Algorithmic Bayesian Persuasion

Algorithm ic Bayesian Persuasio n Shaddin Dughmi ∗ Department of Computer Science Univ ersity of Southern C alifornia shaddin@us c.edu Haifeng Xu † Department of Computer Science Univ ersity of Southern California haifengx@u sc.edu February 16, 2016 Abstract P ersuasion , deﬁned as the act o f exploiting an informationa l advantage in order to effect th e decisions of others, is ub iquitous. In deed, p ersuasive communic ation has been estimated to accou nt fo r almost a third of all economic activity in the US. This pap er examin es per suasion thro ugh a compu tational lens, focusing o n what is perhaps the most b asic and fu ndamen tal model in this space : the celebrated B ayesian persuasion model of Kamenica and G entzkow [3 4]. Here there are tw o players, a sender and a r eceiver . The receiver m ust take on e of a nu mber of ac tions with a -priori unknown pay off, an d the sender h as access to additional in formation regard ing the p ayoffs of the v arious actions for b oth p layers. The send er can co mmit to r ev ealing a noisy sign al regard ing the r ealization of the pay offs of various action s, and would like to do so as to m aximize her own payoff in expectation a ssuming that the receiver ration ally acts to maximize his own p ayoff. When th e payoffs o f various action s follow a joint distribution (the common pr ior), the sender’ s problem is nontr ivial, and its co mputation al complexity depends on the representatio n of this prior . W e examine the sender’ s o ptimization task in three o f the most natural inpu t mod els for this problem, and essentially pin down its computational complexity in each. When the payoff d istributions o f the different actions are i.i.d . and g iv en explicitly , we exhibit a polyno mial-time (exact) algorithmic solution, and a “simple” (1 − 1 /e ) -appr oximation algorithm . Our optimal scheme f or the i.i.d. setting inv olves an analogy to auctio n theory , an d makes use of Border’ s characterizatio n of the space of redu ced-for ms for single-item auctions. When action payoffs are independe nt but non-identical with marginal d istributions giv en explicitly , we show that it is #P-ha rd to comp ute the optim al expected send er utility . In doing so, we rule ou t a generalized Border’ s theo r em , as deﬁned by Go palan et al [30], for this setting . Fi nally , we con sider a g eneral (possibly cor related) joint distribution of a ction payoffs presented by a black b ox sampling oracle , and exhib it a f ully p olyno mial-time appro ximation scheme (FPT AS) w ith a bi-cr iteria guaran tee. Our F PT AS is based on Monte-Carlo sam pling, and its analysis relies on the principle of deferred d ecisions. Moreover, we show th at this result is th e best p ossible in the black -box model fo r informa tion-theo retic reasons. ∗ Supported in part by NSF CAREER A ward CCF-1350900. † Supported by NSF grant CCF-1350900. 1 Introd uction “ One quarter of the GDP is per suasion. ” This is both the title, and the thesis, of a 1995 paper by McCloske y and Klamer [39]. Since then, persua sion as a share of economic acti vity app ears to be gro wing — a more rec ent esti mate place s the ﬁgu re at 30% [4]. As both papers make clear , pers uasion is intrinsi c in most human endea vors. When the tools of “persu asion” are tangible — say go ods, services, or money — t his is the do main of tra ditiona l m ech anism design , which steers the actions of one or many self-inte rested agents tow ards a designe r’ s object i ve. What [39, 4] and much of the rele van t literature refer to as persuasio n, ho wev er , are scenarios in which the po wer to persuade deriv es from an informationa l advanta ge of some party ov er others . T his is also the sens e in whic h w e use the ter m. Such sc enarios are incre asingly common in the informat ion economy , and it is therefo re unsurpr ising that persuasion h as been the subject of a large body of work in recent years, moti v ated by domains as vari ed as auctions [9, 25, 24, 10], adver tising [3, 33, 17], voting [2], security [46, 42], multi- armed ban dits [37, 38], medical resea rch [35], and ﬁnancial re gulation [28, 29]. (For an empiric al surv ey of persuasion , we refer the reader to [21]). What is surprising , ho wev er , is the lack of sys tematic stud y of persua sion through a computa tional lens; this is what we embark on in this paper . In the la r ge body of liter ature de vot ed to persuasion, perhaps no model is more basic a nd f undamen tal than the B ayesia n P ersuasi on model of Kamenica and Gentzk o w [34], gene ralizin g an earlier model by Brocas and Carrillo [14]. Here there are two players , who we call the sender and the r eceiver . The recei ver is fa ced with selectin g one of a number of acti ons , each of which is associa ted w ith an a-pri ori unkno wn payof f to both p layers. The state of na tur e , d escribi ng the payof f to the s ender and receiv er fro m each action, is drawn fr om a prior d istrib ution kno w n to both pl ayers. Howe ver , the se nder po ssesse s an info rmational adv antage, namely access to the r ealize d state of nature prior to th e receiv er choosing his action. In ord er to persua de the rec eive r to tak e a more f av orable action f or he r , the s ender c an co mmit to a policy , ofte n kno wn as an information structur e or signaling sc heme , of releasing information about th e rea lized state of nature to the rece i ver befo re the recei ver make s his cho ice. This poli cy may b e simple, say b y alway s announc ing the payof fs of the variou s actions or always saying nothi ng, or it may be intricate, in vo lving p artial informatio n and added noise. Crucially , the recei ver is aware of the sender’ s committed polic y , and moreov er is rational and Bay esian. W e ex amine the send er’ s algorithmic probl em of impl ementing the optimal signa ling scheme in this p aper . A solution to this proble m, i.e., a sig naling sch eme, is an alg orithm which takes as in put the descri ption of a state of nature and output s a signal, potent ially utilizing some interna l randomnes s. 1.1 T wo Examples T o illu strate the intricac y of Bayesian Persuasion, Kamenica and Gentzk ow [34] use a simple exa mple in which the sender is a prosecutor , the receiv er is a judge, and the state of nature is the guilt or innocence of a defenda nt. The receiv er (judge) has two action s, con viction and acquittal, and wishes to maximize the probability of rendering the correct verdic t. On the other hand, the sender (prose cutor) is interested in maximizing the probabili ty of con viction. As they sh o w , it is easy to construct exampl es in w hich the optimal signali ng sc heme fo r the sender releases noisy partial informatio n regar ding the guilt or innocen ce of th e defendan t. For example , if the d efendant is guilt y with p robabi lity 1 3 , the prosecutor ’ s best strate gy is to claim “guilt” whene ver the defenda nt is guilty , and also claim “guilt” just under half the time w hen the defenda nt is innocent . As a result, the defe ndant will be con victed whene ver the prosec utor c laims “guilt ” (happening with probabil ity just under 2 3 ), assu ming that the judge is fully aware o f the prosecuto r’ s signal ing scheme. W e note that it is not in the prosecu tor’ s interes t to alwa ys claim “guilt”, since a rational judge a ware of su ch a polic y would ascri be no meaning to such a sign al, and render his verd ict based solely on his prior belief — in this case, this would al ways lead to acquittal. 1 1 In other words, a signal is an abstract object with no intrinsic mean ing, and is only imbu ed with meaning by virtue of h ow it is used. In particular , a signal has no meaning beyon d the posterior distribution on states of na ture it induces. 1 A somewhat less artiﬁcial exa mple o f persuasion is in t he context of pro viding ﬁnanci al advic e. Here, the recei ver is an in vesto r , ac tions correspond to stocks, and the sender is a stockbrok er or ﬁnancial adviser with access to stoc k retur n project ions which are a-priori unkno wn to the in vestor . When the advise r’ s commission or return is not aligne d with the in vestor’ s returns, this is a nontri vial Bayesi an persua sion proble m. In fact, interesti ng exa mples exist when stock return s are indepen dent from each other , or ev en i.i.d. Consider th e fol lo wing simpl e example which ﬁts into the i.i.d. model consid ered in Section 3: there are two stock s, eac h of w hich is a-priori equally likely to generate lo w (L), moderate (M), or high (H) short- term returns to the in vestor (independe ntly). W e refer to L/M/H as the types of a stock, and associate them with short-term retur ns of 0 , 1 + ǫ , an d 2 respect i vely . Suppo se, also, that stocks of type L or H are associ ated w ith p oor long-te rm returns of 0 ; i n the cas e of H, high sho rt-term returns might be an indication of vola tility or overv aluatio n, and hence poor long -term perf ormance. This lea ves sto cks of type M as th e only so lid perfo rmers with long- term returns of 1 . Now s uppose that the in vestor is myopic ally intere sted in maximizing short- term returns, wh ereas the fo rward-looki ng ﬁnanci al adviser is concern ed with maximizing long-t erm returns, perhaps due to reputati onal co nsider ations. Simple calculation shows that provi ding full informat ion to the myopi c in vestor results in an exp ected long-t erm rewa rd of 1 3 , as does pro viding no informat ion. An optimal signali ng scheme, which guarantees that the in vestor choose s a stock with type M whe ne ver such a stock e xists, is t he fo llo wing: when e xactly one of the stock s has type M recommend that stock, and otherwise recommend a stock uniformly at random. A simple calculation using Bayes’ rule sho ws that the in vestor prefer s to foll o w the re commendat ions of this pa rtially -informat i ve scheme, and it follo ws that the expected long-term return is 5 9 . 1.2 Results and T e chniques Moti vated by these intricaci es, w e study the computati onal complex ity o f optimal and near- optimal persua- sion in the presence of multiple action s. W e ﬁrst observ e that a linear program with a variab le for each (state- of-natu re, actio n) pair compute s a descript ion of the optimal signa ling scheme. Howe ver , when ac- tion payof fs are distrib uted accordin g to a joint distrib ution — say exhi biting some degr ee of indepe ndenc e across differe nt actions — the numbe r of s tates of nature may be ex ponent ial in th e nu mber of actions; in such settings, both the number of varia bles and constraints of this linear program are expone ntial in the number of actions . It is theref ore unsurprisin g that the computational comple xity of persuasion depends on h o w the pr ior d istribu tion on sta tes of nature is presented as input. W e the refore con sider three natural input models in increasing order of generality , and mostly pin do wn the complexit y of optimal and near - optimal persuas ion in each. Our ﬁrst mod el as sumes that action pa yof fs are drawn i.i.d. from an explici tly descri bed mar ginal distrib ution. Our second model considers indep endent yet non-identic al actions , again with expli citly-d escrib ed marg inals. Our third and m ost general model considers an arbitra ry joint distrib u- tion of action payof fs pre sented by a black-box sampling o racle. In pr ovi ng our results, we draw connec tions to techniq ues and concep ts de veloped in the conte xt of Bayesian mechanism design (BMD ), ex ercising and genera lizing them along the way as needed to prov e our results . W e mention some of these conne ctions brieﬂy here, and elabor ate on the similarit ies and dif ferences from the BMD literature in Appendix A. W e start with the i.i.d mode l, and show two results: a “si mple” and po lynomial -time e − 1 e -appro ximate signal ing scheme, and a poly nomial-time implementation of the opti mal scheme. Both re sults hinge on a “symmetry charact erizatio n” of the o ptimal scheme in the i.i.d. settin g, closely related to the symmetrization result from BM D by [20] bu t with an important diffe rence which we discuss in Appendix A. O ur “simple” scheme decouples the signali ng proble m for the di f ferent a ctions and sig nals ind ependently for each. This result imp lies t hat sign aling in this sett ing can b e “distri b uted” among multip le non-coordi nating persuaders without much loss. Our optimal scheme in v olv es a connec tion to Border’ s characteriz ation of the space of feasible reduced -form auctions [13, 12], as well as its algorithmic properties [15, 1]. This connection in volv es provi ng a correspond ence between “symmetric” signaling schemes and a subset of “symmetric” single -item auction s; one in which action s in persuas ion corresp ond to bidd ers in an auction. 2 Next, we consider Bayes ian p ersuasion with indepe ndent non -identi cal actions. One might e xpect th at the partial correspo ndence between signal ing scheme s and single-ite m auctions in the i.i.d. mod el gen- eralize s here, in which case Border’ s theorem — which exte nds to single-ite m auction s with independ ent non-id entical bidders — would analogou sly lead t o p olynomial time a lgorith m for per suasio n in th is setting. Ho wev er , we surprisin gly sho w that this analogy to single-item auctions ceases to hold for non-id entical action s: we prove that there is no gene rali zed Bor der’ s theor em , in the sense of Gopalan et al. [30], for per - suasio n with indepe ndent actio ns. Spe ciﬁcally , we sho w that it is #P-hard to exactly compute the expec ted sender utility for the op timal scheme, ruling out B order’ s-theo rem-like approaches to this problem unless the polyno m ial hierarch y collapses. O ur proo f starts from the ideas of [30], b ut our reductio n is much more in volv ed and g oes thro ugh the members hip prob lem for an implicit polyto pe which en codes a #P-hard pro b- lem — we ela borate on these d ifferen ces in Appe ndix A. W e note that wherea s we d o rul e out computing an exp licit rep resent ation of t he optimal sche m e which permits ev aluating optimal sender u tility , we do n ot rul e out oth er appro aches which might sa mple the optimal scheme “on t he ﬂy” in the style of Myerson’ s optimal auctio n [41 ]— we lea ve the intriguin g question of whether this is possible as an open problem. Finally , w e consider the black-box m odel with general distrib utions, and prov e essent ially-mat ching pos- iti ve and neg ativ e results. For our p ositi ve resu lt, we exhibi t ful ly po lynomial-time approximat ion scheme (FPT AS) with a bicriteri a guarante e. Speciﬁcally , our sche me loses an additi ve ǫ in both ex pected sender utility and inc entiv e-compatibili ty (as de ﬁ ned i n S ection 2), and runs i n time poly nomial in the number of action s and 1 ǫ . Our negat i ve results sho w tha t thi s is essenti ally the best possi ble for information -theore ti c reason s: any poly nomial-time scheme in the black box model which comes close to optimalit y must signif- icantly sacriﬁce inc enti ve compatibilit y , and vice v ersa. W e n ote that ou r scheme is relate d to some prio r work on BMD with black-box distrib utions [16, 45], bu t is signiﬁcantly simpler and m ore ef ﬁ cient: instead of usin g the ell ipsoid method to op timize ov er “reduced forms”, our sc heme simply sol ves a single linear progra m on a sample fro m th e p rior dis trib ution on s tates of nature. Such simplicit y is possible in our setting due t o the dif ferent no tion of incenti ve compa tibilit y in persu asion, which reduc es to incenti ve compati bility on the sample using the princ iple of deferred decisi ons. W e elaborate on this connection in Appendix A. W e remark that our results suggest that the diffe rences between persuasio n and auction design serve as a double-e dged sword. T his is ev idence d by our nega ti ve result for indepe ndent model and our “simple” positi ve result for the black-b ox model. 1.3 Additional Discussion of Related W ork T o our kno wledge, Brocas and Carrill o [14] were th e ﬁrst to e xplicitly consi der pers uasion through infor ma- tion control. They consid er a sender with the ability to costless ly acquire information rega rding the payof fs of the recei ver’ s actions, with the stipul ation that acquired informatio n is av ailable to both players. This is technically equiv alent to our (and K amenica and Gentzko w’ s [34]) informed sender who commits to a signal ing scheme. Brocas and Carrillo restrict attention to a particu lar s etting with two states of nature and three actions, and c haract erize optimal pol icies fo r the sender an d the ir associat ed payof fs. Kamenic a and Gentzk ow’ s [34] Bayesian Persuasion model naturally generali zes [14] to ﬁnite (or inﬁnite yet compact) states of nature and action spaces. T hey establi sh a number of properties of o ptimal information str ucture s in this mod el; most notably , the y character ize setting s in which sig naling strictly beneﬁts the sender in terms of the con vex ity/con cavity of the sen der’ s payof f as a function of the recei ver’ s posterior belief. Since [14] and [34], an explo sion of interes t in persuasio n problems follo wed. The basic Bayesian persua sion model underl ies, or is closely related to, recent work in a number of differ ent domains : price discrimin ation by B er gemann et al. [ 10], advertisin g by C hakrab orty and H arbaug h [17] , security games by Xu e t al. [46] and Rabino vich et al. [42], multi-armed bandits b y Kremer e t al. [3 7] an d Mansour et al. [38], m edical researc h by K olotili n [35], and ﬁnancial regula tion by G ick and Pausch [28] and Goldstei n and Lei tner [29]. G enerali zation s and v ariants of th e Bayesian per suasio n m odel ha ve also been cons idered : Gentzk ow a nd Kamenica [ 26] consider multiple sen ders, Alo nso and C ˆ amara [2] consider m ultiple recei vers 3 in a v oting setti ng, Gentzk ow an d Kamenica [2 7 ] c onside r costly informat ion acquis ition, Rayo and Sega l [43] consi der an outsid e option for the receiv er , and Kolot ilin et al. [36] conside rs a recei ver with pri vate side informat ion. Optimal persuasio n is a special case of informatio n structur e design in games, also known as signal- ing . The space of informatio n structures, and their induce d equilibr ia, are charact erized by Berg emann an d Morris [8]. Recent wo rk in the CS co mmunity has also exa m ined the design of informatio n structures algo- rithmical ly . W ork b y Emek et al. [24], Mil tersen an d Shef fet [40], Guo and Deligkas [ 32], and Dughmi et al . [23], examine optimal signali ng in a v ariety of auction setting s, and presents polynomial-t ime algorithms and hardne ss re sults. Dugh mi [22] exhibit s h ardness re sults for signaling in two-player zero-sum games, and Cheng et al. [ 18] present an algori thmic frame work a nd ap ply it to a number of dif ferent si gnalin g proble m s. Also related to the Bayesian persuasion model is the extensi ve literat ure on cheap talk start ing with Crawford and Sobel [19]. Cheap talk can be viewed as the analogu e of persuasion w hen the sender cannot commit to an information rev elation policy . Ne vert heless , the commitment assumption in persuas ion has been justiﬁed on the gr ounds that it a rises org anically in r epeate d cheap t alk in teractions with a long horizon — in pa rticula r w hen the sender must balanc e his short term pa yoffs with lon g-term credibil ity . W e ref er the reader to th e dis cussion of this phenomenon in [43 ]. Also to this poin t, Kamenica an d Gentz kow [3 4] mention that an earlier m odel of repeate d 2-player games with asymmetric information by Aumann and Maschler [5] is mathematic ally analogous to Bayesian persua sion. V arious rece nt models on sellin g informatio n in [6, 7, 11] are qu ite similar to Baye sian persuasio n, w ith the mai n dif ference being that the sender’ s utility funct ion is rep laced with r e ven ue. Whereas Babai of f et al. [6] consid er the algorithmic questio n of selling information when states of nature are explic itly giv en as input, the an alogo us algorithmic ques tions to ou rs ha ve not been consi dered in the ir model. W e sp eculate that some of our algorit hmic techniqu es m ight be applica ble to m odels for selling information w hen the prior distr ib ution on st ates of nature is represente d succinct ly . As discusse d pre viously , ou r results in v olv e exe rcising and general izing ideas from prior work in Bayesian mechanis m design. W e view dra wing these connec tions as one of the contrib utions of our paper . In Ap- pendix A, we discus s these connec tions and dif ferences at length. 2 Pr eliminaries In a persua sion game, there ar e two players: a send er and a r eceiver . The receiv er is f aced with selecting an ac tion from [ n ] = { 1 , . . . , n } , with an a-p riori-u nkno wn payof f to each of the sender and receiv er . W e assume pa yoffs are a function of an unkno wn state of na tur e θ , dra wn from an abstr act set Θ of potential realiza tions of nature . S peciﬁcally , the sender and recei ver’ s payof fs are functions s, r : Θ × [ n ] → R , respec ti vely . W e use r = r ( θ ) ∈ R n to denote the recei ver’ s payof f vector as a function of the state of nature, where r i ( θ ) is the receiv er’ s payoff if he tak es action i and the state of nature is θ . Similarly s = s ( θ ) ∈ R n denote s the sender ’ s payof f vector , and s i ( θ ) is the sender’ s payof f if the recei ver take s action i and the state is θ . Wi thout loss of generali ty , we often conﬂate the abstract set Θ inde xing states of nature with the set of realiz able payoff v ector pairs ( s , r ) — i.e., we thin k of Θ as a subset of R n × R n . W e assume that Θ is ﬁnite for notation al co n venience, though this is not neede d for our results in Section 5. In Bayesian persuasio n, it is assumed that the state of nature is a-priori unkno wn to the receiv er , and dra wn from a common-kno wledge prior distrib ution λ support ed o n Θ . The sender , on the other h and, has access to the realizatio n of θ , and can commit to a polic y of partially rev ealing information regardin g its realiza tion be fore the recei ver selects his action. Speciﬁcally , the sender commits to a signaling scheme ϕ , mapping (possi bly randomly ) states of natur e Θ to a family of signa ls Σ . For θ ∈ Θ , we use ϕ ( θ ) to denote the (possibly random) signal selected when the state of nature is θ . M oreo ver , we use ϕ ( θ , σ ) to denote the probab ility of selecting the signal σ gi ven a state of nature θ . A n algorithm implements a signalin g scheme ϕ if it take s as input a state of nature θ , and samples the rand om variab le ϕ ( θ ) . Giv en a sign aling scheme ϕ w ith signals Σ , each signal σ ∈ Σ is realize d with probabi lity α σ = 4 P θ ∈ Θ λ θ ϕ ( θ , σ ) . Cond itioned on the signal σ , the exp ected payo f fs to the receiv er of the v arious action s are summariz ed by the vector r ( σ ) = 1 α σ P θ ∈ Θ λ θ ϕ ( θ , σ ) r ( θ ) . Similarly , the sender’ s payof f as a function of th e recei ver’ s action are summarized by s ( σ ) = 1 α σ P θ ∈ Θ λ θ ϕ ( θ , σ ) s ( θ ) . On receiv ing a signal σ , the recei ver performs a Bayesian update and selects an action i ∗ ( σ ) ∈ argmax i r i ( σ ) with expec ted recei ver utility max i r i ( σ ) . This in duces utility s i ∗ ( σ ) ( σ ) for th e sende r . In t he e vent of ties when se lecting i ∗ ( σ ) , we assume those ties are brok en in fa vor of the sender . W e adopt the perspe cti ve o f a sender looking to design ϕ to maximize her expected utility P σ α σ s i ∗ ( σ ) ( σ ) , in whic h cas e we say ϕ is optimal . When ϕ yields expected send er utility within an additi ve [multipl icati ve] ǫ of the b est possibl e, we say it i s ǫ -optimal [ ǫ -app roximate ] in the additi ve [multiplic ati ve] sense. A simple re velation -princi ple style ar gument [3 4] shows that a n optimal s ignalin g scheme need no t use more than n signal s, w ith o ne r ecommending eac h action. Such a dir ect scheme ϕ ha s signal s Σ = { σ 1 , . . . , σ n } , and satisﬁes r i ( σ i ) ≥ r j ( σ i ) for al l i , j ∈ [ n ] . W e think of σ i as a signal rec ommending act ion i , and the require - ment r i ( σ i ) ≥ max j r j ( σ i ) as a n incentive -compati bility (IC) constrai nt on our signaling sc heme. W e can no w write the sender’ s optimization problem as the follo wing LP with v ariables { ϕ ( θ, σ i ) : θ ∈ Θ , i ∈ [ n ] } . maximize P θ ∈ Θ P n i =1 λ θ ϕ ( θ , σ i ) s i ( θ ) subjec t to P n i =1 ϕ ( θ , σ i ) = 1 , for θ ∈ Θ . P θ ∈ Θ λ θ ϕ ( θ , σ i ) r i ( θ ) ≥ P θ ∈ Θ λ θ ϕ ( θ , σ i ) r j ( θ ) , for i, j ∈ [ n ] . ϕ ( θ , σ i ) ≥ 0 , for θ ∈ Θ , i ∈ [ n ] . (1) For our results in Section 5 , we relax our incenti ve constraints by assuming that the recei ver follo w s the recommend ation so long as it appr oximatel y maximizes h is utility — for a paramet er ǫ > 0 , we relax ou r r e- quiremen t to r i ( σ i ) ≥ max j r j ( σ i ) − ǫ , which translates to the relaxed IC constraints P θ ∈ Θ λ θ ϕ ( θ , σ i ) r i ( θ ) ≥ P θ ∈ Θ λ θ ϕ ( θ , σ i )( r j ( θ ) − ǫ ) in LP (1). W e call such sc hemes ǫ -inc entive compatible ( ǫ -IC) . W e judge the subop timality of an ǫ -IC scheme relati ve to the best (absolute ly) IC scheme ; i.e., in a bi-criteri a sense. Finally , we note that expe cted utilities, incenti ve compatibilit y , and optimality are properties not only of a signal ing scheme ϕ , bu t also of the distrib ution λ over its inputs. When λ is not clear from conte xt and ϕ is supported on a superse t of λ , w e often say that a signaling scheme ϕ is IC [ ǫ -IC] for λ , or optimal [ ǫ -optimal ] for λ . W e also use u s ( ϕ, λ ) to denote the expec ted sender utility P θ ∈ Θ P n i =1 λ θ ϕ ( θ , σ i ) s i ( θ ) . 3 Pe rsuasion with I.I.D. Actions In this sectio n, we assume the payof fs of differe nt actions are indepe ndent ly an d identic ally dis trib uted ( i.i.d.) accord ing to an explicitl y-desc ribed mar ginal distrib ution. Speciﬁcally , each state of nature θ is a vector in Θ = [ m ] n for a parameter m , whe re θ i ∈ [ m ] is the type of action i . A ssocia ted with each type j ∈ [ m ] is a pai r ( ξ j , ρ j ) ∈ R 2 , where ξ j [ ρ j ] is th e payof f to the send er [recei ver] when the recei ver choo ses an action with type j . W e are gi ven a margin al di strib ution over types , describ ed by a vecto r q = ( q 1 , ..., q m ) ∈ ∆ m . W e assume each action’ s type is dra wn indepe ndentl y accordin g to q ; speciﬁcal ly , the prio r dis trib ution λ on states of nature is giv en by λ ( θ ) = Q i ∈ [ n ] q θ i . For con venien ce, we let ξ = ( ξ 1 , ..., ξ m ) ∈ R m and ρ = ( ρ 1 , ..., ρ m ) ∈ R m denote the typ e-index ed vectors of sen der and recei ver payo f fs, respecti vely . W e assume ξ , ρ , and q — the parameter s describing an i.i.d. persu asion instance — are giv en explicitly . Note that t he number of s tates of nat ure is m n , and ther efore the natural repr esenta tion of a signa ling scheme ha s nm n v ariables. Moreov er , the na tural linea r progra m for the persuasion probl em in Section 2 has an expone ntial in n number of both variab les a nd constraint s. Ne ver theless , as mentioned in Section 2 we seek only to implemen t an optimal or near -optimal scheme ϕ as an oracle which take s as input θ and samples a signal σ ∼ ϕ ( θ ) . Our algorithms will run in time polynomial in n and m , and will optimize over a space of succin ct “r educed forms” for signaling schemes which we term signat ur es , to be described next. For a s tate of nature θ , deﬁne t he matrix M θ ∈ { 0 , 1 } n × m so that M θ ij = 1 if a nd only if action i has type j in θ (i.e. θ i = j ). Gi ven an i.i.d prior λ and a s ignalin g scheme ϕ with signals Σ = { σ 1 , . . . , σ n } , for each 5 M σ i = P θ λ ( θ ) ϕ ( θ , σ i ) M θ , for i = 1 , . . . , n. P n i =1 ϕ ( θ , σ i ) = 1 , for θ ∈ Θ . ϕ ( θ , σ i ) ≥ 0 , for θ ∈ Θ , i ∈ [ n ] . Figure 1: Realiza ble Signatures P max P n i =1 ξ · M σ i i s.t. ρ · M σ i i ≥ ρ · M σ i j , for i, j ∈ [ n ] . ( M σ 1 , ..., M σ n ) ∈ P Figure 2: Persua sion in Signature Space i ∈ [ n ] let α i = P θ λ ( θ ) ϕ ( θ , σ i ) den ote the pr obabil ity of se nding σ i , and let M σ i = P θ λ ( θ ) ϕ ( θ , σ i ) M θ . Note that M σ i j k is the joint probability tha t acti on j has type k and the scheme o utputs σ i . Also note that eac h ro w of M σ i sums to α i , and the j th ro w represen ts the un-no rmalized posterior type distri b ution o f action j gi ven signal σ i . W e call M = ( M σ 1 , ..., M σ n ) ∈ R n × m × n the signat ur e of ϕ . The sender’ s obje ctiv e an d recei ver’ s IC constraint s can both be express ed in terms o f the signature . In particular , using M j to denote the j th row of a matrix M , the IC constra ints are ρ · M σ i i ≥ ρ · M σ i j for all i, j ∈ [ n ] , and the sender’ s exp ected utility assuming the recei ver follo ws the scheme’ s recommendati ons is P i ∈ [ n ] ξ · M σ i i . W e say M = ( M σ 1 , ..., M σ n ) ∈ R n × m × n is r ealizable if there exists a signaling scheme ϕ w ith M as its sign ature. Realizable signat ures constitutes a polytop e P ⊆ R n × m × n , which has an e xponential -sized ext ended formul ation as shown Figure 1. Gi ven this ch aracter ization , the sender’ s optimization prob lem can be written as a linea r progra m in the space of signat ures, sho wn in Figure 2: 3.1 Symmetry of the Optimal Signaling Scheme W e no w show that there alway s exists a “symmetric ” optimal scheme when actions are i.i.d. Giv en a signa- ture M = ( M σ 1 , ..., M σ n ) , it will sometimes be con veni ent t o think of it as the set of pairs { ( M σ i , σ i ) } i ∈ [ n ] . Deﬁnition 3 .1. A signaling sc heme ϕ with sign atur e { ( M σ i , σ i ) } i ∈ [ n ] is sy mmetric if ther e exis t x , y ∈ R m suc h that M σ i i = x for all i ∈ [ n ] and M σ i j = y for all j 6 = i . The pair ( x , y ) is the s -signature of ϕ . In othe r words, a symmetric s ignali ng scheme sends eac h signal with equal prob ability || x || 1 , and in- duces only two diff erent posterior typ e distrib utions for actions : x || x || 1 for the recommended action, and y || y | | 1 for th e others. W e call ( x , y ) rea lizable if there ex ists a sig naling sch eme with ( x , y ) as its s -signatu re. T he family of re alizable s -sig natures cons titutes a po lytope P s , and has a n exten ded formulation by a dding the v ariables x , y ∈ R m and constrain ts M σ i i = x and M σ i j = y for all i, j ∈ [ n ] with i 6 = j to the extended formulat ion of (asymmetr ic) realizab le si gnatur es from Figure 1. W e make two simple observ ations rega rding realiza ble s -signatures. F irst, || x || 1 = || y || 1 = 1 n for each ( x , y ) ∈ P s , and this is because both || x || 1 and || y || 1 equal t he pr obabil ity of each of the n signal s. Second, since the signatu re must be consiste nt with prior marg inal distrib ution q , we hav e x + ( n − 1) y = P n i =1 M σ i 1 = q . W e sho w that restricting to symmetric signalin g sc hemes is without loss of generali ty . Theor em 3.2. When the action p ayof fs ar e i.i.d., ther e e xists an optimal and incentive-c ompatibl e sig naling sch eme which is symmetric. Theorem 3.2 is pro ved in Appendix B.1. At a high le vel, we sho w that opti mal signa ling schemes are closed w ith respect to two operati ons: con ve x combinatio n an d permutatio n . Speciﬁcally , a con ve x co m bi- nation of realizable signatures — viewed as vecto rs in R n × m × n — is realized by t he correspond ing “rando m mixture” of sig naling schemes, and this oper ation preserve s optimality . The proof of this fact follo ws easily from th e fact that linea r progr am in Figu re 2 has a con ve x family o f optimal solu tions. Moreov er , giv en a permutat ion π ∈ S n and an opti mal signa ture M = { ( M σ i , σ i ) } i ∈ [ n ] realize d by signaling scheme ϕ , the “permute d” signatu re π ( M ) = { ( π M σ i , σ π ( i ) ) } i ∈ [ n ] — where premul tiplica tion of a matrix by π denotes permutin g th e rows of the m atrix — is realized by the “permuted” s cheme ϕ π ( θ ) = π ( ϕ ( π − 1 ( θ ))) , whic h is als o optimal. The proof of th is fact fol lo ws from the “symmetry” of th e (i.i.d.) prio r distrib ution about the dif ferent actions. Theorem 3.2 is then pro ved constr ucti vely as follo ws: giv en a realizable optimal signa ture M , the “symmetri zed” signatur e M = 1 n ! P π ∈ S n π ( M ) is realizable, optimal, and symmetric. 6 3.2 Implementing the Optimal Signaling Scheme W e no w exhibit a polynomia l-time algorithm for persuasio n in the i.i.d. model. Theorem 3.2 permits re- writing the optimiza tion pro blem in Figure 2 as follo ws, with var iables x , y ∈ R m . maximize n ξ · x subjec t to ρ · x ≥ ρ · y ( x , y ) ∈ P s (2) Problem (2) c annot be solv ed directly , since P s is deﬁned by a n ex tended formulat ion w ith e xponentia lly many vari ables and constrai nts, as describ ed in Section 3.1. Neve rtheless, we make use of a connecti on between symmetric signalin g schemes and single-item auctio ns w ith i.i.d. bidders to solv e (2) using the Ellipsoid method. Spec iﬁcally , we sho w a one-t o-one correspon dence between symmetric signat ures and (a subset of) symmetric reduc ed forms of single- item auction s with i.i.d. bidders, deﬁned as follo ws. Deﬁnition 3.3 ([13]) . Consider a single-item auctio n setting with n i.i.d. bidd ers and m types for each bidder , wher e each bidder ’ s type is distrib uted accor ding to q ∈ ∆ m . An alloca tion rule is a r andomized functi on A mapping a type pr oﬁle θ ∈ [ m ] n to a win ner A ( θ ) ∈ [ n ] ∪ {∗} , wher e ∗ denotes not allocatin g the item. W e say the allocatio n rule has symmetric reduced form τ ∈ [0 , 1] m if for each bidder i ∈ [ n ] and type j ∈ [ m ] , τ j is the condi tional pr obability of i r eceiving the item given she has type j . When q is clear from contex t, we say τ is r ealiza ble if t here exists an all ocatio n rule with τ as its symmetric reduce d form. W e say an alg orithm implements a n alloca tion rule A if it ta kes as input θ , and s amples A ( θ ) . Theor em 3 .4. Conside r the Bayesian P ersuasio n pr oblem w ith n i.i.d. actio ns and m types, wit h par ameter s q ∈ ∆ m , ξ ∈ R m , an d ρ ∈ R m given e xplicitly . An optimal and incentive- compatib le si gnalin g scheme ca n be implemented in p oly( m, n ) time. Theorem 3.4 is a conseq uence of the follo wing set of lemmas. Lemma 3 .5. Let ( x , y ) ∈ [0 , 1] m × [0 , 1] m , and d eﬁne τ = ( x 1 q 1 , ..., x m q m ) . T he p air ( x , y ) is a rea lizabl e s -signat ur e if and only i f (a) || x || 1 = 1 n , (b) x + ( n − 1) y = q , and (c) τ is a r ealizable symmetric r educed form of an allocati on rule with n i.i.d. bidder s, m types, and type distrib ution q . Mor eover , assuming x and y satis fy (a ), (b) and (c), an d giv en blac k-box ac cess to an allocation rule A with s ymmetric r educed form τ , a sign aling scheme with s -signatur e ( x , y ) can be implemented in p oly( n, m ) time. Lemma 3.6. A n op timal re alizab le s -signatur e, as describe d by LP (2) , is computabl e in p oly( n, m ) time. Lemma 3.7. (See [15, 1]) Consi der a single -item auction setting with n i.i.d. bidd ers and m typ es for each bidder , wher e eac h bidder ’ s type is d istrib uted accor ding to q ∈ ∆ m . Give n a r ealizable symmetric r educed form τ ∈ [0 , 1] m , an alloca tion rule with r educed form τ can be implement ed in p oly( n, m ) time. The proofs of L emmas 3.5 and 3.6 can be found in Appendix B .2. The proof of Lemma 3.5 build s a corre spondence between s -signat ures of signaling schemes and certain reduce d-form al locatio n rules. Speciﬁcally , actions correspo nd to bidders , action types correspon d to bidder types, and signal ing σ i cor - respon ds to assigning the item to bidder i . The expre ssion of the reduc ed form in terms of the s-s ignatu re then follo ws from Bayes’ rule. Lemma 3.6 follo ws from Lemma 3.5, the ellipso id method, and the fact that symmetric reduced forms admit an ef ﬁcient separati on oracle (see [13, 12, 15, 1]). 7 Algorithm 1 Indepe ndent Signaling Scheme Input: S ender payo f f vector ξ , recei ver payof f vector ρ , prior distrib ution q Input: S tate of nature θ ∈ [ m ] n Output: An n -d imension al bi nary signal σ ∈ { HIGH , LO W } n 1: Comput e an optimal solution ( x ∗ , y ∗ ) linear program (3). 2: Fo r each action i indepen dently , set component signal o i to HI GH with probabil ity x ∗ θ i q θ i and to LOW otherwis e, where θ i is the type of action i in the input state θ . 3: Retur n σ = ( o 1 , ..., o n ) . 3.3 A Simple (1 − 1 e ) -Ap pr oxi mate Scheme Our next result i s a “simple” si gnalin g scheme which ob tains a (1 − 1 /e ) mul tiplica ti ve approxi mation when payof fs are nonne gati ve. This algor ithm has the d istinct i ve propert y that it signal s independ ently for each action , and therefore implies that approximatel y o ptimal persuasio n can be paral lelized among multiple collud ing senders, each of whom only has access to the type of one or more of the action s. Recall from Section 3.1 that an s-signat ure ( x , y ) satisﬁes || x || 1 = || y || 1 = 1 n and x + ( n − 1) y = q . Our simple scheme, shown in Algo rithm 1, w orks w ith the follo wing e xplicit linear programming re laxatio n of optimiz ation problem (2). maximize n ξ · x subjec t to ρ · x ≥ ρ · y x + ( n − 1) y = q || x || 1 = 1 n x , y ≥ 0 (3) Algorith m 1 has a simple and instructi ve interpreta tion. It compute s the optimal solution ( x ∗ , y ∗ ) to the relaxed problem (3 ), and uses this solutio n as a guide for signali ng inde pendently for each ac tion. The algori thm sele cts, indepen dently for each action i , a component sign al o i ∈ { HIGH , LO W } . I n particular , each o i is chosen so that Pr [ o i = HIGH ] = 1 n , and moreov er the e vents o i = HIGH and o i = LO W induce the poster ior beliefs n x ∗ and n y ∗ , respe cti vely , regardi ng the type of actio n i . The signalin g scheme implemented by Algorithm 1 approxima tely matches the optimal val ue of (3), as sho wn in Theor em 3.8, assuming the re cei ver is rati onal and there fore selects an act ion with a HIGH compone nt signal if one exists. W e note that the scheme of A lgorit hm 1, w hile not a direct scheme as descri bed, can easily be c on verted into one; speciﬁcall y , by recommending an action whos e compone nt sig- nal is HI GH when one exist s (breaking ties arbitrari ly), and recommendi ng an arbitrary action otherwise. Theorem 3.8 follo ws from the f act t hat ( x ∗ , y ∗ ) is an optimal solution to L P (3), the fact that the posterior type distri b ution of an action i is n x ∗ when o i = HIGH and n y ∗ when o i = LOW , and the f act that each compone nt signal is high indep endent ly with pro babili ty 1 n . W e defe r the formal proof to Appendix B.3. Theor em 3.8. Algorithm 1 run s in poly ( m, n ) time, and serves as a (1 − 1 e ) -appr oximate signaling scheme for the Bayesian P ersua sion pr oblem with n i.i.d. actions, m types, and nonne gative payof fs. Remark 3.9. Algorith m 1 signals independ ently for each action . This con vey s an inter esting concep tual messa ge . That is, eve n though the optimal signal ing scheme might induce posterior beliefs which corr elate dif fer ent actions, it is ne vertheless true th at sig naling for eac h acti on inde penden tly y ields an appr oximately optimal signaling scheme . As a consequen ce, coll abor ative persu asion by multiple parties (the sender s), eac h of whom obs erves the payo ff of one or mor e actio ns, is a ta sk that can be p ara llelize d, req uiring no coor dinat ion w hen a ctions ar e identic al and inde pende nt and only an ap pr oximate solutio n is sou ght. W e 8 leave op en the quest ion of whet her th is is possib le when a ction payof fs ar e indepe ndently bu t not identicall y distrib uted. 4 Complexity Barriers to Per suasion with Independ ent Actions In this secti on, we consi der optimal persuas ion with independen t action payof fs as in Section 3, alb eit with action-speci ﬁc mar ginal distrib utions giv en explici tly . Speciﬁcally , for each action i we are giv en a distr ib ution q i ∈ ∆ m i on m i types, and each type j ∈ [ m i ] of action i is associa ted with a sender payof f ξ i j ∈ R and a receiv er payo f f ρ i j ∈ R . The positi ve results of Sect ion 3 draw a c onnec tion between optimal persu asion in the speci al case of identic ally dist rib uted actions and Border ’ s character ization of reduce d-form single-item auctions with i.i.d. bidders. One might expec t this connecti on to generaliz e to the indepe ndent non-iden tical persuasion settin g, since Border’ s theorem e xtends to single-item auction s with indepe ndent non-ident ical bidders. Surprisin gly , we s ho w that this a nalogy to Border’ s ch aracterizatio n fails to gene ralize. W e prov e the follo wing theorem. Theor em 4.1. Consider the Bayesian P ersuasi on pr oblem with indepe ndent actions, with action-spec iﬁc payof f distrib utions given expli citly . It is # P -har d to compute the optimal exp ected sender utility . In vokin g the frame work of Gopalan et al. [30] , this rules out a gener alized Bord er’ s theor em for our setting , in the sense deﬁned by [3 0], unless the pol ynomial hierarch y collapses to P N P . W e v ie w this resu lt as illus trating some of the importan t differe nces between persua sion and mechanism design. The proof of T heorem 4.1 is rather in vol ved . W e defer the full proof to Appendix C, and only present a ske tch here . Our p roof star ts from t he id eas of Gopalan et al. [30], who sho w the #P-hard ness for r e ven ue or welfar e maximiz ation in se veral mechanism des ign pro blems. In one cas e, [30] reduce from the # P - hard proble m of computing the Khintchin e constant of a vect or . Our reductio n also starts from this problem, but is much m ore in v olv ed: 2 First, we e xhibit a poly tope which we t erm the Khi ntchine polytope , and sho w that computin g the Khintchine constant reduce s to linear optimiza tion ov er the Khintchine polytope. Second, we present a reduct ion from the membership problem for the Khintchin e polytope to the computation of optimal sender utilit y in a p articul arly-cr afted instance of persuas ion with independen t action s. In vo king the polyn omial-time equi vale nce between member ship check ing and o ptimization (see, e.g., [31]), we co nclude the #P-hardness of our problem. T he main technical challenge we over come is in the second step of our proof: gi ven a po int x which may or may not be in the Khintchine polyt ope K , we c onstru ct a p ersuas ion instan ce and a thresho ld T so that points in K encode signaling schemes , and the optimal sender utility is at least T if and onl y if x ∈ K and the scheme corre spond ing to x result s in sender utility T . Pr oof Sketch of Theor em 4 .1 The Khintch ine pr oblem , sho wn to be #P-hard in [30], is to compute the K hintc hine const ant K ( a ) of a gi ven vector a ∈ R n , deﬁned as K ( a ) = E θ ∼ {± 1 } n [ | θ · a | ] where θ is drawn uniformly at random from {± 1 } n . T o re late the Khintc hine problem to Bayesia n persuasi on, we beg in with a persuasio n instance with n i.i.d. actions and two action types, whi ch w e refer to as type -1 and type +1 . The state of nature is a uniform random dr aw from the s et {± 1 } n , with the i t h ent ry speci fying the typ e of actio n i . W e cal l this instance the Khintc hine-like persuasio n setting. As in Section 3, we still use the signatur e to capture the payof f-rele vant feature s of a signalin g scheme, bu t we pay special attention to si gnalin g sch emes which use o nly two signals, in which case we represen t them using a two-signal signatur e o f the form ( M 1 , M 2 ) ∈ R n × 2 × R n × 2 . The Khintc hine polyt ope K ( n ) is t hen deﬁned as the (co n ve x) family of all re alizab le two- signal s ignatu res for the Khintchi ne-lik e persuasi on proble m with an additiona l constrai nt: each signal is sent with proba bility exa ctly 1 2 . W e ﬁrst pro ve that general linear optimiz ation ov er K ( n ) is #P-hard by encoding computatio n of 2 In [30], Myerson’ s characterization is used to sho w t hat optimal mechanism design in a public project setting directly encodes computation of the Khintchine constant. No analogous direct connection seems to hold here. 9 the Khintchin e const ant as linear optimizatio n over K ( n ) . In this reduction, the optimal solution in K ( n ) is the signatu re of the two-signa l scheme ϕ ( θ ) = sign ( θ · a ) , which signals + and − each with probabili ty 1 2 . T o reduce the membershi p proble m for the K hintch ine polyt ope to optimal Bayesian persua sion, th e main challenges come from our restriction s on K ( n ) , namely to sch emes with two signals w hich are equa lly probab le. Our r educti on inco rporate s three key ideas. The ﬁrst is to des ign a persuasion inst ance in which the optimal signalin g scheme uses on ly two signals. The instance we deﬁne w ill hav e n + 1 actions. Action 0 is special – it d eterministicall y results in sender ut ility ǫ > 0 (smal l e nough ) and receiv er utility 0 . The other n actions are r e gular . Action i > 0 independen tly results in sender utility − a i and receiv er utilit y a i with probab ility 1 2 (call this type 1 i ), or sende r utility − b i and recei ver utility b i with probabi lity 1 2 (call this type 2 i ), for a i and b i to be set la ter . Note that the sende r and recei ver utilities are zer o-sum for both types. Since the special action is determini stic and the proba bility of its (only ) type is 1 in any signal, we can interpret any ( M 1 , M 2 ) ∈ K ( n ) as a two-signal signature for o ur persuasion instance (the row corres pondi ng to the specia l action 0 is implied). W e sho w that restricting to two-sign al schemes is w ithout loss of generality in this persuasio n instan ce. The proof tr acks the followin g intuitio n: due to the zero-sum n ature of regular action s, any additional info rmation re garding regular actions woul d beneﬁt the recei ver a nd harm the s ender . Consequ ently , sender d oes n ot rev eal an y informa tion which disting uishes between di f ferent regu lar ac tions. Formally , we prove that there al ways exists an optimal signaling scheme with only tw o signals: one si gnal recommend s the specia l action , and the other recommends some regu lar action. W e denote the signal that recommend s the special action 0 by σ + (indic ating that the sender deri ves positi ve utilit y ǫ ), and denote the other signal by σ − (indic ating that the s ender d eri ves nega ti ve utility , as we show). The second key idea concerns choosing appropria te va lues for { a i } n i =1 , { b i } n i =1 for a giv en two- signat ure ( M 1 , M 2 ) to b e tested . W e choos e these v alues to satisfy the follo wing two p ropert ies: (1) For all regular actions, the signaling scheme implementing ( M 1 , M 2 ) (if it exist s) results in the same sender utility − 1 (thus rece i ver utility 1 ) condition ed on σ − and the same sender util ity 0 conditio ned on σ + ; (2) the maximum possible expecte d sen der utility from σ − , i.e., the sender utility condit ioned on σ − multiplie d by th e proba bility of σ − , is − 1 2 . As a res ult of Property (1) , if ( M 1 , M 2 ) ∈ K ( n ) then t he co rresponding signal ing sch eme ϕ is IC and results in expecte d se nder utility T = 1 2 ǫ − 1 2 (since each signal is sent w ith probab ility 1 2 ). Pro perty (2 ) implie s tha t ϕ results in the maximum possible ex pected s ender utility fr om σ − . W e now run into a challenge: the exi stence of a signaling scheme with expec ted sender utility T = 1 2 ǫ − 1 2 does not necessaril y imply that ( M 1 , M 2 ) ∈ K ( n ) if ǫ is lar ge. Our thir d ke y idea is to set ǫ > 0 “sufﬁcien tly small” so that any optimal si gnalin g sch eme must result i n the maximum possible expect ed se nder utility − 1 2 from signa l σ − (see Prope rty (2) abo ve). In other w ords, we must make ǫ so small so tha t the sender prefers to not sacriﬁce an y of her payof f from σ − in order to gain utility fro m the sp ecial actio n rec omm ended by σ + . W e sho w that such an ǫ exis ts with polynomia lly m any bit s. W e prov e its existence by arg uing that the polyto pe of incenti ve-compatib le two-signa l signature s has polynomial bit complexi ty , and therefore an ǫ > 0 that is smaller than the “bit complex ity” of the ve rtices would suf ﬁ ce. As a result of this ch oice of ǫ , if the optimal se nder utility is precisely T = 1 2 ǫ − 1 2 then we know that signal σ + must be sent w ith probabili ty 1 2 since the exp ected sender utility from signal σ − must be − 1 2 . W e sho w that this, together with the speciﬁca lly constru cted { a i } n i =1 , { b i } n i =1 , is suf ﬁcient to guarantee that the optimal signalin g scheme must implement the gi ven two-sig nature ( M 1 , M 2 ) , i.e., ( M 1 , M 2 ) ∈ K ( n ) . When the o ptimal optimal sender utility is strictly greater than 1 2 ǫ − 1 2 , the op timal signali ng scheme does not implement ( M 1 , M 2 ) , b ut we sho w that it can be post-pro cessed into one that does. 5 The General Persuasion Pr oblem W e now turn our attention to the Bayesian Persuasion problem when the payof fs of dif ferent actions are arbitra rily correl ated, and the joint dist rib ution λ is p resente d as a black-bo x sampling o racle. W e a ssume that payo ffs are normalized to lie in the bounde d in terv al, and prov e essentially matching positi ve and neg ati ve r esults . Our positi ve result is a fully polynomial- time appro ximation sch eme for optimal persuasion 10 Algorithm 2 Signaling Scheme for a Black Box Distrib ution Paramet er: ǫ ≥ 0 Paramet er: Inte ger K ≥ 0 Input: P rior distrib ution λ suppo rted on [ − 1 , 1] 2 n , gi ven by a sa m pling oracle Input: S tate of nature θ ∈ [ − 1 , 1] 2 n Output: Signa l σ ∈ Σ , where Σ = { σ 1 , . . . , σ n } . 1: Dra w integer ℓ unifor mly at random from { 1 , . . . , K } , and denot e θ ℓ = θ . 2: Sample θ 1 , . . . , θ ℓ − 1 , θ ℓ +1 . . . , θ K indepe ndently from λ , and let the multiset e λ = { θ 1 , . . . , θ K } denote the empiri cal distrib ution au gmented with the input state θ = θ ℓ . 3: Solv e linear program (4) to obtain the signaling scheme e ϕ : e λ → ∆(Σ) . 4: Outpu t a sample from e ϕ ( θ ) = e ϕ ( θ ℓ ) . maximize P K k =1 P n i =1 1 K e ϕ ( θ k , σ i ) s i ( θ k ) subjec t to P n i =1 e ϕ ( θ k , σ i ) = 1 , for k ∈ [ K ] . P K k =1 1 K e ϕ ( θ k , σ i ) r i ( θ k ) ≥ P K k =1 1 K e ϕ ( θ k , σ i )( r j ( θ k ) − ǫ ) , for i, j ∈ [ n ] . e ϕ ( θ k , σ i ) ≥ 0 , for k ∈ [ K ] , i ∈ [ n ] . (4) Relax ed Empirical Optimal Signaling Problem with a bi-cr iteria guarantee; speciﬁcally , we achie ve approx imate op timality and approx imate incenti ve compatib ility in the additi ve sen se desc ribed in Section 2. Our negat i ve results sho w that such a bi -criteri a loss is ine vitable in the black box model for informat ion-th eoretic rea sons. 5.1 A Bicriteria FPT AS Theor em 5.1. Conside r the Bayesian P er suasion pr oblem in the bla c k-box orac le model w ith n action s and payof fs in [ − 1 , 1] , and let ǫ > 0 be a parameter . A n ǫ -optimal and ǫ -incentive compatible signaling sche m e can be implemented in p oly( n, 1 ǫ ) time. T o prov e T heorem 5.1, we sho w that a simple Monte-Carlo algorithm implements an approximate ly optimal and approximatel y incenti ve compatible scheme ϕ . Notably , our algorith m does not compute a repres entatio n of the ent ire signali ng scheme ϕ as in Section 3, b ut rather merely sa mples its out put ϕ ( θ ) on a gi ven in put θ . At a high lev el, when gi ven as input a st ate o f nat ure θ , our algor ithm ﬁrst takes K = p oly( n, 1 ǫ ) samples from the pri or distrib ution λ w hich, intuiti vely , serve to place the true state of nature θ in co nte xt. Then the al gorith m uses a l inear pro gram to comp ute the optima l ǫ -incent i ve compatible sch eme e ϕ for the empirical distrib ution of samples augmented w ith the input θ . Finally , the algorithm signals as sugge sted by e ϕ f or θ . D etails are in Algorithm 2, which we instantiate with ǫ > 0 and K = ⌈ 256 n 2 ǫ 4 log( 4 n ǫ ) ⌉ . W e note that relaxing incenti ve co mpatibili ty is necessary for con ver gence to the optimal sender utility — we prov e this formally in Section 5.2. This is why LP (4) featur es relaxe d incenti ve compatibilit y constr aints. Instant iating Algorithm 2 with ǫ = 0 resu lts in an e xactly incenti ve compatib le scheme which could be far f rom the optimal send er utility for any ﬁnite number of samples K , as reﬂected in Lemma 5.4. Theorem 5.1 follo ws fr om three lemmas pertain ing to the scheme ϕ implemented by Algorithm 2 . Ap- proximat e inc entiv e compatibi lity for λ (Lemma 5 .2) follo ws from the principle of de ferred deci sions, lin- earity of expect ations , and the f act th at e ϕ is approximately inc enti ve co m patible for the au gmented empirical distrib ution e λ . A similar argumen t, a lso based on the principal of deferred decisions and linearity of expec- tation s, shows tha t the expec ted sender utility from our scheme when θ ∼ λ equals the expec ted optimal v alue of linear program (4), as stated in Lemma 5.3. Finally , we sho w in Lemma 5.4 that the optimal valu e 11 of LP (4) is close to the optimal sende r utility for λ with high probabil ity , and hence also in expecta tion, when K = p oly ( n, 1 ǫ ) is chosen appropria tely; the proof of this fact in v oke s standard tail boun ds as well as structu ral properties of linear progra m (4), and exploit s the fact that LP (4) relax es the incenti ve com- patibi lity constraint. W e prov e all three lemmas in App endix D.1. Even though ou r pro of of Lemma 5.4 is self-co ntained, w e note that it can be sho wn to follo w from [45, Theorem 6] with some additio nal work. Lemma 5.2. A lgorit hm 2 implements an ǫ -incentive compatible signaling scheme for prior distrib ution λ . Lemma 5.3. Assume θ ∼ λ , and assume the re ceiver follows the r ecommendation s of Algorithm 2. The e xpected sender utili ty equa ls the expec ted optimal v alue of the line ar pr ogr am (4) s olved in Step 3. Both e xpectations a re taken over the random input θ as well as internal randomnes s and Monte-Carlo sampling perfor med by the algorit hm. Lemma 5.4. Let O P T denote the expec ted sender utility in duced by the optimal incent ive co mpatible signal ing scheme for distrib ution λ . When Algori thm 2 is instantiat ed with K ≥ 256 n 2 ǫ 4 log( 4 n ǫ ) a nd its input θ is d rawn fr om λ , the e xpected opti mal value of the linear p r ogr am (4) so lved in Step 3 is at least O P T − ǫ . Expectat ion is ove r the random input θ as well as the Monte-Carl o sampling performed by the algorithm. 5.2 Inf ormation-Theor etic Barriers W e no w sh o w that our b i-criteria FPT AS is close t o the best we ca n hope for: there is no bounded -sample signal ing scheme in the black bo x model which guarantee s incenti ve compatibilit y and c -optimality for any consta nt c < 1 , nor i s t here suc h an algo rithm which gu arantees optimality and c -incenti ve compati bility for any c < 1 4 . F ormally , we consider algorit hms which implement d irect signaling schemes . Such an al gorith m tak es as in put a black- box distrib ution λ supporte d on [ − 1 , 1] 2 n and a state of n ature θ ∈ [ − 1 , 1] 2 n , wher e n is the number of actions , and outputs a sig nal σ ∈ { σ 1 , . . . , σ n } recommending an action. W e say such an algori thm is ǫ -ince nti ve compatible [ ǫ -op timal] if for ev ery dist rib ution λ th e sig naling scheme A ( λ ) is ǫ - incent i ve co mpatible [ ǫ -op timal] for λ . W e deﬁne the sample compl e xity S C A ( λ, θ ) as the e xpected numb er of queries made by A to the blackbox gi ven inputs λ and θ , where expectati on is taken the randomness inhere nt in the Monte-Car lo sampling from λ as well as an y other internal coins of A . W e sho w that the worst- case samp le comp lexi ty is not b ounde d by an y fun ction of n and the app roximatio n parameters unle ss we allo w b i-criter ia loss in both optimality and incenti ve compatibili ty . More so, we show a stronger neg ativ e result for e xactly incenti ve compatible algorith ms: the av erage sample complex ity ove r θ ∼ λ is al so not bound ed by a function of n and the subop timality parameter . Wherea s our results imply that we should gi ve up on exact incent i ve compatibi lity , we lea ve open the questio n of whether an optimal and ǫ -incent i ve compatib le al gorithm exist s with p oly( n, 1 ǫ ) a ver age case (b ut unbou nded worst-c ase) sample complexity . Theor em 5.5. The following hold for eve ry algorith m A for Bayesia n P ersuasio n in the blac k-box model: (a) If A is incentive compatible and c -optimal for c < 1 , then for every inte ger K ther e is a distrib ution λ = λ ( K ) on 2 actio ns and 2 states of natur e such tha t E θ ∼ λ [ S C A ( λ, θ )] > K . (b) If A is optimal and c -incent ive c ompatibl e for c < 1 4 , then for ever y inte ger K ther e is a distrib ution λ = λ ( K ) on 3 actio ns and 3 states of natur e, and θ in the support of λ , such tha t S C A ( λ, θ ) > K . Our proof of each par t of this t heorem in volv es constructin g a pair of d istrib utions λ an d λ ′ which are arbitra rily close in statistical distanc e, b ut with the propert y that any algorit hm with the postulated guaran tees must disting uish between λ and λ ′ . W e defe r the proof to Appendix D.2. Ackno wledgments W e thank David Kempe for helpful co mments on an earli er draft of this p aper . W e also thank the anonymou s re viewers for helpf ul feedbac k and suggestions . 12 Refer ences [1] S. Alaei, H. Fu, N. Haghpan ah, J. D. Hartline, and A. Malekian. B ayesia n optimal auction s via multi - to single-a gent reductio n. In B. Faltings , K. Leyton- Bro w n, and P . Ipeiro tis, editors, ACM C onfer ence on Electr onic Commer ce , page 17. 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PhD thesis, Massachu setts Institute of T echnology , 2014. [46] H . Xu, Z. Rabinov ich, S . Dughmi, and M. T ambe. Exploring information asymmetry in two-s tage securi ty games. In AAA I Confe r ence on Artiﬁcial Intellig ence (AA AI) , 2015 . 15 A Additional Discussion of Connections to Bayesian Mechani sm Design Section 3, which cons iders persuasio n with indepe ndent and identical ly-dis trib uted actions, relates to two ideas from auctio n theory . First, o ur sy mmetrizatio n result in Section 3.1 is simil ar to that of Daskalakis and W einber g [20], b ut in volv es an addi tional ingredi ent which is necessary in our ca se: not only is the poster ior type distrib ution for a recommended action (the winning bidder in th e auction analog y) in depen dent of the identi ty of the action, but so is the posterior type distrib ution of an unreco mmended ac tion (losing bidder). Second, ou r algorithm for computing the optimal scheme in Section 3.2 i n v olves a connectio n to Borde r’ s charac terizat ion of the space of feasible reduced- form single-item auctions [13, 12], as well as its algorithmic proper ties [15, 1]. Howe ver , un like in t he case of single-item au ctions , this connectio n hinges cruciall y on the symmetries of the optimal scheme, and fails to generalize to the case of persuasion w ith indepe ndent non-id entical actions (ana logou s to indepe ndent non-id entical bidders) as we sho w in Secti on 4. W e vi e w this as e vidence that persuasion and auction de sign — while bearin g simil arities and technical connectio ns — are importa ntly differe nt. Section 4 sho ws that our Border’ s theorem-based approach in S ection 3 can not be extende d to in- depen dent non-ide ntical act ions. Our starting point are the results of Gopala n et al. [30], w ho rule out Border’ s-theo rem like characteriz ations for a number of mechan ism desi gn settings by sho wing the #P- hardne ss of comp uting the max imum expected reve nue or welfare . Our results similarly sh ow t hat it is #P hard to compute the maximum ex pected sender utility , b ut our reduction is much more in vo lved. Speciﬁ- cally , wherea s w e also reduce from the #P-hard problem of computin g the K hintch ine c onstan t of a vector , unlik e in [30] o ur reduction must go thr ough the membersh ip proble m of a poly tope which we us e to en- code the Khintchine constan t compu tation. This detour seems una voidab le due to the dif ferent nature of the incenti ve-compa tibility constra ints pla ced on a signaling scheme. 3 Speciﬁcally , we present an intricate reduct ion from membership testing in this “Khintchine polytope” to an optimal persuasion problem w ith indepe ndent a ctions . Our alg orithmic result for the blac k box mod el in Section 5 draws i nspira tion from, and is technica lly related to, the work in [15 , 1, 16, 45 ] on algorithmicall y ef ﬁcient mech anisms for multi-dimen sional setting s. Speciﬁcally , an alterna ti ve a lgorith m fo r our problem can be deri ved using the framew ork of r educed forms and virtua l w elfar e of Cai et al. [16] with sig niﬁcant addition al work. 4 For thi s, a dif ferent reduced form is needed which allo w s for an unbounded “type space”, and maintains the correlatio n informati on across action s nec essary for e v aluating the persuasion no tion of incenti ve compat ibility , w hich is importantly dif- ferent from incen ti ve compatibility in mechanism design . Such a reduced form exists , and the result ing algori thm is complex and in v oke s the ellipsoid algorithm as a subroutine. The algorit hm we present here is much simple r and more ef ﬁcient both in te rms of runtime and samples from the distrib ution λ ov er states of nature, with the main computational step being a single explicit linear p rogram which solve s for the op- timal signaling scheme on a sample e λ from λ . The analysi s of our a lgorith m is als o more stra ightfo rward. This is possible in our se tting due to ou r dif ferent notion of in centi ve compatibili ty , which permits reduc - ing incenti ve compa tibility on λ to incenti ve comp atibili ty on the samp le e λ using th e princ iple of deferred decisi ons. 3 In [30], Myerson’ s characterization is used to sho w t hat optimal mechanism design in a public project setting directly encodes computation of the Khintchine constant. No analogous direct connection seems to hold here. 4 W e thank an anonymous rev iewer for po inting out this connection. 16 B Omissions from S ection 3 B.1 Symmetry of the Optimal Scheme (Theor em 3.2) T o prov e T heorem 3.2, we need two closure proper ties of optimal signaling schemes — with respec t to permutat ions and con ve x combinati ons. W e use π to denote a permutatio n of [ n ] , a nd let S n denote the s et of all su ch permutations . W e d eﬁne the permuta tion π ( θ ) of a state of nature θ ∈ [ m ] n so th at ( π ( θ )) j = θ π ( j ) , and similarly t he permutation of a si gnal σ i so that π ( σ i ) = σ π ( i ) . Gi ven a signature M = { ( M σ i , σ i ) } i ∈ [ n ] , we d eﬁne the permuted signature π ( M ) = { ( π M σ i , π ( σ i )) } i ∈ [ n ] , where π M denotes a pplyin g permutatio n π to the rows of a matrix M . Lemma B.1. Assume the action payof fs ar e i.i.d., and let π ∈ S n be an arbitr ary permutat ion. If M is the sign atur e of a sign aling sch eme ϕ , then π ( M ) is the signatur e of the sc heme ϕ π deﬁne d by ϕ π ( θ ) = π ( ϕ ( π − 1 ( θ ))) . Mor eove r , if ϕ is incent ive compatible and optimal, then so is ϕ π . Pr oof. Let M = { ( M σ , σ ) } σ ∈ Σ be the signature of ϕ , as giv en in the statemen t of the lemma. W e ﬁrst sho w that π ( M ) = { ( π M σ , π ( σ )) } σ ∈ Σ is reali zable as the signatur e of the scheme ϕ π . By d eﬁnition , it suf ﬁces to sho w that P θ λ ( θ ) ϕ π ( θ , π ( σ )) M θ = π M σ for an arbitr ary signal π ( σ ) . X θ λ ( θ ) ϕ π ( θ , π ( σ )) M θ = X θ λ ( θ ) ϕ ( π − 1 ( θ ) , σ ) M θ (by deﬁnition of ϕ π ) = π X θ ∈ Θ λ ( θ ) ϕ ( π − 1 ( θ ) , σ )( π − 1 M θ ) (by lineari ty of permutat ion) = π X θ ∈ Θ λ ( θ ) ϕ ( π − 1 ( θ ) , σ ) M π − 1 ( θ ) = π X θ ∈ Θ λ ( π − 1 ( θ )) ϕ ( π − 1 ( θ ) , σ ) M π − 1 ( θ ) (Since λ is i.i.d.) = π X θ ′ ∈ Θ λ ( θ ′ ) ϕ ( θ ′ , σ ) M θ ′ (by renaming π − 1 ( θ ) to θ ′ ) = π M σ (by deﬁnit ion of M σ ) No w , assuming ϕ is incenti ve compatible, we check that ϕ π is incenti ve compatible by verif ying the rele van t inequa lity for its signatu re. ρ · ( πM σ i ) π ( i ) − ρ · ( π M σ i ) π ( j ) = ρ · M σ i i − ρ · M σ i j ≥ 0 Moreo ver , we sho w that the sender’ s utility is the same for ϕ and ϕ π , completin g the proof. ξ · ( π M σ i ) π ( i ) = ξ · ( M σ i ) i Lemma B.2. Let t ∈ [0 , 1] . If A = ( A σ 1 , . . . , A σ n ) is the signat ur e of scheme ϕ A , and B = ( B σ 1 , . . . , B σ n ) is th e sig natur e of a s che me ϕ B , then their con ve x combinatio n C = ( C σ 1 , . . . , C σ n ) w ith C σ i = tA σ i + (1 − t ) B σ i is th e sign atur e of the sc heme ϕ C which , on i nput θ , outputs ϕ A ( θ ) with pr obability t and ϕ B ( θ ) with pr obability 1 − t . Mor eover , if ϕ A and ϕ B ar e both optimal and incentive compatible , th en so is ϕ C . Pr oof. This follo w s almost immedi ately from the fact that the op timization prob lem in Figu re 2 is a li near progra m, with a con vex feasible set and a con ve x fa m ily of optimal solutions . W e omit the straightfo rward details . 17 Pro of of Theorem 3.2 Giv en an optimal and incenti ve compatible signaling scheme ϕ w ith signature { ( M σ i , σ i ) } i ∈ [ n ] , we sho w the exi stence o f a symmetric op timal and in centi ve-co mpatible scheme of the form in Deﬁniti on 3.1. According to Lemma B .1, for π ∈ S n the signatu re { ( π M σ i , π ( σ i )) } i ∈ [ n ] — equiv alently written as { ( π M σ π − 1 ( i ) , σ i } i ∈ [ n ] — corresp onds to the optimal incen ti ve compatible scheme ϕ π . In voki ng Lemma B.2, the signatur e { ( A σ i , σ i ) } i ∈ [ n ] = { ( 1 n ! X π ∈ S n π M σ π − 1 ( i ) , σ i ) } i ∈ [ n ] also cor respon ds to an optimal and inc enti ve compatible scheme, namely th e scheme which dr aws a permu- tation π uniformly at random, then signals according to ϕ π . Observ e that the i th row of the matrix π M σ π − 1 ( i ) is the π − 1 ( i ) th row of the matrix M σ π − 1 ( i ) . Exp ressing A σ i i as a sum ov er permutations π ∈ S n , and groupin g the sum by k = π − 1 ( i ) , we can write A σ i i = 1 n ! X π ∈ S n [ π M σ π − 1 ( i ) ] i = 1 n ! X π ∈ S n M σ π − 1 ( i ) π − 1 ( i ) = 1 n ! n X k =1 M σ k k ·    π ∈ S n : π − 1 ( i ) = k    = 1 n ! n X k =1 M σ k k · ( n − 1)! = 1 n n X k =1 M σ k k , which does not depend on i . S imilarly , the j th row of the matrix π M σ π − 1 ( i ) is the π − 1 ( j ) th row of the matrix M σ π − 1 ( i ) . For j 6 = i , expressi ng A σ i j as a sum ove r per mutation s π ∈ S n , and grouping the sum by k = π − 1 ( i ) and l = π − 1 ( j ) , we can write A σ i j = 1 n ! X π ∈ S n [ π M σ π − 1 ( i ) ] j = 1 n ! X π ∈ S n M σ π − 1 ( i ) π − 1 ( j ) = 1 n ! X k 6 = l M σ k l ·    π ∈ S n : π − 1 ( i ) = k , π − 1 ( j ) = l    = 1 n ! X k 6 = l M σ k l · ( n − 2)! = 1 n ( n − 1) X k 6 = l M σ k l , which does not depen d on i or j . Let x = 1 n n X k =1 M σ k k ; y = 1 n ( n − 1) X k 6 = l M σ k l . 18 The signat ure { ( A σ i , σ i ) } i ∈ [ n ] therefo re de scribes an optimal, incenti ve compatible, and symmetric scheme with s -signa ture ( x , y ) . B.2 The Optimal Scheme Pro of of Lemma 3.5 For t he “only if” directi on, || x || 1 = 1 n and x + ( n − 1) y = q were estab lished in Section 3.1. T o sho w that τ is a r ealizab le symmetric r educed for m for an al locatio n rule, let ϕ be a signa ling scheme with s -signatu re ( x , y ) . Recall from the deﬁnition of an s -signature that, for each i ∈ [ n ] , signal σ i has probabil ity 1 /n , and n x is the posterior distrib ution of action i ’ s type conditioned on signal σ i . Now con sider th e followin g alloca tion rule: Giv en a type proﬁle θ ∈ [ m ] n of the n bi dders, allocate the item to bi dder i with proba bility ϕ ( θ , σ i ) for an y i ∈ [ n ] . By Bayes rule, Pr [ i gets item | i has type j ] = Pr [ i has type j | i gets item ] · Pr [ i gets item ] Pr [ i has type j ] = nx j · 1 /n q j = x j q j Therefore τ is indeed the reduced form of the described allocation rule. For the “if ” directi on, let τ , x , and y be as in the statement of the lemma, and consi der an allo cation rule A with symmetric reduced form τ . Observe that A alway s allocat es the item, since for each player i ∈ [ n ] we hav e Pr [ i gets the item ] = P m j =1 q j τ j = P m j =1 x j = 1 n . W e deﬁne the direct signaling scheme ϕ A by ϕ A ( θ ) = σ A ( θ ) . Let M = ( M σ 1 , . . . , M σ n ) be the signature of ϕ A . Recall that, for θ ∼ λ and arbitra ry i ∈ [ n ] and j ∈ [ m ] , M σ i ij is the probability that ϕ A ( θ ) = σ i and θ i = j ; by deﬁnition , this equals the probability that A allo cates the item to p layer i and h er type i s j , which is τ j q j = x j . As a result, the signat ure M o f ϕ A satisﬁes M σ i i = x for ev ery acti on i . If ϕ A were sy m metric, we would c onclud e that its s -signature is ( x , y ) sinc e ev ery s -signa ture ( x , y ′ ) must sati sfy x + ( n − 1) y ′ = q (see S ection 3.1). Ho wev er , this is not guaran teed w hen the a llocati on rule A ex hibits some asymmetr y . N e vertheles s, ϕ A can be “symmetrize d” into a signali ng scheme ϕ ′ A which ﬁrst draws a random permutation π ∈ S n , and signals π ( ϕ A ( π − 1 ( θ ))) . That ϕ ′ A has s -signat ure ( x , y ) follo ws a similar arg ument to that used in the proof of Theorem 3.2, and we therefo re omit the details here. Finally , observe that the descripti on of ϕ ′ A abo ve is constructi ve assu m ing black-box ac cess to A , with runtime ov erhead that is polynomial in n and m . Pro of of Lemma 3.6 By Lemma 3.5, we can re-write LP (2) as follo ws: maximize n ξ · x subjec t to ρ · x ≥ ρ · y x + ( n − 1) y = q || x || 1 = 1 n ( x 1 q 1 , ...., x m q m ) is a realiz able symmetric reduce d form (5) From [13, 12, 15, 1], we kno w that the family of all the realizab le symmetric reduce d forms consti tutes a polyto pe, and moreov er that this polytope admits an ef ﬁcient separatio n oracle. The runtime of this ora cle is polynomial in m and n , and as a result the abov e l inear program can be solv ed in pol y ( n, m ) time using the Ellipsoid method. 19 B.3 A Simple (1 − 1 /e ) -appr oximate Sch eme Pro of of Theorem 3.8 Giv en a b inary signal σ = ( o 1 , . . . , o n ) ∈ { HIGH , LO W } n , the po sterior typ e distri b ution for an action equals n x ∗ if the correspond ing componen t signal is HI GH , and equals n y ∗ if the component signal is LO W . T his is simply a consequen ce of the independ ence of the action types, the fact that the dif ferent compone nt signal s are chosen indepe ndentl y , and Bayes’ rule. T he constr aint ρ · x ∗ ≥ ρ · y ∗ implies that the receiv er prefers actions i for which o i = HIGH , a ny one of whic h induces a n expected utility of n ρ · x ∗ for the recei ver and n ξ · x ∗ for the sender . The latter quantity matches the optimal valu e of LP (3). The constr aint || x || 1 = 1 n implies that each component signal is HIGH with probabi lity 1 n , indepen dently . Therefore , the proba bility that at lea st one co mponent sign al is HIGH e quals 1 − (1 − 1 n ) n ≥ 1 − 1 e . Sinc e payof fs are nonnega tive , and since a rationa l recei ver selects a HIGH action when one is av ailable, the sender ’ s o verall ex pected utility is at least a 1 − 1 e fractio n of the optimal v alue of LP (3). 20 C Proof of Theor em 4.1 This section is de v oted to provin g Theorem 4.1. Our proof starts from the ideas of Gopalan et al. [30], who show the #P-hardn ess f or rev enue or welfare maximization in se veral mechanism design problems. In one case, [30] reduce from the # P -hard problem of computing the Khintchin e constan t of a ve ctor . Our reduct ion also starts from th is prob lem, b ut is much more in v olve d: First, w e exhibit a polytope whi ch we term Khintchine polytope , an d sh o w that computing the Khintchine constant r educes to li near optimization ov er the Khintchin e polytope. S econd, we presen t a reduction from the membership problem for the Khint- chine polytope t o the comp utation of optimal sender u tility in a particul arly-cr afted instance of persua sion with indepen dent actions. In vok ing the polyno mial-time equi val ence between membership checkin g and optimiza tion (see, e.g., [31]), we conclude the #P-h ardnes s of our pro blem. The m ain technical challenge we ove rcome is in the second step of our proof: gi ven a point x which may or may not be in the Khint- chine p olytop e K , we const ruct a persuasion instanc e and a threshold T so that points in K encode sign aling schemes, and th e optimal send er utility is at leas t T if and only if x ∈ K an d the scheme cor respon ding to x results in sender utility T . The Khintchine Polytope W e start by deﬁning the Khintc hine pr oblem , which is sho w n to be #P-hard in [30]. Deﬁnition C.1. (Khintchin e P r oblem) Given a vector a ∈ R n , comput e the Khintchin e co nstant K ( a ) of a , deﬁne d as follows: K ( a ) = E θ ∼ {± 1 } n [ | θ · a | ] , wher e θ is drawn uniformly at ran dom fr om {± 1 } n . T o rel ate the Khi ntchin e problem to Bayesi an persuas ion, we beg in with a persuasion instan ce with n i.i.d. action s. Moreove r , there are only two ac tion types, 5 which we refer to as type -1 and type +1 . The state of nature is a uniform random draw from the set {± 1 } n , w ith the i th entry specifyin g th e type of action i . It is easy to see that these actions are i.i.d. , with margina l probabili ty 1 2 for each type. W e call this instan ce the Khintc hine-like persuasio n setti ng. As in Section 3, we still use the signatur e to capture the payof f-rele van t features of a signaling scheme. A si gnatur e for the Khintchi ne-lik e persuasion problem is of the form M = ( M 1 , ..., M n ) where M i ∈ R n × 2 for any i ∈ [ n ] . W e pay special at tention to sign aling schemes whi ch u se only two signa ls, in which case we represent them u sing a two-signal sign atur e of the form ( M 1 , M 2 ) ∈ R n × 2 × R n × 2 . Recall tha t suc h a sign ature is rea lizable if there is a sig naling sch eme which uses onl y tw o signals, with the p roperty that M i j t is the joi nt pro bability of t he i th sig nal a nd th e e vent that act ion j has type t . W e no w deﬁne th e Khint chin e polytope , consisting of a con ve x family of two-sig nal signat ures. Deﬁnition C.2. The Khintchin e po lytope is the family K ( n ) of rea lizable two-signal signat ur es ( M 1 , M 2 ) for the Khin tchine-lik e persua sion setting whic h satisfy the additi onal const rain ts M 1 i, 1 + M 1 i, 2 = 1 2 ∀ i ∈ [ n ] . W e sometimes use K to denote the Khintchi ne polytop e K ( n ) when the di mension n is clear from the conte xt. Note that the con straint s M 1 i, 1 + M 1 i, 2 = 1 2 , ∀ i ∈ [ n ] state that the ﬁrst signal sh ould be sen t w ith probab ility 1 2 (hence also the second signal). W e now sho w that optimizing ov er the Khintchin e po lytope is # P -hard by reduc ing the Kintchine proble m to Linear progra m (6). Lemma C.3. General linear optimiza tion over the Khintch ine polytop e K is # P -har d. 5 Recall from Section 3 that each type is associated with a pair ( ξ , ρ ) , where ξ [ ρ ] is the payof f to the sender [receiv er] if the recei ver tak es an action of that type. 21 maximize P n i =1 a i ( M + i, +1 − M + i, − 1 ) − P n i =1 a i ( M − i, +1 − M − i, − 1 ) subjec t to ( M + , M − ) ∈ K ( n ) (6) Linear progra m for computi ng the Khintchin e constant K ( a ) for a ∈ R n Pr oof. Fo r any giv en a ∈ R n , we re duce th e computa tion of K ( a ) – th e Khintch ine constant for a – to a linear o ptimization proble m ov er the Khin tchine polytope K . Since ou r red uction will use tw o signals σ + and σ − which correspond to the sign of θ · a , we will use ( M + , M − ) to denote the two matrices in the signat ure in lieu of ( M 1 , M 2 ) . Moreov er , we use the two action types +1 and − 1 to index the columns of each matrix. For example, M + i, − 1 is th e joi nt p robabi lity of si gnal σ + and the e vent th at th e i th action has type − 1 . W e claim that th e Kintchin e con stant K ( a ) equals the optimal objecti ve v alue of the implicitly- describ ed linear progra m (6). W e denote this op timal objecti ve v alue by O P T ( LP (6) ) . W e ﬁrst prove that K ( a ) ≤ O P T ( LP (6 ) ) . Consider a signaling scheme ϕ in the K intchin e-like persua sion se tting which si mply o ut- puts σ sig n ( θ · a ) for e ach state of na ture θ ∈ {± 1 } n (break ing tie unifo rmly at random if θ · a = 0 ) . Since θ is dra wn unif ormly from {± 1 } n and sign ( θ · a ) = − sig n ( − θ · a ) , this scheme ou tputs each of the signals σ − and σ + with probabil ity 1 2 . Consequentl y , the t wo-sig nal s ignatu re of ϕ is a p oint in K . Moreov er , e valua ting the objecti ve function of LP (6) on the two-sign al signa ture ( M + , M − ) of ϕ yields K ( a ) = E θ [ | θ · a | ] , as sho wn belo w . E θ [ | θ · a | ] = E θ [ θ · a | σ + ] · Pr ( σ + ) + E θ [ − θ · a | σ − ] · Pr ( σ − ) = n X i =1 a i E θ [ θ i | σ + ] · Pr ( σ + ) − n X i =1 a i E θ [ θ i | σ − ] × Pr ( σ − ) = n X i =1  a i [ Pr ( θ i = 1 | σ + ) − Pr ( θ i = − 1 | σ + )] · Pr ( σ + )  − n X i =1  a i [ Pr ( θ i = 1 | σ − ) − Pr ( θ i = − 1 | σ − )] · Pr ( σ − )  = n X i =1  a i [ Pr ( θ i = 1 , σ + ) − Pr ( θ i = − 1 , σ + )]  − n X i =1  a i [ Pr ( θ i = 1 , σ − ) − Pr ( θ i = − 1 , σ − )]  = n X i =1 a i [ M + i, +1 − M + i, − 1 ] − n X i =1 a i [ M − i, +1 − M − i, − 1 ] This concl udes the proof that K ( a ) ≤ O P T ( LP (6) ) . No w we p rove K ( a ) ≥ O P T ( LP (6) ) . T ake any signalin g scheme which us es only tw o signals σ + and σ − , and let ( M + , M − ) be its two-si gnal signature. N otice, ho wev er , that σ + no w is only the “name” of the signal , and does not imply that θ · a is positi ve. Nev ertheless, it is still va lid to rev erse the abov e deriv ation until we reach n X i =1 a i [ M + i, +1 − M + i, − 1 ] − n X i =1 a i [ M − i, +1 − M − i, − 1 ] = E θ [ θ · a | σ + ] · Pr ( σ + ) + E θ [ − θ · a | σ − ] · Pr ( σ − ) . Since θ · a and − θ · a are each no great er than | θ · a | , we ha ve E θ [ θ · a | σ + ] · Pr ( σ + ) + E θ [ − θ · a | σ − ] · Pr ( σ − ) ≤ E θ [ | θ · a | | σ + ] · Pr ( σ + ) + E θ [ | θ · a | | σ − ] · Pr ( σ − ) = E θ [ | θ · a | ] = K ( a ) . 22 That is, the objec ti ve valu e of LP (6) is upper bounded by K ( a ) , as nee ded. Before we procee d to present the reduction from the membership problem for K to optimal persuasi on, we point out an interes ting corollary of Lemma C.3. Cor ollary C.4. Let P be the polyto pe of r ealiza ble signatu res for a per suasion pr oblem with n i.i.d. actio ns and m types (see Sectio n 3). Linear optimization over P is # P -har d, and this holds even when m = 2 . Pr oof. Cons ider the Khintchine- like persuasion setting. It is eas y to see th at the Khint chine polytop e K can be ob tained from P by adding the constrain ts M σ i = 0 fo r i ≥ 3 and M σ 1 i, 1 + M σ 1 i, 2 = 1 2 for i ∈ [ n ] , follo wed by a simple pro jection . Therefore , the membershi p problem for K can be red uced in polyno m ial time to the membership problem for P , since the a dditio nal line ar constraint s can be explici tly c heck ed in polynomia l time. By the polynomial-t ime equiv alence between optimizatio n and membership, it follows that gener al linear optimizat ion ov er P is # P -hard. Remark C .5. It is inter esting to compa re Cor ollary C.4 to single item auctions with i.i.d. bidde rs, wher e the pr oblem does admit a polynomial-ti me separatio n oracle for the polytope of re alizable signat ur es via Bor der’ s Theor em [1 3, 1 2 ] a nd its algor ithmic pr operties [15, 1]. In contrast, th e p olytope of r ealizable sig- natur es for B ayesia n persuasi on is # P -har d to op timize over . Neverthele ss, in Section 3 we wer e indeed able to comput e the optimal signa ling scheme and se nder utility for per suasion with i.i.d. action s. Cor ollary C.4 con ve ys that it was crucia l for our algorithm to e xploit the special structu re of the pers uasion objective and the symmetry of the optimal sche me, since optimizin g a g eneral objectiv e o ver P is #P-har d. Reduction W e now present a reduction from the membership problem for the Khintchine p olytope to the computati on of optimal sender utility for persuas ion wit h indepe ndent actions. As the output of our reduction , we construct a persua sion instance of the follo wing form. There are n + 1 actions. Action 0 is speci al – it d eterministicall y results in sender utility ǫ and receiv er utility 0 . Here, we think of ǫ > 0 as being small enough for our ar guments to go through. The other n actions are re gular . Action i > 0 indep endent ly results in sender utility − a i and rece iver utility a i with proba bility 1 2 (call this the type 1 i ), or sende r utility − b i and recei ver utility b i with prob ability 1 2 (call th is the ty pe 2 i ). Note that th e sende r and recei ver utilit ies are zer o-sum for both types. Notice that, though each regular action’ s type distrib ution is uniform ov er its two types, t he action s here are not id entical bec ause the as sociate d payof fs — speciﬁed by a i and b i for each act ion i — are diff erent fo r dif ferent ac tions. Since the special acti on is deterministic and the p robability of its (onl y) type is 1 in any sig nal, we can interpret any ( M 1 , M 2 ) ∈ K ( n ) as a two-s ignal signature for our persuasio n instan ce (the ro w correspondi ng to the spec ial action 0 is imp lied). Fo r example, M 1 i, 2 is the joint prob ability of the ﬁrst signal and the ev ent t hat action i has type 2 i . Our goal i s to reduce membership checking for K ( n ) to comput ing the opti mal expect ed sender utility for a persuasion instance with care fully chosen paramet ers { a i } n i =1 , { b i } n i =1 , and ǫ . In re lating opti mal persua sion to the Khin tchine pol ytope, there a re two main dif ﬁculties: (1) K con sists of two-sig nal signatures, so there should b e an optimal scheme to ou r per suasio n insta nce w hich uses only two signals; (2 ) T o be consistent with the deﬁnition of K , s uch an optimal scheme should send each signal with proba bility e xactly 1 2 . W e will des ign speciﬁc ǫ, a i , b i to accomplis h both goals. For notational con venience , we will again use ( M + , M − ) to d enote a typical element in K i nstead of ( M 1 , M 2 ) b ecause, as w e will see later , the two constru cted signals will induce posi ti ve and negati ve s ender utiliti es, res pecti vely . Notice that there are only n degrees of freedom in ( M + , M − ) ∈ K . T his is because M + + M − is the al l- 1 2 matrix in R n × 2 , corresp onding to the prio r distrib ution of states of natur e (by the deﬁnitio n of realizable signatu res). Moreove r , M + i, 1 + M − i, 2 = 1 2 for all i ∈ [ n ] (by the deﬁnition of K ). Therefore , we must h a ve M + i, 1 = M − i, 2 = 1 2 − M + i, 2 = 1 2 − M − i, 1 . 23 This implies tha t we can par ametrize signat ures ( M + , M − ) ∈ K by a v ector x ∈ [0 , 1 2 ] n , where M + i, 1 = M − i, 2 = x i and M + i, 2 = M − i, 1 = 1 2 − x i for each i ∈ [ n ] . For an y x ∈ [0 , 1 2 ] n , let M ( x ) denote the signatur e ( M + , M − ) deﬁned by x as just describ ed. W e ca n no w restate the membersh ip problem for K as foll o w s: gi ven x ∈ [0 , 1 2 ] n , determine wheth er M ( x ) ∈ K . When any of the entries of x equals 0 or 1 2 this problem is tri vial, 6 so we assume without loss of genera lity that x ∈ (0 , 1 2 ) n . Moreove r , when x i = 1 4 for some i , it is easy to see that a signa ling scheme with signatu re M ( x ) , if o ne ex ists, must choose its s ignal in depen dently of the typ e of actio n i , and therefo re M ( x ) ∈ K ( n ) if and only if M ( x − i ) ∈ K ( n − 1) . This allo ws us to assume without loss of genera lity that x i 6 = 1 4 for all i . Giv en x ∈ (0 , 1 2 ) n with x i 6 = 1 4 for all i , we construc t sp eciﬁc ǫ and a i , b i for all i such that we can determine whether M ( x ) ∈ K by simply looking at the optimal sender utility in the correspond ing persua sion instance . W e choose parameters a i and b i to satis fy the followin g two equation s. x i a i + ( 1 2 − x i ) b i = 0 . (7) ( 1 2 − x i ) a i + x i b i = 1 2 . (8) W e note that the abo ve linear syste m always has a solut ion when x i 6 = 1 4 , which we assumed prev iously . W e m ake t wo o bserv ations about our c hoice of a i and b i . Firs t, the pr ior expecte d recei ver utility 1 2 ( a i + b i ) equals 1 2 for all actions i (by simply adding Equation (7 ) and (8)). Second, a i and b i are both non-zero, and this follo ws easily from our assump tion that x i ∈ (0 , 1 2 ) . No w we show ho w to d etermine whether M ( x ) ∈ K by only examining the o ptimal sende r utility in the constructed pers uasion instance . W e start by s ho wing that restricting to two-sig nal schemes is witho ut loss of general ity in our instan ce. Lemma C.6. T her e exist s an optimal incentiv e-compatible signa ling scheme which uses at most two signals: one signal r ecommends the special action, and the other r ecommends some r e gular action. Pr oof. Reca ll that an optimal ince ntiv e-compatible scheme u ses n + 1 signa ls, with signal σ i recommend ing action i for i = 0 , 1 , ..., n . F ix such a scheme, and let α i denote the probability of signal σ i . S ignal σ i induce s posterio r expec ted recei ver uti lity r j ( σ i ) and sende r utility s j ( σ i ) for each ac tion j . For a re gular action j 6 = 0 , we ha ve s j ( σ i ) = − r j ( σ i ) for all i due to th e zero-su m nature of our con struction. Notice that r i ( σ i ) ≥ 0 for all regu lar actions i 6 = 0 , since o therwise the recei ver w ould prefer acti on 0 o ver ac tion i . Consequ ently , for each signal σ i with i 6 = 0 , the recei ver deri ves non-ne gati ve utility and the sender deriv es non-p ositi ve utility . W e claim t hat mer ging s ignals σ 1 , σ 2 , . . . , σ n — i.e., modif ying th e sign aling scheme t o output the same signal σ ∗ in lieu of ea ch of them — would not d ecrease the sender’ s exp ected utility . Recall that incenti ve compatib ility impl ies that r i ( σ i ) = max n j =0 r j ( σ i ) . Using Jensen ’ s inequality , we get n X i =1 α i r i ( σ i ) ≥ n max j =0 " n X i =1 α i r j ( σ i ) # . (9) If the m aximum in the right hand side express ion of (9) is attained at j ∗ = 0 , the recei ver will choose the special action 0 when presen ted with the merge d signal σ ∗ . Recalling that s i ( σ i ) is non-p ositi ve for i 6 = 0 , this can only improv e the sender’ s e xpected utility . Otherwise, the receiv er ch ooses a regula r action j ∗ 6 = 0 when presented w ith σ ∗ , resultin g in a total co ntrib ution of P n i =1 α i r j ∗ ( σ i ) to the recei ver’ s e xpecte d 6 If x i is 0 or 1 2 , then M ( x ) ∈ K if and only if x j = 1 4 for all j 6 = i . This i s because the corresponding signaling scheme must choose its signal based solely on the type of action i . 24 utility from the merged signal, do wn from the total contrib ution of P n i =1 α i r i ( σ i ) by the original signals σ 1 , . . . , σ n . Recalli ng the zero-sum na ture of our con struction for re gular actio ns, the me r ged sign al σ ∗ con- trib utes P n i =1 α i s j ∗ ( σ i ) = − P n i =1 α i r j ∗ ( σ i ) to the s ender’ s ex pected utility , up fro m a total con tributi on of P n i =1 α i s i ( σ i ) = − P n i =1 α i r i ( σ i ) by th e origin al signals σ 1 , . . . , σ n . Therefo re, the sender is no t worse of f by merg ing the signal s. Moreo ver , interpre ting σ ∗ as a r ecommenda tion for action j ∗ yields inc enti ve compatib ility . Therefore , in characteriz ing the optimal solution to our construc ted persuasion instance , it suf ﬁces to analyz e two- signal schemes of the the form guaranteed by Lemma C. 6. For such a scheme, we denote the signal that recommend s the specia l action 0 by σ + (indic ating that the sender deri ves positi ve utility ǫ ), a nd d enote th e other s ignal b y σ − (indic ating that the send er deri ves ne gati ve u tility , as we w ill show). For con venien ce, in the follo wing discuss ion we use the exp ression “payof f from a signal ” to signify the exp ected payof f of a player conditioned on that signal multiplied by the probabi lity of that signa l. For exa mple, the sender’ s e xpected payof f fr om signal σ − equals the sen der’ s expected payof f condition ed on signal σ − multiplie d by the overa ll proba bility that the scheme outp uts σ − , assu ming the re ceiv er follo ws the scheme’ s (ince nti ve compatib le) recommendatio ns. W e al so use th e expressi on “pay off from an acti on in a signal ” to signify the posterior exp ected payof f of a player for that action conditi oned on the signal, multiplie d by the probabilit y that the sch eme out puts the signal. For e xample, the r eceiver’ s ex pected pay of f fr om act ion i in signal σ + equals α + · r i ( σ + ) , where r i ( σ + ) i s the r ecei ver’ s posterior ex pected pay of f from action i gi ven signal σ + , and α + is the ov erall probability of signa l σ + . Lemma C. 7. F ix a n incen tive-co mpatible scheme with si gnals σ − and σ + as described abo ve. The sender’ s e xpected payo f f fr om signal σ − is at m ost − 1 2 . M or eover , if the s ender’ exp ected payof f fr om σ − is ex actly − 1 2 , then for each r e gular action i the e xpected p ayof f of both the sen der and the rec eiver fr om action i in signal σ + equals 0 . Pr oof. Assu me t hat signal σ + [ σ − ] is sent with probab ility α + [ α − ] and induces po sterior e xpected receiv er payof f r i ( σ + ) [ r i ( σ − ) ] for each action i . Recall fr om our c onstru ction that the pri or expec ted payof f of each reg ular act ion i 6 = 0 equals 1 2 a i + 1 2 b i = 1 2 . Since the p rior expectati on must eq ual the e xpected po sterior exp ectatio n, it follo ws that α + · r i ( σ + ) + α − · r i ( σ − ) = 1 2 when i is regular . The rec ei ver’ s reward f rom the specia l action is deterministic ally 0 , and therefore incenti ve compatibilit y impli es that r i ( σ + ) ≤ 0 for each reg ular action i . It follo ws that α − · r i ( σ − ) = 1 2 − α + · r i ( σ + ) ≥ 1 2 for regular actions i . In other words, the recei ver’ s exp ected payof f from each regu lar action in signal σ − is at least 1 2 . By th e zero-sum nature of our construction , the sender’ s expected payof f from each re gular action in sign al σ − is at most − 1 2 . Since σ − recommend s a regular action, we conclud e tha t the sender’ s expect ed payof f from σ − is at most − 1 2 . No w assume that the sender’ s expected payof f fr om σ − is exa ctly − 1 2 . By the zero-sum property , incent i ve compatibility , an d the above- establ ished f act that α − · r i ( σ − ) ≥ 1 2 for regu lar actions i , it follo ws that the recei ver’ s expected pa yof f from ea ch re gular action in signal σ − is e xactly 1 2 . R ecallin g that α + · r i ( σ + ) + α − · r i ( σ − ) = 1 2 when i is re gular , w e con clude that the recei ver’ s expected payof f from a regul ar action in signa l σ + equals 0 . By the zero -sum property for regula r action s, the same is true for the sender . The ke y to the remainder of our reduction is to choose a small enough v alue for the paramete r ǫ — the sender’ s utilit y from the specia l action — so that the optimal signaling scheme satisﬁes the proper ty mentione d in L emma C.7: The se nder’ s expected payof f from signal σ − is ex actly equal to its maximum possib le valu e of − 1 2 . In othe r words, we must mak e ǫ so smal l so that the sender prefer s to not sac riﬁce any of her payof f from σ − in order to gain utility from the special action recommended by σ + . Notice that this upper bo und of − 1 2 is indeed achie vable: the uninfor mati ve signali ng sche me which recommends an arbitr ary regular action has this proper ty . W e now sho w that a “small enough ” ǫ indeed exis ts. The 25 ke y idea behind this e xistence proof is the fo llowing: W e start with a signaling scheme which maximiz es the sender’ s payof f from σ − at − 1 2 , and moreov er correspond s to a vertex of the polyto pe of incenti ve- compatib le signature s. When ǫ > 0 is smaller than the “bit comple xity” of the vertice s of this polytope, movin g to a diff erent verte x — one with lower sender payof f from σ − — will result in m ore utility loss from σ − than utility ga in from σ + . W e sh o w that ǫ > 0 wit h polynomially man y bits sufﬁces , and c an be compute d in polyno mial time. Let P 2 be the f amily of all r ealizab le two- signal signatures (agai n, ign oring a ction 0 ). It is easy to see that P 2 is a p olytop e, and importantly , all entries of an y ver tex o f P 2 are inte ger multiple s of 1 2 n . This is becaus e eve ry v ertex of P 2 corres ponds to a determin istic signaling sch eme which par titions t he se t of states of nature, and ev ery s tate of na ture occ urs with probability 1 / 2 n . A s a result, all vertices of P 2 ha ve O ( n ) bit comple xity . T o ease our discussio n, we use a compact repre sentation for poin ts in P 2 . In partic ular , any point in P 2 can be captured by n + 1 variab les: varia ble p denotes the probab ility of sending signal σ + , and v ariable y i denote s the joint probabil ity of signal σ + and the ev ent that action i has type 1 i . It follo w s that joint probability of type 2 i and signal σ + is p − y i , and the probab ilities associat ed with signal σ − are determined by th e constrain t that M + + M − is the all- 1 2 matrix. W ith so m e abu se of notation , we use M ( p, y ) = ( M + , M − ) to denote the signatur e in P 2 corres pondin g to the probab ility p and n-dimension al vec tor y . No w we consider the follo w ing two linear programs. maximize pǫ + u subjec t to M ( p, y ) ∈ P 2 y i a i + ( p − y i ) b i ≤ 0 , for i = 1 , . . . , n. u ≤ − [( 1 2 − y i ) a i + ( 1 2 − p + y i ) b i ] , for i = 1 , . . . , n . (10) maximize u subjec t to M ( p, y ) ∈ P 2 y i a i + ( p − y i ) b i ≤ 0 , for i = 1 , . . . , n. u ≤ − [( 1 2 − y i ) a i + ( 1 2 − p + y i ) b i ] , for i = 1 , . . . , n . (11) Linear progr ams (10) and (11) are identi cal exc ept for the fact that the objec ti ve of LP (10) includ es the additi onal term pǫ . LP (10) computes precisely the o ptimal exp ected sender utility in our constru cted p ersua- sion ins tance: T he ﬁrst se t of inequ ality constrai nts are the in centi ve-co mpatibili ty cons traints for the sign al σ + recommend ing action 0 ; The second set of inequality constraint s state that the sender’ s payof f from signal σ − is the minimum among all act ions, as implied by the zero-sum nature of our cons tructio n; The object i ve is the sum of the sender’ s payof fs from signals σ + and σ − . Notice that the incen tive -compatibility constr aints fo r signal σ − , namely ( 1 2 − y i ) a i + ( 1 2 − p + y i ) b i ≥ 0 for all i 6 = 0 , are imp licitly satisﬁed because 1 2 a i + 1 2 b i = 1 2 by our construc tion and ( 1 2 − y i ) a i + ( 1 2 − p + y i ) b i = 1 2 a i + 1 2 b i − [ y i a i + ( p − y i ) b i ] ≥ 1 2 − 0 > 0 . On the other hand, LP (11) maximizes the sender’ s ex pected payof f from signal σ − . Observ e th at the opti- mal objecti ve valu e of LP (11) is precisely − 1 2 becaus e u ≤ − [( 1 2 − y i ) a i + ( 1 2 − p + y i ) b i ] ≤ − 1 2 for all i 6 = 0 , and equality is attained , for example , at p = 0 and y = 0 . Let f P 2 be the set of all feasi ble ( u, M ( p, y )) for LP (10 ) (and LP (11)). Obv iously , f P 2 is a polytope. W e no w ar gue that all vertices of f P 2 ha ve bit complexit y polynomial in n an d the bit complexit y of x ∈ (0 , 1 2 ) n . In parti cular , deno te the bit complexi ty of x by ℓ . Since a i , b i are compu ted by a two- v ariable two-equ ation linear s ystem in vol ving x i (Equatio ns (7) and (8 ) ), the y each ha ve O ( ℓ ) bit complexity . Conseq uently , all the e xplicit ly describ ed facets of f P 2 ha ve O ( ℓ ) bit comple xity . Moreov er , since each vert ex of P 2 has O ( n ) bit complexity , ea ch fac et of P 2 then has O ( n 3 ) bit comple xity , i.e., the coefﬁcie nts of inequal ities that determin e the f acets ha ve O ( n 3 ) bit comple xity . T his i s due to t he fa ct th at f acet comple xity of a rational polyto pe is upper bou nded by a cubic po lynomial of th e ve rtex complexity a nd vice versa ( see, e.g., [44]). 26 T o sum up, any facet of polytope f P 2 has bit complexity O ( n 3 + ℓ ) , and therefo re any vertex of f P 2 has O ( n 9 ℓ 3 ) bit comple xity . Let the polynomial B ( n, ℓ ) = O ( n 9 ℓ 3 ) be an upper bound on the maximum bit co mplex ity of v ertices of f P 2 . Now we are ready to set the valu e of ǫ . LP (10) alw ays has an optimal vert ex solutio n which we denote as ( u ∗ , M ∗ ) . Recall that u ≤ − 1 2 for a ll points ( u, M ( p, y )) in f P 2 and u = − 1 2 is att ainabl e at so me ver tices. S ince all vertice s of f P 2 ha ve B ( n, ℓ ) bit complex ity , ( u ∗ , M ∗ ) must either satisfy either u ∗ = − 1 2 or u ∗ ≤ − 1 2 − 2 − B ( n,ℓ ) . Therefore, it sufﬁce s to set ǫ = 2 − n · B ( n,ℓ ) , which is a number with polyno m ial bit comple xity . As a result, a ny optimal verte x solut ion t o LP (1 0 ) must satisfy u ∗ = − 1 2 , since t he loss incurred by moving to an y other verte x with u < − 1 2 can ne ver be compensa ted for by the other term pǫ < ǫ . W ith such a small value of ǫ , the sender’ s goal is to send signal σ + with probabi lity as high as possi ble, subjec t to the constraint that her utility from σ − is precisel y − 1 2 . In othe r words, signal σ + must induce exp ected recei ver/sen der utility precisely 0 for each regular action i 6 = 0 (see Lemma C. 7). This character - ization of the opt imal scheme now allows us to d etermine w hether M ( x ) ∈ K by in spectin g the send er’ s optimal expe cted utility . The follo wing Lemma completes our proof of Theorem 4.1. Lemma C. 8. Given the small enou gh value of ǫ descri bed abo ve, the sender ’ s e xpecte d utility in the opt imal signal ing scheme for our construc ted persua sion instance is at least 1 2 ( ǫ − 1) if and only if M ( x ) ∈ K . Pr oof. ⇐ : If M ( x ) ∈ K , then by our c hoice of a i , b i (recall Equations (7) an d (8)), the signaling sch eme implementi ng M ( x ) is incenti ve compatible, the sender ’ s payof f from signal σ + is 1 2 ǫ , an d her payof f from σ − is − 1 2 . Therefo re, the optimal sender utility is at least 1 2 ǫ − 1 2 . ⇒ : Let M ( p, y ) be t he signature of a verte x optimal sig naling sch eme in LP (10). By our cho ice of ǫ we kno w that th e sender payo f f from sign al σ − must be exactly − 1 2 . T herefor e, to achie ve o vera ll sender utility at le ast 1 2 ǫ − 1 2 , signal σ + must be sent with probability p ≥ 1 2 , an d the recei ver’ s payo f f from each reg ular action i 6 = 0 in signal σ + is ex actly 0 . That is, y i a i + ( p − y i ) b i = 0 . By const ructio n, we al so ha ve that x i a i + (0 . 5 − x i ) b i = 0 and a i , b i 6 = 0 , whi ch imp ly that y i x i = p − y i 0 . 5 − x i and, f urthermo re, that y i ≥ x i since p ≥ 1 2 . N o w let ϕ be a signali ng scheme with the sig nature M ( p, y ) . W e c an post-pr ocess ϕ so it has signat ure M ( x ) as follo ws: whenev er ϕ outputs the signal σ + , ﬂip a biased random coin to output σ + with probability 0 . 5 p and output σ − otherwis e. By using the identit y y i x i = p − y i 0 . 5 − x i , it is easy t o see tha t thi s adjust ed signalin g sc heme has signatur e M ( x ) . 27 D Omitted Proofs fr om Section 5 D.1 A Bicriteria FPT AS Pro of of Lemma 5.2 Fix ǫ , K , and λ , and let ϕ denote t he resulting si gnalin g scheme implemented by A lgorith m 2. Let θ ∼ λ denote the input to ϕ , and σ ∼ ϕ ( θ ) denote its output. F irst, we condition on the empirical sample e λ = { θ 1 , . . . , θ K } witho ut condition ing on the inde x ℓ of the inpu t state of na ture θ , and sho w that ǫ -incen tive compatib ility holds subjec t to this condition ing. T he principl e of deferre d decision s impli es that, subject to this conditio ning, θ is uniformly distrib uted in e λ . By deﬁnition of linear program (4), the signaling scheme e ϕ compute d in Step 3 is ǫ -incen tiv e compatible scheme for the empiric al distrib ution e λ . S ince σ ∼ e ϕ ( θ ) and θ is con ditionally distrib uted according to e λ , this implie s that al l ǫ -incen ti ve compatibilit y constrai nts condit ionall y ho ld; formally , th e follo wing holds for each pair of actions i and j : E [ r i ( θ ) | σ = σ i , e λ ] ≥ E [ r j ( θ ) | σ = σ i , e λ ] − ǫ Removin g the conditionin g on e λ and in v oking linea rity of expecta tions sho ws that ϕ is ǫ -incenti ve compatib le fo r λ , completing the proof. Pro of of Lemma 5.3 As in the proof of Lemma 5.2, we condit ion on the empir ical s ample e λ = { θ 1 , . . . , θ K } and obs erve that θ is unifor mly dis tribute d in e λ after this co nditio ning. The condition al ex pectat ion of se nder utility then equals P K k =1 P n i =1 1 K e ϕ ( θ k , σ i ) s i ( θ k ) , where e ϕ is the signaling scheme computed in Step 3 based on e λ . Since this is precisely the o ptimal v alue of the LP (4) solved in Step 3, remo ving th e conditionin g and in vokin g lineari ty of exp ectatio ns comple tes the proof. Pro of of Lemma 5.4 Recall that linea r program (1 ) solv es for the optimal incenti ve compatible scheme for λ . It is easy to see that the linear program (4) solv ed in step 3 is simply the instantia tion of LP (1) for the empirical distri- b ution e λ consis ting of K samples from λ . T o pr ov e the lemma, it would suf ﬁce to sho w that the opti mal incent i ve-c ompatible scheme ϕ ∗ corres pondin g to LP (1) remains ǫ -incenti ve compatible and ǫ -optimal for the distrib ution e λ , with high probability . Unfortu nately , this approach fa ils because polynomia lly-man y samples from λ are not suf ﬁcient to approximat ely preserv e the incenti ve compat ibility cons traints cor - respon ding to low-pro bability signa ls (i.e., signals which are output with probabil ity smaller than in verse polyn omial i n n ). Neve rtheles s, we sho w in Claim D.1 th at there exists an approxi mately optimal solution b ϕ to LP (1) with th e property that e very sig nal σ i is eithe r lar ge , which we deﬁne as bein g output by b ϕ with probab ility at least ǫ 4 n assuming θ ∼ λ , or honest in that only states of nature θ with i ∈ argmax j r j ( θ ) are mappe d to it. It i s eas y to see tha t sampling preser ves incent i ve-c ompatib ility exactly for ho nest signals. As for lar ge signals , we employ tail bounds and th e un ion bound to sh ow that polynomia lly man y samples suf ﬁce to appro ximately preser ve incenti ve compatibility (Claim D.2). Claim D.1. T her e is a signaling sc heme b ϕ which is incentiv e compat ible fo r λ , induces s ender utility u s ( b ϕ, λ ) ≥ O P T − ǫ 2 on λ , and such that e very signal of b ϕ is either lar ge or honest. Pr oof. Let ϕ ∗ be the optimal incenti ve-compatib le scheme for λ — i.e. the optimal solution to LP (1). W e call a signal σ small if it i s output by ϕ ∗ with probabi lity les s than ǫ 4 n , i.e. if P θ ∈ Θ λ θ ϕ ∗ ( θ , σ ) < ǫ 4 n , and otherwis e w e call it lar ge . Let b ϕ be the scheme which is deﬁned as follo ws: on input θ , it ﬁrst samples σ ∼ ϕ ∗ ( θ ) ; if σ is lar ge then b ϕ si mply outp uts σ , and oth erwise it recommends an action maximizi ng recei ver utility in state of nature θ —- i.e., outpu ts σ i ′ for i ′ ∈ argmax i r i ( θ ) . It is easy to see that ev ery signal of b ϕ is either large or honest. Moreov er , si nce ϕ ∗ is incenti ve compatible and b ϕ only replace s recommend ations 28 Rainy Sun ny W alk 1 − δ 1 Driv e 1 0 T able 1: Rece iver ’ s Payof fs in Rain and Shine Example of ϕ ∗ with “h onest” reco m mendatio ns, it is easy to ch eck that b ϕ is incenti ve compatible for λ . Finally , since the total probabi lity of small signal s in ϕ ∗ is at most ǫ 4 , and utilitie s are in [ − 1 , 1] , the sender’ s exp ected utility from b ϕ is no worse than ǫ 2 smaller than her ex pected utility from ϕ ∗ . Claim D.2. Let b ϕ be the signaling scheme fr om Claim D.1. W ith pr obability at least 1 − ǫ 8 ove r the sample e λ , b ϕ is ǫ -incentive compatible for e λ , and mor eove r u s ( b ϕ, e λ ) ≥ u s ( b ϕ, λ ) − ǫ 4 . Pr oof. Reca ll that b ϕ is incenti ve compatible for λ , a nd e very sign al is ei ther lar ge or honest. Since e λ is a set of samples from λ , it i s easy to see tha t incenti ve compa tibility constraints per taining to t he honest signals contin ue to hold ov er e λ . It r emains to sh ow that ince ntiv e compatibili ty co nstraints for lar ge signals, as well as ex pected sender utility , are approximate ly preserve d when repl acing λ with e λ . Recall that incenti ve-c ompatibi lity requi res that E θ [ b ϕ ( θ , σ i )( r i ( θ ) − r j ( θ ))] ≥ 0 for each i, j ∈ [ n ] . Moreo ver , the sende r’ s ex pected utility can be written as E θ [ P n i =1 b ϕ ( θ , σ i ) s i ( θ )] . The left hand side of each incen ti ve compatibi lity constrain t e val uates the expec tation of a ﬁxed fun ction of θ with range [ − 2 , 2] , whereas the sender’ s exp ected utility ev aluates the ex pectation of a function of θ w ith range in [ − 1 , 1] . Standard tail bounds and the union bound , coupl ed with our careful choice of the number of samples K , imply that rep lacing distrib ution λ with e λ appr oximatel y preserv es each of these n 2 + 1 quant ities to within an additi ve error of ǫ 2 4 n with probab ility at least 1 − ǫ 8 . This bound on the additi ve loss translates to ǫ - incent i ve compat ibility for th e lar ge signa ls, and is less than the permitted decre ase of ǫ 4 for e xpected sende r utility . The abo ve claims, coupled w ith the fact that sender payof fs are bounde d in [ − 1 , 1] , imply that the exp ected optimal val ue of linear program (4) is at least O P T − ǫ , a s needed. D.2 Information-T heoretic Barriers Impossibility of Incenti ve Compatibility (Pr o of of Theore m 5.5 (a)) Consider a setting with tw o sta tes of na ture, which we will con venie ntly refer to as r ainy and su nny . The recei ver , who we may think of as a daily commuter , ha s two actions: walk and drive . The recei ver slightly prefers drivin g on a rainy day , and stron gly prefer s walking on a sunny day . W e summarize the receiv er’ s payof f function, parametrize d by δ > 0 , in T able 1. The sender , who we will thi nk of as a municipality with black- box sample acces s to weath er reports drawn from the sa m e dis trib ution as the st ate of nature, strongl y prefers tha t the recei ver ch ooses walk ing regard less of whe ther it is sunny or rainy: we let s w alk = 1 an d s driv e = 0 in both states of nature. Let λ r be the point distrib ution on the rainy state of nature , and let λ s be such that Pr λ s [ rain y ] = 1 1+2 δ and Pr λ s [ sunn y ] = 2 δ 1+2 δ . It i s easy to s ee that th e unique direct ince nti ve-compa tible scheme for λ r alw ays recommend s driv ing, and hence results in exp ected sende r utility of 0 . In contrast, a simple calculation sho ws that alwa ys recommending walking is incenti ve compatible for λ s , and results in exp ected sender utility 1 . If algorithm A is incenti ve compatible and c -optimal for a constant c < 1 , then A ( λ r ) must ne ver recommend walking w hereas A ( λ s ) must recommend walking with constant probabil ity at least (1 − c ) ov erall (in expecta tion o ver the input state of nature θ ∼ λ s as w ell as all other internal randomn ess). Consequ ently , giv en a black box distrib ution D ∈ { λ r , λ s } , ev aluating A ( D , θ ) on a random draw θ ∼ D yields a tester which disting uishes between λ r and λ s with const ant probab ility 1 − c . 29 Pr [ θ 1 ] Pr [ θ 2 ] Pr [ θ 3 ] λ 1 − 2 δ 2 δ 0 λ ′ 1 − 2 δ δ δ T able 2: T wo D istrib utions on Three Actions Since the t otal variati on dist ance be tween λ r and λ s is O ( δ ) , it is well kno wn (and easy to check) th at any black-box algorithm which distin guishe s between the two di strib utions with Ω(1) succ ess probabili ty must take Ω( 1 δ ) samples in expectat ion when presen ted with one of these d istrib utions. As a consequen ce, the a vera ge-cas e sample complexity of A on either of λ r and λ s is Ω ( 1 δ ) . Sin ce δ > 0 can be made arbitrari ly small, this completes the proof. Impossibility of Optimality (Pr oo f of Theor em 5.5 (b)) Consider a settin g with three action s { 1 , 2 , 3 } and three corres pondin g state s of nature θ 1 , θ 2 , θ 3 . In ea ch state θ i , th e recei ver deriv es utilit y 1 from act ion i and utility 0 from the other a ctions. The sender , on th e other hand, deriv es utility 1 from action 3 and utility 0 from actions 1 and 2 . For an arbitrar y parameter δ > 0 , we deﬁne two distrib utions λ an d λ ′ ov er stat es o f nature w ith total variat ion dist ance δ , illus trated in T able 2. Assume al gorith m A is o ptimal and c -in centi ve compatib le for a constant c < 1 4 . The optimal inc enti ve- compatib le scheme for λ ′ results in expe cted sender u tility 3 δ by reco mmending actio n 3 whene ver the state of nature is θ 2 or θ 3 , and with probab ility δ 1 − 2 δ when the state of nature is θ 1 . Some calculation rev eals that in order to match this expected sender utility subjec t to c -incenti ve compatibility , signalin g scheme ϕ ′ = A ( λ ′ ) must s atisfy ϕ ′ ( θ 2 , σ 3 ) ≥ µ for µ = 1 − 4 c > 0 . In other wo rds, ϕ ′ must recommend actio n 3 a co nstant fractio n of the time when giv en state θ 2 as input. In contra st, sin ce c < 1 2 it is e asy to see that ϕ = A ( λ ) ca n ne ver recommend act ion 3 : for an y sign al, the posterior ex pected recei ver reward for a ction 3 is 0 , w hereas o ne of the o ther two action s must ha ve poste rior expected rece i ver re ward at least 1 2 . It f ollo ws that giv en D ∈ { λ, λ ′ } , a c all to A ( D , θ 2 ) yields a tester w hich distingu ishes between λ and λ ′ with constant probab ility µ . Since λ and λ ′ ha ve statistica l distan ce δ , we conc lude that the wo rst case sample c omple xity of A on either of λ or λ ′ is Ω( 1 δ ) . Since δ > 0 can be made arbitrar ily small, this completes the proof. 30

Algorithmic Bayesian Persuasion

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