aa r X i v : . [ ec on . T H ] D ec Local dominance ∗ Emiliano Catonini † and Jingyi Xue ‡ January 1, 2021
Abstract
We define a local notion of dominance that speaks to the true choice prob-lems among actions in a game tree and does not rely on global planning. Whenwe do not restrict players’ ability to do contingent reasoning, a reduced strat-egy is weakly dominant if and only if it prescribes a locally dominant action atevery decision node, therefore any dynamic decomposition of a direct mecha-nism that preserves strategy-proofness is robust to the lack of global planning.Under a form of wishful thinking, we also show that strategy-proofness is ro-bust to the lack of forward-planning. Moreover, from our local perspective, wecan identify rough forms of contingent reasoning that are particularly natural.We construct a dynamic game that implements the Top Trading Cycles alloca-tion under a minimal form of contingent reasoning, related to independence ofirrelevant alternatives.
Keywords: weak dominance, obvious dominance, strategy-proofness, toptrading cycles.
Mechanism design has recently been concerned with the simplicity of the game. Ex-perimental and empirical evidence have shown that players often fail to recognize the ∗ We thank Pierpaolo Battigalli, Shurojit Chatterji, YiChun Chen, Atsushi Kajii, Takashi Kuni-moto, Jiangtao Li, Shuige Liu, Alexander Nesterov, Sergey Stepanov, Satoru Takayashi and AndreyTkachenko for useful comments. † Higher School of Economics, ICEF, [email protected] ‡ Singapore Management University, School of Economics, [email protected]. obvious dominance , which players can verifywithout any contingent reasoning. In this paper, we look for dominance relationsamong the actions available to a player at a decision node, which players can verifywithout having a definite plan for the future, and with a variable degree of contingentreasoning. Our analysis stems from two observations. First, players choose actions, not strate-gies, and the simplicity of a mechanism could be better assessed from the perspectiveof the true decision problems. Second, rough forms of forward planning and con-tingent reasoning, driven by the local perspective, suffice for the choice of actionsin many dynamic mechanisms. Consider for instance a clock auction with privatevalues. Perhaps, a bidder will stay in the auction until the price reaches the own val-uation not because she anticipated it is the best strategy, but because, until then, sherealizes she would rather stay for one more period than leaving immediately. Supposenow that, differently from an English auction, there is no current winner, thus if allbidders leave at the same time, the object will be assigned at random. Then, a biddercould win the object despite leaving immediately, and of course could eventually losedespite staying. Nonetheless, very natural, binary contingent reasoning — what if theothers all leave, and what if they don’t — suffices to recognize the dominant choice.We will formalize this example in Section 2.
Local dominance works as follows. Fix an information set h and a partition S ofthe strategies of the opponents compatible with the information set — the partitionrepresents the “rough contingencies” that our player is able to distinguish. Action a locally dominates action b if for every continuation plan after b , there is a continuationplan after a so that, under each “contingency” S − i ∈ S , the worst payoff after a isnot smaller and sometimes larger than the best payoff after b .We formulate local dominance with respect to a given level of contingent reasoningbut not of forward planning for the following reasons. By varying only the level ofcontingent reasoning, local dominance spans (in a sense we will make precise) fromweak dominance to obvious dominance, and we are mostly interested in discover- In a related fashion, Pycia and Troyan (2019) refine obvious dominance by distinguishing at theoutset between simple histories and not in terms of players’ ability to anticipate their choices. Wedo not refine obvious dominance and do not fix players’ forecasting horizon. a that does the job needs not be “optimal” in any senseand can (or must ) change depending on the continuation plan after b under consid-eration. This flexibility makes local dominance easy to check in many circumstances,and would be lost with any restriction to continuation plans that attempts a formaldiscrimination among dynamic mechanisms based on the required degree of forwardplanning. For instance, in our examples of Section 2 it will be enough to couple thelocally dominant action with just one action for the next period, while in our applica-tion of Section 4 the continuation plan after a that does the job simply imitates theone after b under consideration. Be as it may, speaking to the true decision problemsfaced by players and allowing for a flexible use of tentative continuation plans for ver-ification, local dominance relations that are easy to check for an analyst should alsobe easier to spot for players who do not engage in optimal global/folding-back plan-ning. Therefore, in a normative perspective, an informal recommendation is to designa game tree where the availability of simple albeit possibly suboptimal continuationplans facilitates the local choices.Now consider a situation where choosing one action or the other “makes no differ-ence” for the continuation of the game. As an extreme example, suppose that beforeplaying an obviously strategy-proof mechanism, a player can redeem a gift for havingsigned up. Redeeming the gift cannot be observed by the other players and does notalter the mechanism. Thus, there is no meaningful contingent reasoning to make:the strategies of the opponents are simply irrelevant for the decision problem. Yet,obvious dominance considers players who are always concerned that types and strate-gies of the opponents may matter, but are then unable to conclude they don’t. So,according to obvious dominance, players could refrain from redeeming the gift in fearof somehow altering their prospects for the game. In light of this, we specialize localdominance by only requiring players to distinguish between the contingencies in whichthe current choice “makes a difference” and the contingencies in which it “makes no For this reason, obvious dominance can only be applied in context with (almost) private values. This formof contingent reasoning is related to the independence of irrelevant alternatives, andit is sufficient both in the examples of Section 2 and in our application of Section4. Further, we formalize another partitioning of the contingencies that endogenouslyarises from the local perspective. Suppose that a player asks herself: what will I learntomorrow regarding the opponents’ strategies given my choice of action a or action b ? And given what I will learn, will I still have doubts as to whether a was a betterchoice than b ? (Or put differently: whether I could eventually regret the choice a over b ?) If the answer is no, a locally dominates b under the partition given by theinformation sets or terminal nodes that immediately follow a . The binary form ofcontingent reasoning mentioned above for the clock auction corresponds precisely tothis endogenous partitioning.Consider now the finest and coarsest partitions of the contingencies compatiblewith an information set: the collection of all singletons, and the set of all contingenciesitself. In the first case, we obtain local weak dominance , in the second case local ob-vious dominance . To establish the connections with the corresponding strategic-formnotions, we observe first that a strategy is weakly/obviously dominated if it prescribesa locally weakly/obviously dominated action at an information set it reaches. There-fore, if there is a locally dominant action at an information set, every strategy thatreaches the information set but prescribes another action is dominated. But then,if there is a locally dominant action at every information set that can be reachedby choosing the locally dominant actions themselves, we have a dominant strategy.Conversely, a dominant strategy must prescribe a locally dominant action at everyinformation set it reaches: the dominant continuation plan always does the job. Ourextensive-form characterization of a weakly dominant strategy with local dominancemeans the following: strategy-proofness is robust to the lack of global planning. Thissheds new light on why strategy-proof dynamic mechanisms are often easier to playthan their direct counterparts: the dynamic decomposition of players’ revelation prob-lem never comes at the cost of requiring global planning. More concretely, a playerdoes not need to recognize the existence of a dominant strategy in advance: by just Given their irrelevance, ignoring these contingencies is equivalent to consider them one by oneand conclude that the current choice never matters; this is why we can represent this situation as aspecial case of local dominance. The converse is not true: a strategy can be dominated even without prescribing any locallydominated action. This reflects the lack of global planning in the local notions. Under the finest partition, we call it “wishful dominance”, because it boils downto comparing the two best outcomes after the two actions under each contingency,as if one could always tailor the continuation plan on the unknown strategies of theopponents. (Under the coarsest partition, there is no difference with local obviousdominance.) A possible interpretation of wishful dominance is the lack of understand-ing of the very role of forward planning. It should not be surprising that relaxingplayers’ strategic sophistication leads to weakening the predictions, when players arenot aware of the problem and/or do not take a cautious attitude towards it; in thisvein, obvious dominance strengthens weak dominance, but one could also considerplayers who completely ignore some contingencies, and not for a good reason as whenthey are irrelevant for the current choice. It is instead surprising, but reassuring forthe robustness of our notions, that local weak dominance and wishful dominance, andmore generally local dominance and its weaker counterpart, become equivalent whenthere is a dominant action everywhere. If one buys that real players reasons accordingto wishful dominance, this means that decomposing the direct revelation mechanisminto a sequence of simpler problems with a locally dominant choice never comes at thecost of requiring any forward planning. Thus, an experimental validation of wishfuldominance could shed additional light on the simplicity of dynamic mechanisms: notonly global planning, but even any sort of forward planning is unneeded if playersneglect the potential importance of comparing the current actions under a fixed planfor the future.Consider now a game where, if all players choose locally dominant actions when-ever they exist, they only reach decision nodes where they do exist. With local Surprisingly, the partitions at different informations sets do not need to be related in any way forthe result. We were expecting to need conditions of “perfect recall” or, conversely, of “non-increasingsophisitication”. / The paper is organized as follows. In Section 2 we revisit “guess 2 / Herding game
A shepherd dog has to recall the sheep from the top of the hill forthe night. His goal is to maximize the number of sheep that make it all the way downto the sheepfold before falling asleep. Then, by contract, the dog has to guard thesheep from 2 / When the designer has the ability to quit the game after a detectable deviation and assignsuitable payoffs, the “pruning principle” stated by Li (2017) for obvious strategy-proofness appliesalso here.
6. Then, if some sheep did not make it to the sheepfold, the dog moves up to hisprescribed guarding position.Take now the viewpoint of a sheep at the top of the hill. It is easy to realize thatreaching the dog is a good idea: his guarding position will not be above 67 even incase nobody else moves downhill. Then, the sheep can observe how many others havereached the dog. If all sheep have reached altitude 67, then it is again easy to decideto reach the dog at altitude 45, because his guarding position will not be higher thanthat. And so on.This argument can be formalized with local weak dominance. Consider the pathwhere all sheep arrive to the sheepfold. At any information set along this path,a sheep compares the action of stopping somewhere between her current positionand the dog’s position with the action of reaching the dog. After reaching the dog,the sheep will move again, and it may not be immediate to see what the optimalcontinuation plan will be — actually, if some other sheep will stop along the way,there may not even be a dominant continuation plan. However, to keep it simple, oursheep can tentatively entertain the idea of stopping at the dog’s position and sleepthere. This continuation plan guarantees a higher payoff than the action of stoppingbefore reaching the dog, no matter what the other sheep do: given that all sheephave already reached our sheep’s current altitude, the dog won’t guard them from ahigher altitude than his current barking position. Therefore, if all sheep use localweak dominance, they will reach the sheepfold.Furthermore, along the path where all sheep arrive to the sheepfold, local domi-nance can also be established by just realizing that the strategies of the other sheepsare simply irrelevant for the local decision problem: once concluded that the dog willnot sleep above a given altitude x , moving from altitude y > x to x simply reducesthe distance from the dog’s final position by y − x , no matter what the other sheepdo. Note that the sheep do not have a weakly dominant strategy. As anticipated, if atsome information set a sheep realizes that some other sheep stopped uphill, reachingthe dog’s altitude might not be optimal, because the dog might have to guard the The argument works (with weak incentives under some strategies of the other sheep) also if asheep gets positive utility only when all other sheep are farther from the dog, as in the classic “guess2 / Ascending auctions
Li (2017) considers an English auction with private valuesand two bidders who bid one at a time. He shows that leaving the auction whenthe price surpasses the own valuation is obviously dominant, whereas revealing theown valuation in the second-price, sealed-bid auction is only weakly dominant. Weconsider the following clock auction for one indivisible object. At every round p , thetwo bidders decide simultaneously whether to stay or leave. If one leaves and theother stays, the one who stays wins the object at price p . If they both stay, the priceis increased to p + 1 and the auction goes on. If they both leave, the item is sold atprice p to one of the two bidders selected at random.In this clock auction, it is not obviously dominant to leave at round p = v , where v is the own valuation. Bidder 1 compares the strategy of leaving at round v withthe strategy of leaving at round p < v . The lowest payoff our player can get withthe first strategy is 0. This occurs when bidder 2 stays in the auction until round v included. But the highest payoff bidder 1 can get by leaving at round p is positive,even if bidder 2 is still in the auction at round p : by leaving, bidder 1 can still get v − p , with probability 1 /
2. This occurs when bidder 2 leaves at round p too. Sincethe latter payoff is higher than the former, there is no obvious dominance relationbetween the two strategies.However, we argue, it is not hard for bidder 1 to understand that leaving is nota good idea when p < v , and that it is a good idea when p = v . In our view,when p < v , what makes the choice simple is not the comparison between leavingand the globally optimal strategy, which a player may not have anticipated yet, butthe following local comparison between the actions of leaving and staying. Comparedto leaving immediately, bidder 1 has clearly nothing to lose from staying once moreand leaving at the next round: if bidder 2 is leaving, she will win for sure insteadof with probability 1 / Onthe contrary, when p = v , leaving guarantees a payoff of zero, whereas staying canresult in winning the object at price p = v or higher, in case the bidder 2 stays nowand leaves next. Given these simple considerations, our bidders will leave at the ownvaluation (if reached) without having to realize that it is a dominant strategy. We consider a dynamic game with finite horizon and partial, asymmetric observationof an initial move by nature. We model it as a tree, that is, as a set of nodes endowedwith a precedence relation ≺ . At the root, nature chooses an action θ from the set Θ.Then, at each node, one player from a finite set I chooses an action until a terminalnode is reached. Let Z denote the set of terminal nodes. For each player i ∈ I , the setof the nodes where player i moves is partitioned into information sets. Let H i denotethe collection of i ’s information sets. Each H i satisfies the standard perfect recallassumptions, therefore it inherits from the game tree the partial order ≺ . At eachinformation set h ∈ H i , player i chooses an action from the set A hi . Given our focuson players who do not engage in folding-back planning, we use reduced strategies inplace of full strategies. A reduced strategy of player i (henceforth, just “strategy”)is a map s i that assigns an action a i ∈ A hi to each information set h ∈ H i that canbe reached given the actions assigned to the previous information sets. Let S i denotethe set of strategies of player i , and let S − i := Θ × ( × j = i S j ) denote the set of strategyprofiles of all players excluding i and including nature. For each s ∈ Θ × ( × i ∈ I S i ), let ζ ( s ) denote the induced terminal node. Let u i : Z → R denote the payoff function of This is a way the local perspective offered by the dynamic game and rough contingent reasoninghave a positive interaction: the local problem is so simple that a bipartition yields all the contingentreasoning that is needed. Obvious dominance, not even allowing for a bipartition, forces the playerto compare leaving under the contingency where also the other bidder leaves, which yields a positivepayoff, with staying under the contingency where the other bidder stays longer, which yields a zeropayoff. i .For each information set h ∈ H i , let S i ( h ) and S − i ( h ) denote the subsets of S i and S − i that allow to reach h . For each available action a i ∈ A hi , let S i ( h, a i ) denotethe set of strategies s i ∈ S i ( h ) with s i ( h ) = a i . For each strategy s i ∈ S i , let H i ( s i )denote the set of information sets of i that are consistent with s i . To introduce our notion of dominance, we exogenously fix a partition S h of S − i ( h )for each information h ∈ H i . We do not need to impose any discipline on ( S h ) h ∈ H i for our analysis to be meaningful, although one may want to require a property of“perfect recall” (if h ′ ≻ h , S h ′ should not be coarser than the partition of S − i ( h ′ )induced by S h ) or “no increasing sophistication” (if h ′ ≻ h , S h ′ should not be finerthan the partition of S − i ( h ′ ) induced by S h ). We now introduce local dominancebetween two pure actions. As we are mainly interested in the existence of a dominantaction, this will will suffice for our purposes; the extension to dominance by a mixedaction is straightforward. Definition 1
Fix an information set h ∈ H i and a partition S h of S − i ( h ) . Action a i ∈ A hi locally dominates action a i ∈ A hi given S h if for every s i ∈ S i ( h, a i ) , thereexists s i ∈ S i ( h, a i ) such that ∀ S − i ∈ S h , min s − i ∈ S − i u i ( ζ ( s i , s − i )) ≥ max s − i ∈ S − i u i ( ζ ( s i , s − i )) , (1) ∃ S − i ∈ S h , max s − i ∈ S − i u i ( ζ ( s i , s − i )) > min s − i ∈ S − i u i ( ζ ( s i , s − i )) . (2) We say that a i is locally dominant f it locally dominates every other a i ∈ A hi . A player who wants to establish whether action a i locally dominates action a i should reason as follows. First she fixes a possible continuation plan after a i andlooks for the best outcome she may achieve under each of the “rough contingencies” S − i that she is able to distinguish. Then, she looks for a continuation plan after a i thatguarantees never lower and sometimes higher payoff than the former continuation planunder every rough contingency. Thus, player i does not need to find one continuationplan after a i that beats all continuation plans after a i . Such a continuation plan10ay not even exist, but even when it does, as when there is an obviously dominantstrategy, our player is not assumed to have in mind such definite plan for the future.Rather, the continuation plan after a i conceived by player i can change depending onthe continuation plan after a i player i is considering. This is particularly convenientwhen the game has a recursive structure, and a continuation plan after a i that doesthe job imitates the one after a i under consideration — our TTC game of Section4 is a case in point. Moreover, the continuation plan can be clearly suboptimal butvery simple and convenient for the comparison, as in the examples of Section 2. Localdominance does not allow, instead, to change the continuation plan after a i dependingon the rough contingency under consideration. We will formalize a variation of localdominance with this additional flexibility later.It is easy to see that the finer the partition, the weaker local dominance. Remark 1
Fix two partitions S h , e S h where e S h refines S h . If a i ∈ A hi locally domi-nates a i ∈ A hi given S h , then a i ∈ A hi locally dominates a i ∈ A hi given e S h . Therefore, if we consider the finest and the coarsest partitions of S − i ( h ), we obtainan upper and a lower bound of the local dominance relations we can establish byvarying players’ ability to do contingent reasoning. We formalize these two extremesand call them “local weak dominance” and “local obvious dominance” for the relationswith the corresponding global notions we are going to establish. Definition 2
Fix an information set h ∈ H i .Action a i ∈ A hi locally weakly dominates action a i ∈ A hi if for every s i ∈ S i ( h, a i ) ,there exists s i ∈ S i ( h, a i ) such that ∀ s − i ∈ S − i ( h ) , u i ( ζ ( s i , s − i )) ≥ u i ( ζ ( s i , s − i )) , (3) ∃ s − i ∈ S − i ( h ) , u i ( ζ ( s i , s − i )) > u i ( ζ ( s i , s − i )) . (4) Action a i ∈ A hi locally obviously dominates action a i ∈ A hi if for every s i ∈ S i ( h, a i ) ,there exists s i ∈ S i ( h, a i ) such that min s − i ∈ S − i u i ( ζ ( s i , s − i )) ≥ max s − i ∈ S − i u i ( ζ ( s i , s − i )) , (5)max s − i ∈ S − i u i ( ζ ( s i , s − i )) > min s − i ∈ S − i u i ( ζ ( s i , s − i )) . (6)11 e say that a i is locally weakly/obviously dominant if it locally weakly/obviously dom-inates every other a i ∈ A hi . Remark 2
Action a i locally weakly dominates a i if and only if a i locally dominates a i given the singleton partition S h = {{ s − i } | s − i ∈ S − i ( h ) } .Action a i locally obviously dominates action a i if and only if a i locally dominates a i given the trivial partition S h = { S − i ( h ) } . We are going to discuss in detail the relation between local weak dominance and(global) weak dominance; the very same relation exists between local obvious domi-nance and obvious dominance. In static games, local weak dominance coincides withweak dominance. In dynamic games, if action a i is locally weakly dominated at h ,then every strategy s i that prescribes a i is weakly dominated by the strategy thatprescribes the dominating action a i at h , coincides with the continuation plan thatdoes the job against s i afterwards, and coincides with s i elsewhere. The converse isnot true: a strategy can be weakly dominated even if it does not prescribe any locallyweakly dominated action. This is because a player who reasons according to localweak dominance may choose an action that is locally undominated but, for instance,does not allow to reach the payoff of an outside option she did not take earlier inthe game. This reflects the lack of global planning of our player. However, if thereis a locally weakly dominant action at every information set that follows the choiceof locally weakly dominant actions, all the other actions can only be prescribed byweakly dominated strategies, and a weakly dominant strategy emerges: the one thatprescribes the locally weakly dominant actions. Conversely, a weakly dominant strat-egy always prescribes only locally weakly dominant actions. This is because localweak dominance does not necessarily impose to use the flexibility in the choice ofthe continuation plan, and the one derived from a weakly dominant strategy alwaysdoes the job. Therefore, we obtain an extensive-form characterization of a weaklydominant strategy, and of an obviously dominant strategy as well, provided that weslightly strengthen the original definition of Li (2017) for coherence with our condi-tion (6). Given two strategies s i , s i ∈ S i , let H ( s i , s i ) be the set of their points ofdeparture , that is, the information sets h ∈ H i ( s i ) ∩ H i ( s i ) such that s i ( h ) = s i ( h ). Given our focus on reduced strategies, every point of departure is an earliest point of departurein the sense of Li (2017). efinition 3 Strategy s i ∈ S i weakly dominates strategy s i if ∀ s − i ∈ S − i , u i ( ζ ( s i , s − i )) ≥ u i ( ζ ( s i , s − i )) , (7) ∃ s − i ∈ S − i , u i ( ζ ( s i , s − i )) > u i ( ζ ( s i , s − i )) . (8) We say that strategy s i obviously dominates strategy s i if ∀ h ∈ H ( s i , s i ) , min s − i ∈ S − i ( h ) u i ( ζ ( s i , s − i )) ≥ max s − i ∈ S − i ( h ) u i ( ζ ( s i , s − i )) , (9) ∃ h ∈ H ( s i , s i ) , max s − i ∈ S − i ( h ) u i ( ζ ( s i , s − i )) > min s − i ∈ S − i ( h ) u i ( ζ ( s i , s − i )) . (10) We say that a strategy is weakly/obviously dominant if it weakly/obviously dominatesevery other strategy.
Theorem 1
A strategy s i is weakly dominant if and only if s i ( h ) is locally weaklydominant at every h ∈ H i ( s i ) . A strategy is obviously dominant if and only if s i ( h ) is locally obviously dominant at every h ∈ H i ( s i ) . Proof.
Weak dominance. Only if. Fix the weakly dominant strategy s i . Fix h ∈ H i ( s i ).Fix a i ∈ A hi \ (cid:8) s i ( h ) (cid:9) and s i ∈ S i ( h, a i ). Define s ′ i as s ′ i ( h ) = s i ( h ) for each h (cid:23) h and s ′ i ( h ) = s i ( h ) for each h h . Since s i weakly dominates s ′ i , condition (7) yields ∀ s − i ∈ S − i ( h ) , u i ( ζ ( s i , s − i )) ≥ u i ( ζ ( s ′ i , s − i )) , and since ζ ( s ′ i , s − i ) = ζ ( s i , s − i ) for each s − i S − i ( h ), condition (8) yields ∃ s − i ∈ S − i ( h ) , u i ( ζ ( s i , s − i )) > u i ( ζ ( s ′ i , s − i )) . Since ∀ s − i ∈ S − i ( h ) , ζ ( s ′ i , s − i ) = ζ ( s i , s − i ) , the two conditions above hold also with s i in place of s ′ i , so we get (3) and (4).If. Let s i be the strategy that prescribes the locally weakly dominant action atevery h ∈ H ( s i ). Fix s i = s i . Thus, s i and s i have a point of departure h . Hence, by13onditions (3) and (4), there is s ′ i ∈ S i ( h, s i ( h )) such that ∀ s − i ∈ S − i ( h ) , u i ( ζ ( s ′ i , s − i )) ≥ u i ( ζ ( s i , s − i )) , (11) ∃ s − i ∈ S − i ( h ) , u i ( ζ ( s ′ i , s − i )) > u i ( ζ ( s i , s − i )) . (12)Define s ′′ i as s ′′ i ( h ) = s ′ i ( h ) if h (cid:23) h and s ′′ i ( h ) = s i ( h ) if h h . Thus, ∀ s − i ∈ S − i ( h ) , ζ ( s ′′ i , s − i ) = ζ ( s ′ i , s − i ) , (13) ∀ s − i S − i ( h ) , ζ ( s ′′ i , s − i ) = ζ ( s i , s − i ) . (14)By (13), inequalities (11) and (12) hold with s ′′ i in place of s ′ i , and together with (14),inequality (11) holds for all s − i , thus we get (7) and (8): s i is weakly dominated by s ′′ i . Since every other strategy is weakly dominated, s i is weakly dominant.Obvious dominance. Only if. Fix the obviously dominant strategy s i . Fix h ∈ H i ( s i ). Fix a i ∈ A hi \ (cid:8) s i ( h ) (cid:9) and s i ∈ S i ( h, a i ). Define s ′ i as s ′ i ( h ) = s i ( h ) for each h (cid:23) h and s ′ i ( h ) = s i ( h ) for each h h . Since s i obviously dominates s ′ i and h is theonly point of departure, conditions (9) and (10) yield ∀ s − i ∈ S − i ( h ) , u i ( ζ ( s i , s − i )) ≥ u i ( ζ ( s ′ i , s − i )) , ∃ s − i ∈ S − i ( h ) , u i ( ζ ( s i , s − i )) > u i ( ζ ( s ′ i , s − i )) . Since ∀ s − i ∈ S − i ( h ) , ζ ( s ′ i , s − i ) = ζ ( s i , s − i ) , the two conditions above hold also with s i in place of s ′ i , so we get (5) and (6).If. Let s i be the strategy that prescribes the locally obviously dominant action atevery h ∈ H ( s i ). Fix s i = s i . Thus, s i and s i have a point of departure h . Hence, by(5) and (6), there is s ′ i ∈ S i ( h, s i ( h )) such thatmin s − i ∈ S − i ( h ) u i ( ζ ( s ′ i , s − i )) ≥ max s − i ∈ S − i ( h ) u i ( ζ ( s i , s − i )) , max s − i ∈ S − i ( h ) u i ( ζ ( s ′ i , s − i )) > min s − i ∈ S − i ( h ) u i ( ζ ( s i , s − i )) . Define s ′′ i as s ′′ i ( h ) = s ′ i ( h ) if h (cid:23) h and s ′′ i ( h ) = s i ( h ) if h h . Thus, h is the only14oint of departure between s i and s ′′ i , and ∀ s − i ∈ S − i ( h ) , ζ ( s ′ i , s − i ) = ζ ( s ′′ i , s − i ) . Then, the two conditions above hold with s ′′ i in place of s ′ i and yield (9) and (10): s i is obviously dominated by s ′′ i . Since every other strategy is obviously dominated, s i is obviously dominant. (cid:4) Theorem 1 states the equivalence between our local approach and the strategic-form approach in establishing strategy-proofness and obvious strategy-proofness. Withthis, we have proven that the two forms of strategy-proofness are robust to the lackof global planning by players. By Remark 1, we also obtain that for any intermediatepartitioning of the contingencies, the existence of locally dominant actions “every-where” lies in between strategy-proofness and obvious strategy proofness. Beingendowed with a local notion of dominance, though, we do not need to require thatplayers’ problems are everywhere easy. A natural and equally effective alternative isto require the existence of a locally dominant action only at the information sets thatwill be reached if all players choose their locally dominant actions. In this way, weobtain the weaker notion of on-path strategy-proofness , which is as good as strategy-proofness for implementation purposes and reduces the commitment requirements forthe designer: see the herding example.
Now we consider a simpler notion of local dominance that does not require players tokeep the continuation plans fixed while considering different rough contingencies.
Definition 4
Fix an information set h ∈ H i and a partition S h of S − i ( h ) . Action a i ∈ A hi locally contingency-wise dominates (c-dominates) action a i ∈ A hi given S h if ∀ S − i ∈ S h , max s i ∈ S i ( h,a i ) min s − i ∈ S − i u i ( ζ ( s i , s − i )) ≥ max s i ∈ S i ( h,a i ) max s − i ∈ S − i u i ( ζ ( s i , s − i )) , (15) ∃ S − i ∈ S h , max s i ∈ S i ( h,a i ) max s − i ∈ S − i u i ( ζ ( s i , s − i )) > max s i ∈ S i ( h,a i ) min s − i ∈ S − i u i ( ζ ( s i , s − i )) . (16) We say that a i is locally c-dominant if it locally c-dominates every other a i ∈ A hi . a i change with thecontingencies, while the right-hand sides are not increased when, for local dominance,one is considering the continuation plan after a i that does best given the contingency. Remark 3
If action a i locally dominates action a i given S h , then a i locally c-dominates a i given S h . When S h is the trivial partition, local c-dominance and local dominance, i.e. localobvious dominance, almost coincide: the only difference is that conditions (15) and(16) can be satisfied with different maximizers, while conditions (5) and (6) must besatisfied for the same s i . When S h is the singleton partition, local c-dominanceis sensibly weaker than local dominance, i.e. local weak dominance, and can berewritten as follows. Definition 5
Fix an information set h ∈ H i . Action a i ∈ A hi wishfully dominatesaction a i ∈ A hi if ∀ s − i ∈ S − i ( h ) , max s i ∈ S i ( h,a i ) u i ( ζ ( s i , s − i )) ≥ max s i ∈ S i ( h,a i ) u i ( ζ ( s i , s − i )) , (17) ∃ s − i ∈ S − i ( h ) , max s i ∈ S i ( h,a i ) u i ( ζ ( s i , s − i )) > max s i ∈ S i ( h,a i ) u i ( ζ ( s i , s − i )) . (18) We say that a i is wishfully dominant if it wishfully dominates every other a i ∈ A hi . Remark 4
Action a i wishfully dominates a i if and only if a i locally w-c dominates a i given the singleton partition S h = {{ s − i } | s − i ∈ S − i ( h ) } . The term “wishful” is justified by two observations. First, when the contingenciesare considered one at a time, the cautious component of evaluating the candidatedominating action under the worst-case scenario vanishes. Second, the flexibility ofchanging continuation plan across contingencies translates into “wishful thinking”about the future: the local alternatives are compared according to the best outcomes We allow this additional flexibility for two reasons: for coherence with the idea of local c-wdominance that players do not take forward planning seriously, and for robustness of the equivalenceresult between local dominance and local c-w dominance for “strategy-proofness” to this additionaldifference. However, the definition of local c-w dominance can be easily modified to eliminate thisdifference. not to plan: in general, nosingle plan can achieve the best possible outcome no matter the contingency. Seenfrom this angle, if our players reason according to wishful dominance, the need forforward planning induced by a dynamic mechanism is never an issue.In light of the weakness of wishful dominance, and more generally of local c-dominance, the following equivalence result for implementation is surprising.
Theorem 2
For each strategy s i ∈ S i , s i ( h ) is locally dominant at every h ∈ H i ( s i ) if and only if s i ( h ) is locally c-dominant at every h ∈ H i ( s i ) . Proof.
Only if: trivial.If. Fix a strategy s i that is locally c-dominant at every h ∈ H i ( s i ).Fix h ∈ H i ( s i ) and suppose by induction that for each h ∈ H i ( s i ) with h ≻ (cid:0) h, s i ( h ) (cid:1) , for every s i ∈ S i ( h ) with s i ( h ) = s i ( h ), there is a partition S h of S − i ( h )such that ∀ e S − i ∈ S h , min s − i ∈ e S − i u i ( ζ ( s i , s − i )) ≥ max s − i ∈ e S − i u i ( ζ ( s i , s − i )) . (19)As basis step, observe that the induction hypothesis vacously holds if player i doesnot move anymore after h and s i ( h ).Fix S − i ∈ S h and s ′ i ∈ S i ( h, s i ( h )). Fix s − i ∈ S − i . If ζ ( s ′ i , s − i ) = ζ ( s i , s − i ), thereis h ≻ (cid:0) h, s i ( h ) (cid:1) such that s − i ∈ S − i ( h ) and s ′ i ( h ) = s i ( h ), and then by the inductionhypothesis u i ( ζ ( s i , s − i )) ≥ u i ( ζ ( s i , s − i )). Hence, we obtain ∀ s ′ i ∈ S i ( h, s i ( h )) , min s − i ∈ S − i u i ( ζ ( s i , s − i )) ≥ min s − i ∈ S − i u i ( ζ ( s ′ i , s − i )) , (20) ∀ s ′ i ∈ S i ( h, s i ( h )) , max s − i ∈ S − i u i ( ζ ( s i , s − i )) ≥ max s − i ∈ S − i u i ( ζ ( s ′ i , s − i )) . (21)Then, for every a i ∈ A hi \ (cid:8) s i ( h ) (cid:9) and s i ∈ S i ( h, a i ), condition (15) with equation This is not true in games with perfect information. We conjecture that in this class of games,wishful dominance, local weak dominance, and even local obvious dominance coincide. ∀ S − i ∈ S h , min s − i ∈ S − i u i ( ζ ( s i , s − i )) ≥ max s − i ∈ S − i u i ( ζ ( s i , s − i )) , (22) ∃ e S − i ∈ S h , max s − i ∈ e S − i u i ( ζ ( s i , s − i )) > min s − i ∈ e S − i u i ( ζ ( s i , s − i )) , i.e., conditions (3) and (4): s i ( h ) locally dominates a i . Note also that (22) proves theinduction step. (cid:4) In light of this, the search for a whole plan of locally dominant actions is robustto the contingency-by-contingency use of continuation plans. Hence, for strategy-proofness under given partitions, we do not have to worry about which of the twomodes of reasoning (keeping the continuation plan fixed or changing it across thepartition) better represents real players: their behavior (and our conclusions) do notchange.
Finally, we formalize two ways of endogenizing the partition of the contingencies.First we introduce a “relevant versus irrelevant” partitioning of the contingencies.In many dynamic mechanisms, a local choice may end up having no role in determiningthe own final outcome. In particular, it can be apparent that the choice between twoactions will not make any difference for a vast class of opponents’ strategies. Our TTCgame of Section 4 will provide a good example of this. Here, we formalize the idea thatplayers will simply ignore the contingencies that do not matter for the comparisonbetween two given actions. Ignoring the contingencies when it is appropriate to do so,i.e. when they do not matter, is actually equivalent to considering them one by one; forthis reason, we will obtain an intermediate notion between local dominance and localc-dominance, thus equivalent to both for strategy-proofness under the recognition ofthe irrelevant contingencies. As for the other contingencies, we directly assume thatplayers do not set them apart for the analysis — this will be enough in our TTCgame.We first formalize what we mean by irrelevance of the contingencies. We providea very general notion of irrelevance that accomodates very different situations: the18ases where the choice between a and b makes absolutely no difference, the caseswhere the only difference is a flow payoff given by the choice itself (as in the giftexample), and a variety of similar cases where a and b lead to essentially identicalsubtrees in all non-trivial scenarios — our TTC game will be a case in point. Definition 6
Fix an information set h ∈ H i , an ordered pair of actions ( a i , a i ) ∈ A hi × A hi , and a constant ε ≥ . A subset of opponents’ strategies S − i ⊂ S − i ( h ) is ε -irrelevant for ( a i , a i ) if for every s i ∈ S i ( h, a i ) , there exists s i ∈ S i ( h, a i ) such that ∀ s − i ∈ S − i , u i ( ζ ( s i , s − i )) = u i ( ζ ( s i , s − i )) + ε. We say that S − i is irrelevant for ( a i , a i ) if there exists ε ≥ such that S − i is ε -irrelevant. Our notion of irrelevance refers to an ordered pair of actions because the con-stant difference between the continuation payoffs is required to be positive. Hence,according to our definition, if S − i is ε -irrelevant for ( a i , a i ), it is not irrelevant for thepermutation ( a i , a i ) unless ε = 0. What we have in mind is the viewpoint of a playerwho is wondering whether a i dominates a i , not the other way round, therefore shecan safely focus on the complement of S − i if after a i she can get a payoff constantlyequal to or bigger than after a i .Conditional on the scenario where the opponents are not playing an irrelevantstrategies, things can be complicated, so in the following notion of dominance we donot demand our player to do further contingent reasoning. Definition 7
Fix an information set h ∈ H i . Action a i ∈ A hi locally relevant-dominates (r-dominates) action a i ∈ A hi given S − i ⊆ S − i ( h ) if S − i is irrelevantfor ( a i , a i ) , and there is s i ∈ S i ( h, a i ) such that min s − i ∈ S − i ( h ) \ S − i u i ( ζ ( s i , s − i )) ≥ max s i ∈ S i ( h,a i ) max s − i ∈ S − i ( h ) \ S − i u i ( ζ ( s i , s − i )) , (23)max s − i ∈ S − i ( h ) \ S − i u i ( ζ ( s i , s − i )) > max s i ∈ S i ( h,a i ) min s − i ∈ S − i ( h ) \ S − i u i ( ζ ( s i , s − i )) . (24) We say that a i is locally r-dominant if, for every other action a i ∈ A hi \ { a i } , there is S − i ⊆ S − i ( h ) such that a i locally r-dominates a i given S − i . a i that ensures to player i at least the best payoff she can get after the dominatedaction a i . Condition (24) makes sure that, no matter how player i would continueafter a i , the continuation plan after a i can also yield a strictly higher payoff. Whenwe say that an action is locally r-dominant, we allow the irrelevant set, thus thepartition of the contingencies, to change according to the alternative under consider-ation. So far, only for simplicity of exposition, we talked of locally dominant actionsunder a unique partition of the contigencies, used for all the comparisons with thealternatives. But it is simple and natural to extend also the definitions of a locallydominant or locally c-dominant actions to a map that associates each alternative withthe partition perceived by the player when she does the comparison.Local r-dominance boils down to local weak dominance when the whole S − i ( h ) isirrelevant and to obvious dominance for the degenerate empty irrelevant set. Oth-erwise, as anticipated, local relevant-dominance is intermediate between local domi-nance and local c-dominance given the same endogenous partition. Remark 5
Fix an information set h ∈ H i and actions a i , a i ∈ A hi . Let S − i ⊆ S − i ( h ) be irrelevant for ( a i , a i ) , and let S h be the partition given by S − i and all singletons in S − i ( h ) \ S − i . If a i locally relevant-dominates a i given S h , then a i locally w-c dominates a i given S h ; if a i locally dominates a i given S h , then a i locally relevant-dominates a i given S h . In the herding example, the whole S − i ( h ) is irrelevant for the comparison betweenthe dominant action and any other action “on-path”. In particular, when the sheepcompares moving down from the current position x to the dog’s position y withstopping at some altitude z between x and y , the entire set of strategies of the othersheep where they have all reached x is ( z − y )-irrelevant. The continuation plan thatpins down the z − y payoff difference is the one where the sheep, after reaching y ,stops there for the night: given that at x it is clear that the dog will not guard fromhigher than y , the final distance from the dog is reduced by z − y no matter what theother sheep do after x .Now we introduce the partitioning of the contingencies that derives from lookingat the information sets and terminal nodes that immediately follow the candidatedominating action. We choose the dominating action, as opposed to the dominated20ction, or the join or the meet of the partitions induced by the two actions, for thefollowing reasons. First, it gives a partition player i can keep fixed across all thecomparisons between a candidate dominating action and the alternatives. Second, itseems natural to do so as a player will be especially interested in what she will learnand what she could regret given the choice she is going to make.To introduce this mode of contingent reasoning, we need to introduce informationsets of player i that partition the set of terminal nodes. So, let H i denote the unionof H i with these new information sets, and assume that perfect recall is preserved.Morever, given an information set h ∈ H i and an action a i ∈ A hi , let H ( h, a i ) denotethe set of information sets e h ∈ H i that immediately follow h and a i . Definition 8
Fix an information set h ∈ H i . Action a i ∈ A hi locally one–step domi-nates (1-dominates) action a i ∈ A hi if ∀ e h ∈ H ( h, a i ) , max s i ∈ S i ( h,a i ) min s − i ∈ S − i ( e h ) u i ( ζ ( s i , s − i )) ≥ max s i ∈ S i ( h,a i ) max s − i ∈ S − i ( e h ) u i ( ζ ( s i , s − i )) , (25) ∃ e h ∈ H ( h, a i ) , max s i ∈ S i ( h,a i ) max s − i ∈ S − i ( e h ) u i ( ζ ( s i , s − i )) > max s i ∈ S i ( h,a i ) min s − i ∈ S − i ( e h ) u i ( ζ ( s i , s − i )) . (26) We say that a i is locally r-dominant if it locally r-dominates every other a i ∈ A hi . If all S − i ( e h ) are disjoint, local 1-dominance is a special case of local c-dominance,otherwise it could even be stronger than local dominance, but we are not aware ofany application where information sets correspond to non-nested sets of opponentsstrategies.In our clock auction of Section 2, the choice between leaving and staying when p < v can be naturally analyzed under the partition induced by the terminal nodeand the information set that immediately follow staying. If the terminal node isreached, it means that all opponents of player i left at p . If the information set p + 1is reached, it means that at least one opponent stayed at p . In the first scenario, bystaying player i wins the object at price p < v , whereas by leaving she only wins it atprice p with some probability. So this scenario satisfies (26). In the second scenario,by staying at p and leaving at p + 1 player i gets a non-negative payoff, whereas byleaving at p she gets 0. So also this scenario satisfies (25).It would be interesting to formalize a notion of dominance that captures bothendogenous modes of partitioning we introduced, in the following way. Our player21ould first look at the next information sets, as in local 1-dominance. But then,for some of these information sets, he could realize that the choice at the currentinformation set will be irrelevant, as in local r-dominance. In other words, player i recognizes that, regardless of choosing a i and a i , for a set of opponents’ strategies shewill end up in the same “state”. Under a proper formalization of the notion of state,this local notion of dominance would accurately represent the perspective of a playerin our TTC game of the next section: if after choosing item a over item b player i observes that both a and b are assigned to someone else, the choice between a and b turns out to be irrelevant, as it remains unobserved and does not modify what player i can do and achieve next. The top trading cycles (TTC) algorithm offers an efficient solution to the problemof trading N items among N agents with ownership of one of the items and a strictpreference relation over all items. The algorithm is strategy-proof, in the sense that itis weakly dominant for players to submit their true ranking and let the algorithm dothe assignment. However, it is not easy for a player to recognize this. For instance, aplayer might be tempted to rank an item b above an item a she prefers but believeshaving slim chances to obtain, because she fears she might miss her chance to getitem b while the algorithm insists trying to assign to her item a unsuccessfully. Toavoid these problems, we design a dynamic game, as opposed to an algorithm, withthree simplicity features. First, at each stage players are only asked to name theirfavourite item among the still available ones, and cannot be assigned anything elsethan that. Second, this choice remains secret to the other players, thus our playerneeds not worry that it may (negatively) affect their future choices. Third, playerscan rest reassured that whenever an opportunity for trade pops up, it remains intactthrough time and can be exploited later.For the implementation of the TTC allocation (and other allocation rules), Bo and This also has the advantage that players do not have to figure out their entire ranking (at once),but just recognize their top item from a subset at a time. We do not formalize this dimension ofsimplicity because we consider it orthogonal to the one under investigation in this paper. Bonkoun-gou and Nesterov (2020) investigate the incentive for truthtelling of players when they only havepartial information of their preferences. menu mechanisms for a more general class of problems, and show that private menu mechanisms achieve strategy-proofness. Our game works as follows. For each player, there is a “you name it, you get it”repository with the items she can immediately get. At the beginning of the game,in the repository of each player there is only the own item. Then, players name anitem. Each player’s repository is filled with the items of the players who, directly orindirectly, pointed to our player’s item — indirectly means that they named the itemof a player who named our player’s item, and so on. The players who named an itemin their repository leave the game with that item. The players who named an itemthat was assigned to someone else are asked to name a new available item. The otherplayers wait. At every stage, all players observe the set of still available items, butdo not observe the content of their repositories. Note that, as long as a player isin the game, her repository can only increase: all the players who are directly orindirectly pointing to her can only wait for her to complete a cycle.Now fix an information set h ∈ H i . Given that player i observes the evolution ofavailable items, player i always knows which round the game is at, so an informationset will only collect nodes from the same round (thus, we have a multistage gamewith imperfect observation of past actions). For future reference, let t be the round Mackenzie and Zhou’s mechanism is not designed specifically for TTC, therefore it is naturallymore complicated than ours. While our players will only have to name their favourite availableoutcome, in the menu mechanism players have to pick from menus of allocations, and their finalassignment might not be the last one they picked. It is interesting to note, however, that the privatemenu mechanism achieves strategy proofness in a very similar way to our game, in that the evolutionof the menus of allocations proposed to a player only conveys information about the players wholeft the game, which is precisely all our players can observe. The unobservability of the repositories may raise transparency concerns. Note however that theclassical TTC game requires even more faith in the algorithm: players have to reveal immediately all their private information and typically do not observe how the algorithm uses their preferencesand the preferences of others. It is probably a good idea to communicate this explicitly to players, so a player does not needto realize that every opponent (along a sequence) that points to her own item cannot move as longas she is in the game. h .First we first show that naming the favourite available item is a locally weaklydominant action. We are going to use the characterization of local weak dominancewith wishful dominance. Let a be player i ’s favourite item and let b be any otheravailable item. As long as player i is in the game, her choices have no influence onthe set of available items, thus on what the other players observe, the informationsets they reach, hence the items they name given each s − i ∈ S − i ( h ). Suppose firstthat both a and b are already in player i ’s repository. Then, player i is strictly betteroff by naming a and condition (8) is satisfied. Suppose now that s − i is such thatitem a is or will enter player i ’s repository (unless player i names b and leaves thegame earlier). In this case, by naming a , player i will surely get a and condition (7)is satisfied. Suppose finally that s − i is such that a will be assigned to someone else.Then, whatever item s − i allows player i to get after naming b , she can also get it afternaming a — the repository just keeps growing. Therefore, condition (7) is satisfied.Now we show that naming the favourite available item is also locally r-dominant.The 0-irrelevant set is the same for every comparison of the favourite item a with analternative item b : let S − i be the set of all s − i ∈ S − i ( h ) such that player i will not get a after naming a . Over the complement of S − i , condition (23) is obviously satisfied(player i gets her favourite item), and condition (24) is satisfied as well because bynaming b player i could actually get b . To show that S − i is 0-irrelevant, we formalizethe idea that player i ’s prospects for the game are exactly the same as if she hadnamed b . In the following, keep in mind that, as long as player i is in the game:(i) s − i uniquely determines the evolution of the repository;(ii) s − i uniquely determines the evolution of the available items;(iii) s i and the evolution of the available items uniquely determine the sequenceof information sets reached by player i .Now consider any strategy s i ∈ S i ( h, b ). Take any s i ∈ S i ( h, a ) such that, for each h ∈ H ( s i ) with h ≺ h , s i ( h ) = s i ( e h ), where e h is the latest information set reachedwith s i by round t given the history of available items observed at h . In a nutshell, s i is the strategy that deviates from s i by giving a try at item a at h , and then “catchesup” with s i in case the attempt is unsuccesful.To see that s i is well-defined, proceed as follows. First, e h is uniquely defined byobservation (iii). Second, s i ( e h ) is available at h . To see this, fix s − i ∈ S − i ( h ). If under24 − i and s i player i is still in the game at time t , s i ( e h ) is available at h by observation(ii); otherwise, it means that under s − i and s i player i leaves the game at time e t < t with item s i ( e h ), by observation (i) item s i ( e h ) enters player i ’s repository at round e t also under s i , and cannot be assigned to anybody else.We now show that s i and s i verify the definition of irrelevance of S − i . Fix any s − i ∈ S − i . We show that with s i player i gets the same item she gets with s i . Let h be the last information set along the path induced by s i and s − i . Let t ′ be the roundof h . Thus, with s i player i gets s i ( h ), and by construction we have s i ( h ) = s i ( e h ) forsome e h ∈ H i ( s i ) where the history of available items is the same as, or a prefix ofthe one at h . By observations (i) and (ii), after reaching e h , item s i ( h ) cannot becomeunavailable and must enter player i ’s repository by round t ′ also under s i , thus player i will get s i ( e h ). There only remains to show that player i does reach e h with s i under s − i . This is true because by observation (ii) s − i determines the evolution of availableitems, and given this evolution e h is constructed as one of the information sets reachedwith s i .In the direct mechanism, only very similar reported rankings can be ordered withlocal r-dominance. The true ranking could yield the least preferred item under somereports of the opponents, while another ranking yields the favourite one under otherreports. Moreover, given each profile of opponents’ report, player i could get the sameitem under the two rankings, or a better ranking under the true ranking (no thirdpossibility by strategy-proofness), therefore there is no large-enough set of contingen-cies that satisfies irrelevance. For two rankings that differ only by a swap betweentwo items, instead, the set of irrelevant contingencies includes all those where player i does not get any of the two, and those where player i does not get one of the two eveneven when ranked higher than the other. On the complementary set of contingencies,player i gets the item of the two that is ranked higher, so the ranking that puts firstthe favourite of the two is locally r-dominates the other. Note that, starting fromany ranking, one can reach the true ranking with a sequence of swaps, thus witha sequence of local r-dominance relations. However, the true ranking is not locallyr-dominant, in that it does not locally r-dominates the substantially different ones.This highlights that local r-dominance is not transitive, because the irrelevant sets(must) change with each comparison. 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