Adversarial Selection

Adversarial Selection * Alma Cohen † Alon Klement ‡ Zvika Neeman § Eilon Solan ¶ Mar ch 7, 2026 Abstract In many institutional settings, k items are selected with the goal of rep- resenting the underlying distribution of claims, opinions, or characteristics in a large population. W e study environments with two adversarial par - ties whose prefer ences over the selected items are commonly known and opposed. W e propose the Quantile Mechanism: one party partitions the population into k disjoint subsets, and the other selects one item from each subset. W e show that this procedur e is optimally representative among all feasible mechanisms, and illustrate its use in jury selection, multi-district litigation, and committee formation. [100 words] K E Y W O R D S : Nash implementation, cut-and-choose, Jury selection, MDL, re- districting committees. J E L C O D E S : D82, K40, K14. * W e thank Ron Solan for useful discussions, T amar Itzkovitz for valuable resear ch assistance, and Geoffroy de Clippel, Jack Fanning, Faruk Gul, T eddy Mekonnen, Fedor Sandomirskiy , Roberto Serrano, and Leeat Y ariv , as well as seminar audiences in Berlin, Brown, EUI, Prince- ton, and T el A viv for their comments. Solan acknowledges the support of the Israel Science Foundation, Grant #211/22. † Harvard Law School and Ber glas School of Economics, T el A viv University . ‡ Buchmann Faculty of Law , T el A viv University . § EUI and Berglas School of Economics, T el A viv University . ¶ School of Mathematical Sciences, T el A viv University . 1 Introduction In many institutional settings, a decision maker must select a small subset from a large population. The selected subset is intended to be repr esentative of the underlying distribution of pr efer ences, claims, opinions, or characteristics in the population. Y et the size of the subset is typically constrained by legal, ad- ministrative, or practical considerations, ruling out reliance on the large-sample guarantees of conventional statistical sampling theory . The central challenge is therefor e institutional rather than purely statistical: how to design a selection procedur e that pr oduces a small sample whose empirical distribution closely tracks that of the full population. W e study this problem in environments in which two adversarial parties possess ordinal information about the population and have opposing pr efer - ences over the composition of the selected subset. This structur e arises in sev- eral important contexts, including: (1) jury selection from a pool of eligible ju- rors; (2) Multi-District Litigation (MDL), where the goal is to identify a repr e- sentative sample of legal cases ﬁled against a single defendant, out of a pool of cases ﬁled and consolidated in one court; and (3) the appointment of members to independent redistricting commissions. 1 In each case, a small gr oup must be selected from a population that can be ranked according to its relative support for one of two parties, and the parties disagree over which individuals or cases should be chosen. The problem is how to design a selection procedure that harnesses the parties’ information to produce a sample whose composition best reﬂects the distribution of opinions or characteristics in the population. Formally , we consider a population that consists of n ranked items, but the ranking is unknown. T wo players observe the ranking and have opposing pr ef- erences over the items. One player pr efers higher ranked items, while the other player prefers lower ranked items. The goal is to select a representative sample of size k . The problem can be viewed as one of design behind a veil of ignorance and can be modeled as a Nash implementation problem ( Maskin , 1999 ). The rules 1 Other authors have examined this problem in the context of the selection of arbitrators, panels of judges, and members of citizen assemblies. See the refer ences below . 1 governing the interaction must be chosen ex ante , befor e the parties’ pr efer ences are known, while during the interaction prefer ences are assumed to be common knowledge. The design objective is therefor e to select rules that perform well for every possible realization of pr efer ences. For several reasons, the problem we consider is both tractable and interest- ing from an implementation perspective. First, the applications we consider all involve only two parties, who are engaged in playing a zero-sum game. Thus, it is r easonable to assume they have completely opposed or nearly completely opposed rankings over the population. Second, the applications we consider are all cases where the assumption of complete information, which underlies our theoretical analysis, is reasonable. Indeed, in all three applications, the in- volved parties share the same information with respect to the individuals or items to be selected. Third, in all three applications, the goal is to select a small sample, which is repr esentative of the entir e population. W e evaluate the “repr esentativeness” of the chosen sample by the distance between the cumulative distribution generated by a sampling procedure and that of the full population. W e pr opose a selection rule, which we call the Quan- tile mechanism , that uses only the players’ ordinal ranking information to con- struct a sample that is maximally repr esentative for any possible distribution of items, according to three differ ent distance measures: Kolmogor ov-Smirnov (KS), L 1 , and the Cramér-von Mises statistic (CvM). The Quantile mechanism is inspired by the cut-and-choose procedur e from the cake-cutting literature ( Brams and T aylor , 1996 ). One player partitions the population of n items into k disjoint sets, and the other player selects one item from each set. 2 The Quantile mechanism builds on the observation that the player who partitions the population, say it is the player who prefers high- ranking items, anticipates that the other player , who prefers low-ranking items, will choose the lowest-ranked item from every set. So the ﬁrst player has an incentive to place all the highest-ranking items into the smallest set, the next high-ranking items into the next smallest set, and so on. Similarly , if the player who prefers low-ranking items is chosen to partition the population, then she has the opposite incentive to place all the lowest-ranking items into the small- 2 Throughout, “partition” refers to a collection of disjoint sets; these sets need not cover the entire population. 2 est set, the next low-ranking items into the next smallest set, and so on. Impor- tantly , the Quantile mechanism is symmetric: it yields the same set of choices, or chosen sample, regardless of whether the partitioning is performed by the ﬁrst or second player . Our main result is that the Quantile mechanism produces the most repr e- sentative sample under all three distance measures mentioned above. In par- ticular , the Quantile mechanism is better than all the mechanisms that ar e cur- rently used in the settings mentioned above, including random selection and the strike-out mechanism used in jury selection and in MDLs. 3 Moreover , we show that the sample obtained by the Quantile mechanism is strictly better than any other sample of the same size. The rest of the paper proceeds as follows. In the next section we elabo- rate on the three real-world settings mentioned above: jury selection, multi- district litigation case selection, and the selection of independent redistricting commissions. Section 3 surveys the related theoretical literatur e. Section 4 sets up the model. Section 5 describes three alternative “repr esentativeness” statis- tics: Kolmogorov-Smirnov (KS), L 1 , and the Cramér-von Mises statistic (CvM), and presents the main result. Section 6 contains three extensions of our main result. The ﬁrst result shows that any selection of quantiles can be selected by some cut-and-choose mechanism. The second result extends the main result to the case in which the two players do not have opposing pr eferences, and the third result addresses the case in which ther e is only one player . Finally , Section 7 provides concluding r emarks. All proofs are r elegated to the Appendix. 3 The Quantile mechanism is also superior to the direct mechanism in which both players report the ranking. If the r eports coincide, the mechanism selects the best possible sample given the reported ranking (which as our analysis shows selects the same quantiles as those chosen by the Quantile mechanism); if they differ , it draws a random sample of size k . This is because truthful reporting is not a Nash equilibrium of the direct mechanism if the players prefer ences are diametrically opposed to each other: if one player strictly prefers to report truthfully , then the other player prefers not to, and vice-versa. So the direct mechanism can only implement a selection rule that both players ﬁnd indif ferent to random selection. 3 2 Application to Jury Selection, MDL, and Redis- tricting Committees In this section, we examine three differ ent settings in which the challenge of choosing a small, repr esentative subset from a large population arises: jury se- lection, Multi-District Litigation, and the appointment of members to indepen- dent redistricting commissions. W e brieﬂy outline how selection is curr ently carried out in each of these settings. As we demonstrate in the following sec- tions, in all of these settings, substituting existing pr ocedur es with the Quantile mechanism would enhance the “r epr esentativeness” of the individuals or items selected. J U RY S E L E C T I O N . Pursuant to 28 U.S.C. §1863, each United States district court is requir ed to formulate and implement a written plan governing the random selection of jurors. This plan must be designed to ensure the selection of a fair cross section of the individuals residing within the community of the district or division in which the court is located. The fair cross section requir ement de- rives from the Sixth Amendment to the United States Constitution, as construed by the Supreme Court in T aylor v . Louisiana , 419 U.S. 522 (1975), and Duren v . Missouri , 439 U.S. 357 (1979). The jury selection pr ocess begins with the compilation of a master list. From this master list, a random subset of eligible prospective jurors is drawn and summoned to court for further evaluation. The group of qualiﬁed individuals from which the ﬁnal jury is selected is termed the venir e ( Kovera and Cutler , 2013 ). In the selection process the plaintiff (or , in criminal cases, the prosecution) and the defendant may request that speciﬁc jurors be excluded “for cause.” In addition, each party is allotted a certain number of peremptory challenges, which allow them to strike jur ors without stating a cause, with the dismissed jurors replaced by other prospective jurors who wer e summoned for that same day ( W agner , 1981 ). 4 4 The speciﬁc procedure used to let the parties exercise their challenges varies greatly across jurisdictions and is sometimes left to the discretion of the judge. T wo classes of procedures are most frequently used. In Struck procedures, the parties can observe and extensively question all the jurors who could potentially serve on their trial before exercising their challenges (this questioning pr ocess is known as voir dir e). In contrast, in Strike and Replace procedur es, smaller 4 Empirical evidence consistently indicates that minority groups are under - repr esented in jury pools and venir es. For example, Rose, Casar ez and Gutier- rez ( 2018 ) document pervasive minority underr epr esentation in federal jury pools, with typical “missing” minority counts in venires that are modest in ab- solute magnitude but substantial in probabilistic terms for determining whether any minority jurors are present. In a related vein, Anwar , Bayer and Hjalmars- son ( 2022 ) show that geographic disparities in repr esentation among seated ju- ries closely track disparities already pr esent in the pools of potential jur ors. Misrepr esentation of certain groups is also documented with respect to seated juries. Anwar , Bayer and Hjalmarsson ( 2012 ) ﬁnd that the reduced-form effect of pool composition on case outcomes is substantially larger than what naive correlations based solely on the race of seated jurors would suggest, consis- tent with the interpretation that attorney selection behavior and other strategic responses confound comparisons based on seated-jury characteristics. Anwar , Bayer and Hjalmarsson ( 2014 ) further show that prosecutors disproportionately strike younger potential jurors, whereas defense attorneys disproportionately strike older potential jurors, r esulting in reduced seating pr obabilities at both tails of the age distribution relative to its center . Flanagan ( 2018 ) r eports that the demographic composition of the randomly selected pool causally af fects conviction pr obabilities and that attorneys adapt their per emptory strategies in response. By contrast, Diamond et al. ( 2009 ) ﬁnd that opposing peremptories may partly of fset one another on average, while str uctural featur es such as jury size exert an independent and mor e stable inﬂuence on minority representation. M U LT I - D I S T R I C T L I T I G AT I O N . A Multi-District Litigation (MDL) is a federal procedural mechanism created under 28 U.S. Code §1407. It allows civil law- suits that are pending in dif fer ent federal districts, but shar e common factual questions, to be centralized in a single district court (the transferee court) for co- ordinated pretrial proceedings. MDL cases consist of a signiﬁcant portion of the federal docket, and at times have accounted for more than 40 per cent of all fed- eral civil cases ( Gluck and Burch , 2021 ). Although the transferee court - wher e groups of jur ors are sequentially pr esented to the parties. The parties observe and question the group they are presented with (sometimes a single juror) but must exercise their challenges on that gr oup without knowing the identity of the next potential jurors ( Moro and V an der Linden , 2024 ). 5 all cases are centralized - is in principle r esponsible only for overseeing pr etrial proceedings, in practice more than 98 per cent of these cases end there, typically through settlement or dismissal ( Fallon , 2020 ). Because both the parties and the court need reliable information about the likely value of MDL claims, yet cannot practically conduct a full trial for each one, judges often turn to a bellwether trial process to generate these estimates. The label “bellwether” derives fr om the practice of placing a bell on the lead sheep (a wether) to guide the ﬂock, underscoring that these trials are meant to guide resolution of the remaining cases. The outcomes of bellwether trials do not have preclusive or binding effect on litigants who are not parties to those particular trials and instead serve only as informational inputs. The value of this approach, however , hinges on whether the selected cases ar e truly repre- sentative of the broader pool of claims ( Fallon, Grabill and W ynne , 2007 ; Lahav , 2007 , 2018 ; Whitney , 2019 ). T ransfer ee judges employ bellwether trials to pr omote efﬁcient resolution of complex MDL dockets. These trials serve to collect information about liabil- ity , causation, and defenses in an actual trial setting; to illuminate how juries respond to the evidence and expert testimony; to establish benchmarks or ref- erence points for valuing the r emaining cases in settlement negotiations – often prompting global settlements; and to assist courts in resolving recurring legal and evidentiary questions. The pr ocess of selecting bellwether cases is therefor e crucial, since it hinges on the repr esentativeness of the cases chosen. Courts use a variety of techniques for selecting the bellwether cases, generally grouped into party selection, ran- dom selection, and judicial selection. Under a party selection appr oach, plain- tiffs and defendants each choose a designated number of cases. Alternatively , cases can be drawn at random, either fr om the general pool of cases, or through stratiﬁed sampling. Finally , the court may select a repr esentative sample, of- ten after receiving recommendations from the parties and using some random sampling techniques ( Whitney , 2019 ; Fallon , 2020 ). Although the legal literature articulates substantial concerns regar ding the repr esentativeness of bellwether trials selected through the aforementioned tech- niques, systematic empirical assessment of these selection processes remains limited. Brown, Holian and Ghosh ( 2014 ) analyze the cases chosen in the V ioxx 6 multidistrict litigation (MDL) proceedings and demonstrate that cases selected by plaintiffs’ repr esentatives deviate signiﬁcantly from what would be expected under random selection, whereas no comparable divergence is observed for cases selected by defendants. These results, however , are derived from a single MDL pr oceeding. V illalón ( 2022 ) compares case-selection practices across two MDLs and concludes that, as the size of the consolidated pool incr eases, judges exhibit a stronger pr efer ence for random selection over party-driven selection. S E L E C T I O N O F I N D E P E N D E N T R E D I S T R I C T I N G C O M M I S S I O N S . Redistricting is the pr ocess of r edrawing the lines of legislative districts, gr ouping voters into geographic territories from which they elect their repr esentatives. This process affects districts at all levels of government, including local councils, state legis- latures, and the U.S. House of Representatives. The way these lines are drawn determines which voters are grouped together , dir ectly inﬂuencing which com- munities are r epr esented and which candidates can win elections. The redistricting process typically begins following the federal Census, which is conducted every ten years at the start of a new decade. The legal mandate for this cycle is rooted in the “one person, one vote” constitutional principle estab- lished by the U.S. Supr eme Court in the 1960s. In cases such as Reynolds v . Sims (377 U.S. 533, 1964) and Wesberry v . Sanders (376 U.S. 1, 1964), the Court ruled that legislative districts must contain roughly equal populations. Because populations shift over time, states are practically requir ed to redraw district lines after every Census to ensure this equality is maintained ( Karlan , 1992 ). While lines must be redrawn at least once per decade, some states allow for redistricting to occur more frequently , whereas others pr ohibit mid-decade change ( Cox , 2004 ). States use differ ent procedur es to determine who draws electoral district boundaries. When legislators draw their own district lines, a conﬂict of inter est arises: politicians may effectively choose their voters rather than voters choos- ing their repr esentatives. Incumbents may therefor e engage in gerrymandering, or the manipulation of district boundaries to inﬂuence electoral outcomes ( Ima- mura , 2022 ). T o mitigate this concern, some states, such as Colorado, Michigan, and Cal- 7 ifornia, delegate redistricting to independent citizen commissions. 5 Members of these commissions are selected thr ough procedur es that combine random se- lection with multi-stage vetting by nonpartisan bodies (such as state auditors or judicial panels). These procedur es typically impose constraints on the commis- sion’s partisan composition, for example, equal repr esentation of Democratic and Republican members together with a group of independent or unafﬁliated commissioners ( T orchinsky and Polio ( 2022 )). The goal is to create a commis- sion that r eﬂects the state’s political and demographic diversity and is ther efor e more likely to pr oduce fair district maps. 3 Related Literature A relatively limited body of theoretical literature examines problems closely related to those addressed in the present study . De Clippel, Eliaz and Knight ( 2014 ) analyze the selection of an arbitrator as a Nash implementation pr oblem. They pr opose a pr ocedur e they call shortlisting , in which one party selects a subset of n + 1 2 items out of n and the other party chooses one item from this subset. They show that this mechanism implements a Pareto-ef ﬁcient outcome that Pareto-dominates the median alternative for both parties. 6 When k = 1, their shortlisting pr ocedur e coincides with our Quantile mech- anism. However , our contribution differs along several dimensions. First, we study the selection of k ≥ 1 items rather than a single arbitrator . Second, their objective is Pareto efﬁciency , whereas our focus is on repr esentativeness, mea- sured by the distance between the sample and population distributions. (Im- portantly , the Quantile mechanism is also Par eto ef ﬁcient in our setting.) Thir d, they allow for general prefer ence proﬁles, while we focus on envir onments with strictly opposed prefer ences. 7 5 Colorado Independent Redistricting Commissions, Commissioner Selection Process, https://redistricting.colorado.gov/content/commissioner -selection-pr ocess ; Jason T orchinsky and Dennis W . Polio, How Independent is T oo Independent?: Redistricting Commissions and the Growth of the Unaccountable Administrative State, 20 Geo. JL and Pub. Pol’y 533, 543-50 (2022). 6 See Barberà and Coelho ( 2022 ) for a comparison of variants of this procedur e that allow the parties to determine the role of proposer and the size of the shortlist. See Barberà and Coelho ( 2024 ) for generalizations of this procedur e to the selection of k ≥ 1 alternatives in the context of the selection of arbitrator and judge panels. 7 See Section 6.2 for an extension of our framework to more general pr efer ence proﬁles. 8 More recently , Bogomolnaia, Holzman and Moulin ( 2023 ) have considered a more general version of this problem with possibly more than two agents who need to choose one alternative fr om a given set. They show that any mecha- nism guaranteeing a maximal welfare level to the agents must either combine variants of random dictatorship and voting-by-veto mechanisms or be purely random, depending on the parameters of the model. Their objective is welfare maximization, whereas our criterion is distributional r epr esentativeness. There is also less directly related literature in theoretical computer science which studies the design of repr esentative citizens’ assemblies under demo- graphic constraints. See, for example, Flanigan et al. ( 2021 ) and references therein. The papers in this literature aim to maximize individuals’ selection probabilities subject to the constraint that differ ent (observable) demographic groups are fairly represented in the assembly . In contrast, we consider a set- ting in which the planner does not observe the underlying ranking of items and must rely solely on the or dinal information held by the parties. 4 Model A population consists of n items x = ( x 1 , . . . , x n ) . The items are ranked accord- ing to a complete and transitive ranking ≿ . A social planner , who holds no information about the ranking, would like to select a sample of size k , which is repr esentative of the population, in a way we deﬁne below . The social planner can solicit help from two players, I and II, who are aware of the ranking, but have their own differ ent pr eferences over the selected sample: Player I prefers items whose ranking is higher , while player II prefers items whose ranking is lower . A selection pr ocedure, or a mechanism, is a game-form that selects (possibly randomly) a set of k indices. The mechanism speciﬁes a message set for each one of the players, together with a mapping fr om the players’ chosen messages into sets of k indices (if the mechanism is deterministic) or into distributions over sets of k indices (if the mechanism is random). The mechanism induces a normal-form game in which the players, who are commonly known to know the ranking of the items, choose their messages simultaneously , trying to in- 9 duce the selection of highly and lowly placed items according to the ranking, respectively . W e focus on a Nash-equilibrium of this game. Fixing the population size n and the desired sample size k ≤ n , denote by P ( n , k ) the collection of all subsets of indices { 1, . . . , n } that have size k . A mechanism is formally deﬁned as follows: Deﬁnition 1 (Mechanism) A mechanism is a tuple M = ⟨ A I , A I I , f ⟩ where • A I is a ﬁnite set of messages for Player I, • A I I is a ﬁnite set of messages for Player II, • f is a function that, for every pair of actions a I ∈ A I and a I I ∈ A I I , returns a probability distribution over P ( n , k ) . Remark 1 W e describe mechanisms as simultaneous-move game forms. This r epresen- tation is without loss of generality . Sequential procedur es, including those involving moves by nature, can also be repr esented within this framework using the standard transformation from extensive-form games to normal-form games. The outcome of a mechanism is a probability distribution over the set P ( n , k ) . In order to analyze how the players play in the game that is induced by a mech- anism M , we need to describe the players’ prefer ences over the space of proba- bility distribution over P ( n , k ) . Deﬁnition 2 (Utility function) Let x = ( x 1 , . . . , x n ) be a population, let ≿ be a ranking on x, and let y and y ′ be two samples of k items from x . A utility function of Player I is a function u I that assigns a real number to every sample of k items, such that u I ( y ) ≥ u I ( y ′ ) whenever the sample y is shifted to the right relative to y ′ , or # { j ∈ { 1, . . . , k } : y j ≾ x i } ≤ # { j ∈ { 1, . . . , k } : y ′ j ≾ x i } , ∀ i ∈ { 1, . . . , n } . Similarly , a utility function of Player II is a function u I I that assigns a r eal number to every sample of k items, such that u I I ( y ) ≥ u I I ( y ′ ) whenever y ′ is shifted to the right relative to y. 10 Remark 2 Notice that the statement that the sample “y is shifted to the right relative to y ′ ” is equivalent to the requir ement that the cumulative distribution function (CDF) of the sample y , deﬁned as F y ( x i ) ≡ 1 k · # { j ∈ { 1, . . . , k } : y j ≾ x i } , ∀ i ∈ { 1, . . . , n } , (1) ﬁrst-order stochastically dominates the CDF of y ′ , deﬁned analogously . Equivalently , the sample “y is shifted to the right r elative to y ′ ” holds if and only if F y ( x i ) ≤ F y ′ ( x i ) for every i ∈ { 1, . . . , n } . The assumption that the players’ utility functions are monotone with respect to ﬁrst-order stochastic dominance reﬂects the assumption that Player I prefers items that are ranked higher , while Player II prefers items that are ranked lower . W e do not impose any additional assumptions on the comparison of two samples that are not or der ed by stochastic dominance. Remark 3 Although it is natural in our applications to assume that the two players’ prefer ences ar e exactly opposed, the analysis below does not rely on this assumption. In particular , our results allow for pr efer ence proﬁles in which u I  = − u I I . Since the outcome of a mechanism is a probability distribution over samples, we must specify the players’ utility functions on this space. W e assume that the players’ prefer ences satisfy the von Neumann-Morgenstern axioms, so that they are expected-utility maximizers. That is, a player ’s utility fr om a probability distribution over samples equals the expected utility of the realized sample. The speciﬁcation of a mechanism induces a complete information game be- tween the players, as follows. Deﬁnition 3 (Mechanism game) A mechanism game is a triplet ⟨ M , u I , u I I ⟩ , where M = ⟨ A I , A I I , f ⟩ is a mechanism, and u I and u I I are the two players’ payoff functions. A mechanism game is played as follows. • Fix a population x = ( x i ) n i = 1 and a ranking ≿ over x . 11 • After observing x and ≿ , Player I (resp., Player II) selects a ∈ A I (resp., a I I ∈ A I I ). This selection is made simultaneously . • A set of k indices S ∈ P ( n , k ) is selected randomly according to the prob- ability distribution f ( a I , a I I ) . • The outcome of the game is the vector y : = ( x s ) s ∈ S . A pure strategy for Player I is a function that assigns a message in A I to any possible input x , ≿ , and a mixed strategy for Player I is a function that assigns a probability distribution on A I to any possible input x , ≿ . Pur e and mixed strategies for Player II are deﬁned analogously . W e assume that the players play a Nash equilibrium of the induced game, which is deﬁned as follows: Deﬁnition 4 (Equilibrium) A pair of strategies ( α ∗ I , α ∗ I I ) is a Nash equilibrium if for every two strategies α I of Player I and α I I of Player II we have E α ∗ I , α ∗ I I [ u I ( y ) ] ≥ E α I , α ∗ I I [ u I ( y ) ] , ∀ α I , E α ∗ I , α ∗ I I [ u I I ( y ) ] ≥ E α ∗ I , α I I [ u I I ( y ) ] , ∀ α I I . By Nash’s Theorem, every mechanism game admits a Nash equilibrium in mixed strategies. Note that there are two sour ces of randomization in the play . First, the play- ers select their messages at random, and second, the sample is selected ran- domly according to f . A mechanism game may well have a unique Nash equi- librium in mixed strategies. W e next provide some examples of selection mechanisms that illustrate the model and how mechanisms can be used to select repr esentative samples. The ﬁrst example shows how it is possible to implement the selection of a random sample from the population. Example 1 (Random Sample) Fix a population x = ( x i ) n i = 1 . The mechanism M = ⟨ A I , A I I , f ⟩ that implements the selection of a random sample of size k fr om the popu- lation x is deﬁned as follows: A I and A I I are arbitrary sets. In particular , they can be singletons. The function f ( a I , a I I ) is the uniform distribution over P ( n , k ) , for every 12 a I ∈ A I and a I I ∈ A I I . Since the players’ messages do not affect the outcome of the mechanism, any pair of strategies is an equilibrium of the corresponding mechanism game. Example 2 (Strike and Replace Mechanism) Fix a population x = ( x i ) n i = 1 , a rank- ing ≿ over x , and a sample size k ≤ n . The procedur e selects a random sample of size k, lets player I veto up to c items, and then lets plater I I veto up to c items. Each vetoed item is replaced with a freshly drawn item from the population, without replacement. The pr ocedure outputs the sample that survived this process of elimination. This pro- cedure can be implemented by a mechanism in which the two players’ message sets A I and A I I include all the possible sequential veto strategies. In equilibrium, each player will veto the c items she dislikes the most from the sample (lowest ranking item for Player I; and highest ranking item for Player II) provided these items are worse than a randomly sampled item fr om the population. Example 3 (Median-Sample Mechanism) An interesting improvement to both the Random Sample Mechanism and the Struck Jury Mechanism was recently proposed by Flanagan ( 2025 ). Flanagan suggests drawing 2 c + 1 random samples, and letting each player veto c of these samples. This pr ocedur e can also be implemented by a mechanism. In equilibrium, each player will veto the c samples she dislikes the most, which will produce the median sample as the outcome of the mechanism. Flanagan ( 2025 ) claims that this pr ocedure r educes the variance of the selected sample relative to both random selection and the Struck Jury Mechanism. The next example shows that there exists a mechanism that always outputs the median item from any population x = ( x i ) n i = 1 in equilibrium. Example 4 (The Median Mechanism) Fix a population x = ( x i ) n i = 1 and a ranking ≿ over x . Suppose that k = 1 . The mechanism works as follows: ask Player I to select a subset a I of ⌈ n 2 ⌉ items from the population x ; ask Player II to select one of the items in a I ; select the item that was selected by Player II from Player I’ s selected subset. As mentioned above, the Median Mechanism is equivalent to the shortlisting procedur e proposed by De Clippel, Eliaz and Knight ( 2014 ). In their procedure, one of the players shortlists ⌈ n 2 ⌉ items, from which the other player chooses one item. Since Player II wants to minimize the ranking of the selected item, she will choose the minimal (according to ≿ ) item in a I . Anticipating that, Player I, who wants to 13 maximize the ranking of the selected item, will select the subset of items that contains the top ⌈ n 2 ⌉ items from x (according to ≿ ). Formally , the mechanism M = ⟨ A I , A I I , f ⟩ is described as follows. • A I is the set of all subsets of { 1, . . . , n } that contain ⌈ n 2 ⌉ items. • A I I contains all the functions a I I : A I → { 1, . . . , n } that assign to each subset a I ⊂ { 1, . . . , n } of ⌈ n 2 ⌉ items one of the items in a I . • f ( a I , a I I ) assigns probability one to the implementation of Player II’ s choice on Player I’ s choice, a I I ( a I ) . If the ranking ≿ over x is strict, then the mechanism game admits a unique equilibrium in which Player I selects the set that contains the highest ⌈ n 2 ⌉ items, and Player II selects the function that assigns to each subset a I ⊆ { 1, . . . , n } of ⌈ n 2 ⌉ the item in the set that is smallest according to ≿ . Otherwise, there is a multiplicity of equilibria, but the equilibrium outcome is always taken from the set of items that are equivalent (according to ≿ ) to the item ranked ⌈ n 2 ⌉ . The next example describes a general class of mechanisms that we call cut- and-choose mechanisms. In these mechanisms, one of the players is asked to select k subsets (not necessarily disjoint) from the set of indices { 1, . . . , n } with n 1 , . . . , n k items in each set, r espectively , and the other player is asked to select one index from each subset. The mechanism outputs the k indices selected by the second player . By varying the numbers of indices in each set, n 1 , . . . , n k , and by varying the restrictions on the extent to which these subsets can overlap, it is possible to obtain the general class of cut-and-choose mechanisms. The Median Mechanism described in Example 4 is an example of a cut-and-choose mechanism in which k = 1 and n 1 = ⌈ n 2 ⌉ . Example 5 (Cut-and-Choose Mechanism) Fix a population x = ( x i ) n i = 1 , a sample size k, and a ranking ≿ over x . A cut-and-choose mechanism with k disjoint subsets with sizes n 1 , . . . , n k , ∑ k i n i ≤ n works as follows. Suppose, without loss of generality , that n 1 ≤ · · · ≤ n k . One of the players is asked to partition the set of indices { 1, . . . , n } into k disjoint subsets, with n 1 , . . . , n k indices each, respectively . The other player is asked to choose one index from each one of the k subsets, and the mechanism outputs the chosen k indices. 14 Suppose that Player I is asked to select the subsets (cut), and Player II is asked to choose an index fr om each subset (choose). In this case, the equilibrium of the cut- and-choose mechanism (which is unique up to equivalent outcomes) is that Player I places the indices of the n 1 highest ranked items in the ﬁrst set, the indices of the next n 2 highest largest items in the second set, and so on, up to the k -th set, and Player II chooses the index of the lowest ranked item in every subset. T o see this, note that for player II, choosing the index of the lowest ranked item from each subset of indices chosen by Player I is a dominant strategy . As for Player I, observe that if there are two sets of indices I 1 and I 2 such that the lowest ranked item x 1 from the items indexed by I 1 is ranked below the lowest ranked item x 2 indexed by I 2 , that is, x 1 ≺ x 2 , and if ther e is another item x ′ 1 in I 1 that is ranked above x 2 , then Player I would beneﬁt fr om switching between the two items x ′ 1 and x 2 , because the switch does not affect Player II’ s choice from I 1 , and weakly increases the rank of the item chosen from the set I 2 . This argument implies that the k subsets of indices that ar e selected by Player I ar e ordered in the sense that the lowest ranked item in the i-th subset of indices is ranked above the highest ranked item in the subset i + 1 -th subset. Moreover , by putting the n 1 highest ranked items in the ﬁrst set, the indices of the next n 2 highest largest items in the second set, and so on, Player I shifts the chosen sample as much to the right as possible, which is to his beneﬁt. It follows that in the equilibrium of this cut-and-choose mechanism the outcome con- tains the n − n 1 + 1 -th highest ranked, n − n 1 − n 2 + 1 -th highest ranked item, and so on. If instead Player II was chosen to cut and Player I was chosen to choose, in equi- librium the outcome would have contained the n 1 -th lowest ranked, n 1 + n 2 -th lowest ranked item, and so on, up to the ∑ k i n i -th lowest ranked items from the population. 5 Maximizing Representativeness As explained above, our objective is to identify the mechanism that produces the sample most repr esentative of the population. In this section, we introduce three notions of repr esentativeness, each deﬁned as a distance between the cu- mulative distribution function (CDF) of the sample and that of the population: the Kolmogorov-Smirnov statistic, the L 1 -statistic, and the Cramér-von-Mises 15 statistic. These statistics are widely used in statistical applications. 8 W e show that the Quantile mechanism, which is deﬁned below , produces the most r epr e- sentative sample according to all thr ee measur es. For any population x = ( x i ) n i = 1 and ranking ≿ on x , denote by F x the cumu- lative distribution function (CDF) of x : F x ( x i ) ≡ 1 n · #  j ∈ { 1, . . . , n } : x j ≾ x i  , ∀ i ∈ { 1, . . . , n } . (2) Similarly , for any sample y = ( y i ) k i = 1 of k items from x , recall the deﬁnition of the CDF F y of the sample y fr om Equation ( 1 ). K O L M O G O R O V - S M I R N O V S T A T I S T I C . The Kolmogorov-Smirnov (KS) statistic ( Kolmogorov , 1933 ) compares a population to a sample taken fr om that popula- tion by the maximal differ ence between the sample’s CDF and the population’s CDF . Deﬁnition 5 (KS statistic) Let x be the population, let ≿ be a ranking on x , and let 8 An alternative measure of distance between distributions that is often used in the economics literature is Kullback-Leibler (KL) divergence . KL divergence compares the population distribution (rather than CDF) to the sample distribution. Denoting P x ( x i ) = # { j : x j = x i } and P y ( x i ) = # { j : y j = x i } , for each i ∈ { 1, . . . , n } , KL divergence is given by KL ( x , y ) : = 1 n n ∑ i = 1 log  P x ( x i ) P y ( x i )  . There are two reasons not to use this measure in our setup. First, when the number of equiva- lence classes of items in the population is larger than k , necessarily there will be i ∈ { 1, . . . , n } such that F y ( x i ) = 0, in which case KL divergence is not well-deﬁned. Second, even if the num- ber of equivalence classes of items in the population is at most k , the sample that minimizes KL divergence need not repr esent the population. For example, consider a population that consists of two types of items, red and blue. Suppose that the population consists of one red item and 99 blue items, so that n = 100, and suppose that k = 2. The sample that minimizes the KL divergence is the one that contains one item from each type. However , the sample that contains two blue items better repr esents the population. Another common measure of distance between distributions is p-Wasserstein distance . This measure is applicable when there is a natural distance between items in the population, which is not the case in the applications we consider . Under the following natural distance between items: the distance between items x i and x j is 0 if x i ∼ x j , and 1 plus the number of items that lie strictly between x i and x j otherwise, if p = 1, then this distance measure is equivalent to L 1 -statistic, and hence the Quantile mechanism is also optimal according to this measur e. 16 y be a sample of size k from x. The KS statistic of x and y is KS ( x , y ) ≡ max 1 ≤ i ≤ n | F x ( x i ) − F y ( x i ) | . Accordingly , we say that a sample y is KS-optimal if it minimizes the KS statistic, and that a mechanism is KS-optimal if it pr oduces the KS-optimal sam- ple for every population. Deﬁnition 6 (KS-optimality) A sample y is KS-optimal for the population x if KS ( x , y ) = min y ′ ∈P ( n , k ) KS ( x , y ′ ) . A mechanism M is KS-optimal if for every population x, all (possibly mixed) equilibria of the mechanism game corr esponding to M assign pr obability 1 to KS-optimal samples for x . The deﬁnition of KS-optimality is very strong; it requir es that any sample that may be obtained in an equilibrium of M , is not worse (according to the KS statistic) than any other sample. L 1 S T A T I S T I C . The L 1 statistic compares the population and the sample by measuring the average distance between their cumulative distribution functions. Unlike the KS statistic, which focuses on the maximal deviation, the L 1 statistic aggregates discr epancies across the entire distribution. It is often preferr ed in settings where noise in the tails is important. Deﬁnition 7 ( L 1 statistic) Let x be the population, let ≿ be a ranking on x , and let y be a sample of size k from x . The L 1 statistic of x and y is L 1 ( x , y ) ≡ 1 n n ∑ i = 1 | F x ( x i ) − F y ( x i ) | . Sample and mechanism L 1 -optimality are deﬁned in a similar way to sample and mechanism KS-optimality . Deﬁnition 8 ( L 1 -optimality) A sample y is L 1 -optimal for the population x if L 1 ( x , y ) = min y ′ ∈P ( n , k ) L 1 ( x , y ′ ) . 17 A mechanism M is L 1 -optimal if for every population x , all (possibly mixed) e quilibria of the mechanism game corresponding to M assign probability 1 to L 1 -optimal samples for x . C R A M É R - V O N M I S E S S TA T I S T I C . The Cramér-von Mises (CvM) statistic ( Cramér , 1928 ; von Mises , 2013 ) compares the population and the sample by measuring the average squar e distance between their cumulative distribution functions. It is more sensitive to noise tail than the L 1 statistic. Deﬁnition 9 (The CvM statistic) Let x be the population, let ≿ be a ranking on x , and let y be a sample of size k from x . The CvM statistic of x and y is CvM ( x , y ) ≡ 1 n n ∑ i = 1 ( F x ( x i ) − F y ( x i ) ) 2 . CvM-optimality is deﬁned in a similar way to KS- and L 1 -optimality . Deﬁnition 10 (CvM optimality) A sample y is CvM-optimal for the population x if CvM ( x , y ) = min y ′ ∈P ( n , k ) CvM ( x , y ′ ) . A mechanism M is CvM-optimal if for every population x , all (possibly mixed) equi- libria of the mechanism game corresponding to M assign probability 1 to CvM-optimal samples for x . Remark 4 Both the population and sample CDFs are ordinal concepts. They do not depend on the speciﬁc values that are assigned to the items in the population. If x = ( x i ) n i = 1 and x ′ = ( x ′ i ) n i = 1 are two different populations with two different rankings ≿ , ≿ ′ , respectively , and if x i ≿ x j if and only if x ′ i ≿ ′ x ′ j , then F x ( x i ) = F x ′ ( x ′ i ) for every i ∈ { 1, . . . , n } . It therefore follows that our notion of representativeness depends only on the sampled quantiles of the population, and not on the speciﬁc values of the sampled items. T o illustrate this point, consider two populations that each consist of four natural numbers that are ranked by their order . The ﬁrst population is x = { 1, 10, 12, 100 } , and the second population is e x = { 1, 2, 3, 4 } . The samples y = { 10, 12 } and e y = { 2, 3 } 18 consist of the middle two items in the two populations. Therefor e, KS ( x , y ) = KS ( e x , e y ) , L 1 ( x , y ) = L 1 ( e x , e y ) , and CvM ( x , y ) = CvM ( e x , e y ) . It therefor e follows that whether the sample y repr esents well the population x , does not depend on the absolute values of the smallest and largest (or any other) items in x . The next mechanism, which we call the Quantile Mechanism , is a cut-and- choose mechanism that generates a balanced, or repr esentative sample of the population. Example 6 (Quantile mechanism) Fix a population x = ( x i ) n i = 1 , a sample size k, and a ranking ≿ over x . Suppose that n = ( 2 m + 1 ) k . The Quantile Mechanism is a cut-and-choose mechanism with one subset with m + 1 indices, and k − 1 subsets with 2 m + 1 indices each (notice that m indices are not assigned to any subset). The Quantile Mechanism implements the selection of the ( m + 1 ) -th highest rank item, the m + 1 + ( 2 m + 1 ) -th highest ranked item, and so on up to the m + 1 + ( k − 1 ) ( 2 m + 1 ) -th highest ranked item. The Quantile Mechanism is symmetric: it outputs the same sample r egardless of whether Player I or Player II are chosen as cutter (and Player II, and Player I, respectively , are chosen as chooser). An alternative way to implement the Quantile Mechanism is to let Player I (or Player II, this implementation is also symmetric) eliminate m indices, and let Player II (resp., Player I) select one index, then let Player I (resp., II) eliminate 2 m more indices, and let Player II (resp., I) choose one additional index, and so on with Player I (resp., II) always eliminating 2 m indices, until k indices are selected. Our main result establishes the optimality the Quantile mechanism. It shows that the sample obtained by the Quantile mechanism is strictly better than any other sample (not equivalent to it) according to all three distance measures in- troduced above. Theorem 1 (Characterization of the optimal mechanism) Suppose that n = ( 2 m + 1 ) k . Then, the Quantile mechanism is KS-optimal, L 1 -optimal, and CvM-optimal. Moreover , for every population x and ranking ≿ , every sample y ′ that is not equiva- lent to the sample y that is pr oduced by the Quantile mechanism y KS ( x , y ′ ) > KS ( x , y ) , L 1 ( x , y ′ ) > L 1 ( x , y ) , and CvM ( x , y ′ ) > CvM ( x , y ) . 19 Theorem 1 illustrates the advantage of the Quantile mechanism over any other mechanism, whether deterministic or random. In particular , any deter- ministic mechanism that does not generate an outcome that is equivalent to the outcome of the Quantile mechanism is strictly worse that the Quantile mecha- nism according to the Kolmogorov-Smirnov statistic, the L 1 statistic, and the Cramér-von Mises statistic. A random mechanism can never generate an out- come that is better than the Quantile mechanism, and, if with some positive probability it outputs a sample that is not equivalent to the output of the Quan- tile mechanism, then it is necessarily worse. Remark 5 (The dependence of the statistics on n , m , and k ) For every population x , if the ranking ≿ of the items in the population is strict, then the KS, L 1 , and CvM statistics of the sample y that is pr oduced by the Quantile mechanism are equal to KS ( x , y ) = m n = 1 2 k  1 − 1 n  , L 1 ( x , y ) = m ( m + 1 ) n ( 2 m + 1 ) = 1 4 k 1 −  k n  2 ! , and CvM ( x , y ) = 2 m ( m + 1 ) n 2 = 1 2 k 2 1 −  k n  2 ! , respectively . These bounds describe the discrepancy between the sample generated by the Quantile mechanism and the population as a function of the sample size and the population size, and describe the rate at which the distance between the sample and population CDFs decr eases as a function of the sample and population sizes, k and n , respectively . The intuition for the proof of Theorem 1 is that, according to all three mea- sures, the distance between the population and sample CDFs F x and F y is max- imized right before and right at the chosen quantiles, and decr eases on the dis- tance from the closest chosen quantile. Suppose, without loss of generality that x 1 ≾ · · · ≾ x n . Then, right before the ﬁrst chosen quantile x m + 1 , F x ( x m ) − F y ( x m ) = m n − 0 = m n . And, right at the ﬁrst chosen quantile x m + 1 : F x ( x m ) − F y ( x m ) = m + 1 n − 1 k = m + 1 n − 2 m + 1 n = − m n , 20 because the requir ement that n = ( 2 m + 1 ) k implies that 1 k = 2 m + 1 n . The fact that each additional subset of indices contains 2 m + 1 indices ensures that this calculation is r epeated right befor e and right at all other quantiles. For example, right before the second quantile x 3 m + 2 F x ( x 3 m + 1 ) − F y ( x 3 m + 1 ) = 3 m + 1 n − 1 k = 3 m + 1 n − 2 m + 1 n = m n . And, right at the second chosen quantile x 3 m + 2 : F x ( x m ) − F y ( x m ) = 3 m + 2 n − 2 k = 3 m + 2 n − 4 m + 2 n = − m n , and so on. The calculation above shows that the Quantile mechanism generates a KS statistic of m n . T o see why a mechanism that selects dif fer ent quantiles neces- sarily generate a strictly larger KS statistic, suppose that the smallest quantile chosen by some alternative mechanism x m ′ is smaller than the smallest quantile chosen by the Quantile Mechanism, which is given by x m + 1 , or m ′ < m + 1. In this case, F y ( x m ′ ) − F x ( x m ′ ) = 1 k − m ′ n = 2 m + 1 n − m ′ n > 2 m + 1 n − m + 1 n = m n . And if the statistic x m ′ is larger than x m + 1 , or m ′ > m + 1, then at the point x m ′ − 1 : F x ( x m ′ − 1 ) − F y ( x m ′ − 1 ) = m ′ − 1 n − 0 > m n . A similar ar gument shows that if the smallest quantile chosen by an alternative mechanism is equal to x m + 1 but the next quantile chosen by the mechanism is differ ent from the second quantile chosen by the Quantile Mechanism x 3 m + 2 , then again the KS statistic generated by the alternative mechanism is strictly larger than m n , and do on. The advantage of the Quantile mechanism is illustrated in Figures 1 and 2 below . Figure 1 compares the population CDF to the CDF of the sample that is produced by the Quantile mechanism. As can be seen in the ﬁgure, the sample CDF follows closely the population’s CDF , demonstrating the unique features 21 of the sample produced by the Quantile mechanism – it is symmetric and it repr esents well the population. Figure 1: F x and F y under the Quantile mechanism for n = 972, k = 12, m = 40. Figure 2 compares the KS statistic between the population CDF and the sample CDF , for ﬁve selection procedures: the Quantile Mechanism, the Ran- dom Mechanism (Example 1 ) (with k randomly selected items), the Strike-and- Replace Mechanism (Example 2 ), the Median-Sample Mechanism (Example 3 ), and the Random Mechanism with k = 12 and n ∗ = 259 randomly selected items. 9 As the ﬁgure shows, the KS statistic of the Quantile mechanism is the small- est, and equal to the constant m n . The KS statistic of a Cut-and-Choose mecha- nism in which the player who cuts is r equir ed to partition the population into k equally sized subsets is the constant 2 m n . The ﬁgure also shows the distribution 9 For the simulations, we draw 972 = ( 2 · 40 + 1 ) · 12 observations from a standard normal distribution and treat the resulting empirical CDF as the population CDF . The value n ∗ = 259 is chosen so that the average KS statistic of a random sample of size n ∗ from the population CDF matches, up to a small tolerance, the corresponding average distance generated by the Quantile mechanism. T o implement the Strike and Replace mechanism, we ﬁrst draw a random sample of size k = 12, allow each side to veto up to 3 observations from opposite ends of the sample, and then r eﬁll the str uck observations by random draws from the remaining population. T o im- plement the Median-Sample mechanism, we draw 7 random samples of size k = 12, rank these samples by their sample medians, and let the two sides veto 3 samples each from opposite ends, so that the remaining sample is the sample whose median is the median among the candidate sample medians. W e repeat this simulation procedure 1,000 times and recor d the resulting KS statistic for each method. 22 of the KS statistic of thr ee mechanisms that involve randomization: the median of medians proposed by Flanagan ( 2025 ), the Struck Full Jury , and a random choice of a sample. These three mechanisms perform signiﬁcantly worse than the Quantile mechanism. Figure 2: Comparison of the Quantile, Median-Sample, Strike and Replace, and Random mechanisms with k = 12 and n ∗ = 259 chosen items in terms of KS statistic for sampling 100 times from n = 972, k = 12, m = 40. 6 Extensions 6.1 Any Selection of Quantiles is Possible As mentioned in Remark 4 , the fact that both the population and sample CDFs are ordinal concepts highlights the fact that repr esentativeness depends only on the sampled quantiles of the population, and not on the speciﬁc values of the sampled items. The next theorem shows that any selection of k quantiles can be implemented by some cut-and-choose mechanism. Theorem 2 Fix a population size n and a sample size k ≤ n. Then, any selection of k quantiles ( q 1 , . . . , q k ) ∈ { j 1 n , . . . , j k n : j i ∈ { 1, . . . , n } ∀ i , j 1 < · · · < j k } can be implemented by some Cut-and-Choose Mechanism. 23 The proof of Theor em 2 follows from the pr eceding analysis upon observing that the Cut-and-Choose mechanism in which Player II is asked to select k not necessarily disjoint subsets with q 1 , . . . , q k items each, will lead Player II to select the k subsets with the q 1 lowest ranked items, the q 2 lowest ranked items and so on, which would induce Player I to select the q 1 lowest ranked item, the q 2 lowest ranked item, and so on up to the q k lowest ranked item, as requir ed. Theorem 2 implies that if one believes that the most representative sample consists, say , of the k items which ar e closest the population sample, then there is a cut-and-choose mechanism that can output these, or any other , quantiles. 6.2 Non-Antagonistic Players So far we assumed that the players ar e antagonistic: Player 1 prefers higher- ranked items, while Player 2 prefers lower-ranked items. In this subsection we explore the model where each player i ∈ { 1, 2 } has her own ranking ≿ i on the population, and the utility function u i of each player i is such that u i ( y ) ≥ u i ( y ′ ) whenever F y stochastically dominates F y ′ . The deﬁnition of mechanisms and mechanism games remains as before. Since the mechanism game that corr esponds to the Quantile mechanism is a game of perfect information (it does not include simultaneous moves), the game admits a subgame-perfect equilibrium in pure strategies. The next result shows that the Quantile mechanism still performs well in the following sense. For each population x , denote by y ( x ) the outcome of a pure subgame-perfect equilibrium of the mechanism game that is induced by the Quantile mechanism when the two players’ prefer ences are given by ≿ 1 and ≿ 2 , respectively . Denote by y ∗ 1 ( x ) the outcome of the game that is induced by the Quantile mechanism when the ranking of the population is given by Player 1’s ranking ≿ 1 (that is, when u 1 ( y ) ≥ u 1 ( y ′ ) if and only if F y ﬁrst-order stochas- tically dominates F y ′ and u 2 ( y ) = − u 1 ( y ) for every sample y ). And denote by y ∗ 2 ( x ) the outcome of the game that is induced by the Quantile mechanism when the ranking of the population is given by Player 2’s ranking. Our r esult shows that each player i prefers y ( x ) to y ∗ i ( x ) . That is, when players do not have opposing prefer ences, they both far e better compared to the case in which they each face a player that has opposing preferences to theirs. 24 Intuitively , the player who cuts is better off because they can cut in the same way they would cut if the choosing player had opposing pr eferences and get an outcome that is weakly better for them from every subset. And the player who chooses is better off because a player who has opposing preferences cuts in the worst possible way for them. The fact that both players agree that the outcome is better than what it would have been if they had opposing prefer ences implies that whatever is lost in terms of pur e r epr esentativeness, is mor e than made up by the fact that both players are made better of f. Theorem 3 Given a population x and two rankings ≿ 1 and ≿ 2 , we have i. u 1 ( y ( x ) ) ≥ u 1 ( y ∗ 1 ( x ) ) under ≿ 1 . ii. u 2 ( y ( x ) ) ≥ u 2 ( y ∗ 2 ( x ) ) under ≿ 2 . 6.3 One Player So far , with the exception of Section 6.2 , we have considered the case in which two players have observed the ranking. What if there is only one player , or if one player does not observe the ranking? Such a case may arise, for example, in MDL, if one of the parties holds private information about individual case values. Consider the following mechanism. Deﬁnition 11 (Random Cut-and-Choose Mechanism) Suppose n = k m. The Random Cut-and-Choose Mechanism is the mechanism in which Player 1 who knows the ranking partitions the population into k disjoint subsets, each of size m , and one item is randomly chosen fr om each subset, uniformly and independently among the subsets. The outcome of the random Cut-and-Choose mechanism is a random sam- ple y . Suppose the items in the population are real numbers, and the ranking ≿ ranks these numbers by their cardinality . Given a partition P selected by Player 1, the mean µ P ( y ) and the variance σ 2 P ( y ) of the random sample y ar e random variables, which depend on the partition. W e show that the sample mean µ P ( y ) is independent of the partition, and the partition that minimizes the variance σ 2 P ( y ) is the equilibrium partition of the Random Cut-and-Choose mechanism described in Example 11 . 25 Theorem 4 Suppose that n = k m. For any partition chosen by the player , µ P ( y ) = 1 n n ∑ i = 1 x i . Moreover , the variance E [ σ 2 P ] is minimized by the ordered partition, in which the items are assigned to the k subsets according to their order , so that the largest item in each subset is smaller than the smallest item in the next subset. It follows that the player cannot affect the mean of the chosen sample. If the player is risk-averse and prefers the variance of the sample to be as small as possible, then the random Cut-and-Choose mechanism induces the order ed partition. In this partition, the median items of each subset in the partition coincide with the quantiles selected by the Quantile mechanism, respectively , by their order . 7 Conclusion W e consider the pr oblem of how to choose a small repr esentative sample from a large population. As the three legal examples of jury selection, MDL, and r e- districting commissions illustrate, current methods for making such a selection are unsatisfactory . W e introduce a selection procedur e, the Quantile mechanism, which selects the most repr esentative sample according to three differ ent measures that ar e prevalent in statistics: Kolmogorov-Smirnov , L 1 , and Crámer-von Mises. This procedur e, which is a symmetrized version of the Cut-and-Choose mechanism, is easy to explain and simple to implement. 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Rev . , 55: 501. von Mises, Richard. 2013. Wahrscheinlichkeit statistik und wahrheit. V ol. 7, Springer-V erlag. W agner , W ard. 1981. Art of advocacy: Jury selection. M. Bender . Whitney , Melissa J. 2019. Bellwether trials in MDL proceedings: a guide for trans- feree judges. Federal Judicial Center and Judicial Panel on Multidistrict Litiga- tion. Proof of Theorem 1 Fix a population x = ( x 1 , . . . , x n ) and a ranking ≾ on x . Assume w .l.o.g. that x 1 ≾ x 2 ≾ · · · ≾ x n . As mentioned in Example 6 , when n = ( 2 m + 1 ) k , in all 29 equilibria of the quantile mechanism, the selected sample is equivalent to y = ( x m + 1 , x m + 1 + ( 2 m + 1 ) , . . . , x m + 1 + ( 2 m + 1 ) ( k − 1 ) ) . (3) W e divide the proof into thr ee parts. In Section 7 we prove that the quantile mechanism is KS-optimal, in Section 7 we prove that the quantile mechanism is L 1 -optimal, and in Section 7 we pr ove that the quantile mechanism is C v M - optimal. The quantile mechanism is KS-optimal In Step 1 we will show that KS ( x , y ) ≤ m n , wher e y is deﬁned in Eq. ( 3 ). In Step 2 we will show that for any sample y ′ that is not equivalent to y , we necessarily have KS ( x , y ′ ) > m n . Therefore, KS ( x , y ) < KS ( x , y ′ ) . This in particular holds for any sample y ′ which is possible under some equilibrium in the mechanism game that corresponds to some mechanism M ′ . Since x is arbitrary , this implies that the quantile mechanism is KS-optimal. Step 1: KS ( x , y ) ≤ m n . T o pr ove that KS ( x , y ) ≤ m n , we will show that | F x ( x i ) − F y ( x i ) | ≤ m n , ∀ i ∈ { 1, . . . , n } . (4) Fix i ∈ { 1, . . . , n } . W e will characterize in turn F x ( x i ) and F y ( x i ) . Recall that we assumed w .l.o.g. that x 1 ≾ x 2 ≾ · · · ≾ x n . If x i ≺ x i + 1 , then F x ( x i ) = i n . Otherwise, x i ≈ x i + 1 , and F x ( x i ) is higher than i n . In fact, denote by ℓ the smallest index such that x i ≺ x ℓ , so that x i ≈ x i + 1 ≈ · · · ≈ x ℓ − 1 ≺ x ℓ . W ith this notation, F x ( x i ) = ℓ − 1 n . W e turn to calculate F y ( x i ) . The number of sample elements y j in y that satisfy y j ≾ x i is the maximal r such that m + 1 + ( 2 m + 1 ) ( r − 1 ) < ℓ . In particular , ℓ ≤ m + 1 + ( 2 m + 1 ) r and F y ( x i ) = r k = r ( 2 m + 1 ) n . Note that if r = 0 then x i is strictly lower than all items in y , while if r = m then x i is weakly larger 30 than all items in y . Finally , − m n = m + 1 + ( 2 m + 1 ) ( r − 1 ) n − r ( 2 m + 1 ) n ≤ ℓ − 1 n − r ( 2 m + 1 ) n = F x ( x i ) − F y ( x i ) (5) ≤ m + ( 2 m + 1 ) r n − r ( 2 m + 1 ) n (6) = m n , where Eq. ( 5 ) holds since m + 1 + ( 2 m + 1 ) ( r − 1 ) < ℓ and Eq. ( 6 ) holds since ℓ ≤ m + 1 + ( 2 m + 1 ) r . Thus, Eq. ( 4 ) holds. Step 2: KS ( x , y ′ ) > m n , for every sample y ′ that is not equivalent to y . Fix a sample y ′ = ( x ℓ 1 , . . . , x ℓ k ) that is not equivalent to y . Assume w .l.o.g. that ℓ 1 < · · · < ℓ k . Let r ∈ { 1, . . . , k } be the minimal index such that the r ’th item in y ′ differs fr om the r ’th item in y ; that is, x ℓ r  ≈ x m + 1 + ( 2 m + 1 ) ( r − 1 ) . If x ℓ r ≺ x m + 1 + ( 2 m + 1 ) ( r − 1 ) , then in particular ℓ r < m + 1 + ( 2 m + 1 ) ( r − 1 ) and F x ( x ℓ r ) < m + 1 + ( 2 m + 1 ) ( r − 1 ) n . Moreover , F y ′ ( x ℓ r ) ≥ r k . Therefor e, F y ′ ( x ℓ r ) − F x ( x ℓ r ) > r k − m + 1 + ( 2 m + 1 ) ( r − 1 ) n = m n . If x ℓ r ≻ x m + ( 2 m + 1 ) ( r − 1 ) , then in particular ℓ r > m + ( 2 m + 1 ) ( r − 1 ) . Hence, F x ( x m + ( 2 m + 1 ) ( r − 1 ) ) > m + ( 2 m + 1 ) ( r − 1 ) n . Since x ℓ r ≻ x m + 1 + ( 2 m + 1 ) ( r − 1 ) , we have F y ( x m + 1 + ( 2 m + 1 ) ( r − 1 ) ) = r − 1 k . Therefor e, F x ( x m + ( 2 m + 1 ) ( r − 1 ) ) − F y ′ ( x m + ( 2 m + 1 ) ( r − 1 ) ) > m + ( 2 m + 1 ) ( r − 1 ) n − r − 1 k = m n . In both cases, KS ( x , y ′ ) > m n , as we wanted to prove. The quantile mechanism is L 1 -optimal W e will show that any sample y ′ that is not equivalent to y satisﬁes L 1 ( x , y ′ ) > L 1 ( x , y ) . Since this inequality holds in particular for every mechanism M ′ and every sample y ′ which is possible under some equilibrium of M ′ , it will follow 31 that the quantile mechanism is L 1 -optimal. Let then y ′ = ( x ℓ 1 , x ℓ 2 , . . . , x ℓ k ) be any sample that is not equivalent to y , and assume w .l.o.g. that ℓ 1 < ℓ 2 < · · · < ℓ k . Step 1: The idea of the proof. T o prove that L 1 ( x , y ) < L 1 ( x , y ′ ) , we will construct a sequence y ( 1 ) , y ( 2 ) , . . . , y ( p ) of samples such that y ( 1 ) = y , y ( p ) = y ′ , and L 1 ( x , y ( π ) ) < L 1 ( x , y ( π + 1 ) ) for each π ∈ { 1, . . . , p − 1 } . T o pr ove the existence of such a sequence, we will ar gue that if y ′  ≈ y , then there is another set of k indices ( j 1 , j 2 , . . . , j k ) that satisﬁes the following properties: P1) ( j 1 , j 2 , . . . , j k ) and ( ℓ 1 , ℓ 2 , . . . , ℓ k ) differ by exactly one index. P2) ( j 1 , j 2 , . . . , j k ) is closer to ( m + 1, m + 1 + ( 2 m + 1 ) , . . . , m + 1 + ( 2 m + 1 ) ( k − 1 ) ) than ( ℓ 1 , ℓ 2 , . . . , ℓ k ) in a sense that will be deﬁne shortly . P3) Denoting y ′′ = ( x j 1 , x j 2 , . . . , x j k ) , we have L 1 ( x , y ′′ ) < L 1 ( x , y ′ ) . A recursive application of this result, yields a sequence ( y ( π ) ) p π = 1 as described above. Step 2: Deﬁning the sense in which ( j 1 , j 2 , . . . , j k ) is closer to ( m + 1, m + 1 + ( 2 m + 1 ) , . . . , m + 1 + ( 2 m + 1 )( k − 1 ) ) than ( ℓ 1 , ℓ 2 , . . . , ℓ k ) . W e will say that index j r is closer to m + 1 + ( 2 m + 1 ) ( r − 1 ) than index l r if one of the following conditions holds: • x j r ≈ x m + 1 + ( 2 m + 1 ) ( r − 1 ) and x ℓ r  ≈ x m + 1 + ( 2 m + 1 ) ( r − 1 ) . • x j r  ≈ x m + 1 + ( 2 m + 1 ) ( r − 1 ) , x ℓ r  ≈ x m + 1 + ( 2 m + 1 ) ( r − 1 ) , and the number of items in x that lies strictly between x j r and x m + 1 + ( 2 m + 1 ) ( r − 1 ) is smaller than the number of items in x that lies strictly between x ℓ r and x m + 1 + ( 2 m + 1 ) ( r − 1 ) . By deﬁnition, it cannot be that both j r is closer to x m + 1 + ( 2 m + 1 ) ( r − 1 ) than ℓ r , and ℓ r is closer to x m + 1 + ( 2 m + 1 ) ( r − 1 ) than j r . If j r is not closer to x m + 1 + ( 2 m + 1 ) ( r − 1 ) than ℓ r , and ℓ r is not closer to x m + 1 + ( 2 m + 1 ) ( r − 1 ) than j r , then either (a) x j r ≈ x ℓ r , or (b) x j r  ≈ x ℓ r and the number of items in x that lies strictly between x j r and x m + 1 + ( 2 m + 1 ) ( r − 1 ) is equal to the number of 32 items in x that lies strictly between x ℓ r and x m + 1 + ( 2 m + 1 ) ( r − 1 ) . Note that if (b) holds while (a) does not, then necessarily one among j r and ℓ r is smaller than m + 1 + ( 2 m + 1 ) ( r − 1 ) while the other is larger than m + 1 + ( 2 m + 1 ) ( r − 1 ) . W e will say that the vector ( j 1 , j 2 , . . . , j k ) is closer to ( m + 1, m + 1 + ( 2 m + 1 ) , . . . , m + 1 + ( 2 m + 1 ) ( k − 1 ) ) than the vector ( ℓ 1 , ℓ 2 , . . . , ℓ k ) if for the minimal r such that j r is closer to m + 1 + ( 2 m + 1 ) ( r − 1 ) than index ℓ r is smaller than the minimal r such that ℓ r is closer to m + 1 + ( 2 m + 1 ) ( r − 1 ) than index j r . Step 3: Determining the minimal item in which y ′ and y dif fer . Let i be the largest index such that x ℓ i ≈ x m + 1 + ( 2 m + 1 ) ( i − 1 ) . This is a way to say that up to equivalence, the samples y ′ and y agree in the lower i items. If x ℓ 1  ≈ x m + 1 , we set i : = 0. If i = k , then y ′ ≈ y . Since by assumption y and y ′ are not equivalent, i < k , and hence x ℓ i + 1  ≈ x m + 1 + ( 2 m + 1 ) i . In particular , ℓ i + 1  = m + 1 + ( 2 m + 1 ) i . Step 4: The case ℓ i + 1 > m + 1 + ( 2 m + 1 ) i . W e will show that in this case it is better to r eplace x ℓ i + 1 in y ′ by some x ℓ with ℓ < ℓ i + 1 . Step 4.1: Deﬁning the new set of vertices  j . Let ℓ < ℓ i + 1 be the largest index such that x ℓ ≺ x ℓ i + 1 . In particular , x ℓ ≺ x ℓ + 1 ≈ x ℓ + 2 ≈ · · · ≈ x ℓ i + 1 . Since x ℓ i + 1  ≈ x m + 1 + ( 2 m + 1 ) i , we have ℓ ≥ m + 1 + ( 2 m + 1 ) i . Consider the set of indices  j that is obtained from  ℓ by replacing ℓ i + 1 with ℓ :  j = ( ℓ 1 , ℓ 2 , . . . , ℓ i , ℓ , ℓ i + 2 , . . . , ℓ k ) . In particular , (P1) holds. Since  ℓ and  j coincide in their lower i + 1 items, and since x m + 1 + ( 2 m + 1 ) i ≾ x ℓ ≺ x ℓ i + 1 ,  j is closer to ( m + 1, m + 1 + ( 2 m + 1 ) , . . . , m + 1 + ( 2 m + 1 ) ( k − 1 ) ) than  ℓ , and (P2) holds. Denote by y ′′ the sample induced by the vector of indices  j : y ′′ = ( x ℓ 1 , x ℓ 2 , . . . , x ℓ i , x ℓ , x ℓ i + 2 , . . . , x ℓ k ) . 33 Step 4.2: The difference between L 1 ( x , y ′ ) and L 1 ( x , y ) . Since ℓ < ℓ i + 1 , the statistics L 1 ( x , y ′ ) and L 1 ( x , y ′′ ) differ in the summands that correspond to x ℓ , x ℓ + 1 , . . . , x ℓ i + 1 − 1 : since we changed the index ℓ i + 1 by ℓ < ℓ i + 1 , we add 1 k to the sample’s CDF at the points x ℓ , . . . , x ℓ i + 1 − 1 . The following table describes the quantities F x , F y ′ , and F y ′′ at these points, where F x ( x ℓ i + 1 ) ≥ ℓ i + 1 n : x ℓ x ℓ + 1 . . . x ℓ i + 1 − 1 F x ℓ n F x ( x ℓ i + 1 ) . . . F x ( x ℓ i + 1 ) F y ′ i k i k . . . i k F y ′′ i + 1 k i + 1 k . . . i + 1 k It is thus sufﬁcient to show that     ℓ n − i k     ? >     ℓ n − i + 1 k     (7) and     F x ( x ℓ i + 1 ) − i k     ? >     F x ( x ℓ i + 1 ) − i + 1 k     . (8) Step 4.3: Eq. ( 7 ) holds when ℓ n ≥ i + 1 k . If ℓ n ≥ i + 1 k , then Eq. ( 7 ) holds. Indeed, in this case we can remove the abso- lute values from both sides of Eq. ( 7 ), and from the right-hand side we subtract a larger amount than fr om the left-hand side. Step 4.4: Eq. ( 7 ) holds when ℓ n < i + 1 k . In this case we should then show that ℓ n − i k ? > i + 1 k − ℓ n , (9) which solves to 2 ℓ ? > ( 2 i + 1 ) ( 2 m + 1 ) . However , this inequality holds since ℓ ≥ m + 1 + ( 2 m + 1 ) i . Step 4.5: Eq. ( 8 ) holds when F x ( x ℓ i + 1 ) ≥ i + 1 k . The argument is similar to that in Step 4.3. Step 4.6: Eq. ( 8 ) holds when F x ( x ℓ i + 1 ) < i + 1 k . 34 In this case we need to show that F x ( x ℓ i + 1 ) − i k ? > i + 1 k − F x ( x ℓ i + 1 ) . (10) Since Eq. ( 9 ) holds, and since F x ( x ℓ i + 1 ) > ℓ n , Eq. ( 10 ) holds as well. Indeed, the left-hand side in Eq. ( 10 ) is larger than the left-hand side in Eq. ( 9 ), while the right-hand side in Eq. ( 10 ) is smaller than the right-hand side in Eq. ( 9 ). Step 5: The case ℓ i + 1 < m + 1 + ( 2 m + 1 ) ( i + 1 ) . Note that in this case x ℓ i + 1 ≺ x m + 1 + ( 2 m + 1 ) ( i + 1 ) , and it may happen that x m + 1 + ( 2 m + 1 ) ( i + 1 ) lies in the sample y ′ . W e will construct y ′′ by r eplacing x ℓ i + 1 with an item x ℓ with ℓ > ℓ i + 1 . In fact, ℓ will be the smallest index that is larger than ℓ i + 1 and does not appear among the indices that deﬁne y ′ . Step 5.1: Deﬁning the new set of vertices  j . Let ℓ > ℓ i + 1 be the minimal index such that (a) x ℓ  ≈ x ℓ i + 1 , and (b) ℓ  ∈ { ℓ i + 2 , ℓ i + 3 , . . . , ℓ k } . W e will show that in this case it is better to r eplace x ℓ i + 1 in y ′ by x ℓ . W e ﬁrst argue that such an index ℓ exists. Indeed, since m ≥ 1, and since ℓ i + 1 < m + 1 + ( 2 m + 1 ) ( i + 1 ) , the number of indices in { 1, . . . , n } lar ger than ℓ i + 1 is larger than 2 ( k − i ) . However , the number of items in y ′ larger than x ℓ i + 1 is k − i − 1. Hence there is at least one index ℓ > ℓ i + 1 that satisﬁes (a) and (b). Let  j be the vector that is derived fr om  ℓ by r eplacing ℓ i + 1 with ℓ , so that (P1) holds. Since ℓ i + 1 < ℓ , this vector is closer to ( m + 1, m + 1 + ( 2 m + 1 ) , . . . , m + 1 + ( 2 m + 1 ) ( k − 1 ) ) than  ℓ in the lexicographic order , and (P2) holds. W e will show that the output vector y ′′ : = y ′ \ { x ℓ i + 1 } ∪ { x ℓ } satisﬁes L 1 ( x , y ′ ) > L 1 ( x , y ′′ ) , so that (P3) holds as well. Step 5.2: The difference between L 1 ( x , y ′ ) and L 1 ( x , y ) . Since ℓ i + 1 < ℓ , the statistics L 1 ( x , y ′ ) and L 1 ( x , y ′′ ) differ in the summands that correspond to all r with x r ≈ x ℓ i + 1 , and in the summands that correspond to ℓ i + 1 + 1, ℓ i + 1 + 2,. . . , x ℓ − 1 . Since to create y ′′ from y ′ we changed x ℓ i + 1 to x ℓ , 35 and since ℓ > ℓ i + 1 , we have F y ′ ( x j ) = F y ( x j ) + 1 k for any j such that x j ≈ x ℓ i + 1 and for ℓ i + 1 + 1, ℓ i + 1 + 2,. . . , x ℓ − 1 . The following table describes the quantities F x , F y ′ , and F y ′′ in these summands: x r ≈ x ℓ i + 1 x ℓ i + 1 + 1 x ℓ i + 1 + 2 . . . x ℓ − 1 F x ( · ) F x ( x ℓ i + 1 ) F x ( x ℓ i + 1 + 1 ) F x ( x ℓ i + 1 + 2 ) . . . F x ( x ℓ ) F y ′ ( · ) F y ′′ ( x ℓ i + 1 ) + 1 k F y ′′ ( x ℓ i + 1 + 1 ) + 1 k F y ′′ ( x ℓ i + 1 + 2 ) + 1 k . . . F y ′′ ( x ℓ − 1 ) + 1 k F y ′′ ( · ) F y ′′ ( x ℓ i + 1 ) F y ′′ ( x ℓ i + 1 + 1 ) F y ′′ ( x ℓ i + 1 + 2 ) . . . F y ′′ ( x ℓ − 1 ) W e will show that in all these summands, the contribution to L 1 ( x , y ′′ ) is smaller than the contribution to L 1 ( x , y ′ ) . Step 5.3: Comparing summands. T ake any q ≥ 0 such that ℓ i + 1 + q < ℓ . W e will show that     F x ( x ℓ i + 1 + q ) − F y ′′ ( x ℓ i + 1 + q ) − 1 k     ? >    F x ( x ℓ i + 1 + q ) − F y ′′ ( x ℓ i + 1 + q )    . (11) If F x ( x ℓ i + 1 + q ) ≤ F ′′ ( x ℓ i + 1 + q ) , then Eq. ( 11 ) as in Step 4.3. Assume then that F x ( x ℓ i + 1 + q ) > F ′′ ( x ℓ i + 1 + q ) . T o prove Eq. ( 11 ) it is sufﬁcient to show that F y ′′ ( x ℓ i + 1 + q ) + 1 k − F x ( x ℓ i + 1 + q ) ? > F x ( x ℓ i + 1 + q ) − F y ′′ ( x ℓ i + 1 + q ) . (12) W e ﬁrst relate F x ( x ℓ i + 1 + q ) to F x ( x ℓ i + 1 ) , and F y ′′ ( x ℓ i + 1 + q ) to F y ′′ ( x ℓ i + 1 ) . Let D ≥ 0 be the number of indices j such that x ℓ i + 1 ≺ x j ≾ x ℓ i + 1 + q . Then F x ( x ℓ i + 1 + q ) = F x ( x ℓ i + 1 ) + D n . (13) Since by the deﬁnition of ℓ , for every such j , x j is in y ′′ , we have F y ′′ ( x ℓ i + 1 + q ) = F y ′′ ( x ℓ i + 1 ) + D k . (14) Substituting Eqs. ( 13 ) and ( 14 ) in Eq. ( 12 ) and simplifying the resulting equation, 36 we obtain that we need to show that 2 F y ′′ ( x ℓ i + 1 ) + 2 D + 1 k ? > 2 F x ( x ℓ i + 1 ) + 2 D n . (15) However , F y ′′ ( x ℓ i + 1 ) ≥ i k , and F x ( x ℓ i + 1 ) ≤ m + ( 2 m + 1 ) i . Hence the left-hand side in Eq. ( 15 ) is ( 2 i + 2 D + 1 ) ( 2 m + 1 ) n , while the right-hand side is at most 2 m + 2 i ( 2 m + 1 )+ 2 D n , and the former is always larger than the latter . The quantile mechanism is C v M -optimal The proof is analogous to that presented in Section 7 . W e here detail the differ- ences. Step 1: The case ℓ i + 1 > m + ( 2 m + 1 ) ( i + 1 ) . When ℓ i + 1 > m + ( 2 m + 1 ) ( i + 1 ) , the analog of Eq. ( 7 ) is  ℓ n − i k  2 ? >  ℓ n − i + 1 k  2 . This equation reduces to 2 ℓ n k ? >  i k + 1 k  2 −  i k  2 , which solves to 2 ℓ ? > ( 2 i + 1 ) ( 2 m + 1 ) . However , this equation holds since ℓ ≥ m + 1 + ( 2 m + 1 ) i . Step 2: The case ℓ i + 1 < m + ( 2 m + 1 ) ( i + 1 ) . The analog of Eq. ( 11 ) is  F x ( x ℓ i + 1 + q ) − F y ′′ ( x ℓ i + 1 + q ) − 1 k  2 ? >  F x ( x ℓ i + 1 + q ) − F y ′′ ( x ℓ i + 1 + q )  2 , which simpliﬁes to 1 k 2 ? > 2 k  F x ( x ℓ i + 1 + q ) − F y ′′ ( x ℓ i + 1 + q )  . (16) 37 Multiplying both sides of Eq. ( 16 ) by k 2 and taking into account Eqs. ( 13 ) and ( 14 ), Eq. ( 16 ) translates to 1 ? > 2 k  F x ( x ℓ i + 1 ) − F y ′′ ( x ℓ i + 1 ) + D n − D k  , which is equivalent to 2 k F y ′′ ( x ℓ i + 1 ) + 2 D + 1 ? > 2 k F x ( x ℓ i + 1 ) + 2 k D n . However , this inequality is equivalent to Eq. ( 15 ). Proof of Theorem 3 W e start by proving Claim (i). For every partition P ∈ P ( n , k ) , let y ( P ) be the sample that contains, for each element in P , an item that is maximal according to ≾ 2 . The sample y ( P ) is the outcome when Player 1 partitions x according to P , Player 2’s ranking is ≾ 2 , and Player 2 best responds to P . Let P ∗ ∈ P ( n , k ) be Player 1’s optimal partition under ≾ 1 . Note that y ( P ∗ ) stochastically dominates y ∗ 1 . Indeed, both samples are derived from the parti- tion P ∗ ; however , y ∗ 1 is derived assuming Player 2’s ranking is ≿ 1 – the worst from Player 1’s point of view . Hence, u 1 ( y ( P ∗ ) ) ≥ u 1 ( y ∗ 1 ) under ≾ 1 . Therefore, in equilibrium, Player 1’s utility is at least u 1 ( y ( P ∗ ) ) , and (i) follows. W e turn to prove Claim (ii). W e will show that F y ∗ 2 stochastically dominates F y ( x ) under ≾ 2 , from which (ii) will follow . Fix then i ∈ { 1, . . . , n } , and suppose that # { j ∈ { 1, . . . , n } : x j ≾ 2 x i } = r . In the sample y ∗ 2 ( x ) , there are ⌈ r − m 2 m + 1 ⌉ items that are lower than or equivalent to x i under ≾ 2 . Whatever be the partition of Player 1 under the equilibrium, the r items in { x j : x j ≾ 2 x i } lie in at least ⌈ r 2 m + 1 ⌉ elements of the partition. Since in equilib- rium Player 2 selects the minimal item accor ding to ≾ 2 from each element of the partition, there are at least ⌈ r 2 m + 1 ⌉ items in y ( x ) that are lower than or equiva- lent to x i under ≾ 2 . Since ⌈ r 2 m + 1 ⌉ ≥ ⌈ r − m ⌉ 2 m + 1 , we have F y ( x ) ( x i ) ≥ F y ∗ 2 ( x ) according to ≾ 2 . Since i is arbitrary , F y ∗ 2 stochastically dominates F y ( x ) , as claimed. 38 Proof of Theorem 4 Fix for a moment a partition P = ( Q 1 , . . . , Q k ) ∈ P ( n , k ) selected by Player 1. Since all elements of P include m items, the probability that item x i is selected to the sample is 1 m . By the linearity of the expectation operator , µ P ( y ) = 1 k k ∑ ℓ = 1 1 m ∑ x i ∈ Q ℓ x i ! = 1 n n ∑ i = 1 x i . Denote by ( j 1 , . . . , j k ) the random indices of the items in the sample; that is, j ℓ is selected uniformly from Q ℓ , for every 1 ≤ ℓ ≤ k . The sample’s variance is σ 2 P ( y ) = ∑ P ∈ P ( n , k )   1 k k ∑ ℓ = 1 x 2 j ℓ − 1 k k ∑ ℓ = 1 x j ℓ ! 2   . Because there are m k differ ent samples, and they are all equally likely , the ex- pected variance is E [ σ 2 P ( y ) ] = 1 n n ∑ i = 1 x 2 i − 1 m k ∑ P ∈ P ( n , k ) 1 k k ∑ ℓ = 1 x j ℓ ! 2 . Since the ﬁrst summand is independent of P , to minimize E [ σ 2 P ( y ) ] it is suf ﬁcient to maximize the second summand. Since the function x 7 → x 2 is convex, by Jensen’s inequality , the second summand is maximized when P is the partition of the population into k blocks of size m , wher e each block contains consecutive items according to ≾ . 39

Adversarial Selection

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