Obviously Strategy-Proof Mechanisms

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1 Obvously Strategy-Proof Mechansms Shengwu L Workng Paper Frst uploaded: 3 Feb Ths verson: 17 July Abstract In mechansm desgn, strategy-proofness s often sad to be desrable because t makes t easy for agents to decde what to do. However, some strategy-proof mechansms are easer to understand than others. In ths paper, I defne what t means for a mechansm to be Obvously Strategy Proof (OSP). Ths s a refnement of strategy-proofness that apples to extensve game forms. Usng a formal model of cogntve lmtatons, I show that a mechansm s OSP ff the optmalty of truth-tellng can be deduced wthout contngent reasonng. I show that a choce rule s OSP-mplementable ff t can be carred out by a socal planner under a partcular regme of partal commtment. Fnally, I characterze the set of OSP mechansms n a canoncal settng, that encompasses prvate-value auctons wth unt demand and bnary publc good problems. 2 Introducton Strategy-proof mechansms are often sad to be desrable. They reduce partcpaton costs and cogntve costs, by makng t easy for agents to decde what to do. 1 They protect unsophstcated agents from strategc errors. I thank especally my advsors, Paul Mlgrom and Murel Nederle. I thank Nck Arnost, Douglas Bernhem, Gabrel Carroll, Matthew Jackson, Fuhto Kojma, Mchael Ostrovsky, Alvn Roth, and Ilya Segal for ther nvaluable advce. All errors reman my own. shengwu@stanford.edu 1 Vckrey (1961) wrtes that, n second-prce auctons: Each bdder can confne hs efforts and attenton to an apprasal of the value the artcle would have n hs own hands, at a consderable savng n mental stran and possbly n out-of-pocket expense. 1

2 They prevent waste from rent-seekng esponage. The resultng outcome does not depend senstvely on each agent s belefs about other agents preferences or nformaton. These benefts largely depend on agents recognzng that the mechansm s strategy-proof (SP). Only then can they conclude that they need not engage n detaled reasonng about ther opponents strateges. However, some strategy-proof mechansms are smpler for real people to understand than others. For nstance, ascendng clock auctons and secondprce sealed-bd auctons are somorphc under current theores. 2 Nonetheless, laboratory subjects are substantally more lkely to play the weakly domnant strategy under a clock aucton than under sealed bds. (Kagel et al., 1987) Theorsts have also expressed ths ntuton: Some other possble advantages of dynamc auctons over statc auctons are dffcult to model explctly wthn standard economcs or game-theory frameworks. For example,... t s generally held that the Englsh aucton s smpler for real-world bdders to understand than the sealed-bd second-prce aucton, leadng the Englsh aucton to perform more closely to theory. (Ausubel, 2004) In ths paper, I model explctly what t means for a mechansm to be obvously strategy-proof. Ths s a refnement of strategy-proofness, that apples to extensve game forms. Ths approach nvokes no new prmtves. It dentfes a set of mechansms as obvously strategy-proof, whle remanng as parsmonous as the standard theory of extensve games. A mechansm s obvously strategy-proof (OSP) f t has an equlbrum n obvously domnant strateges. A strategy S s obvously domnant f, for any devatng strategy S, startng from any earlest nformaton set where S and S dsagree, the best possble outcome from S s no better than the worst possble outcome from S. Note that ths requrement apples only to earlest nformaton sets where S and S dsagree. Ths requrement therefore depends on the extensve game form. Ascendng clock auctons are OSP. Suppose Bdder 1 values the object at $10. In an ascendng clock aucton, the prce rses monotoncally. If the current prce s below $10, then the best possble outcome from quttng now 2 To be precse, f we restrct our attenton to cut-off strateges n ascendng clock auctons, then ascendng clock auctons and second-prce sealed-bd auctons have dentcal reduced normal form representatons. 2

3 Fgure 1: Smlar mechansms from 1 s perspectve. s no better than the worst possble outcome from stayng n the aucton (and quttng at $10). If the prce s above $10, then the best possble outcome from stayng n the aucton s no better than the worst possble outcome from quttng now. Second-prce sealed-bd auctons are SP, but not OSP. Consder the strateges bd $10 and bd $11. The earlest nformaton set where these dsagree s the pont where Bdder 1 submts her bd. If Bdder 1 bds $11, she mght wn the object at some prce strctly below $10. If Bdder 1 bds $10, she mght not wn the object. The best possble outcome from devatng s better than the worst possble outcome from truth-tellng. Ths captures an ntuton expressed by expermental economsts: The dea that bddng modestly n excess of x only ncreases the chance of wnnng the aucton when you don t want to wn s far from obvous from the sealed bd procedure. (Kagel et al., 1987) I produce two characterzaton theorems, whch suggest two nterpretatons of OSP. Frst, I propose a formal model of a cogntvely lmted agent. I show that a strategy S s obvously domnant f and only f such an agent can recognze S as weakly domnant. Consder the mechansms n Fgure 1. Suppose Agent 1 has preferences: A B C D. In (), t s a weakly domnant strategy for 1 to play L. All three mechansms are ntutvely smlar, but t s not a weakly domnant strategy for Agent 1 to play L n () and (). In order for Agent 1 to recognze that t s weakly domnant to play L n mechansm (), he must thnk through hypothetcal scenaros case-by-case: 3

4 If Agent 2 plays l, then I should play L, snce I prefer A to B. If Agent 2 plays r, then I should play L, snce I prefer C to D. Therefore, I should play L, no matter what agent 2 plays. Ths mental process s called contngent reasonng. Notce that the quoted nferences are vald n (), but not vald n () and (). Suppose Agent 1 s unable to engage n contngent reasonng. That s, he knows that playng L mght lead to A or C, and playng R mght lead to B or D. However, he does not understand how, case-by-case, the outcomes after playng L are related to the outcomes after playng R. Then t s as though he cannot dstngush (), (), and (). Ths dea can be made formal and general. I defne an equvalence relaton on the space of mechansms: The experence of agent at hstory h records the nformaton sets where was called to play, and the actons that took, n chronologcal order. 3 Two mechansms G and G generate the same experences for f there s a bjecton from s nformaton sets and actons n G, onto s nformaton sets and actons n G, such that: 1. G can produce for some experence f and only f G can produce for ts bjected partner experence. 2. An experence mght result n some outcome n G f and only f ts bjected partner mght result n that same outcome n G. Wth ths relaton, we can partton the set of all mechansms nto equvalence classes. For nstance, all three mechansms n Fgure 1 generate the same experences for Agent 1. Wthn a mechansm, a partton of hstores nto nformaton sets represents mperfect nformaton about past actons. Smlarly, a partton of mechansms nto equvalence classes represents mperfect understandng about the propertes of each mechansm. Any such partton s, mplctly, a model of a cogntvely lmted agent. The partton defned by the relaton G and G generate the same experences for rules out contngent reasonng. Suppose an agent s unable to dstngush games that generate the same experences. He retans substantal knowledge about the structure of the game. He knows the nformaton sets at whch he may be called to play, and the actons avalable at each nformaton set. He knows, for any experence, what outcomes may result. However, he s unable to reason case-by-case about hypothetcal scenaros. 3 An experence s a standard concept n the theory of extensve games; experences are used to defne perfect recall. 4

5 The frst characterzaton theorem states: A strategy S s obvously domnant n G f and only f t s weakly domnant n every G that generates the same experences for as G. Ths shows that obvously domnant strateges are those that can be recognzed as weakly domnant wthout contngent reasonng. An obvously domnant strategy s weakly domnant n any game that generates the same experences. In that sense, such a strategy s robustly domnant. The Prsoner s Dlemma s a specal case of game () n Fgure 1; playng defect s not obvously domnant. On the other hand, f Agent 1 s nformed of Agent 2 s acton before makng hs decson, then playng defect s obvously domnant. Shafr and Tversky (1992) fnd that laboratory subjects n a Prsoner s Dlemma are more lkely to play the weakly domnant strategy when they are nformed beforehand that ther opponent has cooperated (84%) or when they are nformed beforehand that ther opponent has defected (97%), compared to when they are not nformed of ther opponent s strategy (63%). 4 Ths volates the Sure Thng Prncple (Savage, 1954), but s predcted by the cogntve model that I have just exposted. The second characterzaton theorem for OSP relates to the problem of mechansm desgn under partal commtment. In mechansm desgn, we usually assume that the Planner can commt to every detal of a mechansm, ncludng the events that an ndvdual agent does not drectly observe. For nstance, n a sealed-bd aucton, t s assumed that the Planner can commt to the functon from all sealed bds to allocatons and payments, even though each agent only drectly observes hs own bd. In some settngs, ths s unrealstc. If agents cannot ndvdually verfy the detals of a mechansm, the Planner may be unable to commt to t. Mechansm desgn under partal commtment s a pressng problem. Auctons run by central brokers over the Internet account for bllons of dollars of economc actvty. (Edelman et al., 2007) In such settngs, bdders may be unable to verfy that the other bdders exst, let alone what actons they have taken. As another example, some wreless spectrum auctons use computatonally demandng technques to solve complex assgnment problems. 4 82% of subjects defect both when nformed that ther opponent has cooperated, and when nformed that ther opponent has defected. Note that random choosers would play the weakly domnant strategy 50% of the tme. In Shafr and Tversky s data, swtchng from con flps to human bengs wth weakly domnant strateges yelds a 13% ncrease n the rate that the domnant strategy s played. Swtchng from weakly domnant strateges to obvously domnant strateges yelds a further ncrease of 19%. Ths suggests that, n addton to socal preferences, cogntve lmtatons have a role to play n explanng the data. 5

6 In these settngs, ndvdual bdders may fnd t dffcult and costly to verfy the output of the auctoneer s algorthm. (Mlgrom and Segal, 2014) For the second characterzaton theorem, I consder a metagame where the Planner prvately communcates wth agents, and eventually decdes on an outcome. The Planner chooses one agent, and sends a prvate message, along wth a set of acceptable reples. That agent chooses a reply, whch the Planner observes. The Planner can then ether repeat ths process (possbly wth a dfferent agent) or announce an outcome and end the game. The Planner has partal commtment power: For each agent, she can commt to use only a subset of her avalable strateges. However, the subset she offers to agent must be measurable wth respect to s observatons n the game. We call ths a blateral commtment. Suppose, for robustness, that each agent could hold any belefs about the Planner s strategy, subject to the constrants of hs blateral commtment. What choce rules can be mplemented n ths metagame? The second characterzaton theorem states: A choce rule can be supported by blateral commtments f and only f that choce rule s OSPmplementable. Thus, n addton to formalzng a noton of cogntve smplcty, OSP also captures the set of choce rules that can be carred out wth only blateral commtments. After defnng and characterzng OSP, I apply ths new concept to the envronment of bnary allocaton problems. In ths envronment, there s a set of agents N wth contnuous sngle-dmensonal types θ [θ, θ ]. An allocaton y s a subset of N. An allocaton rule f y s a functon from type profles to allocatons. We augment ths wth a transfer rule f t, whch specfes money transfers for each agent. Each agent has utlty equal to hs type f he s n the allocaton, plus hs net transfer. u (θ, y, t) = 1 y θ + t (1) Bnary allocaton problems encompass several canoncal settngs. They nclude prvate-value auctons wth unt demand. They nclude procurement auctons wth unt supply; not beng n the allocaton s wnnng the contract, and the bdder s type s hs cost of provson. They also nclude bnary publc good problems; the feasble allocatons are N and the empty set. Mechansm desgn theory has extensvely nvestgated SP-mplementaton n ths envronment. f y s SP-mplementable f and only f f y s monotone n each agent s type (.e. 1 fy(θ) s weakly ncreasng n θ ). (Spence, 1974; 6

7 Mrrlees, 1971; Myerson, 1981) If f y s SP-mplementable, then the accompanyng transfer rule f t s essentally unque. (Green and Laffont, 1977; Holmström, 1979) f t, (θ, θ ) = 1 fy(θ) Y nf{θ : f y (θ, θ )} + r (θ ) (2) where r s some arbtrary determnstc functon of the other agents preferences. What are analogues of these canoncal results, f we requre OSP-mplementaton rather than SP-mplementaton? Are ascendng clock auctons specal, or are there other OSP mechansms n ths settng? I develop an analogous essental unqueness theorem for extensve game forms: Every mechansm that OSP-mplements a prvate-value allocaton rule s essentally a monotone prce mechansm, whch s a generalzaton of ascendng clock auctons. Ths theorem mples that when we desre OSP-mplementaton n ths envronment, we need not search the (very large) space of all extensve game forms. We can focus our attenton on the class of monotone prce mechansms. Next, I characterze the set of OSP-mplementable allocaton rules. For ths part, I assume that the lowest type of each agent s never n the allocaton, and s requred to have a zero transfer. 5 Gven an allocaton rule, I show how to dentfy subsets of R N that contan vable prce paths for a monotone prce mechansm. I provde a necessary and suffcent condton for an allocaton rule to be OSP-mplementable. As a second applcaton, I consder a generalzaton of the Edelman et al. (2007) onlne advertsng envronment. In ths settng, agents bd for advertsng postons, each worth a certan number of clcks. Each agent s type s a vector of per-clck values, one for each poston. I show that f preferences satsfy a sngle-crossng condton, then we can OSP-mplement the effcent allocaton and the Vckrey payments. As a thrd applcaton, I produce an mpossblty result for a classc matchng algorthm: Wth 3 or more agents, there does not exst a mechansm that OSP-mplements Top Tradng Cycles. (Shapley and Scarf, 1974) The rest of the paper proceeds n the usual order. Proofs omtted from the man text are n Appendx C. 1981) 5 Ths assumpton s borrowed from the Revenue Equvalence Theorem. (Myerson, 7

8 3 Related Lterature It s generally acknowledged that ascendng auctons are smpler for real bdders than sealed-bd auctons. (Ausubel, 2004) Laboratory experments have nvestgated and corroborated ths clam (Kagel et al., 1987; Kagel and Levn, 1993). More generally, Charness and Levn (2009) and Esponda and Vespa (2014) document that laboratory subjects fnd t dffcult to reason case-by-case about hypothetcal scenaros. Ths mental process s often called contngent reasonng, but has receved lttle formal treatment n economc theory. 6 There s also a strand of lterature, ncludng Vckrey s semnal paper, that observes that sealed-bd auctons rase problems of commtment. (Vckrey, 1961; Rothkopf et al., 1990; Cramton, 1998) For nstance, t may be dffcult to prevent shll bddng wthout thrd-party verfcaton. Rothkopf et al. (1990) argue that robustness n the face of cheatng and of fear of cheatng s mportant n determnng aucton form. Ths paper formalzes and unfes both these strands of thought. It shows that mechansms that do not requre contngent reasonng are dentcal to mechansms that can be run under blateral commtment. Ths paper also relates to the planned US aucton to repurchase televson broadcast rghts. In ths settng, complex underlyng constrants have the result that Vckrey prces cannot be computed wthout large approxmaton errors. Mlgrom and Segal (2014) propose the use of a clock aucton to repurchase broadcast rghts. They recommend ths over an equvalent sealed-bd procedure, argung that clock auctons make strategy-proofness self-evdent even for bdders who msunderstand or mstrust the auctoneer s calculatons. The Mlgrom-Segal clock aucton uses advanced computatonal technques to solve a challengng allocaton problem. However, t s obvously strategy-proof. In mult-unt combnatoral aucton problems, fndng the optmal soluton s NP-hard, so the Vckrey-Clarke-Groves mechansm may be computatonally nfeasble. Bartal et al. (2003) propose an approxmatng mechansm that runs n polynomal tme. In ths mechansm, bdders are processed sequentally. Each bdder s presented wth a prce vector and chooses hs most-preferred bundle, and hs choce s used to adjust the prce vectors 6 There s a theory lterature that studes agents that do not fully account for other agents prvate nformaton. (Eyster and Rabn, 2005; Crawford and Irberr, 2007; Esponda, 2008) Ths s a related noton, but conceptually dstnct from mstakes n contngent reasonng. In partcular, these models predct no devatons from classcal play n strategy-proof mechansms. 8

9 for the remanng bdders. One prevously unmodeled advantage of such mechansms s that they are obvously strategy-proof. OSP requres equlbrum n obvously domnant strateges. Ths s dstnct from O-solvablty, a soluton concept used n the computer scence lterature on decentralzed learnng. (Fredman, 2002, 2004) Strategy S s sad to overwhelm S f the worst possble outcome from S s strctly better than the best possble outcome from S. O-solvablty calls for the terated deleton of overwhelmed strateges. One dfference between the two concepts s that O-solvablty s for normal form games, whereas OSP nvokes a noton of an earlest pont of departure, whch s only defned n the extensve form. 7 4 Defnton and Characterzaton The planner operates n an envronment consstng of: 1. A set of agents, N {1,..., n}. 2. A set of outcomes, X. 3. A set of preference profles over X, Θ N Θ. 4. A set of game forms wth consequences n X, G. An extensve game form wth consequences n X s a tuple N, H, P, δ c, (I ) N, g, where: 1. N s a set of agents. 2. H s a set of hstores (sequences of actons, ncludng the null sequence h ). (a) Let Z be the termnal hstores. z Z ff there does not exst h H such that z s a proper subhstory of h. 8 (b) Let A(h) be the actons avalable at h. 3. P s a player functon. P : H \ Z N c 7 Ascendng clock auctons and sequental random seral dctatorshp are OSP, but not O-solvable. O-solvablty s too strong for our current purpose. 8 As usual, ths mples that nfntely long hstores are termnal hstores. 9

10 4. δ c s the chance functon. For any h where P (h) = c, δ c specfes a probablty measure on A(h) I s a partton of {h : P (h) = } wth the property that A(h) = A(h ) whenever h and h are n the same member of the partton. (a) For any I I, we denote: P (I ) = P (h) for any h I. A(I ) = A(h) for any h I. (b) If (a, a ) are such that a A(I ), a A(I j ), I I j then we regard a and a as dstnct. 6. g s an outcome functon. It assocates each termnal hstory wth an outcome. g : Z X A strategy S for agent n game G specfes what agent does at every one of her nformaton sets. S (I ) A(I ). A strategy profle S = (S ) N s a set of strateges, one for each agent. When we want to refer to the strateges used by dfferent types of, we use S θ to denote the strategy assgned to type θ. Let z G (h, S, δ c ) be the lottery over termnal hstores that results n game form G when we start from h and play proceeds accordng to (S, δ c ). z G (h, S, d c ) s the result of one realzaton of the chance moves under δ c. We sometmes wrte ths as z G (h, S, S, d c ). u (x, θ ) R s the utlty to agent from outcome x gven preferences θ. Let u G (h, S, S, d c, θ ) u (g(z G (h, S, S, d c )), θ ). Ths s the utlty to agent n game G, when we start at hstory h, play proceeds accordng to (S, S, d c ), and the resultng outcome s evaluated accordng to preferences θ. We requre that δ c has full support on the avalable moves A(h) when t s called to play. 10 Ths s wthout loss of generalty, snce f some acton s not n the support of the chance functon, then we can smply delete that acton from the game. Defnton 1. ψ (h) s the experence of agent along hstory h. ψ (h) s an alternatng sequence of nformaton sets and actons. It s constructed as follows: Let h 1,..., h K be the subhstores of h where P (h) =, n order. 9 We requre the probablty dstrbutons to be ndependent across dfferent hstores. 10 Ths ensures that obvous domnance has a useful nvarance property: Namely, that the set of obvously domnant strateges does not change when we add new chance moves that occur wth zero probablty. Appendx A defnes full support n a way that accommodates nfnte and uncountably nfnte acton sets. 10

11 1. The (2k 1)th element n ψ (h) s I : h k I. 2. If h k s a proper subhstory of h, then the (2k)th element n ψ (h) s the acton that mmedately follows h k n h. We use Ψ to denote the set {ψ (h) : h H and ψ (h) { }}. An extensve game form has perfect recall f for any nformaton set I, for any two hstores h and h n I, ψ (h) = ψ (h ). We use ψ (I ) to denote ψ (h) : h I. Defnton 2. G s the set of all extensve game forms wth consequences n X and perfect recall, where δ c has full support. A choce rule s a functon f : Θ X. If we consder stochastc choce rules, then t s a functon f : Θ X. 11 A soluton concept S s a set-valued functon wth doman G Θ. It takes values n the set of strategy profles. Defnton 3. f s S-mplementable f there exsts 1. G G 2. (S θ ) θ Θ ((S θ ) N) θ Θ such that, for all θ Θ 1. S θ S(G, θ). 2. f(θ) = g(z G (, S θ, δ c )) Note that our concern s wth weak mplementaton: We requre that S θ S(G, θ), not {S θ } = S(G, θ). Ths s to preserve the analogy wth canoncal results for strategy-proofness, many of whch assume weak mplementaton. (Myerson, 1981; Saks and Yu, 2005) We use (G, (S θ ) θ Θ ) S-mplements f to mean that (G, (S θ ) θ Θ ) fulfls the requrements of Defnton 3. We use G S-mplements f to mean that there exsts (S θ ) θ Θ such that (G, (S θ ) θ Θ ) fulfls the requrements of Defnton For readablty, we generally suppress the latter notaton, but the clams that follow hold for both determnstc and stochastc choce rules. Addtonally, the set X could tself be a set of lotteres. The nterpretaton of ths s that the planner can carry out one-tme publc lotteres at the end of the mechansm, where the randomzaton s observable and verfable. 11

12 Defnton 4 (Weakly Domnant). In G for agent wth preferences θ, S s weakly domnant f: S : S : E δc [u G (h, S, S, d c, θ )] E δc [u G (h, S, S, d c, θ )] (3) Let α(s, S ) be the set of earlest ponts of departure for S and S. That s, α(s, S ) contans the nformaton sets where S and S have made dentcal decsons at all pror nformaton sets, but are makng a dfferent decson now. We defne ths recursvely as follows: 12 Defnton 5 (Earlest Ponts of Departure). I α(s, S ) f and only f: 1. S (I ) S (I ) 2. There exsts h I, S, d c such that h s a subhstory of z G (h, S, S, d c ) There does not exst (h, h, I ) such that: (a) h s a proper subhstory of h (b) h I (c) h I (d) I α(s, S ) Defnton 6 (Obvously Domnant). In G for agent wth preferences θ, S s obvously domnant f: S : I α(s, S ) : sup u G (h, S, S, d c, θ ) nf u G h I,S,dc h I,S (h, S, S, d c, θ ) (4),dc 12 The perfect recall assumpton s necessary for ths to work. Suppose a game of mperfect recall, wth one agent, wth hstores {, L, R, RL, RR}, wth nformaton set I 1 = {, R}. Then the strateges S 1(I 1) = L and S 1(I 1) = R have no earlest pont of departure. Ths recalls Pccone and Rubnsten s paradox of the absentmnded drver. (Pccone and Rubnsten, 1997) 13 Ths s just sayng that S does not tself rule out I from the path of play. Even though ths requrement s not stated symmetrcally, for all (S, S ), α(s, S ) = α(s, S ). 12

13 Compare Defnton 4 and Defnton 6. Weak domnance s defned usng h, the hstory that begns the game. Consequently, f two extensve games have the same normal form, then they have the same weakly domnant strateges. Obvous domnance s defned wth hstores that are n nformaton sets that are earlest ponts of departure. Thus two extensve games wth the same normal form may not have the same obvously domnant strateges. Swtchng to a drect revelaton mechansm may not preserve obvous domnance, so the standard revelaton prncple does not apply. In what sense s obvous domnance obvous? Defnton 4 contans the quantfer S. To assess that S weakly domnates S, one has to keep track of as many nequaltes as there are combnatons of opponent strateges. By contrast, Defnton 6 does not contan S. To assess that S obvously domnates S requres only one nequalty at each earlest pont of departure. Thus, for agents who fnd case-by-case reasonng dffcult, obvous domnance may be easer to assess than weak domnance. Defnton 7 (Strategy-Proof). S SP(G, θ) f for all, S s weakly domnant. Defnton 8 (Obvously Strategy-Proof). S OSP(G, θ) f for all, S s obvously domnant. Consder the case where there s a set of N objects and each agent s type s a vector of per-object values. Sequental Random Seral Dctatorshp refers to the procedure where, n a random order, each agent takes one object from the set that remans. Proposton 1. Sequental Random Seral Dctatorshp s obvously strategyproof. Proof. By nspecton. Proposton 2. If G s obvously strategy-proof, then G s weakly groupstrategy-proof. Proof. Take any type profle θ. Take S OSP(G, θ). Suppose there was a coalton ˆN N that could jontly devate to strateges (Ŝ) ˆN and all be strctly better off. Fx (S ) (N\ ˆN) and d c such that all agents n the coalton are strctly better off. Along the resultng termnal hstory, there must be a frst agent n the coalton to devate from S to Ŝ. That frst devaton happens at some nformaton set I α(s, Ŝ). Snce S OSP(G, θ), agent cannot strctly gan from devatng to Ŝ; a contradcton. Corollary 1. If G s obvously strategy-proof, then G s strategy-proof. 13

14 4.1 Cogntve lmtatons We now defne an equvalence relaton between mechansms. In words, G and G generate the same experences for f there exsts a bjecton from s nformaton sets and actons n G onto s nformaton sets and actons n G, such that: 1. ψ s an experence n G ff ψ s bjected partner s an experence n G. 2. Outcome x could follow experence ψ n G ff x could follow ψ s bjected partner n G Defnton 9. Take any G, G G, wth nformaton parttons I, I and experence sets Ψ, Ψ. G and G generate the same experences for f there exsts a bjecton λ from I A(I ) onto 14 I A (I ) such that: 1. ψ Ψ ff λ(ψ ) Ψ 2. z Z : g(z) = x, ψ (z) = ψ ff z Z : g (z ) = x, ψ (z ) = λ(ψ ) where we use λ(ψ ) to denote {λ(ψ k)}t k=1, where T N. For G and G that generate the same experences, we defne λ(s ) to be the strategy that, gven nformaton set I n G, plays λ(s (λ 1 (I ))). Theorem 1. For any, θ : Consder the equvalence classes on G defned by the relaton G and G generate the same experences for. S s obvously domnant n G f and only f for every G n the equvalence class of G, λ(s ) s weakly domnant n G. The f drecton permts a constructve proof. Suppose S s not obvously domnant n G. Then we can take a sequence of experence-preservng transformatons of G, to produce some G where λ(s ) s not weakly domnant. The only f drecton proceeds as follows: Suppose there exsts some G n the equvalence class of G, where λ(s ) s not weakly domnant. Then we can use λ 1 to locate an nformaton set n G, and a devaton S, that do not satsfy the obvous domnance nequalty. Appendx C provdes the detals. 14 Ths defnton entals that λ maps I onto I and A(I ) onto A (I ). If an nformaton set n G was mapped onto an acton n G, then any experence nvolvng that nformaton set would, when passed through the bjecton, result n a sequence that was not an experence, and pso facto not an experence of G. 14

15 4.2 Supported by blateral commtments Suppose the followng game: As before we have a set of agents N, outcomes X, and preference profles N Θ. However, there s one player n addton to N: Player 0, the Planner. The Planner has an uncountably nfnte message space M. At the start of the game, each agent N prvately observes θ. Play proceeds as follows: 1. The Planner chooses one agent N and sends a query m M, along wth a set of acceptable reples R M 2. observes (m, R), and chooses a reply r R. 3. The Planner observes r. 4. The Planner ether selects an outcome x X, or chooses to send another query. (a) If the Planner selects an outcome, the game ends. (b) If the Planner chooses to send another query, go to Step 1. We allow the Planner s strategy S 0 to assgn outcomes to nfntely long sequences of queres and reples. For N, s strategy specfes what reply to gve, as a functon of her preferences, the past sequence of queres and reples between her and the Planner, and the current (m, R). That s S (θ, (m k, R k, r k ) t 1 k=1, m t, R t ) R t (5) We use S θ to denote the strategy played by type θ of agent. Let Ψ 0 be the set of all Planner strateges. The standard full commtment paradgm s equvalent to allowng the Planner to commt to a unque S 0 Ψ 0. Instead, we assume that for each agent, the Planner can commt to a subset ˆΨ 0 Ψ 0 that s measurable wth respect to that agent s observatons n the game. γ s some full sequence of messages and responses seen by player, (m k, R k, r k ) T k=1 for T N, along wth the resultng outcome. Γ denotes a communcaton event, whch s some set of such objects. Let µ(γ, S 0, S N ) be the probablty that some γ Γ occurs when the strateges are ( S 0, S N ). Defnton 10. ˆΨ 0 s a blateral commtment f there exsts Γ such that: S 0 ˆΨ 0 f and only f S N : µ(γ, S 0, S N ) = 0 (6) 15

16 A blateral commtment s (equvalently) specfed by a prohbted set of observatons Γ for player. Blateral commtments are those such that, f the planner reneges, then agent mght detect ths renegng (.e. observe some γ Γ ) wth postve probablty. Defnton 11. A choce rule f s supported by blateral commtments ( ˆΨ 0 ) N, f there exsts ( S 0, S N ) such that: 1. For all θ: ( S 0, S N ) results n f(θ). 2. For all N: S0 ˆΨ 0 3. For all N, for all θ, for all S N\, for all S 0 ˆΨ 0 : Sθ s a best response to S N\ and S 0 The frst requrement states that the strategy profle must result n the outcome prescrbed by the choce rule. The second requrement s that the Planner s strategy s compatble wth the commtment she offers to each agent. The thrd requrement s that each agent s strategy S s weakly domnant, when we consder the Planner as just another player whose strateges are confned to ˆΨ 0. Supported by blateral commtments s just one of many partal commtment regmes. Ths one requres that the commtment offered to each agent s measurable wth respect to events that she can observe. In realty, contracts are seldom enforceable unless each party can observe breaches. Thus, supported by blateral commtments s a natural case to study. Theorem 2. f s OSP-mplementable f and only f there exst blateral commtments ( ˆΨ 0 ) N that support f. The ntuton behnd the proof s as follows: A blateral commtment ˆΨ 0 s almost equvalent to the Planner commttng to run only games n some equvalence class of G, for the equvalence classes n Theorem 1. To be precse, a blateral commtment s equvalent to commttng to run mechansms n the closure of such an equvalence class, where the dstance functon s wth respect to the total varaton dstance of chance moves. In defnng G, we assumed that δ c has full support on the avalable actons when t s called to play. A blateral commtment permts the Planner to run a mechansm where some chance actons are never played. Such a mechansm 16

17 does not generate the same experences as the full-support versons, but we can get arbtrarly close to t wthout leavng the equvalence class. 15 Consequently, we can fnd a set of blateral commtments that support f y ff we can fnd some (G, (S θ ) θ Θ ) such that, for every, for every θ, for every G that generates the same experences for, λ(s θ ) s weakly domnant n G. By Theorem 1, ths holds ff f y s OSP-mplementable. Appendx C provdes the detals. 5 Applcatons 5.1 Bnary Allocaton Problems We now consder a canoncal envronment, (N, X, Θ, G). N and G reman general as before. Let Y 2 N be the set of feasble allocatons, wth representatve element y Y. An outcome conssts of an allocaton y Y and a transfer for each agent, X = Y R n. t (t ) N denotes a profle of transfers. Preferences are quaslnear. Θ = N Θ, where Θ = [θ, θ ], for 0 θ < θ <. For θ Θ u (θ, y, t) = 1 y θ + t (7) For nstance, n a prvate value aucton wth unt demand, y ff agent receves at least one unt of the good under allocaton y. In a procurement aucton, y ff does not ncur costs of provson under allocaton y. θ s agent s cost of provson (equvalently, beneft of non-provson). In a publc goods game, Y = {, N}. An allocaton rule s f y : Θ Y. A choce rule s thus a combnaton of an allocaton rule and a payment rule, f = (f y, f t ), where f t : Θ R n. Smlarly, for each game form G, we dsaggregate the outcome functon, g = (g y, g t ). In ths part, we concern ourselves only wth determnstc allocaton rules and payment rules, and thus suppress notaton nvolvng δ c and d c. Defnton 12. An allocaton rule f y s S-mplementable f there exsts f t such that (f y, f t ) s S-mplementable. Defnton 13. f y s monotone f for all, for all θ, 1 fy(θ) s weakly ncreasng n θ. 15 One temptng modfcaton of the theory s to defne equvalence classes to nclude ther closures wth respect to chance moves. Ths would result n a non-transtve equvalence relaton. 17

18 In ths envronment, f y s SP-mplementable f and only f f y s monotone. Ths result s mplct n Spence (1974) and Mrrlees (1971), and s proved explctly n Myerson (1981). 16 Moreover, f an allocaton rule f y s SP-mplementable, then the accompanyng payment rule f t s essentally unque. f t, (θ, θ ) = 1 fy(θ) Y nf{θ : f y (θ, θ )} + r (θ ) (8) where r s some arbtrary determnstc functon of the other agents preferences. Ths follows easly by arguments smlar to those n Green and Laffont (1977) and Holmström (1979). We are nterested n how these results change when we requre OSPmplementaton. In partcular: 1. What condton on f y characterzes the set of OSP-mplementable allocaton rules? 2. For OSP-mplementaton, s there an analogous essental unqueness result on the extensve game form G? Defnton 14 (Prunng). Take any G = N, H, P, (I ) N, g, and (S θ ) θ Θ. P(G, (S θ ) θ Θ ) N, H, P, (Ĩ) N, g s the prunng of (G, (S θ ) θ Θ ), constructed as follows: 1. H = {h H : θ : h s a subhstory of z G (, S θ )} 2. For all, f I I then (I H) Ĩ ( P, g) are (P, g) restrcted to doman H. We now defne a monotone prce mechansm. Informally, a monotone prce mechansm s such that, for every, 1. Ether (a) We present wth a fxed transfer assocated wth not beng n the allocaton. 16 These monotoncty results for are for weak SP-mplementaton rather than full SPmplementaton mplementaton. Weak SP-mplementaton requres S θ SP(G, θ). Full SP-mplementaton requres {S θ } = SP(G, θ). There are monotone allocaton rules for whch the latter requrement cannot be satsfed. For example, suppose two agents wth unt demand. Agent 1 receves one unt ff v 1 >.5. Agent 2 receves one unt ff v 2 > v Note that the empty hstory h s dstnct from the empty set. That s to say, (I H) = does not ental that {h } Ĩ. 18

19 2. Or (b) And a gong transfer assocated wth beng n the allocaton. (a) We present wth a fxed transfer assocated wth beng n the allocaton. (b) And a gong transfer assocated wth not beng n the allocaton. 3. The gong transfer falls monotoncally. (Equvalently, the gong prce rses.) 4. Whenever the gong transfer strctly falls, has the opton to qut, takng the fxed transfer. 5. If the gong transfer could fall n future, has a unque non-quttng acton. The ether clause contans ascendng clock auctons as a specal case. The or clause contans descendng prce procurement auctons; agents that do not wn the contract receve a fxed zero transfer. There s a postve payment assocated wth wnnng the contract (.e. not beng n the allocaton), whch starts at a hgh level and counts downwards. Defnton 15 (Monotone Prce Mechansm). A game G s a monotone prce mechansm f, for every N, at every earlest nformaton set I such that A(I ) > 1: 1. Ether: There exsts a real number t 0, a functon t 1 : {I : I ψ (I )} R, and a set of actons A 0 such that: (a) For all a A 0, for all z such that a ψ (z): / g y (z) and g t, (z) = t 0. (b) A 0 A(I ). (c) For all I, I {I : I ψ (I )}.. If I ψ (I ), then t 1 (I ) t 1 (I ).. If I s the penultmate nformaton set n ψ (I ) and t 1 (I ) > t 1 (I ), then A0 A(I ).. If I ψ (I ) and t 1 (I ) > t 1 (I ), then A(I ) \ A0 = 1. (d) For all z where I ψ (z):. Ether: / g y (z) and g t, (z) = t 0. 19

20 . Or: g y (z) and g t, (z) = t 1 (I ), where I s the last nformaton set n ψ (z). 2. Or: There exsts a real number t 1, a functon t 0 : {I : I ψ (I )} R, and a set of actons A 1 such that: (a) For all a A 1, for all z such that a ψ (z): g y (z) and g t, (z) = t 1. (b) A 1 A(I ). (c) For all I, I {I : I ψ (I )}.. If I ψ (I ), then t 1 (I ) t 1 (I ).. If I s the penultmate nformaton set n ψ (I ) and t 1 (I ) > t 1 (I ), then A1 A(I ).. If I ψ (I ) and t 0 (I ) > t 0 (I ), then A(I ) \ A1 = 1. (d) For all z where I ψ (z):. Ether: g y (z) and g t, (z) = t 1.. Or: / g y (z) and g t, (z) = t 0 (I ), where I s the last nformaton set n ψ (z). Notce what ths defnton does not requre. The gong transfer need not be equal across agents. Whether and how much one agent s gong transfer changes could depend on other agents actons. Some agents could face a procedure consstent wth the ether clause, and other agents could face a procedure consstent wth the or clause. Indeed, whch procedure an agent faces could depend on other agents actons. Theorem 3. If (G, (S θ ) θ Θ ) OSP-mplements (f y, f t ), then G P(G, (S θ ) θ Θ ) s a monotone prce mechansm, and ( G, ( S θ ) θ Θ ) OSP-mplements (f y, f t ), where ( S θ ) θ Θ s (S θ ) θ Θ restrcted to G. The next theorem characterzes the set of OSP-mplementable allocaton rules. It nvokes two addtonal assumptons. Frst, we assume that f y admts a fnte partton, whch means that we can partton the type space nto a fnte set of N -dmensonal ntervals, wth the allocaton rule constant wthn each nterval. Ths assumpton s largely techncal. It s requred because OSP s defned for dscrete-tme extensve game forms. OSP s not defned for contnuous-tme auctons, although we can approxmate some of them arbtrarly fnely Smon and Stnchcombe (1989) show that dscrete tme wth a very fne grd can be a good proxy for contnuous tme. However, n ther theory, players have perfect nformaton about past actvty n the system. Adaptng ths to our theory, where G ncludes all dscrete-tme game forms wth mperfect nformaton, s far from straghtforward. 20

21 Second, we assume that the lowest type of each agent s never n the allocaton, and has a zero transfer. Ths assumpton s borrowed from the Revenue Equvalence Theorem (Myerson, 1981). However, t s a substantve restrcton, and rules out certan cases of nterest. Defnton 16. f y admts a fnte partton f there exsts K N such that, for each, there exsts {θ k}k k=1 such that: 1. θ = θ 1 < θ2 <... < θk = θ. 2. For all θ, θ, for all θ, f there does not exst k such that θ θ k < θ, then f y (θ, θ ) = f y (θ, θ ) The use of a sngle K for all agents s wthout loss of generalty. All vector nequaltes n the followng theorem are n the product order. That s, v v ff for every ndex, v v. Smlarly, v > v ff for every ndex, v > v. Theorem 4. Assume that: 1. f y admts a fnte partton. 2. For all, for all θ, / f y (θ, θ ). There exsts G and f t such that: 1. G OSP-mplements (f y, f t ) 2. For all, for all θ, f t, (θ, θ ) = 0 f and only f 1. f y s monotone. 2. For all A N, for all θ N\A, for Θ A (θ N\A ) A closure({θ A : θ A\ θ A\ : / f y (θ, θ A\, θ N\A)}) (9) (a) Θ A (θ N\A ) s connected. (b) There exsts A such that, f θ A > sup{ Θ A (θ N\A )}, then f y (θ A, θ N\A ). The sets defned by Equaton 9 are jon-semlattces. 19 Snce ther supremum s also the supremum of a fnte set of partton coordnates, t s well defned. 19 For a proof, see Lemma 8 n Appendx C. 21

22 5.2 Onlne Advertsng Auctons We now study an onlne advertsng envronment, whch generalzes Edelman et al. (2007). There are n bdders, and n 1 advertsng postons. 20 Each poston has an assocated clck-through rate α k, where α 1 α 2... α n 1 > 0. For convenence, we defne poston n wth α n = 0. Each bdder s type s a vector, θ (θ k)n k=1. A bdder wth type θ who receves poston k and transfer t has utlty: u (k, t, θ ) = α k θ k + t (10) The margnal utlty of movng to poston k from poston k, for type θ, s m(k, k, θ ) α k θ k α k θ k (11) We make the followng assumptons on the type space Θ: A1. Fnte: Θ < (12) A2. Hgher slots are better: A3. Sngle-crossng: 21 k n 1 : θ Θ : N : m(k, k + 1, θ ) 0 (13) k n 2 : θ, θ Θ :, j N : If m(k, k + 1, θ ) > m(k, k + 1, θ j), then m(k + 1, k + 2, θ ) > m(k + 1, k + 2, θ j). (14) A1 s a techncal assumpton to accommodate extensve game forms that move n dscrete steps. A2 and A3 are substantve assumptons. Edelman et al. (2007) assume that for all k, k, θ k = θ k, whch entals A2. If α 1 > α 2 >... > α n 1 > 0, then ther assumpton also entals A3. In ths envronment, the Vckrey-Clarke-Groves (VCG) mechansm selects the effcent allocaton. Suppose we number each buyer accordng to the slot he wns. Then bdder has VCG payment: 20 It s trval to extend what follows to fewer than n 1 advertsng postons, but dong so would add notaton. 21 Ths assumpton s not dentcal to the sngle-crossng assumpton n Yenmez (2014). For nstance, ther condton permts the second nequalty n Equaton 14 to be weak. 22

23 n 1 t = m(k, k + 1, θ k+1 ) (15) k= Edelman et al. (2007) produce a generalzed Englsh aucton that ex post mplements the effcent allocaton rule n onlne advertsng auctons. The generalzed Englsh aucton has a unque perfect Bayesan equlbrum n contnuous strateges. It s not SP, and therefore s not OSP. Here we produce an alternatve ascendng aucton that OSP-mplements the effcent allocaton rule. Proposton 3. Assume A1, A2, A3. There exsts G that OSP-mplements the effcent allocaton rule and the VCG payments. Proof. We construct G. Set p n 1 := 0, A n 1 = N. For l = 1,..., n 1: 1. Start the prce at p n l. 2. Rase the prce n small ncrements. If the current prce s p n l, the next prce s: p n l := nf {m(n l, n l + 1, θ ) : m(n l, n l + 1, θ ) > p n l } (16) θ Θ, N 3. At each prce, query each agent n A n l (n an arbtrary order), gvng her the opton to qut. 4. At any prce p n l, f agent quts, allocate her slot n l+1, and charge every agent n A n l \ the prce p n l. 5. Set p n l 1 := nf {m(n l 1, n l, θ ) : m(n l, n l + 1, θ ) p n l } (17) θ Θ, N A n l 1 := A n l \ (18) It s an obvously domnant strategy for agent to qut ff the prce n round l s weakly greater than m(n l, n l + 1, θ ). 23

24 Consder any round l. Payments from prevous rounds are sunk costs. Quttng yelds slot n l + 1 at no addtonal cost, and removes the agent from future rounds. Consder devatons where the earlest pont of departure nvolves quttng. The current prce p n l s weakly less than m(n l, n l + 1, θ ). If the truth-tellng strategy has the result that quts n round l, ths outcome s at least as good for as quttng now. If the truth-tellng strategy has the result that does not qut n round l, then s charged some amount less than hs margnal value for movng up a slot, and the next startng prce s p n l 1 m(n l 1, n l, θ ), so the argument repeats. Consder devatons where the earlest pont of departure nvolves stayng n. The current prce p n l s weakly greater than m(n l, n l + 1, θ ), so ths ether has the same result as quttng now, or rases s poston at margnal cost weakly above s margnal utlty. Ths s trvally true for the current round. Consder the next round, l + 1. If the startng prce p n l 1 s strctly less m(n l 1, n l, θ ), then there exsts some θ and j such that m(n l 1, n l, θ ) > m(n l 1, n l, θ j ). And m(n l, n l + 1, θ ) p n l m(n l, n l + 1, θ j ), whch contradcts A3. Repeatng the argument suffces to prove the clam for all rounds l l. By nspecton, ths mechansm and the specfed strategy profle result n the effcent allocaton and the VCG payments. Internet transactons conducted by a central auctoneer rase commtment problems, and bdders may be legtmately concerned about shll bddng. If we consder such auctons as repeated games, reputaton can amelorate commtment problems, but the set of equlbra can be very large and prevent tractable analyss. Proposton 3 shows that, even f we do not consder such auctons as repeated games, there sometmes exst robust mechansms that rely only on blateral commtments. In the case of advertsng auctons, the speed of transactons may requre bdders to mplement ther strateges usng automata. 5.3 Top Tradng Cycles We now produce an mpossblty result for OSP-mplementaton n a classc matchng envronment. (Shapley and Scarf, 1974) There are n agents n the market, each endowed wth an ndvsble good. An agent s type s a vector θ R n. Θ s the set of all n by n matrces of 24

25 real numbers. An outcome assgns one object to each agent. If agent s assgned object k, he has utlty θ k. There are no money transfers. Followng Roth (1982), we assume that the algorthm n queston has an arbtrary, fxed way of resolvng tes. Gven preferences θ and agents R N, a top tradng cycle s a set R R whose members can be ndexed n a cyclc order: R = { 1, 2,..., r = 0 } (19) such that each agent k lkes k+1 s good more than any other good n R, resolvng tes accordng to the fxed order. Defnton 17. f s a top tradng cycle rule f, for all θ, f(θ) s equal to the output of the followng algorthm: 1. Set R 1 := N 2. For l = 1, 2,,...: (a) Choose some top tradng cycle R R l. (b) Carry out the ndcated trades. (c) Set R l+1 := R l \ R. (d) Termnate f R l+1 =. Proposton 4. If f s a top tradng cycle rule, then there exsts G that SP-mplements f. Ths result s proved n Roth (1982). Proposton 5. If n 3 and f s a top tradng cycle rule, then there does not exst G that OSP-mplements f. Proof. Saks and Yu (2005) note that SP-mplementablty s a heredtary property of functons. That s, f f s SP-mplementable gven doman Θ, then the subfuncton f = f wth doman Θ Θ s SP-mplementable. By nspecton, the same s true for OSP-mplementablty. Thus, to prove Proposton 5, t suffces to produce a subfuncton that s not OSP-mplementable. Consder the followng subset of Θ Θ. Take agents a, b, c, wth endowed goods A, B, C. a has only two possble types, θ a and θ a, such that Ether B a C a A a... or C a B a A a... (20) 25

26 We make the symmetrc assumpton for b and c. We now argue by contradcton. Take any G pruned wth respect to the truthful strategy profles, such that G OSP-mplements f = f for doman Θ. 22 Consder some hstory h at whch P (h) = a wth a non-sngleton acton set. Ths cannot come before all such hstores for b and c. Suppose not, and suppose B a C. If a chooses the acton correspondng to B a C, and faces opponent strateges correspondng to C b A and B c A, then a receves good A. If a chooses the acton correspondng to C a B, and faces opponent strateges correspondng A c B, then a receves good C. Thus, t s not an obvously domnant strategy to choose the acton correspondng to B a C. So a cannot be the frst to have a non-sngleton acton set. By symmetry, ths argument apples to b and c as well. So all of the acton sets for a, b, and c are sngletons, and G does not OSP-mplement f, a contradcton. Proposton 5 mples that no mplementaton of top tradng cycles can be deduced to be strategy-proof wthout contngent reasonng. Ths s well, snce few would clam that the man result of Roth (1982) s obvous. 6 Concluson In ths paper, we produced a compact defnton of obvously strategy-proof mechansms. Ths uses a novel noton: We defne the earlest ponts of departure of two strateges. Appendx B shows that ths noton s latent n our usual defnton of strategy-proofness. Usng a formal model of cogntve lmtatons, we proved that a strategy s obvously domnant f and only f t can be deduced to be weakly domnant wthout contngent reasonng. We proved that a choce rule s OSP-mplementable f and only f t can be supported by blateral commtments. For bnary allocaton problems, we characterzed the OSP mechansms and the OSP-mplementable allocaton rules. We produced one possblty result for a case wth mult-mnded bdders, and one mpossblty result for a classc matchng algorthm. Much remans to be done. There are many classc results for SP-mplementaton, where OSP-mplementaton s an open queston. In ths paper, we consdered a cogntve model that rules out contngent reasonng. It remans to be 22 Lemma 2 establshes that, f f s OSP-mplementable, then such a pruned G exsts. 26

27 seen whether there exsts a tractable model of agents who can use contngent reasonng, but of lmted complexty. Fnally, ths paper has characterzed one natural partal commtment metagame, but there exst others; and t would be valuable to fnd a way to thnk rgorously and generally about partal commtment. References Ausubel, L. M. (2004). An effcent ascendng-bd aucton for multple objects. Amercan Economc Revew, pages Bartal, Y., Gonen, R., and Nsan, N. (2003). Incentve compatble mult unt combnatoral auctons. In Proceedngs of the 9th conference on Theoretcal aspects of ratonalty and knowledge, pages ACM. Charness, G. and Levn, D. (2009). The orgn of the wnner s curse: a laboratory study. Amercan Economc Journal: Mcroeconomcs, pages Cramton, P. (1998). Ascendng auctons. European Economc Revew, 42(3): Crawford, V. P. and Irberr, N. (2007). Level-k auctons: Can a nonequlbrum model of strategc thnkng explan the wnner s curse and overbddng n prvate-value auctons? Econometrca, 75(6): Edelman, B., Ostrovsky, M., and Schwarz, M. (2007). Internet advertsng and the generalzed second-prce aucton: Sellng bllons of dollars worth of keywords. The Amercan Economc Revew, pages Esponda, I. (2008). Behavoral equlbrum n economes wth adverse selecton. The Amercan Economc Revew, 98(4): Esponda, I. and Vespa, E. (2014). Hypothetcal thnkng and nformaton extracton n the laboratory. Amercan Economc Journal: Mcroeconomcs, 6(4): Eyster, E. and Rabn, M. (2005). Cursed equlbrum. Econometrca, 73(5): Fredman, E. J. (2002). Strategc propertes of heterogeneous seral cost sharng. Mathematcal Socal Scences, 44(2):