Obviously Strategy-Proof Mechanisms

Transcription

1 MPRA Munch Personal RePEc Archve Obvously Strategy-Proof Mechansms Shengwu L Harvard Unversty 2017 Onlne at MPRA Paper No , posted 4 May :24 UTC

2 Obvously Strategy-Proof Mechansms Shengwu L Frst uploaded: 3 February Ths verson: 17 Aprl Abstract A strategy s obvously domnant f, for any devaton, at any nformaton set where both strateges frst dverge, the best outcome under the devaton s no better than the worst outcome under the domnant strategy. A mechansm s obvously strategy-proof (OSP) f t has an equlbrum n obvously domnant strateges. Ths has a behavoral nterpretaton: A strategy s obvously domnant ff a cogntvely lmted agent can recognze t as weakly domnant. It also has a classcal nterpretaton: A choce rule s OSP-mplementable ff t can be carred out by a socal planner under a partcular regme of partal commtment. 1 Introducton Domnant-strategy 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 protect agents from strategc errors. 2 They Domnant-strategy mechansms prevent waste from rent-seekng esponage, snce spyng on other players yelds no strategc advantage. I thank especally my advsors, Paul Mlgrom and Murel Nederle. I thank Nck Arnost, Roland Benabou, Douglas Bernhem, Gabrel Carroll, Paul J. Healy, Matthew Jackson, Fuhto Kojma, Roger Myerson, Mchael Ostrovsky, Matthew Rabn, Alvn Roth, Ilya Segal, and four anonymous referees for ther nvaluable advce. I thank Paul J. Healy for hs generosty n allowng my use of the Oho State Unversty Expermental Economcs Laboratory, and Luyao Zhang for her help n runnng the experments. Ths work was supported by the Kohlhagen Fellowshp Fund, through a grant to the Stanford Insttute for Economc Polcy Research. All errors reman my own. shengwu l@fas.harvard.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. 2 For nstance, school choce mechansms that lack domnant strateges may harm parents who do not strategze well (Pathak and Sönmez, 2008). 1

3 Moreover, the resultng outcome does not depend senstvely on each agent s hgher-order belefs. These benefts largely depend on agents understandng that the mechansm has an equlbrum n domnant strateges;.e. that t s strategy-proof (SP). Only then can they conclude that they need not attempt to dscover ther opponents strateges or to game the system. 3 However, some strategy-proof mechansms are smpler for real people to understand than others. For nstance, choosng when to qut n an ascendng clock aucton s the same as choosng a bd n a second-prce sealed-bd aucton (Vckrey, 1961). The two formats are strategcally equvalent; they have the same reduced normal form. 4 Nonetheless, laboratory subjects are substantally more lkely to play the 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 secondprce aucton, leadng the Englsh aucton to perform more closely to theory. (Ausubel, 2004) In ths paper, I model what t means for a mechansm to be obvously strategy-proof. Ths approach nvokes no new prmtves. Thus, t dentfes a set of mechansms as smple to understand, whle remanng as parsmonous as standard game theory. A strategy S s obvously domnant f, for any devatng strategy S, startng from any earlest nformaton set where S and S dverge, the best possble outcome from S s no better than the worst possble outcome from S. A mechansm s obvously strategyproof (OSP) f t has an equlbrum n obvously domnant strateges. By constructon, OSP depends on the extensve game form, so two games wth the same normal form may dffer on ths crteron. Obvous domnance mples weak domnance, so OSP mples SP. Ths defnton dstngushes ascendng auctons and second-prce sealed-bd auctons. Ascendng auctons are obvously strategy-proof. Suppose you value the object at $10. If the current prce s below $10, then the best possble outcome from quttng now s 3 Polcymakers could announce that a mechansm s strategy-proof, but that may not be enough. If agents do not understand the mechansm well, then they may be justfably skeptcal of such declaratons. For nstance, Google s advertsng materals for the Generalzed Second-Prce aucton appeared to mply that t was strategy-proof, when n fact t was not (Edelman et al., 2007). 4 Ths equvalence assumes that we restrct attenton to cut-off strateges n ascendng auctons. 2

4 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 strategy-proof, but not obvously strategy-proof. Consder the strateges bd $10 and bd $11. The earlest nformaton set where these dverge s the pont where you submt your bd. If you bd $11, you mght wn the object at some prce strctly below $10. If you bd $10, you 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 obvous strategy-proofness. The frst nterpretaton s behavoral: Obvously domnant strateges are those that can be recognzed as domnant by a cogntvely lmted agent. The second nterpretaton s classcal: Obvously strategy-proof mechansms are those that can be carred out by a socal planner wth only partal commtment power. Frst, I model an agent who has a smplfed mental representaton of the world: Instead of understandng every detal of every game, hs understandng s lmted by a coarse partton on the space of all games. 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 mechansm (), t s a weakly domnant strategy for 1 to play L. Both mechansms are ntutvely smlar, but t s not a weakly domnant strategy for Agent 1 to play L n mechansm (). In order for Agent 1 to recognze that t s weakly domnant to play L n mechansm (), he must use contngent reasonng. That s, he must thnk through hypothetcal scenaros: 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. Notce that the quoted nferences are vald n (), but not vald n (). 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, state-by-state, the outcomes after playng L are related to the outcomes after playng R. Then t s as though he cannot dstngush () and (). 3

5 Fgure 1: Smlar mechansms from 1 s perspectve. 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. 5 Two mechansms G and G are -ndstngushable 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 the mage (under the bjecton) of that experence. 2. An experence mght result n some outcome n G f and only f ts mage mght result n that same outcome n G. Wth ths relaton, we can partton the set of all mechansms nto equvalence classes. For nstance, the mechansms n Fgure 1 are 1-ndstngushable. Our agent knows the experences that a mechansm mght generate, and the resultng outcomes. Thus, at any of hs nformaton sets, for any contnuaton strategy, he knows all the outcomes that mght result from that strategy. However, ths does not nal down every detal of the mechansm. In partcular, our agent s unable to reason contngently about hypothetcal scenaros. At any nformaton set, f S and S prescrbe dfferent actons, our agent s unable to make nferences of the form, If playng S leads to outcome x, then playng S leads to outcome x. For nstance, n a second-prce aucton, our agent knows that for each bd, he wll ether wn the object at a prce weakly less than that bd, or not wn (and pay zero). 5 An experence s a standard concept n the theory of extensve games; experences are sometmes used to defne perfect recall. 4

6 However, he cannot make the nference, If bddng $11 would cause me to wn the object at $8, then bddng $10 would cause me to wn the object at $8. Snce the agent s unable to make these nferences about the second-prce aucton, he s unable to correctly dentfy hs domnant strategy. In general, what strateges can our agent recognze as weakly domnant, when he s constraned by ths coarse partton? These are all (and only) the obvously domnant strateges. 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 s -ndstngushable from G. 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, we assume that the Planner can commt to the functon from all bd profles to allocatons and payments, even though each agent only drectly observes hs own bd. Sometmes ths assumpton s too strong. 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, and the auctoneer may not be permtted to publcly dsclose all the bds. In these settngs, ndvdual bdders may fnd t dffcult and costly to verfy the output of the auctoneer s algorthm (Mlgrom and Segal, 2015). 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 promses to Agent must be measurable wth respect to s observatons n the game. That s, f the Planner plays a strategy not n that subset, then there exsts some agent strategy profle such that Agent detects (wth certanty) that the Planner has devated. We call ths a blateral commtment. 5

7 Suppose we requre that each agent s strategy be optmal, for any strateges of the other agents, and for any Planner strategy compatble wth the commtment made to. 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 OSP-mplementable. Consequently, 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 concept to several mechansm desgn envronments. For the frst applcaton, I consder 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 prvatevalue 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 (Spence, 1974; Mrrlees, 1971; Myerson, 1981). If f y s SP-mplementable, then the requred transfer rule f t s essentally unque (Green and Laffont, 1977; Holmström, 1979). 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 envronment? I prove the followng theorem: Every mechansm that OSP-mplements an allocaton rule s essentally a personal-clock aucton, whch s a new generalzaton of ascendng auctons. Moreover, ths s a full characterzaton of OSP mechansms: For any personalclock aucton, there exsts some allocaton rule that t OSP-mplements. These results mply that when we desre OSP-mplementaton n a bnary allocaton problem, we need not search the space of all extensve game forms. Wthout loss of 6

8 generalty, we can focus our attenton on the class of personal-clock auctons. 6 As a second 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). I conduct a laboratory experment to test the theory. In the experment, I compare three pars of mechansms. In each par, both mechansms mplement the same choce rule. One mechansm s obvously strategy-proof. The other mechansm s strategyproof, but not obvously strategy-proof. Standard theory predcts that both mechansms result n domnant strategy play, and have dentcal outcomes. Instead, subjects play the domnant strategy at sgnfcantly hgher rates under the OSP mechansm, compared to the mechansm that s just SP. Ths effect occurs for all three pars of mechansms, and perssts even after playng each mechansm fve tmes wth feedback. The rest of the paper proceeds n the usual order. Secton 2 revews the lterature. Secton 3 provdes formal defntons and characterzatons. Secton 4 covers applcatons. Secton 5 reports the laboratory experment. Secton 6 concludes. Proofs omtted from the man text are n Appendx B. 2 Related Lterature It s wdely acknowledged that ascendng auctons are smpler for real bdders than second-prce sealed-bd auctons (Ausubel, 2004). Laboratory experments have nvestgated and corroborated ths clam (Kagel et al., 1987; Kagel and Levn, 1993). More generally, laboratory subjects fnd t dffcult to reason state-by-state about hypothetcal scenaros (Charness and Levn, 2009; Esponda and Vespa, 2014; Ngangoue and Wezsacker, 2015). Ths mental process, often called contngent reasonng, has receved lttle formal treatment n economc theory. 7 There s also a strand of lterature, ncludng Vckrey s semnal paper, that observes that sealed-bd auctons rase problems of commtment (Vckrey, 1961; 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 these two strands of thought. It shows that mechansms that do not requre contngent reasonng are dentcal to mechansms that can be 6 Of course, f we do not mpose the addtonal structure of a bnary allocaton problem, then there exst OSP mechansms that are not personal-clock auctons. Ths paper contans several examples. 7 In subsequent work, Esponda and Vespa (2016) nvestgate axoms governng contngent reasonng n sngle-agent decson problems. 7

9 run under blateral commtment. For combnatoral auctons, Vckrey-Clarke-Groves mechansms can be strategcally complex and computatonally nfeasble. Consequently, there has been substantal nterest n desgnng smple mechansms that perform well, such as deferred-acceptance clock auctons (Mlgrom and Segal, 2015) and posted-prce mechansms (Bartal et al., 2003; Feldman et al., 2014; Düttng et al., 2016). These have the (prevously unmodeled) advantage of beng obvously strategy-proof. OSP s dstnct from O-solvablty, a soluton concept used n the computer scence lterature on decentralzed learnng (Fredman, 2002, 2004). Strategy S overwhelms 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. O-solvablty s too strong for our current purposes, because almost no games studed n mechansm desgn are O-solvable. 8 There s a small lterature that bulds on ths paper. In school choce settngs, there exst school prortes such that no mechansm OSP-mplements the deferred acceptance algorthm for those prortes (Ashlag and Gonczarowsk, 2015). Bade and Gonczarowsk (2016) characterze OSP mechansms for house matchng, and n socal choce envronments wth sngle-peaked preferences. Pyca and Troyan (2016) characterze OSP mechansms n general settngs wth no transfers and rch preferences, and propose a stronger soluton concept (strong obvous strategy-proofness). Zhang and Levn (2017) provde decson-theoretc foundatons for obvous domnance, and propose a weaker soluton concept (equlbrum n partton domnant strateges). L (2017) defnes obvous ex post equlbrum for settngs wth nterdependent values. 3 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 type profles, Θ N N Θ. 4. A utlty functon for each agent, u : X Θ R 8 For nstance, nether ascendng clock auctons nor second-prce sealed-bd auctons are O-solvable. 8

10 A mechansm s an extensve game form wth consequences n X. 9 Ths s an extensve game form where each termnal hstory z results n some outcome g(z) X. We restrct attenton to game forms wth perfect recall 10 and fnte depth 11. The full defnton s famlar to most readers, so we relegate t to Appendx A. G denotes the set of all such game forms, wth representatve element G. Useful notaton s compled n Table 1. Table 1: Notaton for Extensve Game Forms Name Notaton Representatve element hstores H h precedence relaton over hstores mmedate successors of h σ(h) ntal hstory h termnal hstores Z z outcome resultng from z g(z) player (agent or chance) called to play at h P (h) nformaton sets for agent I I actons avalable at I A(I ) most recent acton at h A(h) probablty measure for chance moves δ c realzaton of chance moves We wrte I I f there exst hstores h I and h I such that h h. We wrte I h f there exsts h I such that h h. h I s defned symmetrcally. We use to denote the correspondng weak order. A strategy S ( ) chooses an acton at every nformaton set for agent, S (I ) A(I ). A strategy profle S N = (S ) N specfes a strategy for each agent. A type-strategy S ( ) specfes a strategy for every type of agent, where S (θ ) denotes the strategy assgned to type θ. A type-strategy profle S N = (S ) N specfes a typestrategy for each agent. Let z G (h, S N, 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 N, δ c ). z G (h, S N, d c ) s the 9 By adoptng ths paradgm, we assume that the mechansm descrbes the entre strategc nteracton between the agents. For nstance, after the mechansm has concluded, agent 1 cannot send money to agent 2, and agent 2 cannot throw a brck through agent 1 s wndow. Such post-game moves often mply that agents do not have domnant strateges, let alone obvously domnant strateges. Savage (1954) notes that the use of modest lttle worlds, talored to partcular contexts, s often a smplfcaton, the advantage of whch s justfed by a consderable body of mathematcal experence wth related deas. 10 G has perfect recall f for any nformaton set I, for any two hstores h and h n I, ψ (h) = ψ (h ), where ψ(h) s the experence of agent at hstory h (see Defnton 8). 11 That s, for each game form, there exsts some number k such that no hstory has more than k predecessors. Ths restrcton s not essental to the man results, but smplfes the metagame n Theorem 2. d c 9

11 result of one realzaton of the chance moves under δ c. We sometmes wrte ths as z G (h, S, S, d c ). 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 form 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. Defnton 1. Gven G and 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 )] (2) 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 choose dfferent actons, that can be on the path of play gven S and gven S. Defnton 2 (Earlest Ponts of Departure). I α(s, S ) f and only f: 1. S (I ) S (I ) 2. There exst S, d c such that I z G (h, S, S, d c ). 3. There exst S, d c such that I z G (h, S, S, d c ). Ths defnton can be extended to deal wth mxed strateges 12, but pure strateges suffce for our current purposes. Defnton 3. Gven G and θ, S s obvously domnant f: S : I α(s, S ) : sup u G (h, S, S, d c, θ ) nf u G (h, S, S, d c, θ ) h I,S,d c h I,S,d c (3) In words, S s obvously domnant f, for any devatng strategy S, condtonal on reachng any earlest pont of departure, the best possble outcome under S s no better than the worst possble outcome under S. 13 Compare Defnton 1 and Defnton 3. 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 12 Three modfcatons are necessary: Frst, we change requrement 1 to be that both strateges specfy dfferent probablty measures at I. Second, we adapt requrements 2 and 3 to hold for some realzaton of the mxed strateges. Fnally, we nclude the recursve requrement, There does not exst I I such that I α(s, S ). 13 Obvous domnance s related to condtonal domnance (Shmoj and Watson, 1998); S condtonally domnates S at I f, for any S consstent wth reachng I, the payoff under S s no better than the payoff under S. 10

12 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. 14 Swtchng to a drect revelaton mechansm may not preserve obvous domnance, so the standard revelaton prncple does not apply. 15 Weak domnance treats chance moves and other players asymmetrcally. Suppose G has a Bayes-Nash equlbrum, and we replace all the agents n N \ 1 wth chance moves drawn from the Bayes-Nash equlbrum dstrbuton. Then agent 1 has a weakly domnant strategy. players symmetrcally. By contrast, obvous domnance treats chance moves and other A choce rule s a functon f : Θ N X. If we consder stochastc choce rules, then t s a functon f : Θ N X. 16 A soluton concept C( ) s a set-valued functon; for each G, t specfes a set of type-strategy profles C(G), whch may be an empty set. Defnton 4. (G, S N ) C-mplements f f: 1. S N C(G). 2. θ N Θ N : f(θ N ) = g(z G (h, (S (θ )) N, δ c )) G C-mplements f f there exsts S N that satsfes the above requrements. f s C-mplementable f there exst (G, S N ) that satsfy the above requrements. Note that our concern s wth weak mplementaton: We requre that S N C(G), not {S N } = C(G). Ths s to preserve the analogy wth canoncal results for strategyproofness, many of whch assume weak mplementaton (Myerson, 1981; Saks and Yu, 2005). Defnton 5 (Strategy-Proof). S N SP(G) f for all and for all θ, S (θ ) s weakly domnant. 14 Two extensve games have the same reduced normal form f and only f they can be made dentcal usng a small set of elementary transformatons (Thompson, 1952; Elmes and Reny, 1994). Whch of these transformatons does not preserve obvous domnance? Elmes and Reny (1994) propose three such transformatons, INT, COA, and ADD, whch preserve perfect recall. Bref nspecton reveals that obvous domnance s nvarant under INT and COA, but vares under ADD. 15 Glazer and Rubnsten (1996) argue that extensve games can be easer to domnance-solve than normal-form games, because backward nducton provdes gudance about the correct order to delete strateges. The standard revelaton prncple does not apply to ther soluton concept ether. 16 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

13 Defnton 6 (Obvously Strategy-Proof). S N OSP(G) f for all and for all θ, S (θ ) s obvously domnant. A mechansm s weakly group-strategy-proof f there does not exst a coalton that could devate and all be strctly better off ex post. Defnton 7 (Weakly Group-Strategy-Proof). S N WGSP(G) f there does not exst a coalton ˆN N, type profle θ N, devatng strateges Ŝ ˆN, non-coalton strateges S N\ ˆN and chance moves d c such that: For all ˆN: u G (h, Ŝ ˆN, S N\ ˆN, d c, θ ) > u G (h, S ˆN(θ ˆN), S N\ ˆN, d c, θ ) (4) Obvous strategy-proofness mples weak group-strategy-proofness. 17 Proposton 1. If S N OSP(G), then S N WGSP(G). Proof. Suppose S N / WGSP(G). Then there s a coalton ˆN wth types ˆθ ˆN 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 s a frst agent n the coalton to devate from S (θ ) to Ŝ. That frst devaton happens at some nformaton set I α(s (θ ), Ŝ). devaton, so S N / OSP(G). 18 Corollary 1. If S N OSP(G), then S N SP(G). Agent strctly gans from that Proposton 1 suggests a queston: Is a choce rule OSP-mplementable f and only f t s WGSP-mplementable? Proposton 5 shows that ths s not so. 3.1 Cogntve lmtatons In what sense s obvous domnance obvous? Intutvely, to see that S s weakly domnated by S, the agent must understand the entre functon u G, and check that for all opponent strategy profles S, the payoff from S s no better than the payoff from S. By contrast, to see that S s obvously domnated by S, the agent need only know the range of the functons u G (, S, ) and ug (, S, ) at any earlest pont of departure. Thus, obvous domnance can be recognzed even f the agent has a smplfed mental model of the world. We now make ths pont rgorously. 17 Barberà et al. (2016) note that many well-known ndvdually strategy-proof mechansms are also group strategy-proof, even f the latter s n prncple a much stronger condton than the former. 18 I thank Ilya Segal for suggestng ths concse proof. 12

14 Defnton 8. ψ (h) denotes s experence along hstory h; t s the sequence of nformaton sets where was called to play, and the actons that took, n order. construct ths by startng at h, and movng step-by-step through predecessors of h. At each h h, f P (h ) =, then we add the current nformaton set to the sequence. If has just played an acton at h, then we add that acton to the sequence. We use Ψ to denote the set {ψ (h) h H} ψ, where ψ s the empty sequence. 19 We defne ψ (I ) := ψ (h) h I. We defne an equvalence relaton between mechansms. We In words, G and G are -ndstngushable 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 mage s an experence n G. 2. Outcome x could follow experence ψ n G ff x could follow ψ s mage n G. Defnton 9. Take any G, G G, wth nformaton parttons I, I and experence sets Ψ, Ψ. G and G are -ndstngushable f there exsts a bjecton λ G,G from I A(I ) to I A (I ) such that: 1. ψ Ψ ff λ G,G (ψ ) Ψ 2. z Z : g(z) = x, ψ (z) = ψ ff z Z : g (z ) = x, ψ (z ) = λ G,G (ψ ) where we use λ G,G (ψ ) to denote the sequence produced by passng every element of ψ through λ G,G. For G and G that are -ndstngushable, we defne λ G,G (S ) to be the strategy that, gven nformaton set I n G, plays λ G,G (S (λ 1 G,G (I ))). The next theorem states that obvously domnant strateges are the strateges that can be recognzed as weakly domnant, by an agent who has ths smplfed mental model of the world. Theorem 1. For any, θ : S s obvously domnant n G f and only f for every G that s -ndstngushable from G, λ G,G (S ) s weakly domnant n G. The f drecton permts a constructve proof. Suppose S s not obvously domnant n G. Then there s some devatng strategy S and some earlest pont of departure I where the obvous domnance nequalty does not hold. We construct a game form 19 Mandatng the ncluson of the empty sequence has the followng consequence: By lookng at the set Ψ, t s not possble to nfer whether P (h ) =. 13

15 G that s -ndstngushable from G, such that, for some opponent strateges, f plays λ G,G (S ) then play proceeds accordng to the worst-case scenaro (consstent wth reachng I ), and f devates at λ G,G (I ), then play proceeds accordng to some best-case scenaro (consstent wth reachng I ). The key s to fnd a general constructon that always remans n the same equvalence class. The only f drecton proceeds as follows: Suppose there exsts some G n the equvalence class of G, where λ G,G (S ) s not weakly domnant. There exsts some earlest nformaton set n G where could gan by devatng. We then use λ 1 G,G to locate an nformaton set n G, and a devaton S, that do not satsfy the obvous domnance nequalty. Appendx B provdes the detals. One nterpretaton of Theorem 1 s that obvously domnant strateges are those that can be recognzed as domnant gven only a partal descrpton of the game form. Another nterpretaton of Theorem 1 s that obvously domnant strateges are robust to local msunderstandngs. Suppose an agent could mstake any G for any - ndstngushable G. He has some belef about how hs opponents are playng n G, and best responds to that (mstaken) belef. For nstance, when faced wth a second-prce aucton, he could beleve that he faces a thrd-prce aucton 20, n whch case bddng above hs value s a best response to the (symmetrc) Nash equlbrum opponent strateges (Kagel and Levn, 1993). When s hs (true) domnant strategy stll n hs set of best responses, for any such local mstake? By Theorem 1, ths holds f and only f t s obvously domnant. Many soluton concepts n behavoral game theory specfy that agents understand the game form, but best-respond to mstaken belefs about opponent strateges. 21 Snce a domnant strategy s always a best response, such theores predct no mstakes n strategy-proof mechansms. 22 the game form, and s thus orthogonal to these theores. Obvous domnance captures msunderstandngs about 3.2 Supported by blateral commtments Suppose the followng augmented game form G wth consequences n X: 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. 20 In both second-prce and thrd-prce auctons, the bdder submts a sealed bd. For any sealed bd, ether he receves the object and pays a prce less than or equal to hs bd, or does not receve the object and makes no payments. Thus, both mechansms are -ndstngushable. 21 Such concepts nclude level-k equlbrum (Stahl and Wlson, 1994, 1995; Nagel, 1995), cogntve herarchy equlbrum (Camerer et al., 2004), cursed equlbrum (Eyster and Rabn, 2005; Esponda, 2008), and analogy-based expectaton equlbrum (Jehel, 2005). 22 Wth the excepton of level-0 agents n level-k and cogntve herarchy models. 14

16 The Planner has an arbtrarly rch 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 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. For N, s type-strategy specfes what reply to gve, as a functon of hs preferences, the past sequence of queres and reples between hm 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) Analogously, a strategy S depends on ((m k, R k, r k ) t 1 k=1, m t, R t ), but not on θ. We use S θ to denote the strategy played by type θ of agent. We abbrevate ( S θ ) N S θ N N. S 0 denotes a pure strategy for the Planner; where S 0 denotes the set of all pure strateges. We restrct the Planner to send only fntely many queres. The standard full commtment paradgm s equvalent to allowng the Planner to commt to a unque S 0 S 0 (or some probablty measure over S 0 ). Instead, we assume that for each agent, the Planner can commt to a subset Ŝ 0 S 0 that s measurable wth respect to that agent s observatons n the game. Ths s formalzed as follows: Each ( S 0, S N ) results n some observaton o (o C, ox ), consstng of a communcaton sequence between the Planner and agent, o C = (m k, R k, r k ) T k=1 for T N, as well as some outcome o X X. 24 O s the set of all possble observatons (for agent ). We defne φ : S 0 S N O, where φ ( S 0, S N ) s the unque observaton resultng from ( S 0, S N ). Next we defne, for any Ŝ0 S 0 : Φ (Ŝ0) {o S 0 Ŝ0 : S N : o = φ ( S 0, S N )} (6) 23 Ths game could be made smpler wthout alterng Theorem 2; the Planner could send only a set of acceptable reples R M. Any nformaton contaned n the query could smply be wrtten nto every acceptable reply. However, dstngushng queres and reples makes the exposton more ntutve. 24 The communcaton sequence mght be empty, whch we represent usng T = 0. 15

17 For any Ô O : Φ 1 (Ô) { S 0 S N : φ ( S 0, S N ) Ô} (7) Defnton 10. Ŝ0 s -measurable f there exsts Ô such that: Ŝ 0 = Φ 1 (Ô) (8) Intutvely, the -measurable subsets of S 0 are those such that, f the Planner devates, then there exsts an agent strategy profle such that agent detects the devaton; that s, has an observaton that s not compatble wth any strategy n the promsed set. Formally, the -measurable subsets of S 0 are the σ-algebra generated by Φ (where we mpose the dscrete σ-algebra on O ). 25 Defnton 11. A mxed strategy of fnte length over Ŝ0 specfes a probablty measure over a subset S 0 Ŝ0 such that: There exsts k N such that: For all S0 S 0 and all SN : ( S 0, S N ) results n the Planner sendng k or fewer total queres. We use Ŝ0 to denote the mxed strateges of fnte length over Ŝ0. S 0 element of such a set. denotes an Defnton 12. A choce rule f s supported by blateral commtments (Ŝ 0 ) N f 1. For all N: Ŝ 0 s -measurable. 2. There exst S 0, and S N such that: (a) For all θ N : ( S 0, S θ N N ) results n f(θ N ). (b) S 0 N Ŝ 0 (c) For all N, θ, S N\, S 0 Ŝ 0 : S θ preferences θ. s a best response to ( S 0, S N\ ) gven Requrement 2.a s that the Planner s mxed strategy and the agent s pure strateges result n the (dstrbuton over) outcomes requred by the choce rule. Requrement 2.b s that the Planner s strategy s a (possbly degenerate) mxture over pure strateges 25 Notce that an observaton for player s smply a sequence of messages and responses, followed by an outcome. It does not nclude addtonal nformaton about calendar tme. A consequence of ths formulaton s that the Planner s unable to commt to the order n whch she approaches the players. 16

18 compatble wth every blateral commtment c s that each agent s assgned strategy s weakly domnant, when we consder the Planner as a player restrcted to playng mxtures over strateges n Ŝ 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 he 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 (Ŝ 0 ) N that support f. The ntuton behnd the proof s as follows: A blateral commtment Ŝ 0 s essentally equvalent to the Planner commttng to run only games n some -ndstngushable equvalence class of G. Consequently, we can fnd a set of blateral commtments that support f y f and only f we can fnd some (G, S N ) such that, for every, for every θ, for every G that s -ndstngushable from G, λ G,G (S (θ )) s weakly domnant n G. By Theorem 1, ths holds f and only f f y s OSP-mplementable. Appendx B provdes the detals. 3.3 The prunng prncple When seekng SP-mplementaton, we can wthout loss of generalty restrct attenton to the class of drect revelaton mechansms, by the revelaton prncple. The standard revelaton prncple does not hold for OSP mechansms. As the second-prce aucton llustrates, there exst OSP-mplementable choce rules that cannot be mplemented va a drect revelaton mechansm. However, there s a weaker prncple that substantally smplfes the analyss. Here we defne the prunng of a mechansm wth respect to a type-strategy profle. If, for all type profles, a hstory s never reached, then we delete that hstory (and redefne the other parts of the tuple so that what remans s well-formed). Defnton 13 (Prunng). Take any G = H,, A, A, P, δ c, (I ) N, g, and S N. P(G, S N ) H,, Ã, Ã, P, δ c, (Ĩ) N, g s the prunng of G wth respect to S N, constructed as follows: 1. H = {h H θn : d c : h z G (h, (S (θ )) N, d c )} 26 Ths requrement prevents the Planner from extractng arbtrary nformaton by makng promses and breakng them. Otherwse, the Planner could promse to mplement some constant outcome, ask every agent to report ther type, and then do whatever she pleased wth that nformaton. 17

19 2. For all, f I I then (I H) Ĩ (, Ã, Ã, P, δ c, g) are (, A, A, P, δ c, g) restrcted to H. It turns out that, f some mechansm OSP-mplements a choce rule, then the prunng of that mechansm wth respect to the equlbrum strateges OSP-mplements that same choce rule. Thus, whle we cannot restrct attenton to drect revelaton mechansms, we can restrct attenton to mnmal mechansms, such that no hstores are off the path of play. Ths s used both n ths paper and n the subsequent lterature 28 to state clean results. Proposton 2 (The Prunng Prncple). Let G P(G, S N ), and S N be S N restrcted to G. If (G, S N ) OSP-mplements f, then ( G, S N ) OSP-mplements f. 4 Applcatons 4.1 Bnary Allocaton Problems We now consder a canoncal envronment, (N, X, Θ N, (u ) N ). 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 = N Θ, where Θ = [θ, θ ], for 0 θ < θ <. For θ Θ, u (θ, y, t) = 1 y θ + t (9) For nstance, n a prvate value aucton wth unt demand, y f and only f agent receves at least one unt of the good under allocaton y. In a procurement aucton, y f and only f does not ncur costs of provson under allocaton y. θ s agent s cost of provson (equvalently, beneft of non-provson). In a bnary publc goods game, Y = {, N}. An allocaton rule s a functon 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. 27 Note that the ntal hstory h s dstnct from the empty set. That s to say, (I H) = does not ental that {h } Ĩ. 28 Ashlag and Gonczarowsk (2015); Bade and Gonczarowsk (2016); Pyca and Troyan (2016). 18

20 Defnton 14. An allocaton rule f y s C-mplementable f there exsts f t such that (f y, f t ) s C-mplementable. G C-mplements f y f there exsts f t such that G C-mplements (f y, f t ) In a bnary allocaton problem, f y s SP-mplementable f and only f f y s monotone. 29 Ths result s mplct n Spence (1974) and Mrrlees (1971), and s proved explctly n Myerson (1981). 30 Moreover, f an allocaton rule f y s SP-mplementable, then the accompanyng transfer rule f t s essentally unque. f t, (θ, θ ) = 1 fy(θ) nf{θ f y (θ, θ )} + r (θ ) (10) where r s some arbtrary functon of the other agents preferences. Ths follows easly by arguments smlar to those n Green and Laffont (1977) and Holmström (1979). In ths standard envronment, what happens when we requre OSP-mplementaton? In partcular, what restrctons does OSP-mplementaton place on the transfer rules and the extensve game form? In bnary allocaton problems, every OSP mechansm s essentally a personal-clock aucton. It s personal n two respects: Frstly, the clock prce and closng rule could vary across agents. Secondly, the clock prce could be assocated wth dfferent consequences for dfferent agents. The full statement of the defnton s novel, but we wll buld t out of famlar parts. Consder how the ascendng aucton appears to a sngle bdder: At each pont, there s a gong transfer assocated wth beng n the allocaton (wnnng the object). The agent ether plays qut or contnue; f he plays qut, then he s out of the allocaton and makes no payments. If he plays contnue, then ether he s n the allocaton and has the gong transfer, or the gong transfer falls (the gong prce rses) and he faces the same decson agan. We can generalze ths procedure a bt, whle stll ensurng that the agent has an obvously domnant strategy. The gong transfer could fall n any ncrements, and how much t falls could depend on other agents actons. The transfer assocated wth beng out of the allocaton could be non-zero, as long as t s some (known) fxed amount. Even f the agent chooses contnue, we could sometmes mandate that he quts, n whch case he s out of the allocaton (and receves the fxed transfer). The agent need not choose 29 f y s monotone f for all, for all θ, 1 fy(θ,θ ) s weakly ncreasng n θ. 30 These monotoncty results for are for weak SP-mplementaton rather than full SP-mplementaton mplementaton. Weak SP-mplementaton requres S N SP(G). Full SP-mplementaton requres S N = SP(G). 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 1. 19

21 between contnue and qut at every nformaton set; he need only be offered a choce when the gong transfer strctly falls. There could be multple actons that qut, all havng the same consequence for hm. 31 There could be multple actons that contnue, provded that the gong transfer wll not fall n future and at least one such acton guarantees that he s n the allocaton. The agent could receve arbtrary nformaton about the hstory of play. We call ths generalzed procedure In-Transfer Falls. Consder how a descendng-prce procurement aucton appears to a sngle suppler (holdng one unt of an ndvsble good): At each pont, there s a gong transfer assocated wth beng out of the allocaton. The agent ether plays qut or contnue; f he plays qut, then he s n the allocaton (keeps the good) and receves nothng. If he plays contnue, then ether he s out of the allocaton (sells the good) and receves the gong transfer, or the gong transfer falls and he faces the same decson agan. Notce that now the transfer assocated wth beng n the allocaton s fxed and the transfer assocated wth beng out of the allocaton falls monotoncally. We could generalze ths n the same ways as before, and call the resultng procedure Out-Transfer Falls. In a personal-clock aucton, startng from any pont an agent frst has a non-sngleton nformaton set, that agent ether faces In-Transfer Falls or Out-Transfer Falls. Defnton 15. G s a personal-clock aucton f, for every N, at every earlest nformaton set I such that A(I ) > 1: 1. Ether (In-Transfer Falls): There exsts a fxed transfer t R, a gong transfer t : {I I I } R, and a set of quttng actons A q such that: (a) For all z where I z:. Ether: / g y (z) and g t, (z) = t.. Or: g y (z) and g t, (z) = (b) For all a A q, for all z such that a ψ (z): / g y (z) (c) A q A(I ). (d) For all I, I {I I I }:. If I I, then t (I ) t (I ).. If I I, t (I ) > t (I I I, then Aq A(I ). nf t (I ) (11) I :I I z ), and there does not exst I such that I 31 Although they could affect what happens to other agents - t s ncentve compatble for the agent to reveal any nformaton about hs type at the pont when he quts, and the allocaton rule could n prncple depend on ths nformaton. 20

22 . If I I and t (I ) > t (I ), then A(I ) \ Aq = 1. v. If A(I ) \ Aq > 1, then there exsts a A(I ) such that: For all z such that a ψ (z): g y (z). 2. Or (Out-Transfer Falls): As above, but we substtute every nstance of g y (z) wth / g y (z) and vce versa. 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 In-Transfer Falls, and other agents could face Out-Transfer Falls (a two-sded clock aucton). 32 faces could even depend on other agents past actons. Whch procedure an agent Theorem 3. If (G, S N ) OSP-mplements f y, then P(G, S N ) s a personal-clock aucton. If G s a personal-clock aucton, then there exst S N and f y such that (G, S N ) OSPmplements f y. For any normal-form mechansm, there are typcally many equvalent extensve forms. The theory of mechansm desgn seldom provdes general crtera to choose between them. In bnary allocaton problems, obvous domnance pns down many extensve-form detals of the ascendng aucton, and provdes an answer to the queston, Why are ascendng auctons so common? 4.2 Top Tradng Cycles We now produce an mpossblty result 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 N s the set of all n by n matrces of 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. 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 } (12) such that each agent k lkes k+1 s good at least as much as any other good n R. Followng Roth (1982), we assume that the algorthm n queston has an arbtrary, fxed way of resolvng tes. 32 Loertscher and Marx (2017) nvestgate two-sded clock auctons that are pror-free, asymptotcally optmal, and obvously strategy-proof. 21

23 Defnton 16. 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 =. The above algorthm s of economc nterest, because t fnds a core allocaton n an economy wth ndvsble goods (Shapley and Scarf, 1974). Proposton 3. If f s a top tradng cycle rule, then there exsts G that SP-mplements f. (Roth, 1982) Proposton 4. If f s a top tradng cycle rule, then there exsts G that WGSPmplements f. (Brd, 1984) Proposton 5. If f s a top tradng cycle rule and n 3, then there does not exst G that OSP-mplements f. Proof. OSP-mplementablty s a heredtary property of functons. That s, f f s OSPmplementable gven doman Θ N, then the subfuncton f = f wth doman Θ N Θ N s OSP-mplementable. Thus, to prove Proposton 5, t suffces to produce a subfuncton that s not OSP-mplementable. Consder the followng subset Θ N Θ N. 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... (13) 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 (by Proposton 2) G OSP-mplements f = f for doman Θ. 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. 22

24 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 the OSP-mplementable choce rules are not dentcal to the WGSP-mplementable choce rules. 5 Laboratory Experment Are obvously strategy-proof mechansms easer for real people to understand? The followng laboratory experment provdes a straghtforward test: We compare pars of mechansms that mplement the same choce rule. One mechansm n each par s SP, but not OSP. The other mechansm s OSP. Standard game theory predcts that both mechansms wll produce the same outcome. We are nterested n whether subjects play the domnant strategy at hgher rates under OSP mechansms. 5.1 Experment Desgn The experment s an across-subjects desgn, comparng three pars of games. There are four players n each game. For the frst par, we compare the second-prce aucton (2P) and the ascendng clock aucton (AC). In both these games, subjects bd for a money prze. Subjects have nduced afflated prvate values; f a subject wns the prze, he earns an amount equal to the value of the prze, mnus hs payments from the aucton. For each subject, hs value for the prze s equal to a group draw plus a prvate adjustment. The group draw s unformly dstrbuted between $10 and $110. dstrbuted between $0 and $20. The prvate adjustment s unformly All money amounts n these games are n 25-cent ncrements. Each subject knows hs own value, but not the group draw or the prvate adjustment I use afflated prvate values for two reasons. Frst, n strategy-proof auctons wth ndependent prvate values, ncentves for truthful bddng are weak for bdders wth values near the extremes. Afflaton strengthens ncentves for these bdders. Second, Kagel et al. (1987) use afflated prvate values, and the frst part of the experment s desgned to replcate ther results. 23

25 2P s SP, but not OSP. In 2P, subjects submt ther bds smultaneously. The hghest bdder wns the prze, and makes a payment equal to the second-hghest bd. Bds are constraned to be between $0 and $ AC s OSP. In AC, the prce starts at a low value (the hghest $25 ncrement that s below the group draw), and counts upwards, up to a maxmum of $150. Each bdder can qut at any pont. 35 When only one bdder s left, that bdder wns the object at the current prce. Prevous studes comparng second-prce auctons to ascendng clock auctons have small sample szes, gven that when the same subjects play a sequence of auctons, these are planly not ndependent observatons. Kagel et al. (1987) compare 2 groups playng second-prce auctons to 2 groups playng ascendng clock auctons. Harstad (2000) compares 5 groups playng second-prce auctons to 3 groups playng ascendng clock auctons. (The comparson s not the man goal of ether experment.) Other studes fnd smlar results for second-prce auctons (Kagel and Levn, 1993) and for ascendng clock auctons (McCabe et al., 1990), but these do not drectly compare the two formats wth the same value dstrbuton and the same subject pool. When we compare 2P and AC, we can see ths as a hgh-powered replcaton of Kagel et al. (1987), snce we now observe 18 groups playng 2P and 18 groups playng AC. 36 For the second par, we compare the second-prce plus-x aucton (2P+X) and the ascendng clock plus-x aucton (AC+X). Subjects values are drawn as before. However, there s an addtonal random varable X, whch s unformly dstrbuted between $0 and $3. Subjects are not told the value of X untl after the aucton. 2P+X s SP, but not OSP. In 2P+X, subjects submt ther bds smultaneously. The hghest bdder wns the prze f and only f hs bd exceeds the second-hghest bd plus X. If the hghest bdder wns the prze, then he makes a payment equal to the second-hghest bd plus X. Otherwse, no agent wns the prze, and no payments are made. In ths game, t s a domnant strategy to submt a bd equal to your value. AC+X s OSP. In AC+X, the prce starts at a low value (the hghest $25 ncrement that s below the group draw), and counts upwards. Each bdder can qut at any pont. 37 When only one bdder s left, the prce contnues to rse for another X dollars, and then 34 In both 2P and AC, f there s a te for the hghest bd, then no bdder wns the object. 35 Durng the aucton, each bdder observes the number of actve bdders. 36 I am not aware of any prevous laboratory experment that drectly compares second-prce and ascendng clock auctons, holdng constant the value dstrbuton and subject pool, wth more than fve groups playng each format. 37 As n AC, each bdder observes the number of actve bdders. However, f the number of actve bdders s 1 or 2, then the computer dsplay nforms bdders that the number of actve bdders s 1 or 2. 24

26 freezes. If the hghest bdder keeps bddng untl the prce freezes, then she wns the prze at the fnal prce. Otherwse, no agent wns the prze and no payments are made. In ths game, t s an obvously domnant strategy to keep bddng f the prce s strctly below your value, and qut otherwse. Some subjects mght fnd 2P or AC famlar, snce such mechansms occur n some natural economc envronments. Dfferences n subject behavor mght be caused by dfferent degrees of famlarty wth the mechansm. 2P+X and AC+X are novel mechansms that subjects are unlkely to fnd famlar. 2P+X and AC+X can be seen as perturbatons of 2P and AC; the underlyng choce rule s made more complex whle preservng the SP-OSP dstncton. Thus, comparng 2P+X and AC+X ndcates whether the dstncton between SP and OSP mechansms holds for novel and more complcated aucton formats. In the thrd par of games, subjects may receve one of four common-value money przes. The four prze values are drawn, unformly at random and wthout replacement, from the set {$0.00, $0.25, $0.50, $0.75, $1.00, $1.25}. Subjects observe the values of all four przes at the start of each game. In a strategy-proof random seral dctatorshp (SP-RSD), subjects are nformed of ther prorty score, whch s drawn unformly at random from the ntegers 1 to 10. They then smultaneously submt ranked lsts of the four przes. Players are processed sequentally, from the hghest prorty score to the lowest. Tes n prorty score are broken randomly. Each player s assgned the hghest-ranked prze on hs lst, among the przes that have not yet been assgned. It s a domnant strategy to rank the przes n order of ther money value. SP-RSD s SP, but not OSP. 38 In an obvously strategy-proof random seral dctatorshp (OSP-RSD), subjects are nformed of ther prorty score. Players take turns, from the hghest prorty score to the lowest. When a player takes hs turn, he s shown the przes that have not yet been taken, and pcks one of them. It s an obvously domnant strategy to pck the avalable prze wth the hghest money value. SP-RSD and OSP-RSD dffer from the auctons n several ways. The auctons are prvate-value games of ncomplete nformaton, whereas SP-RSD and OSP-RSD are common-value games of complete nformaton. In the auctons, subjects face two sources of strategc uncertanty: They are uncertan about ther opponents valuatons, and they are uncertan about ther opponents strateges (a functon of valuatons). By contrast, n SP-RSD and OSP-RSD, subjects face no uncertanty about ther 38 SP-RSD s SP, but not OSP snce, f a player swaps the order of the hghest and second-hghest przes, he mght wn the second-hghest prze. If he reports hs true rank-order lst, he mght wn the thrd-hghest prze. 25

27 Table 2: Mechansms n each treatment 10 rounds 10 rounds 10 rounds Treatment 1 AC AC+X OSP-RSD Treatment 2 2P 2P+X SP-RSD Treatment 3 AC AC+X SP-RSD Treatment 4 2P 2P+X OSP-RSD opponents valuatons. Unlke the auctons, SP-RSD and OSP-RSD are constant-sum games, such that one player s acton cannot affect total player surplus. Any effect that perssts n both the auctons and the seral dctatorshps s dffcult to explan usng socal preferences, snce such theores typcally make dfferent predctons for constant-sum and non-constantsum games. Thus, n comparng SP-RSD and OSP-RSD, we test whether the SP-OSP dstncton has emprcal support n mechansms that are very dfferent from auctons. At the start of the experment, subjects are randomly assgned nto groups of four. These groups persst throughout the experment. Consequently, each group s play can be regarded as a sngle ndependent observaton n the statstcal analyss. Each group ether plays 10 rounds of AC, followed by 10 rounds of AC+X, or plays 10 rounds of 2P, followed by 10 rounds of 2P+X. 39 At the end of each round, subjects are shown the aucton result, ther own proft from ths round, the wnnng bdder s proft from ths round, and the bds (n order from hghest to lowest). Notce that subjects have 10 rounds of experence wth a standard aucton, before beng presented wth ts unusual +X varant. Thus, the data from +X auctons record moderately experenced bdders grapplng wth a new aucton format. Next, groups are re-randomzed nto ether 10 rounds of OSP-RSD or 10 rounds of SP-RSD. At the end of each round, subjects see whch prze they have obtaned, and whether ther prorty score was the hghest, or second-hghest, and so on. Table 2 summarzes the desgn. Subjects had prnted copes of the nstructons, and the expermenter read aloud the part pertanng to each 10-round segment just before that segment began. The nstructons (correctly) nformed subjects that ther play n earler segments would not affect the games n later segments. The nstructons dd not menton domnant strateges or provde recommendatons for how to play, so as to prevent confounds from the expermenter demand effect. Instructons for both SP and 39 If a stage game wth domnant strateges s repeated fntely many tmes, then the resultng repeated game typcally does not have a domnant strategy. The same holds for obvously domnant strateges. Consequently, n nterpretng these results as nformng us about domnant strategy play, we nvoke an mplct narrow framng assumpton. The same assumpton s made for other experments n ths lterature, such as Kagel et al. (1987) and Kagel and Levn (1993). 26

28 OSP mechansms are of smlar length and smlar readng levels 40, and can be found n the onlne appendx. In every SP mechansm, each subject had 90 seconds to make hs choce. Each subject could revse hs choce as many tmes as he desred durng the 90 seconds, and only hs fnal choce would count. For OSP mechansms, mean tme to completon was seconds n AC, seconds n AC+X, and 40.5 seconds n OSP-RSD. However, the rules of the OSP mechansms mply that not every subject was actvely choosng throughout that tme. 5.2 Admnstratve detals Subjects were pad $20 for partcpatng, n addton to ther profts or losses from every round of the experment. On average, subjects made $37.54, ncludng the partcpaton payment. Subjects who made negatve net profts receved just the $20 partcpaton payment. I conducted the experment at the Oho State Unversty Expermental Economcs Laboratory n August 2015, usng z-tree (Fschbacher, 2007). I recruted subjects from the student populaton usng an onlne system (Grener, 2015). I admnstered 16 sessons, where each sesson nvolved 1 to 3 groups. Each sesson lasted about 90 mnutes. In total, the data nclude 144 subjects n 36 groups of 4 (wth 9 groups n each treatment) Statstcal Analyss The data nclude 4 dfferent aucton formats, wth 180 auctons per format, for a total of 720 auctons. 42 One natural summary statstc for each aucton s the dfference between the secondhghest bd and the second-hghest value. Ths s, equvalently, the dfference between that aucton s closng prce, and the closng prce that would have occurred f all bdders played the domnant strategy. Fgure 2a dsplays hstograms of the second-hghest bd mnus the second-hghest value, for AC and 2P. Fgure 2b does the same for AC+X and 2P+X. If all agents are playng the domnant strategy n an aucton, then the hstogram for that aucton wll be a pont mass at zero. 40 Both sets of nstructons are approxmately at a ffth-grade readng level accordng to the Flesch- Kncad readablty test, whch s a standard measure for how dffcult a pece of text s to read (Kncad et al., 1975). 41 In two cases, network errors caused crashes whch prevented a group from contnung n the experment. I recruted new subjects to replace these groups. 42 In 2 out of 720 auctons, computer errors prevented bdders from correctly enterng ther bds. We omt these 2 observatons, but ncludng them has lttle effect on any of the results that follow. 27

29 (a) Standard auctons (b) +X auctons Fgure 2: Hstogram: 2nd-hghest bd mnus 2nd-hghest value There s a substantal dfference between the emprcal dstrbutons for OSP and SP mechansms. If we choose a random aucton from the data, how lkely s t to have a closng prce wthn $2.00 of the equlbrum prce? An aucton s 31 percentage ponts more lkely to have a closng prce wthn $2.00 of the equlbrum prce under AC (OSP) compared to 2P (SP). An aucton s 28 percentage ponts more lkely to have a closng prce wthn $2.00 of the equlbrum prce under AC+X (OSP) compared to 2P+X (SP). Closng prces under 2P+X are systematcally based upwards (p =.0031) 43. Table 3 dsplays the mean absolute dfference between the second-hghest bd and the second-hghest value, for the frst 5 rounds and the last 5 rounds of each aucton. Ths measures the magntude of errors under each mechansm. (Alternatve measures of errors are n Appendx C.) Errors are systematcally larger under SP than under OSP, and ths dfference s sgnfcant n both the standard auctons and the novel +X auctons, and n both early and late rounds. To buld ntuton for effect szes, consder that the expected proft of the wnnng bdder n 2P and AC s about $4.00 (gven domnant strategy play). Thus, the average errors under 2P are larger than the theoretcal predcton for total bdder surplus. There s some evdence of learnng n 2P; errors are smaller n the last fve rounds compared to the frst fve rounds (p =.045, pared t-test). For the other three aucton formats, there s no sgnfcant evdence of learnng. 44 To compare subject behavor under SP-RSD and OSP-RSD, we compute the pro- 43 For each group, we take the mean dfference between the second-hghest bd and the second-hghest value. Ths produces one observaton per group playng 2P+X, for a total of 18 observatons, and we use a t-test for the null that these have zero mean. 44 p =.173 for AC, p =.694 for 2P+X, and p =.290 for AC+X. 28