1 and ≻ 2 under µ 123 to their 2-conjugates (the pair formed b y sw apping their terminal segments cdef and ef cd ). Lab el these resulting preferences ≻ ′ 1 and ≻ . Then, transfer all mass on ≻ and ≻ 3 to their 4- conjugates (the pair obtained from exc hanging ef and f e ) to yield µ ′ 123 . ■ 4 Supp ort Restrictions and Extreme P oin ts P erhaps the simplest class of restrictions are those in whic h an analyst constrains the set of preferences held in a p opulation. W e sa y M ⊆ ∆( L ) deﬁnes a supp ort restriction if: M =  µ ∈ ∆( L ) : supp( µ ) ⊆ S  , for some set of preferences S ⊆ L . Such restrictions are natural when a mo deler wishes to a priori restrict the set of preferences, but not their relativ e frequencies. 25 T o see this, note an y Ryser swap b etw een an y pair of these six preferences is either trivial or places non-zero mass on some preference outside the set. 10 Example 4. Let X consist of the following four monetary lotteries: a = 1 2  $10  + 1 2  $20  , b = 1 2  $5  + 1 2  $25  c = 1 4  $10  + 3 4  $20  , d = 1 4  $5  + 3 4  $25  . Supp ose w e wish to consider only those preferences S ⊆ L whic h are consistent with some (monotone) exp ected utility represen tation. T o determine this subset, note c ≿ F O S D a and d ≿ F O S D b , hence any preference in S ranks c ≻ a and d ≻ b . 26 Moreo ver, c = 1 2  a  + 1 2  $20  and d = 1 2  b  + 1 2  $25  , th us every preference in S which ranks b ≻ a , also ob eys d ≻ c . These are all the restrictions imp osed b y monotonicit y and the exp ected utilit y axioms, hence S consists of the ﬁv e orders: ≻ 1 ≻ 2 ≻ 3 ≻ 4 ≻ 5 d d d c c c c b d a b a c a d a b a b b listed here. ■ While Example 4 highligh ts the case of exp ected utilit y preferences, one could lik ewise consider other restrictions, from parametric sub-families such as CARA or CRRA, to broader classes of risk preferences, suc h as b et weenness or disapp ointmen t av ersion. Similarly , many other environmen ts suggest natural restrictions, such as exponentially discounted preferences o ver consumption streams, or sub jective exp ected utilit y preferences o v er acts. The question w e no w consider is how suc h restrictions aﬀect the set of rationalizing distributions, and when the probability weigh ts on these and other families of orders can b e uniquely reco vered from c hoice frequency data alone. Let ≻ 1 , . . . , ≻ N b e a (ﬁnite) sequence of preferences, allo wing rep etition. F or eac h 1 ≤ k ≤ | X | , let Π k denote the partition of { 1 , . . . , N } suc h that i and j b elong to the same partition elemen t if and only if ≻ i and ≻ j are ( k − 1)-compatible. 27 A rearrangemen t is a collection of p erm utations { σ k } | X | k =1 of { 1 , . . . , N } , such that σ k ﬁxes eac h partition element of Π k . In other words, a rearrangemen t is a choice of p erm utations, where eac h σ k exc hanges 26 Here, ≿ F OS D corresp onds to ﬁrst order sto chastic dominance. 27 In particular, as ev ery pair of preferences are trivially 0-compatible, w e alwa ys ha ve Π 1 =  { 1 , . . . , N }  . 11 the lab els of preferences in ≻ 1 , . . . , ≻ N , but only among those which agree on the set of ( k − 1) b est alternativ es. Let x k n denote the k -th most preferred alternativ e of ≻ n . W e sa y tw o sequences ≻ 1 , . . . , ≻ N and ≻ ′ 1 , . . . , ≻ ′ N are rearrangemen t equiv alent if there exists a rearrangement { σ k } | X | k =1 suc h that, for all 1 ≤ n ≤ N : ≻ ′ n = x 1 σ − 1 1 ( n ) x 2 ( σ 2 ◦ σ 1 ) − 1 ( n ) · · · x | X | ( σ | X | ◦ ··· ◦ σ 1 ) − 1 ( n ) , Rearrangemen t equiv alence describ es when one sequence of preferences can b e transformed in to the other b y iteratively replacing compatible pairs with their conjugates. As the follo w- ing example illustrates, this has a simple visual intuition: tw o sequences are rearrangemen t equiv alent precisely when the strings of one can b e ‘braided together’ to form the other. Example 5. Suppose X = { a, b, c, d, e, f } , and consider again the sequence of preferences ≻ 1 , ≻ 2 , ≻ 3 from Example 3 given b y: ≻ 1 ≻ 2 ≻ 3 a b c b a d c e b d f a e c f f d e Π ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ σ − → ≻ ′ 1 ≻ ′ 2 ≻ ′ 3 a b c b a d e c b f d a c f e d e f W e also plot the partitions Π 1 , . . . , Π 6 of { 1 , 2 , 3 } deﬁned ab ov e, where ∗ denotes any single- ton partition elemen t. The sequence ≻ ′ 1 , ≻ ′ 2 , ≻ ′ 3 is rearrangement equiv alen t to ≻ 1 , ≻ 2 , ≻ 3 , under the collection of p erm utations where σ 1 , σ 2 , σ 4 , and σ 6 are the identit y , and σ 3 (resp. σ 5 ) swaps the tw o entries of the unique non-singleton partition elemen t of Π 3 (resp. Π 5 ). T o see this, note that σ 1 and ( σ 2 ◦ σ 1 ) are the iden tity permutation (1 , 2 , 3), while ( σ 3 ◦ σ 2 ◦ σ 1 ) = ( σ 4 ◦ · · · ◦ σ 1 ) = (2 , 1 , 3). 28 Finally , ( σ 5 ◦ · · · ◦ σ 1 ) = ( σ 6 ◦ · · · ◦ σ 1 ) = (3 , 1 , 2). Thus, for example, the ﬁfth-b est alternativ e under ≻ ′ 2 is the same as for ≻ 3 , as ( σ 5 ◦ · · · ◦ σ 1 ) = (3 , 1 , 2) and therefore ( σ 5 ◦ · · · ◦ σ 1 ) − 1 = (2 , 3 , 1). ■ When M is a supp ort restriction, ( 2 ) implies the set of measures in M observ ationally equiv alent to some µ ∈ ∆( L ) forms a p olytop e. Our next result shows that an observ ationally 28 W e use one-line notation for p erm utations, e.g., (2 , 3 , 1) represents the p ermutation σ (1) = 2, σ (2) = 3, and σ (3) = 1. In particular, this means σ − 1 ( n ) corresp onds to which comp onent of the vector has v alue n . 12 equiv alent µ ′ ∈ M is an extreme point of this set if and only if its supp ort con tains no pair of distinct, rearrangemen t-equiv alen t sequences. 29 In particular, such a pair of sequences exist if and only if there are a pair of distinct rationalizations in M whic h av erage to µ ′ . Theorem 2. Supp ose M is a supp ort restriction. Then µ ∈ M is an extreme p oint of the iden tiﬁed set ( µ + R ) ∩ M if and only if supp( µ ) con tains no pair of distinct, rearrangemen t- equiv alent sequences. Whereas Theorem 1 c haracterizes the identiﬁed sets of the random utility model in terms of Ryser sw aps, in the case of supp ort restrictions, Theorem 2 pro vides a complementary description via their extreme p oin ts. 30 These extremal distributions are of natural in terest, as they constitute the primitiv e, irreducible rationalizations from whic h an y others may b e constructed. In practice, they often corresp ond to the simplest, or most parsimonious, p opulation mo dels whic h explain the data. F or example, single-crossing ( Ap esteguia et al. 2017 ), progressive ( Filiz-Ozbay and Masatlioglu 2023 ), and swap-progressiv e ( Section 5 ) random utilit y represen tations are alwa ys extreme p oin ts of their iden tiﬁed sets. 31 Theorem 2 also resolv es an op en question in the theory of rational joint c hoice b ehavior. 32 Kashaev et al. ( 2024 ) sho w that, while the revealed preference implications of such mo dels are generally in tractable, they can b e simply axiomatized under any restriction to sets of preferences with linearly independent choice functions. 33 T o date, no characterization of these sets has b een kno wn. Ho w ever, a k ey ingredien t in our proof is showing they are precisely the sets whic h contain no pair of distinct, rearrangemen t-equiv alen t sequences. Th us our results also describ e when testing for sto chastically rational joint b eha vior is easy . In addition, this allows us to completely describ e identifying supp ort restrictions. Corollary 1. A supp ort restriction deﬁned b y S ⊆ L is identifying if and only if S con tains no pair of distinct, rearrangemen t-equiv alent sequences. T o illustrate, consider again the support restriction S consisting of the four preferences in Example 1 . As noted before, the only admissible p erm utation in an y rearrangement that can generate new preferences is σ 2 , thus tw o sequences are rearrangemen t equiv alent if and only if one can b e obtained from the other b y shuﬄing terminal segments s 2 ↓ . As a consequence, 29 That is, one cannot construct t w o such sequences by dra wing (with replacement) from supp( µ ′ ). 30 When M is a supp ort restriction, Theorem 1 and Theorem 2 provide dual characterizations of the iden- tiﬁed p olytop es, in terms of their faces and extreme points, resp ectiv ely . W e note, ho w ever, that Theorem 1 remains v alid without assumptions on M . 31 See Prop osition 2 . Ho wev er, there exist iden tiﬁed sets whose extreme p oints do not all arise in this manner; see Prop osition 3 . 32 E.g. Chambers et al. ( 2024 ). 33 Here, we are identifying c hoice functions with their representation as 0 − 1 vectors in P . 13 while S is not identifying, any prop er subset is. In particular, a distribution is an extreme p oin t of its identiﬁed set if and only if it do es not hav e full supp ort; see Figure 1 . Similarly , Corollary 1 also implies that the restriction to monotone exp ected utility pref- erences in Example 4 is identifying. As in Example 1 , it once again suﬃces to consider only rearrangemen ts with σ 2 the sole non-trivial p ermutation. As a consequence, proving iden tiﬁ- cation reduces to v erifying one cannot construct tw o distinct sequences of preferences from S with iden tical initial 2-segments, but whose corresp onding sequences of terminal 2-segmen ts are distinct. But such sequences cannot exist, essen tially due to the fact each preference in S whic h ranks b ≻ a do es not also rank c ≻ d . Thus S is iden tifying. 5 Order-Based Restrictions In man y economic en vironments, the underlying set of alternatives carries a natural order structure. F or example, monetary lotteries may b e ordered by v arious measures of risk or sto c hastic dominance, vehicles b y gas mileage, or gov ernmen t p olicies by lev el of public go o d pro vision. In such contexts, b eha vioral types often p ossess an analogous ordering, e.g. by risk a version, en vironmen tal conscientiousness, or civic-mindedness. Throughout this section, we will assume that ⊵ is a linear order on X . F ollowing Ap esteguia et al. ( 2017 ), a distribution µ ∈ ∆( L ) is a single-crossing represen tation if supp( µ ) can b e ordered ≻ 1 , . . . , ≻ N suc h that if (i) x ⊵ y and (ii) x ≻ j y , then for all i > j , x ≻ i y as well. When a single-crossing representation exists, Ap esteguia et al. ( 2017 ) show is necessarily unique; ho w ever there exist choice frequencies whic h are consisten t with the random utilit y mo del but incompatible with any single-crossing represen tation. An alternativ e approac h to identiﬁcation due to Filiz-Ozba y and Masatlioglu ( 2023 ) considers a broader class of primitives. Let C denote the set of all choice functions, including those not rationalizable by any strict preference. A distribution µ ∈ ∆( C ) is said to b e a progressiv e random c hoice representation if supp( µ ) can b e ordered c 1 , . . . , c N suc h that i > j implies c i ( A ) ⊵ c j ( A ) for all men us ∅ ⊊ A ⊆ X . In con trast to single-crossing represen tations, progressive random choice rules exist and are unique for any choice shares. Ho wev er, they ma y not b e supp orted on the rational c hoice functions, and hence fail to b e a random utility representation, ev en for data compatible with the random utilit y mo del. Our notion of compatibilit y provides a no vel, order-based iden tifying restriction whic h preserv es the b est features of b oth the single-crossing and progressiv e frameworks. W e sa y that a random utility represen tation µ ∈ ∆( L ) is sw ap-progressive if supp( µ ) can b e 14 ordered ≻ 1 , . . . , ≻ N suc h that, whenever ≻ i and ≻ j are k -compatible, 0 ≤ k ≤ | X | − 1, then i > j implies x k +1 i ⊵ x k +1 j . 34 Our next result sho ws sw ap-progressive representations are unique, and exist if and only if the data admit a random utilit y rationalization. Th us the empirical conten t of swap- progressivit y coincides with that of the baseline random utilit y model. Moreo ver, when either single-crossing or progressiv e random utility represen tations exist, b oth do, and b oth coincide with the unique sw ap-progressive representation. 35 Theorem 3. Let ⊵ b e a linear order on X . Then: (i) A swap-progressiv e rationalization exists if and only if the data are compatible with the random utility mo del. Moreov er, if suc h a represen tation exists, it is unique. (ii) F or any order ⊵ , ev ery single-crossing represen tation is sw ap-progressiv e. Conv ersely , a sw ap-progressive represen tation is single-crossing if and only if it is progressiv e. In light of Theorem 3 , we interpret swap-progressivit y as a principled reﬁnement of b oth notions: it agrees with the random utility predictions of b oth single-crossing and progressivit y whenev er these exist, but generates unique predictions ev en when these other frameworks cannot. This provides a practical, but general, metho d for selecting rationalizations, that systematically incorporates the order structure of the environmen t. In particular, in man y applications it is v aluable to obtain rationalizations whose supp ort is ‘1-dimensional,’ or able to b e linearly ordered according to some economic criterion. 36 As the follo wing example illustrates, this is alw a ys true of sw ap-progressive rationalizations. 37 Example 6. Consider again the setting of Example 1 , and supp ose p olicies are rank ed according to their ﬁscal cost a ⊵ b ⊵ d ⊵ c . Theorem 3 guarantees the p olicy mak er is able to recov er a unique sw ap-progressive rationalization from straw-poll data, even though inﬁnitely man y rationalizations exist. The p ow er of swap progressivit y is that it severely limits the supp ort of a rationalization. F or example, no sw ap-progressive distribution can place p ositive w eight on b oth ≻ 1 = abcd and ≻ 2 = badc . As b oth are trivially 0-compatible, a ⊵ b implies ≻ 1 m ust b e ranked higher 34 Recall that x k n denotes the k -th most preferred alternative of ≻ n , and note x k +1 i ⊵ x k +1 j explicitly allows for the equality of x k +1 i and x k +1 j . 35 The ﬁrst part of this claim is Remark 1 in Filiz-Ozba y and Masatlioglu ( 2023 ). 36 See, e.g., Chiapp ori et al. ( 2019 ); Barseghy an et al. ( 2021 ). F or further examples and a discussion of these ideas, see Filiz-Ozbay and Masatlioglu ( 2023 ). 37 F or a more complex example of the sw ap-progressiv e representation of data which do not admit a single- crossing representation, see Online App endix D.2 . 15 within the supp ort than ≻ 2 . Ho wev er, as b oth are also 2-compatible, d ⊵ c implies ≻ 2 m ust also b e ranked ab ov e ≻ 1 , an imp ossibility . In this example, this observ ation already suﬃces to pin down the distribution: the ⊵ - sw ap progressive rationalization is precisely ¯ µ = (0 , 0 , 5 8 , 3 8 ). In particular, ¯ µ is supp orted on bacd and abdc . These corresp ond to the tw o preferences, among those held in some rationalization, which (resp.) agree least and most with ⊵ . As a consequence, w e can regard the support of ¯ µ as b eing ordered by ﬁscal conserv ativ eness. ■ 6 P arametric Restrictions In practical applications, restrictions on random utility , and more general mo dels of sto c has- tic choice, are often sp eciﬁed parametrically . If Θ ⊆ R d denotes a set of parameters, we refer to any one-to-one and onto map F : Θ → M ⊆ ∆( L ) as a parametric restriction . 38 By Theorem 1 , a parametric restriction is identifying if and only if: F ( θ ′ ) ∈ F ( θ ) + R = ⇒ θ = θ ′ , i.e. if M = range( F ) in tersects eac h translate of the Ryser subspace at most once. In principle, this is a complete c haracterization of identiﬁcation. Ho wev er, in practice, it is often unclear ho w to verify this from only knowledge of F . Moreov er, man y times restrictions are deﬁned b y sp ecifying c hoice probabilities directly in terms of parameters, lea ving F only implicit. W e now consider a simple, practical test for iden tiﬁcation in suc h cases. W e will consider only smo oth parametric models. How ev er, as it will essentially b e costless to do so, w e will allo w for arbitrary mo dels of parametric sto chastic c hoice, rather than solely random utility . Recall P denotes the set of all random choice rules. A parametric sto c hastic c hoice mo del is a smo oth mapping φ : Θ → P . 39 In general, it is intractable to directly analyze the full v ector of functions φ in an y realistically-sized setting. 40 T o determine inv ertibilit y in practice, one t ypically restricts to subsystems with equal num b ers of parameters and choice probabilities. W e say ¯ φ : Θ → R d deﬁnes a smo oth submo del if it is obtained by selecting any d comp onen ts of the vector-v alued map G ◦ φ , where G is an y smo oth and smo othly inv ertible 38 This is distinct from the case in which a supp ort constraint is deﬁned by the restriction of some para- metric family of utilities to X . There, parameters corresp ond to the preferences of the mo del, and hence rationalizations to distributions ov er parameters. Here, we consider the case in which the allow able p opula- tion distributions themselves b elong to some parametric family M = F (Θ) ⊆ ∆( L ). 39 P arametric random utilit y corresp onds to the case where φ = Φ ◦ F , for some one-to-one F : Θ → ∆( L ). 40 F or example, when | X | = 6, a parametric sto chastic c hoice mo del is a vector of 129 functions. 16 transformation of choice probabilities. As there are t ypically many fewer parameters than c hoice probabilities, this en tails a meaningful reduction. While our deﬁnition allo ws for general transformations, in practice, G often corresp onds to simply normalizing by certain c hoice probabilities, see Example 7 and Example 8 . A smo oth submo del ¯ φ is parametrically identiﬁed if it admits a smo oth inv erse ¯ φ (Θ) → Θ. P arametric identiﬁcation of a submo del ¯ φ implies the same of the full, un- restricted mo del φ . While in full generalit y the conv erse need not b e true, in practice it is typically straightforw ard to use an uniden tiﬁed ¯ φ to show the full mo del φ fails to b e parametrically iden tiﬁed. 41 Our next result sho ws a smo oth submo del is parametrically identiﬁed precisely when it is so locally , and it satisﬁes a mild, b oundary-preserv ation condition. This reduces the diﬃcult, global problem of verifying iden tiﬁcation of a submo del to a simple analytic test. Theorem 4. Let Θ ⊆ R d b e a con tractible, op en set of parameters. 42 A smooth map ¯ φ : Θ → R d is parametrically identiﬁed if and only if: (i) The Jacobian matrix J ¯ φ is ev erywhere in v ertible; and (ii) F or any sequence ( θ n ) ∞ n =1 ⊂ Θ such that eac h compact K ⊂ Θ contains at most ﬁnitely man y terms, the same is true of  ¯ φ ( θ n )  ∞ n =1 for an y compact K ′ ⊂ ¯ φ (Θ). Relativ e to existing global in verse results, Theorem 4 has several key adv an tages. First, unlik e approaches relying on monotonicity , whic h typically yield suﬃcien t but not necessary conditions, it pro vides a complete c haracterization of parametric iden tiﬁcation. F or example, when studying the in v ertibility of a smo oth demand system, Theorem 4 remains v alid ev en when go o ds may be complements, and is applicable in settings when the law of demand may fail. 43 Second, unlike many existing results, Theorem 4 allows for b oth the domain and range of ¯ φ to b e distinct, prop er subsets of R d , and requires structural assumptions only on the domain , rather than range of ¯ φ . 44 Giv en that the range of a general vector-v alued function can be diﬃcult to determine, this property makes Theorem 4 particularly v aluable in practice. 41 See, e.g., Example 8 and Online App endix D.1 . 42 A set Θ is contractible if there exist a contin uous map H : Θ × [0 , 1] → Θ such that for eac h θ ∈ Θ, H ( θ , 0) = θ , and H ( θ , 1) = θ 0 for some ﬁxed vector θ 0 . This includes all con v ex (or star-shap ed) sets. 43 In particular, Theorem 4 remains applicable even in economic environmen ts not amenable to classic demand inv ersion results, e.g. Gale and Nik aido 1965 ; Berry et al. 2013 ; Allen 2022 . 44 C.f. Palais 1959 ; McKenzie 1967 ; Gordon 1972 ; Krantz and P arks 2002 ; Komunjer 2012 . 17 Example 7. Let X = { x 0 , . . . , x K } . A sto c hastic choice rule b elongs to the Luce mo del if there exist weigh ts w 0 ≡ 1 and w k > 0 for all k > 0, such that, for an y x ∈ A ⊆ X : φ ( w 1 , . . . , w K ) ( x i ,A ) = ρ ( x i , A ) = w i P j ∈ A w j . Th us Θ = R K ++ , with each θ corresp onding to the vector of normalized weigh ts ( w 1 , . . . , w K ). Consider the map: ¯ φ ( w 1 , . . . , w K ) =     ρ ( x 1 ,X ) ρ ( x 0 ,X ) . . . ρ ( x K ,X ) ρ ( x 0 ,X )     =     w 1 . . . w K     , noting this deﬁnes a smo oth submodel. 45 Since (i) the Jacobian of ¯ φ is ev erywhere the iden tity , and (ii) some comp onen t of ¯ φ ( θ n ) trivially tends to zero or inﬁnity whenev er the same is true of θ n , it follo ws from Theorem 4 that the submo del ¯ φ , and hence the full Luce mo del, is parametrically identiﬁed. ■ Our results can also b e used to detect identiﬁcation failures in more complex settings. T o illustrate, we establish a nov el result, that the dynamic, logit habit-formation mo del of T uransick ( 2025 ) is not parametrically identiﬁed from unconditional choice data. Example 8. Supp ose X = { x 0 , . . . , x N } , where x 0 denotes an outside option which is alw ays a v ailable. A menu is an y non-empty subset of X whic h con tains x 0 . A sto chastic choice rule represents the unconditional choice probabilities of the habit-formation logit mo del of T uransick ( 2025 ) if it takes the form: ρ ( x i , A ) = v i  P j ∈ A \ i v j + v i c i  P j ∈ A v j  P k ∈ A \ j v k + v j c j  , for any menu A , where v 0 = c 0 ≡ 1, and v i , c i > 1 for all i = 1 , . . . , N . Here, the parameters v i capture the desirabilit y of consuming each alternativ e, while c i sp eaks to how addicting, or habit-forming, consumption of x i is in a given p erio d. Deﬁne a submo del as follows. F or i = 1 , . . . , N , let: ¯ φ i ( v 1 , c 1 , . . . , v N , c N ) = ρ  x 0 , { x 0 , x i }  ρ  x i , { x 0 , x i }  = 1 + v i v i (1 + v i c i ) , 45 Here, the map G tak es eac h comp onen t of φ and normalizes it by φ ρ ( x 0 ,X ) ; ¯ φ is then obtained b y selecting the d = K comp onen ts corresp onding to normalized choice probabilities for x 1 , . . . , x K from X . 18 and for i = N + 1 , . . . , 2 N : ¯ φ N + i ( v 1 , c 1 , . . . , v N , c N ) = ρ  x 0 , { x 0 , x i , x i +1 }  ρ  x i , { x 0 , x i , x i +1 }  = 1 + v i + v i +1 v i + v i v i +1 + c i v 2 i , where v N +1 ≡ v 1 , and c N +1 ≡ c 1 . W e now sho w this submo del fails to satisfy condition (ii) of Theorem 4 , and hence fails to b e parametrically identiﬁed. T o do so, we ﬁrst deﬁne the function: c ( v i ) = 1 + v i 10 v 2 i − 1 v i . Note that, for any i = 1 , . . . , N , we ha ve ¯ φ i ( . . . , v i , c ( v i ) , . . . ) = 1 10 . Consider now the restriction of ¯ φ to the curve  v , c ( v ) , . . . , v , c ( v )  . The last N comp onen ts of ¯ φ are iden tical, and equal to: ¯ φ N + i  v , c ( v ) , . . . , v , c ( v )  = 1 + 2 v v 2 + 10(1 + v ) . As v → 1 along this curv e, ¯ φ N + i → 1 7 . But this is precisely the v alue attained by ¯ φ N + i at v = 3. Since the ﬁrst N comp onen ts of ¯ φ are constant in v , w e ha ve shown that for an y sequence of parameters of the form  v n , c ( v n ) , . . . , v n , c ( v n )  with v n → 1, w e obtain a violation of condition (ii) of Theorem 4 , and hence ¯ φ is not parametrically identiﬁed. It is then straightforw ard to use the insight from this submo del to construct a distinct pair of parameter v ectors that yield identical choice probabilities in the full mo del. 46 ■ 7 Bey ond Random Utilit y Th us far, we ha ve considered primarily the random utilit y mo del. Ho wev er, our core notion of Ryser sw ap c haracterizes the iden tiﬁed sets of man y other s to c hastic c hoice mo dels. In eac h of these additional settings, exact analogs of our main results obtain. W e illustrate sev eral such extensions b elo w. Random Choice The simplest extension is to the unrestricted class of random choice models, p ossibly sub ject to limited observ abilit y . Here, X is a ﬁnite set of alternatives, and Σ ⊆ 2 X \ { ∅ } an arbitrary collection of men us on whic h c hoice frequencies are observed. Let C denote the collection of all choice functions on Σ. 47 A random choice mo del is a probability measure µ ∈ ∆( C Σ ) 46 See Online App endix D.1 for an explicit example. 47 That is, C Σ = Q A ∈ Σ A . 19 suc h that, for all a ∈ A ∈ Σ: ρ ( a, A ) = X c ∈C µ ( c ) 1 { a = c ( A ) } . Here, a Ryser swap is a signed measure R ∈ R C Σ of the form: R = 1 { c ′ 1 ,c ′ 2 } − 1 { c 1 ,c 2 } , where c ′ 1 , c ′ 2 ∈ C Σ are the c hoice functions obtained by swapping the choices made by c 1 and c 2 on a single men u A ∈ Σ. Similarly , we deﬁne the Ryser subspace for the random c hoice mo del R rc ⊂ R C Σ as the linear span of all such v ectors. Once again, the subspace R rc c haracterizes identifying restrictions. Theorem 5. Fix Σ ⊆ 2 X \ { ∅ } . Two distributions µ, µ ′ ∈ ∆( C Σ ) generate identical c hoice probabilities on ev ery menu in Σ if and only if µ − µ ′ ∈ R rc . In particular, for any M ⊆ ∆( C Σ ) the set of distributions generating iden tical choice probabilities to µ on every men u in Σ is precisely  µ + R rc  ∩ M . Exact analogues of Theorem 2 and Theorem 3 also obtain in this setting. 48 Notably , since C Σ con tains the rational choice functions on Σ, the random utilit y mo del can b e viewed as a supp ort restriction within this more general framew ork. Th us Theorem 5 , and the random c hoice analogue of Theorem 2 , allow our results to extend to the case of incompletely observ ed data, i.e. when Σ ⊊ 2 X \ { ∅ } . F rame Dep enden t Choice Our results also extend naturally to mo dels of b oundedly rational b ehavior. W e illustrate this on a simple v ersion of the frame-dep endent random utilit y mo del of Cheung and Masatlioglu ( 2024 ). 49 Here, individuals face a ﬁxed, ﬁnite choice set X = { x 1 , . . . , x K } , but receiv e v arying suggestions from a black-box recommendation system that highligh ts some subset A ⊆ X , rendering them more desirable. Here, the empiricist observ es fr ame-dep endent sto c hastic c hoice data ρ : X × 2 X → [0 , 1], where: X x ∈ X ρ ( x, A ) = 1 , 48 In particular, the analogue of our notion of swap-progressivit y coincides precisely with the progressivity condition of Filiz-Ozba y and Masatlioglu ( 2023 ); the analogue of our Theorem 3 in this setting corresp onds their Theorem 1. 49 W e extend this analysis to the full mo del of Cheung and Masatlioglu ( 2024 ) in Online App endix B.2 . 20 for eac h A ⊆ X , interpreted as the observed probability of the the agen t c ho osing x from X when faced with the recommendation A . Deﬁne X ∗ = { x N 1 , . . . , x N K , x F 1 , . . . , x F K } , where w e regard x i when framed ( x F i ) as distinct from x i when it is not ( x N i ). Given a recommendation set A ⊆ X , the sub ject ma y b e regarded as choosing from the virtual menu: A ∗ = { x N : x ∈ X \ A } ∪ { x F : x ∈ A } ⊂ X ∗ . Since framing mak es any alternativ e strictly more preferred, no alternative b elo w the most desirable x N can ev er aﬀect observ ed choice. Thus, let L ∗ denote the set of trunc ate d strict preferences on X ∗ , consisting only of the ranking up to, and including, the b est, non-framed alternativ e, and whic h satisfy x F ≻ ∗ x N for all x ∈ X . 50 The data are compatible with a frame-dep enden t random utility represen tation µ ∈ ∆( L ∗ ) if: ρ ( x, A ) = X ≻ ∗ ∈L ∗ µ ( ≻ ∗ ) 1 { ˆ x is ≻ ∗ -maximal in A ∗ } , where ˆ x = x N if x ∈ A , and x F if x ∈ A . Given tw o truncated preferences in ≻ ∗ 1 , ≻ ∗ 2 ∈ L ∗ w e once again say they are compatible if they agree up on the set of k -b est alternatives, though not necessarily their ranking. A Ryser sw ap for the frame-dependent model is simply a signed measure in R L ∗ whic h places unit mass on a k -compatible pair ≻ ∗ 1 and ≻ ∗ 2 and mass negative one on the pair of truncated preferences obtained b y sw apping their k -initial segments. 51 Let R f d ⊂ R L ∗ denote the linear span of these vectors. Theorem 6. Tw o distributions µ, µ ′ ∈ ∆( L ∗ ) are observ ationally equiv alent if and only if µ − µ ′ ∈ R f d . In particular, for an y M ⊆ ∆( L ∗ ), its iden tiﬁed set is precisely  µ + R f d  ∩ M . As b efore, analogues of Theorem 2 and Theorem 3 also obtain in this setting. In particu- lar, as was true for the random utility mo del, a swap-progressiv e fr ame-dep endent represen- tation exists whenever the data are rationalizable, and will itself alwa ys b elong to ∆( L ∗ ). Dynamic Discrete Choice Our metho dology is also applicable to the problem of identifying distributions o ver dynamic c hoice functions from Mark ovian conditional choice probabilities. Let X denote a ﬁnite set of alternatives, and T a ﬁnite time horizon. In p erio d t = 1, the analyst observes 50 F or example, an (unrestricted) preference x F 1 ≻ x F 2 ≻ x N 1 ≻ x N 2 w ould corresp ond to the truncated preference x F 1 ≻ ∗ x F 2 ≻ ∗ x N 1 . 51 Note this is well-deﬁned, as these swaps necessarily b elong to L ∗ . 21 unconditional choice probabilities ρ 1 ( x ). F or each p erio d t = 2 , . . . , T , the analyst observ es probabilities of time t c hoice, conditional the prior p erio d’s choice, ρ t ( y | x ). T ogether, w e refer to { ρ 1 , . . . , ρ T } as a system of conditional choice probabilities. An analyst seeks to decompose a system of conditional c hoice probabilities into a measure o ver dynamic c hoice functions in C dc = X T . A system of conditional c hoice probabilities is compatible with a dynamic discrete c hoice represen tation µ ∈ ∆( C dc ) if: ρ 1 ( x ) = X c ∈C dc µ ( c ) 1 { x = c (1) } and ρ t ( y | x ) = P c ∈C dc µ ( c ) 1 { x = c ( t − 1) , y = c ( t ) } P c ∈C dc µ ( c ) 1 { x = c ( t − 1) } for all t = 2 , . . . , T and x, y ∈ X . W e sa y tw o c hoice functions c 1 , c 2 ∈ C dc are t -compatible if c 1 and c 2 mak e the same c hoice at time t − 1. A Ryser swap is a signed measure R ∈ R C dc of the form 1 { c ′ 1 ,c ′ 2 } − 1 { c 1 ,c 2 } , where c ′ 1 and c ′ 2 are the choice functions obtained b y sw apping the ﬁrst t − 1 choices of some t -compatible pair of c hoice functions c 1 , c 2 ∈ C dc . Let R dc ⊂ R C dc denote the linear span of these measures. Theorem 7. Tw o distributions µ, µ ′ ∈ ∆( C dc ) are observ ationally equiv alent if and only if µ − µ ′ ∈ R dc . In particular, for an y M ⊆ ∆( C dc ), its iden tiﬁed set is precisely  µ + R dc  ∩ M . Finally , as in eac h previous setting, analogues of Theorem 2 and Theorem 3 also obtain. In particular, a sw ap-progressive rationalization exists and is unique for an y rationalizable data, and will alwa ys b e supp orted on C dc . While w e consider here dynamic discrete choice, our results also extend to more complex, ﬁnite dynamic programming mo dels, including those inv olving state-dep endent utilit y and men u v ariation. This allo ws for further application to mo dels such as Rust ( 1987 ). 8 Related Literature The ﬁrst explicit example of identiﬁcation failure in the random utility mo del is due to Fish- burn ( 1998 ). His coun terexample forms the basis of our notion of a (non-trivial) compatible pair. T uransick ( 2022 ) c haracterizes data sets which p ossess a unique random utility ratio- nalization. W e obtain this characterization as a straigh tforward consequence of the pro of of our Theorem 1 . 52 52 See also Doignon and Saito ( 2023 ). 22 A num b er of pap ers consider restrictions on the random utilit y mo del for purp oses of obtaining iden tiﬁcation. Manzini and Mariotti ( 2018 ) show that in a tw o-state mo del, dis- tributions o v er preferences can generically b e reco v ered. Suleymano v ( 2024 ) recov ers iden- tiﬁcation b y requiring certain indep endence prop erties of the rationalization; Honda ( 2021 ) considers a random cra vings mo del where identiﬁcation obtains under monotonicit y restric- tions. By directly characterizing the iden tiﬁed sets of the general random utility mo del, w e instead pro vide necessary and suﬃcient conditions for an y restriction to b e iden tifying. When alternativ es are ordered, a well-kno wn class of restrictions rely on v arious single- crossing type conditions on the supp ort of rationalizations to ac hieve uniqueness. This idea is originally due to Ap esteguia et al. ( 2017 ); Filiz-Ozbay and Masatlioglu ( 2023 ) considered an extension to general sto c hastic choice mo dels that guaran tees existence while preserving uniqueness, at the cost of the unique rationalization p ossibly failing to b elong to the random utilit y mo del, even when the data is rationalizable b y such a distribution. 53 Our notion of sw ap-progressivit y builds on these con tributions. Applied to the random utilit y mo del, it generalizes single-crossing and provides unique predictions, even for data whic h fail the cen trality axiom of Apesteguia et al. ( 2017 ). 54 In the con text of general random choice, the analogue of our Theorem 3 is precisely Theorem 1 of Filiz-Ozbay and Masatlioglu ( 2023 ). Moreov er, sw ap-progressiv e representations exist, are unique, and b elong to the mo dels of interest for eac h of the environmen ts considered in Section 7 . In practice, it is common to sp ecify random utility model parametrically , e.g. Luce ( 1959 ). 55 More recently , Chambers et al. ( 2025 ) considers an extension of the Luce mo del including b oth salience and utilit y in its parameterization which preserves iden tiﬁcation. In general, Luce t yp e structure allo ws for simple identiﬁcation argumen ts. Ho w ever, in more complex en vironments, Theorem 4 provides a practical to ol for establishing similar results. While our pap er fo cuses on the case of ﬁnite en vironments, there are n umerous p ositiv e iden tiﬁcation results when domains are inﬁnite. Gul and Pesendorfer ( 2006 ) sho ws that the random exp ected utility mo del is identiﬁed, as is the random quasilinear utility mo del ( Williams , 1977 ; Daly and Zac hary , 1979 ; Y ang and Kopylo v , 2023 ). Metho dologically , our Theorem 1 builds on w ork in the mathematical discipline of to- 53 Yildiz ( 2023 ) studies the structure of such ordered mo dels and shows that a unique ordered representa- tions is p ossible only when the types in the mo del form a lattice. On the other hand, Petri ( 2022 ) considers an extension of these ordered mo dels to allow for multi-v alued choice. 54 In particular, in this setting, swap progressivity has the same explanatory p ow er as the baseline random utilit y mo del, while also b eing iden tiﬁed. 55 This mo del is observ ationally equiv alent to the logit mo del of McF adden ( 1974 ). See also Sandomirskiy and T am uz ( 2023 ) for a recent characterization. 23 mograph y ( Ryser 1957 ; Fishburn et al. 1991 ; Kong and Herman 1999 ). 56 W e show that b oth classic tomographic applications, as well as related questions ab out certain, restricted sets of measures with prescrib ed marginals, can b e b e fruitfully enco ded as ﬂow problems on directed graphs. In this general setting, w e provide a uniﬁed notion of Ryser swap, which sp ecializes to the forms seen in Section 3 and Section 7 , and sho w it c haracterizes the abstract analogue of our identiﬁed sets. Similarly , to the extent of our kno wledge, Theorem 4 do es not app ear in the economic or mathematical literature. Relative to classical results of Hadamard ( Gordon 1972 ; Krantz and Parks 2002 ), we sho w that when the domain of a non-linear system is con vex (or, more generally , con tractible) and the system con tains an equal n umber of equations and unkno wns, the classical, diﬃcult-to-v erify conditions on the system’s range which characterize inv ert- ibilit y are alw a ys satisﬁed. Our pro of relies on a ﬁxed p oint theorem due to Smith ( 1934 ) whic h we b elieve has not found prior economic application. 9 Conclusion When analysts observ e only aggregate choice frequency data, many work-horse mo dels of sto c hastic c hoice are unidentiﬁed. In such instances, explicit descriptions of the identiﬁed sets are of direct practical imp ortance, as they allo w analysts not only to understand the range of p ossible aggregate eﬀects of a policy in terven tion, but also its distributional implications. Despite this, little has b een known about the structure of these sets, even for many widely used models. W e characterize the iden tiﬁed sets of a wide range of sto chastic choice mo dels. Our k ey insigh t is that an y transformation of a distribution ov er primitives that preserves all c hoice probabilities can b e represen ted as a ﬁnite sequence of simple, mass-swapping op erations that we term Ryser swaps . Using this, we giv e explicit characterizations of b oth the deﬁning inequalities and extreme p oints of these mo dels’ identiﬁed sets. In eac h of these settings, our results pro vide tight b ounds on the p opulation cross-sections compatible with observ ed c hoice frequencies. This yields v aluable insigh ts, e.g., for assessing the eﬀects of policies whic h may diﬀeren tially aﬀect v arious subsets of a p opulation. W e also pro vide sp ecialized characterizations of iden tiﬁcation for a n umber of common, structured classes of restriction, suc h as constrain ts on the supp ort of the rationalizing distri- 56 Hec kman and Pinto ( 2018 ) use Ryser’s theorem to sho w identiﬁcation in a diﬀeren t con text. A diﬀeren t strand of literature uses tensor-decomp osition tec hniques to study identiﬁcation; see Dardanoni et al. ( 2023 ) and Kops et al. ( 2026 ). 24 bution, or when c hoice frequencies are assumed to v ary smo othly in some v ector of parame- ters. These provide nov el to ols for practitioners to emplo y when ev aluating the identiﬁcation prop erties of v arious restrictions going forward. Our results suggest several fruitful directions for further inquiry . One would be to under- stand how our metho dological tec hniques generalize to the case of contin uous consumption spaces. 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(2023): “F oundations of self-progressiv e choice mo dels,” arXiv pr eprint arXiv:2212.13449 . 30 A Graphical Results A.1 Deﬁnitions & Preliminaries A graph is a pair ( N , E ) where N is a ﬁnite set of no des , and E a ﬁnite collection of ordered pairs in N × N , where we explicitly allow for rep etition. 57 W e in terpret an ordered pair ( x, y ) ∈ E as directed from x (the ‘tail’) to y (the ‘head’). W e will sometimes denote the ordered pair ( x, y ) ∈ E by x → y . F or a given no de n ∈ N , we write E ↓ n (resp. E n ↓ ) for the sub-collection of all edges in E whose head (resp. tail) is n . A graph is said to b e acyclic if there is no subset of edges with the prop erty that an y no de app earing in some edge in the set do es so precisely once as the head of some edge, and once as a tail. A directed acyclic graph (DA G) is a tuple ( N , E , s, t ), where ( N , E ) is an acyclic graph, and s and t denote sp ecialized no des in N , resp ectiv ely the source and sink , such that E ↓ s = E t ↓ = ∅ . W e will assume that s and t are the unique no des with these prop erties. Flo ws & P ath Decomp ositions Giv en a directed acyclic graph, a ﬂo w is a function f : E → R + suc h that (i) at ev ery non-source/sink no de n , total in-ﬂow equals total out-ﬂow, and (ii) the total out-ﬂow from the source is normalized to unit y: (i) X e ∈E ↓ n f ( e ) = X e ∈E n ↓ f ( e ) and (ii) X e ∈E s ↓ f ( e ) = 1 . (4) A function f : E → R + is a quasi-ﬂow if it satisﬁes only condition (i). A path P is a sequence of edges such that (i) the source s and sink t eac h app ear in exactly one edge, and (ii) every other n ∈ N that app ears in any edge app ears exactly twice, once as a head and once as a tail. A path P is said to pass through a no de n if n app ears in some edge in P , and for any node n ∈ N app earing in P , w e let P ↓ n (resp. P n ↓ ) denote the unique edge in P whose head (resp. tail) is n . 58 Let P denote the collection of all paths. 59 W e say a non-negativ e 57 That is, E is a m ulti-set, hence our deﬁnition of graph explicitly allows for multigr aphs . 58 F urther, we use s ↑ k ( P ) (resp. s k ↓ ( P )) to denote the ﬁrst k (resp. last | P | − k ) edges in a path P . W e will also sometimes use s ↑ n and s n ↓ to denote the initial/terminal segments of a path up to/starting from some n ∈ N . 59 In particular, throughout the app endix we will use P to denote the path set of a graph, rather than the set of random choice rules as used in the b o dy . 31 measure π ∈ R P + deﬁnes a path decomp osition of a quasi-ﬂo w f if, for all e ∈ E : f ( e ) = X { P ∈P : e ∈ P } π ( P ) . A path decomposition b elongs to ∆( P ) if and only if it decomp oses a ﬂo w. A map f : E → R + admits at least one path decomp osition if and only if it is a quasi-ﬂow (e.g. Ahuja et al. 1993 ). Compatibilit y & Ryser Sw aps Tw o paths P 1 and P 2 are n - compatible if they pass through some common no de n ∈ N . A pair of paths P ′ 1 , P ′ 2 are n - conjugate to an n -compatible pair P 1 , P 2 if they are the paths obtained b y follo wing P 1 un til n , then P 2 thereafter, and vice-v ersa. 60 Tw o path decomp ositions π , π ′ ∈ ∆( P ) diﬀer b y a w eighted Ryser swap if: π − π ′ = ε  1 P 1 + 1 P 2 − 1 P ′ 1 − 1 P ′ 2  , (5) where P ′ 1 and P ′ 2 are conjugate to some compatible pair P 1 , P 2 . 61 Note that if π ′ and π diﬀer b y a weigh ted Ryser sw ap, they are path decomp ositions of the same quasi-ﬂo w. W e deﬁne the Ryser subspace R ⊆ R P as the span of all weigh ted Ryser swaps with R denoting a t ypical Ryser sw ap. Rearrangemen ts Let P 1 , . . . , P K b e a sequence of paths, explicitly allo wing rep etition. F or an y n ∈ N , let Π n ⊆ { 1 , . . . , N } denote the subset of indices corresp onding to paths P i passing through n . A rearrangemen t of P 1 , . . . , P K is a choice of p ermutation σ n : Π n → Π n for eac h n ∈ N . An y rearrangement of P 1 , . . . , P K recursiv ely deﬁnes a new sequence of paths P ′ 1 , . . . , P ′ K via: P ′ k : n 0 → · · · → n M where n 0 ≡ s , n M ≡ t , and: n i +1 = head [ P ( σ n i ◦ ··· ◦ σ n 0 ) − 1 ( k ) ] n i ↓ , for all 0 ≤ i ≤ M − 1. In other words, P ′ k is formed b y initially following P σ − 1 n 0 ( k ) to its ﬁrst non-source no de, n 1 , then following the outgoing edge from n 1 b elonging to P ( σ n 1 ◦ σ n 0 ) − 1 ( k ) 60 That is, P ′ 1 agrees with P 1 up to n , then agrees with P 2 thereafter, and vice-versa. 61 F ormally , we refer to any v ector in R P of the form taken by the right-hand side of ( 5 ) to b e a weigh ted Ryser swap. 32 to n 2 , and so forth. Tw o sequences are said to b e rearrangemen t equiv alent if they can b e obtained from one another b y a rearrangemen t. The Random Utility Graph Our main technical contribution is to c haracterize the sets of path decomp ositions of arbitrary quasi-ﬂo ws, for general directed acyclic graphs. Eac h of our economic results then obtains simply by applying these abstract theorems to the particular family of graphs enco ding a sp eciﬁc sto chastic c hoice theory . As our exp osition has fo cused on the random utility model, w e introduce its graphical represen tation, due to Fiorini ( 2004 ), here. In Online App endix B , we collect similar constructions for the v arious other sto c hastic c hoice theories mentioned in the text. The random utility graph is deﬁned by N = 2 X and E =  ( A, B ) | B = A \ { a } , a ∈ A  , where s = X and t = ∅ . Any rationalizable (i.e. consistent with the random utility mo del) random c hoice rule ρ induces a unique ﬂo w on this graph as follows: f  A → A \ { a }  = X A ⊆ B ( − 1) | B \ A | ρ ( a, B ) . (6) The quan tities f  A → A \ { a }  are often referred to as Blo ck-Marschak p olynomials . 62 Fiorini ( 2004 ) sho ws this f indeed deﬁnes a ﬂow, i.e. satisﬁes b oth conditions of ( 4 ) and, con versely , that ev ery ﬂow on this graph ma y be regarded as arising via ( 6 ) from some unique, rationalizable random c hoice rule. Moreov er, there is a bijective corresp ondence b etw een the path set P and strict preferences on X , b y regarding a path: X → X \ { x 1 } → · · · → X \ { x 1 , . . . , x N − 1 } → ∅ as a sequence of nested low er contour sets, whic h uniquely deﬁnes the preference x 1 ≻ · · · ≻ x N . As a consequence, w e may iden tify ∆( P ) with ∆( L ), and, under this iden tiﬁcation, Fiorini ( 2004 ) shows a distribution in ∆( P ) rationalizes the data ρ if and only if it is a path decomp osition of the ﬂow ( 6 ). Th us the partial identiﬁcation prop erties of the random utilit y mo del are precisely equiv alent to the path decomp ositions of ﬂows on ( N , E , s, t ). A.2 Graphical Results and Pro ofs Lemma 1 (Zipp er Lemma 1) . Consider tw o path decomp ositions π , π ′ ∈ ∆( P ) of a quasi- ﬂo w f . Let P ∈ supp( π ), { P u } U u =1 ⊆ supp( π ′ ) and, for some ﬁxed 1 ≤ ¯ k < | P | , supp ose 62 F or the origin of ( 6 ), see Blo ck and Marschak ( 1959 ). 33 s ↑ ¯ k ( P ) = s ↑ ¯ k ( P u ) for eac h u . 63 Then, if π ( P ) ≤ P U u =1 π ′ ( P u ), there exists a ﬁnite sequence of weigh ted Ryser sw aps ( c v R v ) V v =1 and a ﬁnite set of paths { P w } W w =1 satisfying s ↑ ¯ k +1 ( P ) = s ↑ ¯ k +1 ( P w ) for all w , such that: π ( P ) ≤ W X w =1 π ′ ( P w ) + V X v =1 c v ⟨ R v , e P w ⟩ ! , where e P w denotes the P w -th standard Euclidean basis v ector of R P . Pr o of. Let ( t ¯ k +1 , h ¯ k +1 ) denote the ( ¯ k + 1)-st edge of the path P . First observ e that eac h path in { P u } U u =1 ⊆ supp( π ′ ) passes through no de t ¯ k +1 since t ¯ k +1 is the head of the ¯ k -th edge in P and s ↑ ¯ k ( P ) = s ↑ ¯ k ( P u ) for eac h u . Now, since π and π ′ are path decomp ositions of the same quasi-ﬂo w, w e hav e b y hypothesis: X P ′ ∈P π ′ ( P ′ ) 1 { ( t ¯ k +1 ,h ¯ k +1 ) ∈ P ′ } = X P ′ ∈P π ( P ′ ) 1 { ( t ¯ k +1 ,h ¯ k +1 ) ∈ P ′ } ≥ π ( P ) , i.e. both π and π ′ put at least π ( P ) mass on paths whic h pass through ( t ¯ k +1 , h ¯ k +1 ) ∈ P . Let { P v } V v =1 denote those paths in supp( π ′ ) that contain ( t ¯ k +1 , h ¯ k +1 ) as an edge. Since eac h P v and each P u passes through no de t ¯ k +1 , eac h pair ( P u , P v ) is compatible at t ¯ k +1 . F urther, as s ↑ ¯ k ( P ) = s ↑ ¯ k ( P u ) for each u and ( t ¯ k +1 , h ¯ k +1 ) ∈ P v ∩ P for eac h v , at least one path in the t ¯ k +1 -conjugate of each ( P u , P v ) agrees with P on s ↑ ¯ k +1 . Let θ b e the minimum b etw een (i) the total amount of mass π ′ puts on paths in { P u } U u =1 and (ii) the total mass π ′ puts on paths in { P v } V v =1 . By the preceding discussion, θ ≥ π ( P ). F or eac h pair of paths ( P u , P v ), let c uv = π ′ ( P u ) P U u ′ =1 π ′ ( P u ′ ) ! π ′ ( P v ) P V v ′ =1 π ′ ( P v ′ ) ! θ . By construction, w e get that P U u =1 P V v =1 c uv = θ . F urther, P U u =1 c uv = π ′ ( P v ) θ P V v ′ =1 π ′ ( P v ′ ) ≤ π ′ ( P v ), and analogously for P u . No w let R uv b e the Ryser swap whic h puts − 1 mass on paths P u and P v and +1 mass on their t ¯ k +1 -conjugate. F urther, b y previous argumen ts, the signed measure resulting from P U u =1 P V v =1 c uv R uv puts θ mass on paths which agree with P on S ↑ ¯ k +1 ( P ). Hence w e obtain a ﬁnite sequence of weigh ted Ryser swaps which shifts θ mass on to paths { P w } W w =1 satisfying s ↑ ¯ k +1 ( P ) = s ↑ ¯ k +1 ( P w ), whic h giv es: W X w =1 π ′ ( P w ) + V X v =1 c v ⟨ R v , e P w ⟩ ! ≥ θ ≥ π ( P ) 63 W e use | P | to denote the num b er of edges in P . 34 as desired while maintaining non-negativit y of the resulting measure. Lemma 2 (Zipp er Lemma 2) . Supp ose π and π ′ are path decomp ositions of some quasi-ﬂo w f . Fix a path P in the supp ort of π . Then there exists a ﬁnite sequence of weigh ted Ryser sw aps with p ositive co eﬃcien ts { c i R i } , c i > 0, suc h that π ′ + P i c i R i = π ′′ , π ′′ ( P ) ≥ π ( P ), and π ′′ is a path-decomp osition. Pr o of. Fix P and denote its ﬁrst edge ( s, n 1 ). Since π and π ′ are path decomp ositions of f , X P ′ ∈P π ( P ) 1 { ( s,n 1 ) ∈ P ′ } = X P ′ ∈P π ′ ( P ′ ) 1 { ( s,n 1 ) ∈ P ′ } ≥ π ( P ) > 0 . As a consequence, the set of paths in supp( π ′ ) whic h include ( s, n 1 ) is non-empt y; en umerate it { P u } U u =1 . Then by the ab o ve we ha ve π ( P ) ≤ P U u =1 π ′ ( P u ), and hence the requirements of Lemma 1 are satisﬁed. Ho w ever, note that the enumerated set of paths { P w } W w =1 whose existence Lemma 1 asserts also satisfy the requirements of Lemma 1 (for the same ﬁxed choice of P ), but this time for s ↑ 2 ( P ). Rep eating this pro cess | P | times yields a path decomp osition π ′′ diﬀering from π b y a sum of w eighted Ryser sw aps, and an enumerated set of paths { P l } L l =1 b elonging to the supp ort of π ′′ suc h that (i) s ↑ | P | ( P ) = s ↑ | P | ( P l ) and (ii) π ( P ) ≤ P L l =1 π ′′ ( P l ), but (i) implies L = 1 and P l = P , hence (ii) reduces to π ′′ ( P ) ≥ π ( P ) as desired. Lemma 3. Both π and π ′ are path decomp ositions of the same quasi-ﬂow f if and only if there exists a ﬁnite sequence of Ryser swaps with p ositive co eﬃcien ts { c i R i } , c i > 0, suc h that π + P i c i R i = π ′ . Pr o of. ( ⇐ =): Supp ose π + P i c i R i = π ′ , where { c i R i } is a ﬁnite sequence of weigh ted Ryser swaps. If t wo path decomp ositions diﬀer by a single weigh ted Ryser swap, they are necessarily path decomp ositions of the same quasi-ﬂow. By induction, w e obtain π ′ and π are path decomp ositions of the same quasi-ﬂo w f . (= ⇒ ): T ow ard the con v erse, w e provide an explicit algorithm to construct π from π ′ via a sequence of weigh ted Ryser swaps. 1. Initialize b y en umerating the set of paths in the supp ort of π ′ via i ∈ { 1 , . . . , I } and set i = 1. Set π 1 = π and set π ′ 1 = π ′ . 2. π i and π ′ i are path decomp ositions of the quasi-ﬂo w f i . 64 As such, by Lemma 2 , w e obtain a path decomp osition π ′′ i of the same quasi-ﬂow as π i , π ′ i , but satisfying π ′′ i ( P i ) ≥ π i ( P i ). 64 Here, f 1 = f and for all i > 1, f i ( e ) = P { P ∈P : e ∈ P } π i ( P ). 35 3. Set i = i + 1. Set π ′ i +1 ( P ) = π ′′ i ( P ) − π ( P ) 1 { P = P i } , and π i +1 ( P ) = π i ( P ) − π ( P ) 1 { P = P i } . 4. If π i = 0 , terminate the algorithm. If not, return to step 2. A t each step, the algorithm applies only ﬁnitely man y w eigh ted Ryser sw aps. As it terminates with π i = 0 , it entails a ﬁnite sequence of weigh ted Ryser swaps which transforms π ′ in to π , as desired. Pro of of Theorem 1 and Prop osition 1 Theorem 1 follows from applying the follo wing abstract result to the random utilit y graph deﬁned at the start of this app endix. Prop osition 1 then follows as an immediate corollary . Theorem 8. A pair π , π ′ ∈ ∆( P ) are path decomp ositions of the same quasi-ﬂo w if and only if π − π ′ ∈ R . In particular, for an y π ∈ M ⊆ ∆( P ), the set of decomp ositions of the same ﬂo w as π corresp onds to  π + R  ∩ M . Pr o of. The ﬁrst claim follo ws directly from Lemma 3 . F or the latter, observe that { P i c i R i | c i ∈ R } deﬁnes R . So π + R is the set { π + P i c i R i | c i ∈ R ++ } , as the set of (unw eigh ted) Ryser sw aps is closed under multiplication by − 1. Thus again b y Lemma 3 the claim follows. A.2.1 Graphical Supp ort Restrictions Theorem 9. Let P 1 , . . . , P K and P ′ 1 , . . . , P ′ K b e ﬁnite sequences of paths, allowing for rep e- tition. P 1 , . . . , P K and P ′ 1 , . . . , P ′ K are rearrangemen t equiv alen t if and only if the sums: K X k =1 1 P k and K X k =1 1 P ′ k diﬀer b y a ﬁnite sum of un w eighted Ryser sw aps. Before pro ving Theorem 9 , we require a preliminary lemma. Lemma 4. F or an y directed acyclic graph ( N , E , s, t ), there is an enumeration of the set of no des from 1 to |N | such that, for each i , there is no j ≥ i suc h that ( n j , n i ) ∈ E . Pr o of. Let d : N → Z + denote the function which assigns to each no de n ∈ N the length (i.e. num b er of edges) of the longest sequence of edges in E of the form: s → · · · → n connecting s to n . As ( N , E , s, t ) is ﬁnite and acyclic, d is w ell-deﬁned. Let ⪰ d denote the w eak order on N represen ted b y d , and ⪰ ∗ an y linear order extension of its asymmetric comp onent. Let n i denote the unique node such that   { n ∈ N : n i ⪰ ∗ n }   = i . 36 It is straigh tforward to verify this gives an ordering n 1 , . . . , n |N | . W e claim this ordering has the desired monotonicity prop erties. T o see this, supp ose for sak e of con tradiction that ( n j , n i ) ∈ E , where j ≥ i . Since ( N , E , s, t ) is acyclic, j > i . F or an y sequence of edges s → · · · → n j , b y app ending the edge n j → n i w e obtain a sequence terminating at n i , hence d ( n i ) > d ( n j ) and n i ≻ d n j , and therefore n i ≻ ∗ n j , whic h implies i > j , a contradiction. W e now mo v e to our pro of of Theorem 9 . Pr o of. T o b egin, supp ose that P 1 , . . . , P K and P ′ 1 , . . . , P ′ K are rearrangemen t equiv alen t. Cho ose some en umeration of the no des of our graph satisfying the conditions of Lemma 4 . F or a path P that passes through no de n , let s ↑ n ( P ) denote the subset of edges in P that precede no de n (and thus ends with an edge of form ( m, n )). Similarly , for a path P whic h passes through no de n , let s n ↓ ( P ) denote the subset of edges in P that come after no de n (and thus b egins with and edge of form ( n, m )). W e no w prop ose an algorithm that tak es a rearrangemen t and constructs Ryser swaps that induce the rearrangement. 1. Initialize at j = 1, and deﬁne the signed measure µ 0 = P K k =1 1 P k , and sequence of paths P 0 k ≡ P k , for all k = 1 , . . . , K 2. F or no de n j , let Π j ⊆ { 1 , . . . , K } consist of the indices of the paths in P j − 1 1 , . . . , P j − 1 k whic h pass through n j . F or any i ∈ Π j , let R j i = 1 { ˜ P i , ˜ P σ − 1 n j ( i ) } − 1 { P i ,P σ − 1 n j ( i ) } , where ˜ P i = s ↑ n j ( P i ) s n j ↓ ( P σ − 1 n j ( i ) ) and ˜ P σ − 1 n j ( i ) = s ↑ n j ( P σ − 1 n j ( i ) ) s n j ↓ ( P i ). Note every R j i , b y deﬁnition of a rearrangement, is a v alid Ryser swap. Deﬁne µ j = µ j − 1 + P i ∈ Π j R j i .This creates a new, Z + -v alued signed measure in R P whic h we may identify with a sequence of paths P j 1 , . . . , P j K , where: P j k =    P j − 1 k if k ∈ Π j s ↑ n j ( P j k ) s n j ↓  P j σ − 1 n j ( k )  else. 3. Set j = j + 1. If n j = t , terminate the algorithm. If n j  = t , return to step 2. By construction, the algorithm terminates with a set of paths P |N | 1 , . . . , P |N | k suc h that, if P |N | k is written as s → n 1 → · · · → t , we ha ve n i +1 = head  P ( σ n i ◦ ··· ◦ σ s ) − 1 ( k )  n i ↓ and hence P |N | 1 , . . . , P |N | k is precisely the sequence P ′ 1 , . . . P ′ K . Therefore, P K k =1 1 P ′ k = P K k =1 1 P k + P |N | j =1 P i ∈ N j R j i as desired. Con versely , supp ose P K k =1 1 P ′ k − P K k =1 1 P k = P L l =1 R l . W e no w show their underlying se- quences P 1 , . . . , P K and P ′ 1 , . . . , P ′ K are rearrangement equiv alent b y induction on L . Supp ose 37 ﬁrst that L = 1. By relab eling, without loss we ma y supp ose the sequences P 1 , . . . , P K to P ′ 1 , . . . , P ′ K are identical but for t wo paths in eac h, P i , P j and P ′ i ′ , P ′ j ′ , and R 1 m ust correspond to sw apping the outgoing terminal segmen ts s n ∗ ↓ ( P i ) and s n ∗ ↓ ( P j ) to yield P ′ i ′ = s ↑ n ∗ ( P i ) s n ∗ ↓ ( P j ) and P ′ j ′ = s ↑ n ∗ ( P j ) s n ∗ ↓ ( P i ). In particular, s ↑ n ∗ ( P ′ i ′ ) = s ↑ n ∗ ( P i ) and s ↑ n ∗ ( P ′ j ′ ) = s ↑ n ∗ ( P j ). Then deﬁne a rearrangemen t where σ s ﬁxes every index except i, j, i ′ , j ′ and maps i ↔ i ′ and j ↔ j ′ , σ n ∗ ﬁxes every element in Π n ∗ except i, j and maps i ↔ j , and σ n is the identit y for all n ∈ N \ { s, n ∗ } . By construction, this deﬁnes a rearrangemen t taking P 1 , . . . , P K to P ′ 1 , . . . , P ′ K . No w, supp ose that for L ≤ ¯ L the claim holds, and let P K k =1 1 P ′ k and P K k =1 1 P k diﬀer by a sum of ¯ L + 1 w eighted Ryser sw aps. Consider ﬁrst the case in which there exists some R l , without loss R ¯ L +1 , such that P K k =1 1 P k + R ¯ L +1 ≥ 0. Then P K k =1 1 P k + R ¯ L +1 diﬀers from P K k =1 1 P ′ k b y a sum of ¯ L un weigh ted Ryser sw aps, and hence identifying P K k =1 1 P k + R ¯ L +1 with a sequence ˜ P 1 , . . . , ˜ P K , by h yp othesis this sequence is rearrangemen t equiv alent to P ′ 1 , . . . , P ′ K . Moreo ver, since 1 ≤ ¯ L , we obtain that similarly P 1 , . . . , P K is rearrangement equiv alent to ˜ P 1 , . . . , ˜ P K . Denote these rearrangemen ts b y { σ 1 n } n ∈N and { σ 2 n } n ∈N resp ectiv ely . Then deﬁning, for each n ∈ N , σ n = σ 1 n ◦ σ 2 n deﬁnes a rearrangemen t under which P 1 , , . . . , P K and P ′ 1 , . . . , P ′ K are equiv alent. On the other hand, supp ose for every R l , the sum P K k =1 1 P k + R l ≥ 0 . Deﬁne Q as the set of paths whic h receiv e negative mass under P K k =1 1 P k + R ¯ L +1 . Then P K k =1 1 P k + R ¯ L +1 + P P ∈Q 1 P and P K k =1 1 P k + P P ∈Q 1 P diﬀer b y a single Ryser swap, and the former diﬀers from P K k =1 1 P ′ k + P P ∈Q 1 P b y ¯ L Ryser swaps. By the preceding argument, the sequence represen tations of these last tw o expressions are rearrangemen t equiv alen t. W e may regard the path(s) in Q as the last path(s) in each sequence; if this rearrangemen t do es not ﬁx the paths corresp onding to these last indices, then the path(s) in Q app earing in the non-prime sequence corresp ond to some P ′ i or P ′ i and P ′ j . In such a case, deﬁne a new rearrangement that is the iden tity at every no de except s , and which at s maps (resp ectively) i ↔ K + 1 or i ↔ K + 1 and j ↔ K + 2, and ﬁxes ev ery other index. Comp osing this new rearrangement with the former then giv es a rearrangemen t betw een the augme n ted sequences whic h ﬁxes the terms in the sequence corresp onding to Q , and hence restricts to a rearrangement b etw een P 1 , . . . , P K and P ′ 1 , . . . , P ′ K as desired. Giv en a directed acyclic graph, its set of paths P , and some subset P ′ , let R P P ′ denote the set of vectors in R P whic h take on v alue zero in the dimensions indexed by paths P ∈ P \ P ′ . By minor abuse of notation, let ∆( P ′ ) denote the set of path decomp ositions whose supp ort is a subset of P ′ ⊆ P . Theorem 10. Fix a graph ( N , E , s, t ). The following are equiv alen t. 38 1. F or eac h π ∈ ∆( P ′ ), there is no path decomp osition π ′  = π with π ′ ∈ ∆( P ′ ) that induces the same ﬂow as π . 2. F or all π ∈ ∆( P ′ ),  π + R  ∩ ∆( P ′ ) = { π } . 3. P ′ con tains no distinct, rearrangemen t-equiv alen t sequences of paths. 4. F or ev ery ﬁnite sequence of unweighte d Ryser sw aps { R i } K i =1 (allo wing for rep etition): K X i =1 R i ∈ R P P ′ ⇐ ⇒ K X i =1 R i = 0 . Pr o of. As ∆( P ′ ) ⊆ ∆( P ), the equiv alence b etw een (1) and (2) is an immediate consequence of Theorem 8 . W e now sho w (3) implies (1). By Lemma 9 , (1) holds if and only the restriction M = ∆( P ′ ) ∩ Q P is identifying. Supp ose, for purp oses of con trap osition, that ∆( P ′ ) ∩ Q P is not identifying. Then there exist distinct π , π ′ ∈ ∆( P ′ ) ∩ Q P whic h decomp ose the same ﬂo w. By Lemma 3 , π and π ′ diﬀer b y a ﬁnite sum of Ryser swaps, π − π ′ = P K i =1 R i . Since b oth π and π ′ tak e only rational v alues, by multiplying b y a suﬃciently large positive integer, w e obtain Z + -v alued signed measures ¯ π , ¯ π ′ . W e ma y identify these signed measures with sequences of paths P 1 , . . . , P K and P ′ 1 , . . . , P ′ K , where ¯ π = P K k =1 1 P k and ¯ π ′ = P K k =1 1 P ′ k . As these signed measures diﬀer by a sum of Ryser swaps (by repeatedly summing copies of P i R i ), by Theorem 9 , π N and π ′ N b eing rearrangement-equiv alent. Th us ¬ (1) implies ¬ (3). W e no w show that (4) implies (3). Once more, b y means of contraposition, suppose P 1 , . . . , P k and P ′ 1 , . . . , P ′ k are distinct, rearrangement-equiv alent sequences of paths. Then b y Theorem 9 , P N i =1 R i = P K k =1 1 P k − 1 P ′ k  = 0 is a ﬁnite sum of unw eighted Ryser sw aps. F urthermore, since P 1 , . . . , P k and P ′ 1 , . . . , P ′ k con tain only paths in P ′ , the sum P K i =1 R i is also supp orted on P ′ . Thus P k i =1 1 P i − P k i =1 1 P ′ i = P K i =1 R i  = 0 and yet P k i =1 1 P i − P k i =1 1 P ′ i = P K i =1 R i ∈ R P P ′ . Th us ¬ (3) implies ¬ (4). Finally , w e show (1) implies (4). T o this end, supp ose (4) is false. Thus there exists some ﬁnite sequence of un w eighted Ryser swaps such that P K i =1 R i  = 0 with P K i =1 R i ∈ R P P ′ . Since each Ryser sw ap has t wo en tries equal to one, t w o en tries equal to negativ e one, and all other entries equal to zero, P K i =1 R i puts equal mass on its negative en tries and its p ositiv e entries. W e can then renormalize and treat the mass on the negative en tries as a path decomp osition and the mass on the p ositive en tries as a path decomp osition. By construction, these diﬀer b y a ﬁnite sum of (p otentially w eighted) Ryser sw aps. By Lemma 3 , they induce the same ﬂow and, by construction, they hav e distinct supp orts and are th us 39 distinct. It immediately follo ws that condition (1) of Theorem 10 fails. Thus together we ha ve shown that ¬ (1) = ⇒ ¬ (3) = ⇒ ¬ (4) = ⇒ ¬ (1). Corollary 2. F or an y P ′ ⊆ P , the ﬂows { 1 { e ∈ P } } P ∈P ′ are linearly indep enden t if and only if P ′ con tains no distinct, rearrangemen t-equiv alen t sequences of paths. Pr o of. ( ⇐ =): Supp ose, by wa y of contraposition, that these ﬂo ws are linearly dep endent. Th us there exist scalars { w P } P ∈P ′ , not all zero, such that P P ∈P ′ w P 1 { e ∈ P } = 0. Deﬁne mass functions π + ( P ) = max { 0 , w P } and let π − ( P ) = max { 0 , − w P } in R P . As ev ery path leav es the source from a unique edge, and π + and π − put the same total mass on each edge, π + and π − ha ve equal total mass. F urther, as P P ∈P [ π ( P ) + 1 { e ∈ P } − π − ( P ) 1 { e ∈ P } ] = 0, i.e. the zero quasi-ﬂow, w e obtain that π + and π − are path decomp ositions of the same quasi-ﬂow, whic h is supp orted only on the edge set of paths in P ′ . By Theorem 10 , P ′ then con tains a pair of distinct, rearrangement-equiv alen t sequences, as desired. (= ⇒ ) : Conv ersely , supp ose { 1 { e ∈ P } } P ∈P ′ are linearly indep endent. Th us these v ectors span a simplex, and any path decomp osition is necessarily unique. Then, by equiv alence of (1) and (3) in Theorem 10 , P ′ con tains no pair of distinct, rearrangement-equiv alent sequences. Theorem 11. Fix a graph ( N , E , s, t ). As a sligh t abuse of notation, let ∆( P ′ ) denote the set of path decompositions whose supp ort is a subset of P ′ ⊆ P . The follo wing are equiv alen t for π ∈ ∆( P ′ ). 1. π is an extreme p oin t of ∆( P ′ ) ∩  π + R  . 2. The set { 1 { e ∈ P } : π ( P ) > 0 } P ∈P ′ is linearly indep endent. 3. supp( π ) con tains no distinct, rearrangemen t-equiv alen t sequences of paths. 4. F or ev ery ﬁnite sequence of unweighte d Ryser sw aps { R i } K i =1 (allo wing for rep etition): K X i =1 R i ∈ R P supp( π ) ⇐ ⇒ K X i =1 R i = 0 . Pr o of. The set ∆( P ′ ) ∩  π + R  deﬁnes a momen t set in the sense of Winkler ( 1988 ), i.e. it is a set of probabilit y measures whic h satisfy some auxiliary , ﬁnite system of (linear) equalit y constrain ts. By Prop osition 2.1.a of Winkler ( 1988 ), π is an extreme point of ∆( P ′ ) ∩  π + R  if and only if the set { 1 { e ∈ P } : π ( P ) > 0 } P ∈P ′ is linearly indep endent. This shows the equiv alence of (1) and (2). The equiv alence b etw een (2), (3), and (4) follo ws directly from Corollary 2 and Theorem 10 . 40 Pro of of Theorem 2 and Corollary 1 Our Theorem 2 corresp onds to the equiv alence b etw een (1) and (3) in Theorem 11 applied to the random utilit y graph. Similarly , Corollary 1 follo ws from the equiv alence b et w een conditions (1) and (3) in Theorem 10 applied to the random utility graph. A.2.2 Graphical Ordered Results Fix a graph and let π b e a path decomp osition of quasi-ﬂo w f . Supp ose that ⊵ is a linear order ov er the edges of the ﬁxed graph. W e say that π is swap-progressiv e (with resp ect to ⊵ ) if supp( π ) can b e ordered P 1 , . . . , P N suc h that, whenever P i and P j are compatible at some no de n , then i > j implies that ( n, m i ) ⊵ ( n, m j ) where ( n, m i ) and ( n, m j ) are the edges of P i and P j , resp ectiv ely , lea ving no de n . Theorem 12. Fix a graph, a quasi-ﬂo w function f , and a linear order ⊵ ov er the edges of our graph. There is some sw ap-progressiv e path decomp osition of f and it is unique. Pr o of. W e ﬁrst show uniqueness. Supp ose, to w ard a con tradiction, that the quasi-ﬂo w f admits t wo distinct swap-progressiv e path decomp ositions, π 1 and π 2 . Consider the zero- sum signed measure π 1 − π 2 . Since π 1 and π 2 decomp ose the same quasi-ﬂow, π 1 − π 2 = P N i =1 c i R i where c i R i ∈ R for all i = 1 , . . . , N . Deﬁne the signed measures π + , π − ∈ R P via π + ( P ) = max { 0 , π 1 ( P ) − π 2 ( P ) } and π − ( P ) = max { 0 , π 2 ( P ) − π 1 ( P ) } . As π 1 , π 2 ≥ 0 , supp( π + ) ⊆ supp( π 1 ) and similarly supp( π − ) ⊆ supp( π 2 ). By restriction of the ordering of the supp orts of π 1 and π 2 , the supp orts of π + and π − still satisfy the deﬁnition of swap- progressivit y and moreo v er: X P ∈ supp( π + ) π + ( P ) 1 { e ∈ P } = X P ′ ∈ supp( π − ) π − ( P ′ ) 1 { e ∈ P ′ } for ev ery e ∈ E and th us are path decompositions of the same quasi-ﬂo w. Let P + b e the highest ordered path in the supp ort of π + . As π − equals π + mo dulo a ﬁnite sequence of w eigh ted Ryser sw aps, there is some path P − in the supp ort of π − and some num b er k > 0 such that s ↑ k ( P − ) = s ↑ k ( P + ). Let P − b e the highest ordered path in the supp ort of π − suc h that s ↑ k ( P − ) = s ↑ k ( P + ) for some k > 0. F urther, s k ↓ ( P − )  = s k ↓ ( P + ) for the corresp onding k by disjoin tedness of the supp orts of π + and π − . Let e + and e − denote the ﬁrst edges at which P + and P − disagree. Since π − can b e reac hed from π + via a sequence of w eighted Ryser sw aps, we get that there is some path, P ′ + , in the supp ort of π + suc h that e − ∈ P ′ + . By P + b eing the highest ordered path in the supp ort of π + , we get that e + ⊵ e − in our exogenous ordering. Once again, b y π + b eing π − plus a sequence of Ryser sw aps, we 41 get that there is some path in the supp ort of π − , P ′ − , which has e + ∈ P ′ − . There are tw o cases. 1. s ↑ k ′ ( P ′ − ) = s ↑ k ′ ( P + ) for some k ′ > 0. In this case, we ha ve P − > P ′ − b y P − b eing the highest ordered path for whic h s ↑ k ( P − ) = s ↑ k ( P + ) for some k > 0. Ho w ever, w e ha v e that e + ⊵ e − , which gives us P ′ − > P − as b oth e + and e − originate from the same no de. This is a contradiction to swap- progressiv eness. 2. s ↑ k ( P ′ − )  = s ↑ k ( P + ) for all k > 0. Let e ′ + and e ′ − denote the edges where P ′ − and P + ﬁrst disagree. W e now hav e e ′ + ∈ P + ∩ P − (b y s ↑ k ( P − ) = s ↑ k ( P + ) for some k > 0 and e ′ + b eing the ﬁrst edge of P + ) and e ′ − ∈ P ′ − . Once again, applying the Ryser sw ap theorem, w e get that there is some path in the supp ort of π + , P ′ + , suc h that e ′ − ∈ P ′ + . This, plus P + b eing the highest ordered path, giv es us that e ′ + ⊵ e ′ − . Ho wev er, e + ⊵ e − means that P ′ − > P − and e ′ + ⊵ e ′ − means that P − > P ′ − . This is a contradiction to sw ap-progressiveness. By con tradiction in our t w o cases, we get that there cannot b e t wo diﬀerent swap-progressiv e path decompositions for the same ﬂo w function, and so we are done. W e now show that every quasi-ﬂow function can b e induced b y a sw ap-progressive path decomp osition. W e present an algorithm which tak es as input a quasi-ﬂow and returns the unique sw ap-progressive path decomp osition. 1. Initialize at k = 0. Set f 0 ( e ) = f ( e ) and π 0 ( P ) = 0 for all P ∈ P . 2. No w, let k = k + 1. (a) Set l = 1, n 1 = s , and ˜ P k = ∅ . Let e b e the highest ranked edge according to ⊵ at no de n l suc h that f k ( e ) > 0. Set ˜ P k = ˜ P k ∪ { e } and m = n l +1 where m is the head of e . (b) Set l = l + 1. By prop erty (i) of ( 4 ), at every interior no de, we hav e that there is some edge e whose tail is n l suc h that f k ( e ) > 0. 65 Let e b e the highest ranked edge according to ⊵ whose tail is n l satisfying f k ( e ) > 0. Set ˜ P k = ˜ P k ∪ { e } and m = n l +1 where m is the head of e . (c) If n l +1 = t , pro ceed to the Step 3. Otherwise, return to 2.(b). 3. W e ha ve now deﬁned a path P k = ˜ P k . Set π k ( P k ) = min e ∈ P k f k ( e ). 65 W e note that the algorithm starts with prop erty (i) of ( 4 ) holding. At ev ery iteration of the algorithm, w e subtract out a vector proportional to a ﬂow along a single path, thus preserving prop erty (i). 42 4. F or e ∈ P k , deﬁne f k +1 ( e ) = f k ( e ) − π k ( P k ). F or all other e , deﬁne f k +1 ( e ) = f k ( e ). 5. If f k +1 ( e ) = 0 for all e ∈ E , terminate, else return to step 2. When this algorithm terminates, it pro duces a swap-progressiv e path decomp osition. T o see that it yields a path decomp osition, note that at eac h iteration (i.e. for eac h k ), w e subtract min e ∈ P k f k ( e ) from each f k ( e ) for e ∈ P k . This means that ﬁrstly , f k ≥ 0 at each step of the algorithm. Second, at each iteration, w e are zeroing the mass on some edge whic h previously had strictly p ositive w eight. As the underlying graph is ﬁnite, we obtain that after ﬁnitely many steps, ev ery edge will satisfy f k ( e ) = 0. W e now argue this path decomp osition is swap-progressiv e. Note that in each iteration of the algorithm, the path w e construct alwa ys c ho oses the highest ranked av ailable (i.e. with strictly p ositive weigh t) edge at eac h no de it visits. F urther, as discussed in the previ- ous paragraph, the set of edges with strictly p ositive weigh ts prop erly decreases with each iteration. It follo ws that, for k < l , whenever P k and P l visit the same no de in the graph, P k lea ves on a w eakly higher edge than P l according to ⊵ . This corresp onds precisely to sw ap-progressivity . Pro of of Theorem 3 Pr o of. In the random utility graph, each edge is of the form ( A, A \ { x } ). Assign edges of this form to the alternativ e x . Given the linear order o v er alternatives ⊵ X , choose any linear order ⊵ E o ver edges in the random utilit y graph suc h that x ⊵ X y implies that ( A, A \ { x } ) ⊵ E ( B , B \ { y } ). P art (i) from Theorem 3 no w follo ws immediately from applying Theorem 12 to the random utility graph using the linear order ⊵ E . 66 W e no w prov e claim (ii). First, w e show that a single-crossing represen tation is also a sw ap-progressive representation. Suppose that ⊵ X is the linear order o v er alternatives for b oth the single-crossing and swap-progressiv e represen tations. Supp ose tow ards a con- tradiction that ≻ i and ≻ j , b oth in the support of the single-crossing represen tation, are non-trivially compatible, i > j , and that x k +1 j ⊵ x k +1 i . Note that x k +1 j ≻ j x k +1 i . This, the single-crossing prop ert y , x k +1 j ⊵ x k +1 i , and i > j tells us that x k +1 j ≻ i x k +1 i . How ev er, w e ha ve that x k +1 i ≻ i x k +1 j , whic h is a con tradiction. Thus any single-crossing represen tation is also sw ap-progressive. No w, it follo ws from Filiz-Ozba y and Masatlioglu ( 2023 ) that a progressiv e represen tation is a random utility represen tation if and only if it is single-crossing. Th us, b y the preceding paragraph, if a sw ap-progressive representation is single-crossing, it is also progressiv e. Since 66 Note that w e could hav e used any order ov er edges for this construction. This translates to part (i) from Theorem 3 holding even in the case of a menu dep enden t order ov er alternatives. 43 ev ery sw ap-progressive represen tation is a random utilit y represen tation, it follo ws that ev ery sw ap-progressive represen tation which is also progressiv e m ust b e a random utilit y represen- tation and thus single-crossing. A.3 P arametric In v ertibilit y Results A.3.1 T op ological Preliminaries Let X , Y b e top ological spaces. A con tinuous map p : X → Y is a cov ering map if, for ev ery y ∈ Y there exists a connected op en neighborho o d V such that p − 1 ( V ) is a disjoint union of op en sets { U i } i ∈I in X eac h of which p maps homeomorphically on to V . When Y is connected, the cardinalit y of the set I do es not dep end on the choice of y ∈ Y , and |I | is referred to as the n umber of sheets of the cov ering p . A homeomorphism D : X → X is called a dec k transformation if ( p ◦ D )( x ) = p ( x ) for all x ∈ X . The set of all deck transformations for a cov ering map form a group under comp osition. Fix x 0 ∈ X and y 0 ∈ Y . The structure of a cov ering map is intimately tied to the fundamen tal groups π 1 ( X , x 0 ) and π 1 ( Y , y 0 ). 67 W e gather sev eral relev an t results b elow. Lemma 5 ( Bredon 2013 , Lemma I I I.4.4) . Supp ose D is a dec k transformation for p such that D ( x ) = x for some x ∈ X . Then D is the iden tity . 68 Lemma 6 ( Bredon 2013 , Corr. I I I.5.3) . If π 1 ( X , x 0 ) is trivial, then the cardinality of π 1 ( Y , y 0 ) equals the num b er of sheets of p . Lemma 7 ( Bredon 2013 , Corr. I I I.6.10) . If π 1 ( X , x 0 ) is trivial, the group of dec k transfor- mations of p is isomorphic to π 1 ( Y , y 0 ). W e say that a con tinuous map b etw een top ological spaces is prop er if the preimage of an y compact set is compact. The following shows condition (ii) of Theorem 4 gives a simple c haracterization of prop erness. Lemma 8 ( Lee 2012 Prop. 2.17) . The map ¯ φ satisﬁes condition (ii) of Theorem 4 if and only if it is prop er. The ﬁnal ingredient we require is a ﬁxed-p oint theorem for p erio dic functions with prime p erio d. 69 67 F or the basic deﬁnitions and theory of fundamental groups, see, e.g. Bredon ( 2013 ) Chapter I I I. 68 The statemen t here is consequence of Lemma I I I.4.4 in Bredon ( 2013 ) where, in his notation, W = X and f = p . Alternatively , our statement here app ears as an unlab eled remark at the top of p. 148. 69 While Smith ( 1934 ) do es not explicitly state the requiremen t that n b e prime, this requirement is implicit in his pro of. See also Eilenberg ( 1940 ), F o otnote 2. 44 Theorem ( Smith 1934 ) . Supp ose X ⊆ R n is contractible, and h : X → X a homeomorphism suc h that the n -fold comp osition ( h ◦ · · · ◦ h ) = id for prime n ∈ N . Then h has a ﬁxed p oin t. A.3.2 Pro of of Theorem 4 Pr o of. Necessity of (i) is standard, whereas (ii) follows from Lemma 8 . W e no w consider suﬃciency . Condition (i) implies that ¯ φ is a lo cal diﬀeomorphism ( Lee 2013 , Prop. 4.8). In particular, this implies ¯ φ is an op en map, hence ¯ φ (Θ) ⊆ R d is op en and therefore a smo oth manifold. F urthermore, Lemma 8 implies ¯ φ is prop er. Finally , as X is contractible, it is connected, hence so to o is ¯ φ (Θ). Then Lee ( 2013 ) Prop. 4.46 implies ¯ φ : Θ → ¯ φ (Θ) is a smo oth co vering map. Fix some vector y 0 ∈ ¯ φ (Θ). As ¯ φ is a cov ering map, the pre-image ¯ φ − 1 ( y 0 ) is discrete; since ¯ φ is prop er, ¯ φ − 1 ( y 0 ) is ﬁnite, and by the connectedness of ¯ φ (Θ), the preimage of every y ∈ ¯ φ (Θ) has the same cardinality . Now, as Θ con tractible, it is simply connected, th us Lemma 6 implies π 1  ¯ φ (Θ) , y 0  is ﬁnite, and by Lemma 7 , π 1  ¯ φ (Θ) , y 0  acts on Θ via deck transformations such that the identit y 1 ∈ π 1  ¯ φ (Θ) , y 0  is the only element mapp ed to the iden tity deck transform Θ → Θ. 70 Supp ose now, for sake of contradiction, that ¯ φ (Θ) is not simply connected. Then there exists some α ∈ π 1  ¯ φ (Θ) , y 0  , where α  = 1 . Since π 1  ¯ φ (Θ) , y 0  is ﬁnite, α must b e of ﬁnite order, hence for some n > 1, we hav e α n = 1 , and α m  = 1 for all m < n . Without loss of generalit y , we ma y supp ose n is prime. 71 Then under the action of π 1  ¯ φ (Θ) , y 0  on Θ, α corresp onds to some non-identit y dec k transformation D α : Θ → Θ of prime p erio d n . But by Smith’s Theorem , D α m ust ha v e a ﬁxed p oint, thus Lemma 5 implies D α is the identit y , a con tradiction. Thus π 1  ¯ φ (Θ) , y 0  = { 1 } and Lemma 6 implies the preimage ¯ φ − 1 ( y 0 ), and hence ¯ φ − 1 ( y ) for ev ery y ∈ ¯ φ (Θ), is singleton; in particular this means ¯ φ is injective. By Lee ( 2013 ) Prop. 4.33.b, ¯ φ is then a diﬀeomorphism, as desired. 70 This action is simply the isomorphism whose existence is asserted by Lemma 7 . Since the kernel of any group isomorphism is trivial, the claim follows. 71 If n is not prime, let m b e any prime factor of n and consider α n m . 45 Online App endix for “Iden tiﬁcation in Sto c hastic Choice” P eter Caradonna ∗ and Christopher T uransic k † F ebruary 24, 2026 B Bey ond Random Utilit y B.1 Random Choice Mo dels The pro of of Theorem 5 , as w ell as the corresp onding analogues of Theorem 2 and Theorem 3 , follo w from Theorem 8 , Theorem 11 , and Theorem 12 applied to the following graphical represen tation of random c hoice rules. Let Σ ⊆ 2 X \ { ∅ } denote the collection of observ able men us, and index these A 1 , . . . , A | Σ | . Deﬁne a graph with N = Σ ∪ { ∅ } , and for every 1 ≤ i ≤ | Σ | and for eac h x ∈ A i there is a unique edge A i → A i +1 in E , where w e deﬁne A | Σ | +1 = ∅ . 1 F or the edge e ∈ E of the form A i → A i +1 asso ciated with the alternative x ∈ A i , w e deﬁne f ( e ) = ρ ( x, A i ). Since ρ ≥ 0 and P x ∈ A i ρ ( x, A i ) = 1, b y ( 4 ), f is a ﬂow. In the random choice graph, each path corresp onds to a sequence of the form ( x 1 , A 1 ) , ( x 2 , A 2 ) , . . . , ( x | Σ | , A | Σ | ). This path exactly corresp onds to the c hoice function whic h chooses x i from A i . Thus there is a one-to-one corresp ondence betw een P and C Σ , and hence b etw een path decomp ositions of f and random c hoice represen tations. Moreov er, a function f : E → R + admits a path decomp osition if and only if it is a (quasi-)ﬂow, hence path decomp ositions on this graph precisely corresp ond to the identiﬁed sets of the random choice mo del. W e now pro ceed with the pro of of Theorem 5 . ∗ Division of the Humanities and So cial Sciences, Caltech. Email: ppc@caltech.edu . † Departmen t of Decision Sciences, Bo cconi Universit y . Email: christopher.turansick@unibocconi. it . 1 Th us E is a multiset. 1 Pr o of. As an application of Lemma 3 to the graphical representation describ ed ab ov e, w e ha ve observ ational equiv alence b et ween tw o distributions o ver choice functions, ν and µ , if and only if ν = µ + P n i =1 c i R ′ i where R i do not corresp ond to the Ryser sw aps in Theorem 5 but the Ryser sw aps from Lemma 3 . F or the purp oses of this pro of, call the Ryser swaps from Theorem 5 pseudo-Ryser sw aps. W e no w sho w that every pseudo-Ryser swap can b e generated by a series of Ryser sw aps and ev ery Ryser sw ap can b e generated by a series of pseudo-Ryser sw aps. Consider an unw eigh ted Ryser swap. Given tw o choice functions ( c 0 , c 1 ) it creates choice functions ( c 2 , c 3 ) where c 2 agrees with c 0 strictly ab o v e A i (according to the indexing of sets used in our graphical representation) and agrees with c 1 w eakly b elo w A i . F urther, c 3 agrees with c 1 strictly ab ov e A and agrees with c 0 w eakly b elo w A i . Now our aim is to construct a pseudo-Ryser swap out of Ryser sw aps. W e consider a Ryser sw ap b etw een c 0 and c 1 at c hoice set A i . Start with t wo choice functions ( c 0 , c 1 ). Ryser sw ap these choice functions at c hoice set A i . This induces c hoice functions ( c 2 , c 3 ). Consider the c hoice set A i +1 . No w Ryser sw ap ( c 2 , c 3 ) at choice set A i +1 . The resulting choice functions ( c 4 , c 5 ) satisfy the following. 1. c 4 agrees with c 0 on ev ery set B  = A i and agrees with c 1 at A i 2. c 5 agrees with c 1 on ev ery set B  = A i and agrees with c 0 at A i Th us ( c 4 , c 5 ) are the resultant c hoice functions of a pseudo-Ryser swap on ( c 0 , c 1 ). No w we sho w that a Ryser sw ap can b e constructed from pseudo-Ryser swaps. A Ryser sw ap swaps the choices b et w een c 0 and c 1 at every c hoice set A j satisfying j ≥ i for some c hoice set A i . Simply apply a pseudo-Ryser swap at each of these c hoice sets. This sequence of pseudo-Ryser swaps results in a Ryser sw ap. The prior equiv alence sho ws that the span of Ryser sw aps and the span of pseudo-Ryser swaps is the same. As suc h, by Lemma 3 , we get that Theorem 5 holds. B.2 F rame Dep endent Random Utility As mentioned in Section 7 , there is a more general v ersion of the frame-dep endent random utilit y mo del. Instead of an alternative b eing framed and not framed, there are no w multiple frames. W e lab el these frames as { 1 , . . . , M } . In addition, there is a second version of the mo del where the frames are lab eled as { 0 , 1 , . . . , M } and the zero frame corresp onds to an alternativ e not b eing av ailable. F or now, we will fo cus on the v ersion of the mo del without the zero frame. Let x m denote alternative x under frame m . Mirroring our notation from Section 7 , deﬁne X ∗ = { x 1 1 , . . . , x 1 K , . . . , x M 1 , . . . , x M K } denoting the set of alternativ e-frame pairs. A 2 men u corresp onds to some subset A ⊊ X ∗ satisfying, for eac h i ∈ { 1 , . . . , K } , there is exactly one m suc h that x m i ∈ A . Thus a men u contains every alternative x i with eac h alternativ e under some alternativ e-dep endent frame m . The frame-dep endent random utility mo del considers strict preferences ov er the set X ∗ under one additional assumption. The mo del restricts to preferences ≻ suc h that, if m > m ′ , then x m i ≻ x m ′ i . Giv en a preference ≻ , observe that if, for m > 1 and arbitrary i  = j , x 1 j ≻ x m i , then there is no menu A in which x m i is maximal. With this in mind, we now deﬁne a truncated preference ≻ ∗ . Giv en a preference ≻ , let ≻ ∗ b e the ranking whic h agrees with ≻ up to, and including, the b est alternativ e under frame 1. That is, ≻ ∗ is formed b y taking ≻ and remo ving all alternatives x m i whic h rank b elo w the highest alternative of the form x 1 j . Let L ∗ denote the set of truncated strict preferences. The data are compatible with the frame-dep endent random utilit y mo del if there exists some distribution µ ∈ ∆( L ∗ ) suc h that: ρ ( x, A ) = X ≻ ∗ ∈L ∗ µ ( L ∗ ) 1 { x is ≻ ∗ -maximal in A } for all men us A and eac h x ∈ A . Here, x corresp onds to some alternative x m i and men u A satisﬁes the assumption we hav e made in this section. As b efore, we say that t wo truncated preferences ≻ ∗ 1 , ≻ ∗ 2 ∈ L ∗ are k -compatible if they agree on their k -b est alternatives, though not necessarily their ranking. A Ryser sw ap for the frame-dependent model is simply a signed measure in R L ∗ whic h places unit mass on a k -compatible pair ≻ ∗ 1 and ≻ ∗ 2 and mass negative one on the pair of truncated preferences obtained by sw apping their k -initial segments. Let R f d ⊂ R L ∗ denote the linear span of these v ectors. Theorem 6 still holds under this form ulation of the frame-dep enden t random utility mo del and R f d . The graphical represen tation of the frame dep endent random utility mo del is due to Cheung and Masatlioglu ( 2024 ) although w e make a slight alteration to ﬁt exactly in to our framework. Let X denote the collection of all menus A ⊊ X ∗ satisfying, for eac h i ∈ { 1 , . . . , K } , there is exactly one m such that x m i ∈ A . Consider the directed acyclic graph where N = X ∪ { t } and the edge set is describ ed as follo ws. • F or each A ∈ X and eac h x m i ∈ A with i > 1, there is an edge from A to ( A \ { x m i } ) ∪ { x m − 1 i } , A → ( A \ { x m i } ) ∪ { x m − 1 i } • F or eac h A ∈ X and eac h x 1 i ∈ A , there is an edge from A to t , A → t Here, the source no de corresp onds to the no de A ∗ where A ∗ is the menu giv en b y { x M 1 , . . . , x M K } . In order to describ e our ﬂow function, ﬁrst observ e that each menu A can b e represented as a v ector in ¯ A ∈ R X . Each dimension of ¯ A is indexed b y some elemen t x ∈ X . The dimension 3 of ¯ A corresp onding to alternativ e x is giv en b y the frame associated with x in menu A . Th us, if x m i ∈ A then the i -th dimension of ¯ A is m . Under this observ ation, the v ectors ¯ A can b e partially ordered b y the greater than or equal to relationship. As an abuse of notation, we rank menus A ≥ B if ¯ A ≥ ¯ B . Given the frame-dep enden t random choice rule ρ ( x, A ), we can implicitly deﬁne a new ob ject: ρ ( x, A ) = X A ≤ B : ¯ A x = ¯ B x y ( x, B ) , where ¯ A x denotes the en try in vector ¯ A in the dimension indexed by x . Cheung and Masatli- oglu ( 2024 ) provides a closed form expression for y ( x, A ), but, to keep things simple, w e only presen t the recursive deﬁnition here. F or our graphical representation, we use y ( x, A ) as the ﬂo w assigned to the edge leaving no de A which corresp onds to alternativ e x . Cheung and Masatlioglu ( 2024 ) sho ws that, whenev er the frame-dep enden t random c hoice rule has a frame-dependent random utility represen tation, this constitutes a ﬂow as in Equation 4 . Our in terpretation of the frame-dep endent random utility graph relies on our notion of a truncated preference. Consider the path giv en b y A ∗ → B → · · · → t . Observe that B diﬀers from A ∗ b y a single elemen t and only by the frame of that element. Sp eciﬁcally , A ∗ has x M 1 while B has x M − 1 1 . This generalizes to successiv e no des in our path. W e now rewrite our path with sup erscripts on each → corresp onding to the diﬀerence b et ween these t wo successiv e no des: A ∗ → x M 1 B → x m i · · · → x 1 j t. This path is then asso ciated with the truncated preference x M 1 ≻ ∗ x m i ≻ ∗ · · · ≻ ∗ x 1 j . Just as in the previous cases, a path decomp osition putting mass on a path corresp onds to a probabilit y distribution putting mass on its corresp onding truncated preference. Theorem 6 is then recov ered b y applying Theorem 8 to the frame-dep endent random utilit y graph. As we men tioned earlier, there is a second version of the frame-dep enden t random utilit y mo del which allows for v ariation in whic h alternativ es are a v ailable. This can b e enco ded as a zero frame whic h places suﬃcien tly negativ e utilit y on an alternative in order to make it nev er b e c hosen. The graphical represen tation whic h Cheung and Masatlioglu ( 2024 ) pro vides for this version of the mo del satisﬁes the assumptions w e make on the graphs we consider in Theorem 8 . As suc h, our results can be applied directly out of the b o x to the frame-dep endent random utility mo del with av ailabilit y v ariation. In this case, types corresp ond to full, strict preferences (i.e. not truncated) ov er X ∗ . 4 B.3 Dynamic Discrete Choice In this section, w e extend the setup w e describ ed in Section 7 to allow each p erio d’s menu to dep end on the previous p erio d’s choice. There is a sequence t = 1 , . . . , T of discrete time p erio ds. In each, an agent c ho oses from some men u ∅ ⊊ A t ⊆ X . W e assume A 1 is exogenously given and, for all 2 ≤ t ≤ T − 1, that the menu A t faced by the agen t at time t is determined by: A t ( x t − 1 ) ≡ g t − 1 ( x t − 1 ) , where x t − 1 ∈ X denotes the agent’s choice at t − 1, and g t − 1 : X → 2 X \ { ∅ } is a ﬁxed ev olution function, describing ho w the agen t’s choice at t − 1 determines their c hoice set at time t . W e consider an empiricist who observes a system of conditional choice probabilities. F or- mally , this consists of (i) a distribution of time-one c hoice probabilities ρ 1 ∈ ∆( X ) supp orted on A 1 , and (ii) a family of functions ρ t : X × X → [0 , 1], for 2 ≤ t ≤ T , suc h that: X a ∈ A t ( b ) ρ t ( a | b ) = 1 , whenev er ρ t − 1 ( b | · ) is not everywhere zero, and ρ t ( a | b ) = 0 for all x ∈ X otherwise. Giv en suc h data, w e seek to describ e the set of compatible distributions o ver choice histories . T o represent this problem as one of describing ﬂo w functions, we deﬁne N recursively , as the set of all pairs in ( t, x ) suc h that either (i) t = 1 and x ∈ A 1 , or (ii) 2 ≤ t ≤ T , and x ∈ A t ( x ′ ) for some ( t − 1 , x ′ ) ∈ N , as well as an abstract source and sink, s and t . Similarly , w e let E consist of all directed edges of the form ( t − 1 , x ′ ) → ( t, x ), where x ∈ A t ( x ′ ), as w ell as edges from the source s to each pair in { 1 } × A 1 , and edges from eac h pair in N with t = T to the sink t . Giv en such a graph, we recursively deﬁne the ﬂo w from the data ρ t . F or the edge connecting the source no de to (1 , x ), w e assign ρ 1 ( x ) as the ﬂo w assigned to that edge. F or edges connecting ( t − 1 , x ′ ) to ( t, x ), w e normalize ρ t ( x | x ′ ) by multiplying by the total in-ﬂo w in to no de ( t − 1 , x ′ ) and assign this new v alue as the ﬂow for this edge. Finally , for the edge connecting ( T , x ) to the sink, w e simply assign the total in-ﬂo w in to no de ( T , x ) as the ﬂo w. It is straigh tforward to verify that our construction deﬁnes a ﬂow. In fact, our construction recov ers the absolute frequency of conditional choices in each p erio d. Eac h path in our constructed graph corresp onds to a full history of c hoices and a ﬂo w function of this graph assigns frequencies to eac h full history . Theorem 7 can b e recov ered by applying Lemma 3 to this graphical representation under the assumption that the choice set is ﬁxed across time p erio ds and g t − 1 ( · ) is equal to our ﬁxed 5 c hoice set. A v ersion of Theorem 7 can b e recov ered for the more general setup describ ed here. W e simply need to restrict ourselves to considering sw aps b etw een histories which are p ossible giv en g t − 1 ( · ). B.4 Correlated Random Utilit y In addition to the three t yp es of mo dels we prop ose in Section 7 , our techniques can also b e used to study identiﬁcation in the correlated random utility model of Chambers et al. ( 2024 ) as well as the consumption dep endent random utility mo del of T uransick ( 2024 ). Supp ose that an analyst has access to a random joint c hoice rule, ρ : X 2 × (2 X \ { ∅ } ) 2 → [0 , 1], whic h satisﬁes, for all non-empty A, B ⊆ X , X x ∈ A X y ∈ B ρ ( x, y , A, B ) = 1 . The correlated random utilit y mo del asks that there is distribution µ ∈ ∆( L 2 ) such that ρ ( x, y , A, B ) = X ≻∈L X ≻ ′ ∈L µ ( ≻ , ≻ ′ ) 1 { x ≻ A \ x, y ≻ ′ B \ y } . The correlated random utilit y mo del represents c hoice in tw o p erio ds where choices in the t wo p erio ds may be correlated through the distribution ov er preference pairs, but, conditional on this dra w, the agent’s choice in the ﬁrst p erio d has no impact on their choice in the second p erio d. On the other hand, the consumption dep enden t random utilit y mo del asks that there is a distribution µ ∈ ∆( L ) as w ell as distributions for eac h alternativ e preference pair, ( x, ≻ ), ν ( x, ≻ ) ∈ ∆( L ) such that ρ ( x, y , A, B ) = X ≻∈L µ ( ≻ ) 1 { x ≻ A \ x } X ≻ ′ ∈L ν ( x, ≻ ) ( ≻ ′ ) 1 { y ≻ ′ B \ y } . The consumption dep endent random utilit y mo del represents c hoice in tw o p erio ds where an agen t’s preference, and thus choice, in the second p erio d dep ends b oth on their preference and c hoice in the ﬁrst p erio d. In eac h of these mo dels, ﬁrst p erio d choices are w ell deﬁned in- dep enden tly of the second p erio d’s choice set. As such, w e use ρ 1 ( x, A ) = P y ∈ B ρ ( x, y , A, B ) for an y c hoice of non-empty B . Both of these mo dels share the same graphical representation, but, while every path on this graph corresp onds to some type in the consumption dep enden t random utilit y mo del, the correlated random utility mo del constitutes a supp ort restriction on this graph. The graphical representation for these t w o mo dels consists of a series of graphs, one represen ting 6 c hoice in the ﬁrst p erio d and the rest, one for eac h edge of the ﬁrst perio d graph, representing c hoice in the second perio d. These series of graphs can then b e made into a single graph b y replacing eac h edge of the ﬁrst p erio d graph with the corresp onding second p erio d graph. T o b egin, the ﬁrst p erio d graph is exactly the random utilit y graph with ﬂo ws coming from ρ 1 and Equation 6 . Eac h second p erio d graph has the same no de and edge set as the random utilit y graph. That is N = 2 X and E =  ( A, B ) | B = A \ { a } , a ∈ A  , where s = X and t = ∅ . The second p erio d graphs diﬀer from the ﬁrst p erio d graphs in their quasi-ﬂo w function. F or the second p erio d graph asso ciated with the edge ( A → A \ { a } ) in the ﬁrst p erio d graph, the random joint c hoice rule ρ induces a unique ﬂow as follo ws: f ( B → B \ { b } ) = X A ⊆ A ′ X B ⊆ B ′ ( − 1) | A ′ \ A | + | B ′ \ B | ρ ( a, b, A ′ , B ′ ) . (7) The function f given in Equation 7 deﬁnes a quasi-ﬂo w as shown in Chambers et al. ( 2024 ). In the consumption dep enden t random utilit y mo del, a t yp e corresp onds to a preference in the ﬁrst p erio d and | X | preferences in the second p erio d. Eac h of these | X | preferences corresp ond to the type’s preference in the second perio d conditional on choosing alternative x in the ﬁrst p erio d. As is the case with the random utility graph, in the ﬁrst p erio d graph, as w ell as each of the second p erio d graphs, each path is bijectiv ely associated with a preference. Recall that this bijection is giv en b y regarding a path: X → X \ { x 1 } → · · · → X \ { x 1 , . . . , x N − 1 } → ∅ as a sequence of nested low er contour sets, whic h uniquely deﬁnes the preference x 1 ≻ · · · ≻ x N . With this bijection in mind, the path on the ﬁrst p erio d graph deﬁnes the preference whic h dictates the agen t’s ﬁrst p erio d choice. Given this path and edge ( A, A \ { a } ) in this path, a path on the second p erio d graph asso ciated with edge ( A, A \ { a } ) on the ﬁrst p erio d graph corresp onds to the agen t’s second p erio d preference conditional on c ho osing a in the ﬁrst p erio d. The consumption dep endent random utility mo del puts no restriction on whic h paths are allow ed in the second p erio d graph while the correlated random utility mo del asks that, for a given t yp e, all paths on the second p erio d graphs are the same. This graphical construction is most easily understoo d via a ﬁrst perio d graph and a series of second p erio d graphs. W e can directly apply our result ab out path decomp osition to eac h of these graphs to discuss uniqueness in the correlated and consumption dep endent random utilit y mo dels. As men tioned earlier, w e can alternatively replace eac h edge in the ﬁrst p erio d graph with its corresp onding second p erio d graph to get a single graph which acts as a graphical represen tation for these tw o mo dels. In this case, we need only apply our 7 graphical results to this single graph. While our fo cus in this section was to discuss ho w to extend the graphical represen tation of the random utilit y mo del in order to capture m ultidimensional analogues of the random utilit y mo del, this pro cess generalizes to other mo dels whic h can b e represented b y path decomp ositions of quasi-ﬂo ws on directed acyclic graphs. This allows for discussion of b oth correlation of types/paths across dimension, as in the correlated random utilit y mo del, as w ell as dep endence of later types/paths on earlier types-choice/path-edge pairs, as in the consumption dependent random utility mo del. B.5 Maximization of Other T yp es of Orders In the main b o dy of the pap er, we fo cused on agents who maximize linear orders. How ev er, an analyst ma y be w ary of this assumption and instead only ask that agents maximize a w eak order, an interv al order, or a semiorder. Under diﬀering data assumptions than what w e use for the random utilit y mo del, Da vis-Stob er et al. ( 2018 ) provides graphical represen tation for mo dels which consider a distribution ov er agents maximizing each of these three t yp es of orders. Doignon and Saito ( 2023 ) also studies the graphical representation of these mo dels in order to characterize adjacency of extreme p oints in their corresp onding p olytop es. The tec hniques we dev elop here can b e applied to discuss uniqueness in eac h of these mo dels. C F urther Results C.1 T ec hnical Results Lemma 9. Let R denote the subspace of R P spanned b y v ectors of the form: 1 { P,P ′ } − 1 { P ′′ ,P ′′′ } where ( P , P ′ ) are compatible and ( P ′′ , P ′′′ ) are their corresp onding conjugate, and by minor abuse of notation let ∆( P ′ ) denote the face of ∆( P ) ⊆ R P spanned by the abstract simplex ∅ ⊊ P ′ ⊆ P . Supp ose there exists t ∈ R P suc h that: dim   R + t  ∩ ∆( P ′ )  > 0 . Then there exists t ′ ∈ R P suc h that  R + t ′  ∩ ∆( P ′ ) contains a pair of distinct p oin ts in Q P . 8 Pr o of. Let V denote the subspace of R P giv en by: span  e P − e P ′  P,P ′ ∈P ′ , where e i denotes the i th standard Euclidean basis vector. By hypothesis, K = dim R ∩ V > 0 . F rom their deﬁnitions, b oth R and V admit bases { q i R } dim( R ) i =1 and { q j V } dim( V ) j =1 whic h b elong to Q P . Deﬁne the |P | × dim( R ) and |P | × dim( V ) matrices: Q R = h q 1 R · · · q dim( R ) R i and Q V = h q 1 V · · · q dim( V ) V i , resp ectiv ely , and let: Q = h Q R − Q V i Since each of the ab o ve matrices consists exclusiv ely of rational elemen ts, their resp ective column spaces trivially admit a bases in Q P . This implies that, by Gaussian elimination, the annihilators of their column spaces admit a bases in Q P ; in particular the annihilator of the column space of Q admits a rational basis { r K } K k =1 , where each vector r k = [ r k R | r k V ]. Let ¯ q k = Q R r k R  = Q V r k V  . By construction, { ¯ q k } k ⊂ Q P and, by construction { ¯ q k } k form a basis for R ∩ V . Since dim( R ∩ V ) > 0, there exists some non-zero ¯ q k ∈ R ∩ V ∩ Q P . Then letting: t ′ = 1 |P ′ | 1 P ′ , w e ha v e t ′ and t ′ + α ¯ q k , for small enough α ∈ Q ++ , are distinct rational v ectors in  R + t ′  ∩ ∆( P ′ ) as desired. C.2 Order-Based Restrictions and Extreme P oin ts W e b egin b y noting that every swap-progressiv e path decomp osition of a quasi-ﬂow is an extreme point of the set of path decomp ositions for the giv en quasi-ﬂo w. Prop osition 2. Consider a graph with quasi-ﬂow function f . Each sw ap-progressiv e path decomp osition of f is an extreme p oin t of the set of path decomp ositions of f . Pr o of. Supp ose w e hav e a swap-progressiv e path decomp osition π for some quasi-ﬂow func- 9 tion f . Note that any path decomp osition with the same supp ort is also a sw ap-progressiv e path decomp osition as swap-progressivit y is a prop erty of the supp ort. By Theorem 12 , w e kno w that a swap-progressiv e path decomp osition is unique. This means that no other path decomp osition whose supp ort is con tained within supp( π ) decomp oses the same quasi- ﬂo w. Th us, the set of ﬂows { 1 { e ∈ P } } P ∈ supp( π ) is linearly indep endent. By Corollary 2 and Theorem 11 , we get that π is an extreme p oint of the set of path decomp ositions inducing quasi-ﬂo w f . No w we note that our notion of sw ap-progressivity , for b oth graphs and random utilit y , is a special case of a more general notion of an ordered restriction. Consider a graph ( N , E , s, t ) and the set of paths P . Consider some linear order ov er P , ⊵ P . F or ease, ⊵ P is equiv alen t to an en umeration of P where P i ⊵ P P j ⇐ ⇒ i ≤ j . Given an ordering ov er edges ⊵ E , we can induce some ordering ov er paths ⊵ P . W e do this b y starting at no de s and sa ying that a path P ⊵ P P ′ if ( s, n ) ⊵ E ( s, n ′ ) where ( s, n ) and ( s, n ′ ) are the ﬁrst edges of P and P ′ resp ectiv ely . Generally , this will not yet lea ve us with a full linear ov er paths so w e iterate on this construction. Now mo v e onto an y no de n for which each edge of the form ( m, n ) has already b een considered. At no de n , each path P whic h passes through no de n will already b e partially ordered. F urther, by construction, incomparability by this partial order is an equiv alence relation among those paths which pass through no de n . F or eac h of these equiv alence classes, say that P ⊵ P P ′ if ( n, m ) ⊵ E ( n, m ′ ) where ( n, m ) and ( n, m ′ ) are edges in P and P ′ resp ectiv ely . By iterating this pro cess through eac h no de, we are left with a linear order o ver paths, ⊵ P . Recall that our construction of a swap-progressiv e pro ceeds as follo ws. 1. Set i = 1. 2. Put as m uc h mass on P i as p ossible (i.e. the minim um of f ( e ) among edges in P 1 ). Call this amount π ( P i ). 3. Set f ( e ) = f ( e ) if e ∈ P i . Set f ( e ) = f ( e ) − π ( P i ) if e ∈ P i . 4. Set i = i + 1. If i > |P | , terminate the construction. Otherwise, return to step 2. Notably , the construction w orks for any linear order o ver paths ⊵ P and, given a quasi-ﬂow function f , the induced path decomp osition of the quasi-ﬂow function will b e unique giv en ⊵ P . W e call a path decomp osition induced by suc h a construction an ordered path de- comp osition . Single-crossing, sw ap-progressiv e, and progressiv e represen tations are ordered path decomp ositions of the correctly chosen graph, but there are also ordered path decom- p ositions whic h are not single-crossing, swap-progressiv e, or progressive. While ordered path 10 decomp ositions share man y of the nice iden tiﬁcation prop erties of sw ap-progressivity , ordered path decomp ositions are not suﬃciently general enough to reco ver ev ery extreme p oin t of the set of path decomp ositions for ev ery quasi-ﬂo w. Prop osition 3. There exist graphs ( N , E , s, t ), ﬂows f , and extreme p oin ts of the set of path decomp ositions of f suc h that an y ordered decomp osition of f do es not induce the extreme point. F urther, the random utility graph has this prop ert y . Pr o of. Consider the graph in Figure 2 of Chambers and T uransick ( 2025 ) and the uniform distribution ov er the eight paths considered in Example 4.1 of the same pap er, i.e. putting 1 8 mass on eac h of the eight paths. Cham b ers & T uransic k sho w the indicators of these eight paths form a linearly indep endent set of ﬂo ws. As suc h, b y Winkler ( 1988 ), the uniform distribution is an extreme p oint of the set of edge decompositions inducing the corresp onding ﬂo w. No w, ﬁx an y ordered path decomp osition π . By deﬁnition (i.e. the construction in the paragraph preceding Prop osition 3 ), the highest ranked path in supp( π ) must contain some edge unique to it among paths in supp( π ). How ev er, as every path in Example 4.1 of Chambers and T uransic k ( 2025 ) shares each of its edges with at least one other path in the example, the uniform distribution ov er these paths cannot arise as an ordered path decomp osition, for any choice of order ⊵ E ; as the graph in this example is obtained as a supp ort restriction of the random utilit y graph, w e are done. D Examples and Computations Omitted F rom T ext D.1 F ailure of Iden tiﬁcation in Example 8 Let X = { x 0 , x 1 , x 2 } and consider any random c hoice rule ρ which satisﬁes: (i) ρ ( x 1 , x 0 x 1 ) = ρ ( x 2 , x 0 x 2 ) = 1 11 and (ii) ρ ( x 1 , x 0 x 1 x 2 ) = ρ ( x 2 , x 0 x 1 x 2 ) . (Recall here we only consider menus which include the outside option x 0 ∈ X , hence this completely sp eciﬁes ρ up to the c hoice frequency of x 0 from X .) Consider again the submo del s deﬁned in Example 8 . F or i = 1 , 2 w e ha ve: s i ( v 1 , c 1 , v 2 , c 2 ) = 1 + v i v i (1 + v i c i ) = 10 , or, simplifying: v 2 i c i = 10 11 for all i = 1 , 2. Similarly , by h yp othesis w e ha v e: s i +2 ( v 1 , c 1 , v 2 , c 2 ) = 1 + v i + v i +1 v i + v i v i +1 + c i v 2 i = K (8) for eac h i = 1 , 2 and some K > 0, hence simplifying yields: v 1 = v 2 . Th us, there are only tw o pieces of information w e ha v e not determined y et: the v alue of v 1 (without loss), and the v alue of K in ( 8 ), which pins do wn the c hoice probability of the outside option from the men u X . F rom ( 8 ), we obtain: 1 + 2 v 1 v 1 + v 2 1 + 10 = K , or: K  10 + v 1 + v 2 1 ) = 1 + 2 v 1 . When, e.g., K = 0 . 3 this has m ultiple solutions, i.e. v 1 = 5 3 and v 1 = 4. ■ D.2 A Non-Single Crossing Example Let X = { a, b, c, d } , and consider the random c hoice rule ρ , which chooses uniformly from an y men u. F or any linear ordering ⊵ of alternatives, ρ fails to satisfy the cen tralit y axiom of Ap esteguia et al. ( 2017 ), and hence do es not admit a single-crossing represen tation. How ev er, ρ is clearly consistent with the random utility mo del: for example it is rationalized by the uniform distribution on L . By Theorem 3 , the random choice rule ρ admits a swap-progressiv e rationalization for an y order ⊵ . T o see how swap-progressivit y pins do wn a unique represen tation, ﬁrst note that the uniform distribution on L places p ositiv e (and equal) mass on six distinct, non- o verlapping collections of four preferences, eac h consisting of a 2-compatible pair and its 2-conjugate. 2 Giv en ⊵ , the unique swap-progressiv e representation places (equal) mass on only t wo of the four preferences in eac h collection, namely those satisfying: (i)  x 1 ⊵ x 2 and x 3 ⊵ x 4  or (ii)  x 2 ⊵ x 1 and x 4 ⊵ x 3  . In other w ords, on each collection, the sw ap-progressiv e representation uses the order ⊵ to 2 These collections eac h consist of, for any { x, y } ⊂ X , the four preferences whose initial strings are xy or y x . 12 select one of the t wo compatible pairs to receive full mass. 3 Concretely , if a ⊵ b ⊵ c ⊵ d , the unique swap-progressiv e represen tation places equal mass on: ≻ 1 : dcba ≻ 2 : dbca ≻ 3 : dabd ≻ 4 : cdab ≻ 5 : cbda ≻ 6 : cadb ≻ 7 : bdac ≻ 8 : bcad ≻ 9 : badc ≻ 10 : adbc ≻ 11 : acbd ≻ 12 : abcd. Th us, even when a single-crossing representation may fail to exist, sw ap-progressivit y pro- vides a natural criteria for selecting rationalizations in a manner that incorp orates the natural ordering of the environmen t. ■ 3 See also Example 6 . 13

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