PPICK-AN-OBJECT MECHANISMS
IN ´ACIO B ´O AND RUSTAMDJAN HAKIMOV
Abstract.
We introduce a new family of mechanisms for one-sided matching markets,denoted pick-an-object (PAO) mechanisms. When implementing an allocation rule viaPAO, agents are asked to pick an object from individualized menus. These choices may berejected later on, and these agents are presented with new menus. When the procedureends, agents are assigned the last object they picked. We characterize the allocationrules that can be sequentialized by PAO mechanisms, as well as the ones that can beimplemented in a robust truthful equilibrium. We justify the use of PAO as opposed todirect mechanisms by showing that its equilibrium behavior is closely related to the onein obviously strategy-proof (OSP) mechanisms, but includes commonly used rules, suchas Gale-Shapley DA and Top Trading Cycles, which are not OSP-implementable. We runlaboratory experiments comparing truthful behavior when using PAO, OSP, and directmechanisms to implement different rules. These indicate that individuals are more likelyto behave in line with the theoretical prediction under PAO and OSP implementationsthan their direct counterparts.
JEL classification : C78, C73, D78, D82.
Keywords : Market Design, Matching, Sequential Mechanisms, Experiments, obvious strategy-proofness. 1.
Introduction
The literature of market design, and its applications, have grown and evolved greatlyover recent years. Even if we restrict our attention to the design of centralized matchingmarkets, the instances in which theoretical and empirical contributions have influencedthe way resources are allocated seem to be continuously expanding. Examples includethe design of school choice procedures (Abdulkadiro˘glu and S¨onmez, 2003), centralizedcollege admissions (Balinski and S¨onmez, 1999), matching of resident doctors to hospitals(Roth and Peranson, 1999), organs to patients (Roth et al. , 2004), refugees to localities(Jones and Teytelboym, 2017), and more. By carefully choosing how to determine these
Date : September, 2020.
B´o : University of York, Department of Economics and Related Studies, York, United Kingdom; website: ; e-mail: [email protected].
Hakimov : Department of Economics, University of Lausanne, Switzerland and WZB Social CenterBerlin, Germany. e-mail: [email protected] thank Samson Alva, Lars Ehlers, Dorothea K¨ubler, Andrew Mackenzie, Alexander Nesterov, MarekPycia, Madhav Raghavan, and Andrew Schotter for helpful comments. We thank Jenny Rontganger forcopy editing and Nina Bonge for support in running the experiments. Hakimov acknowledges financialsupport from the Swiss National Science Foundation (project a r X i v : . [ ec on . T H ] D ec ICK-AN-OBJECT MECHANISMS 2 allocations as a function of information such as participants’ preferences, priority struc-tures, and fairness concerns, these procedures can lead to allocations that satisfy certaindesirable properties.One of the crucial challenges faced by the designer of these mechanisms is the fact thatthe determination of allocation satisfying the desired characteristics depends on informa-tion that is known by the participants but not by the designer. These are, often, theirpreferences over outcomes, but may also include other relevant information, such as theirsocioeconomic status. Usually when it comes to real-life applications, this issue is solvedby the combination of two tools: dominant strategy implementation and the revelationprinciple. These guarantee that if the designer wants the strategic simplicity provided bydominant strategy implementation, it suffices to consider revelation mechanisms, in whichparticipants are simply asked to report their private information, and they can safely betruthful when doing so. In fact, the vast majority of the literature focuses solely on directrevelation, and often strategy-proof, mechanisms.However, recent experimental and empirical evidence has raised concerns about theability of market participants to understand the incentive properties of these mechanisms.Many participants use dominated strategies, thus distorting the market allocations. Thisphenomenon was documented in many laboratory experiments and in empirical papers. These present a new challenge: is there an alternative to the implementation of allocationrules via the standard strategy-proof direct mechanisms that would result in behaviorthat is more often in line with the theoretical predictions?One recent and celebrated attempt to formalize an alternative to strategy-proofnessthat accounts for the extent to which participants can easily understand the incentivesinduced by mechanisms was the concept of obvious strategy-proofness (OSP). “A strategyis obviously dominant if, for any deviation, at any information set where both strategiesfirst diverge, the best outcome under the deviation is no better than the worst outcomeunder the dominant strategy, and a mechanism is obviously strategy-proof (OSP) if ithas an equilibrium in obviously dominant strategies.” (Li, 2017). OSP is therefore arefinement of the notion of strategy-proofness, in that obvious dominance implies weakdominance. This concept could help explain why, for example, laboratory experimentsindicate that individuals are more likely to bid truthfully under a clock auction than undera sealed-bid second-price auction (Kagel et al. , 1987). While both implement the samerule in truthful dominant strategies, the former is also obviously dominant as opposed tothe latter. See Aygun and B´o (2019). See Hakimov and K¨ubler (2019) for an extensive survey of the experimental matching literature. See our related literature section. The author also presents the results of laboratory experiments comparing behavior and outcomes understrategy-proof and OSP implementations of the random serial dictatorship rule, obtaining similar results.Note, however, that Breitmoser and Schweighofer-Kodritsch (2019) raise questions as to whether thedifference can be attributed to obvious strategy-proofness.
ICK-AN-OBJECT MECHANISMS 3
Pycia and Troyan (2019) characterize all OSP mechanisms in the domain of objectallocation. They show that every OSP-implementable mechanism could be presented asa “millipede game.” In millipede games, at every decision node each player has a choicebetween leaving with an object among those in a given menu (“clinching action”), and atmost one “passing” action. At every decision node, the obviously dominant strategy inthese games consists of choosing the passing action unless the current decision node con-tains the most-preferred object among all objects presented in a menu in any continuationhistory. When that is the case, the strategy would require clinching that object. Thisresult could help explain the behavioral attractiveness of OSP mechanisms: participantsdo not have to engage in counterfactual reasoning regarding the possible actions fromother participants, but simply have to consider the set of allocations that are still feasibleat the decision nodes. Note, however, for some OSP strategies participants need to haveperfect foresight to correctly predict feasible actions, which can be demanding. In orderto address this concern, the authors also define an even stronger concept, strong OSP,which does not require foresight from participants. In strong OSP, decision nodes do notinclude passing actions. That is, participants are called only once, and they choose theirallocation from the menu given.One important shortcoming of OSP, especially for practical purposes, is that it is a veryrestrictive concept. Rules that are commonly considered for object allocation problems,such as top trading cycles, and stable rules, cannot be implemented via OSP mechanisms(Li, 2017; Ashlagi and Gonczarowski, 2018). This, therefore, leaves a large set of problemswithout this kind of behavioral guidance.On the other hand, recent laboratory experiments show that the dynamic implementa-tion of the deferred-acceptance rule (DA), in which the equilibrium behavior also consistsof choosing the most-preferred object from menus, leads to higher rates of truthful behav-ior than its standard direct revelation counterpart (B´o and Hakimov, 2020; Klijn et al. ,2019). The results are especially surprising given that, while the standard direct DA isstrategy-proof, truthful behavior is not a dominant strategy in its dynamic counterparts.If one considers that the main driver behind the behavior more in line with the theory inOSP mechanisms is the fact that strategies are obviously dominant, then the forces behindthe experimental results in Li (2017), B´o and Hakimov (2020), and Klijn et al. (2019)would be unrelated, since in the latter the equilibrium strategy is not even dominant.In this paper we conjecture that the main driver behind the observed behavior more inline with the theoretical predictions is the simple mechanics of the equilibrium strategy,in which agents “pick” the object they would like to have from a menu, as opposedto submitting a ranking of objects representing their preferences. This would providea unified and alternative explanation to these experimental results. Based on this, weintroduce a class of sequential revelation mechanisms which implement various object Bade and Gonczarowski (2016) also provide characterizations of OSP mechanisms, both for generaldomains and for Pareto efficient object allocation problems.
ICK-AN-OBJECT MECHANISMS 4 allocation rules via an equilibrium behavior with closely related mechanics. We denotethem pick-an-object mechanisms (PAO).In a PAO mechanism, agents are asked to pick an object from individualized menus.These choices may be rejected later on, and these agents are then presented with newmenus containing strict subsets of the previous menus from which the previous choiceshave been redacted. When the procedure ends, agents are assigned the last object theypicked, if any. A PAO mechanism implements an allocation rule if it always results inthe unique allocation consistent with preference profiles that could rationalize the choicesmade by the agents. Therefore, if agents simply choose their most-preferred object whengiven a menu, then the object they hold once the procedure ends is the one that theallocation rule determines given their true preferences. Notice, therefore, that a truthfulequilibrium behavior in OSP and PAO mechanisms are closely related. While in theformer it can be expressed as “Wait until you can pick your best feasible object,” in thelatter it would be “Pick your best feasible object and wait to see if you can keep it.”The simple mechanics involved in a PAO mechanism induces a trade-off: since infor-mation about an agent’s preferences can only be obtained through choices from menus,obtaining more information about her preferences requires ruling out her last choice.This, in turn, restricts the set of allocation rules which can be “sequentialized” via PAOmechanisms. We characterize such rules (Theorem 1), via a new property that we denote monotonic discoverability . Many familiar rules, such as the Gale-Shapley DA and toptrading cycles, satisfy it (Proposition 1). We characterize the rules that can be imple-mented in truthful strategies via a robust equilibrium (robust ordinal perfect BayesianEquilibrium) as being those that are strategy-proof and satisfy monotonic discoverability(Theorem 2). Finally, we show that every non-bossy OSP-implementable rule is imple-mentable in weakly dominant strategies via PAO mechanisms (Theorem 3). In otherwords, the set of allocation rules that are implementable in a robust equilibrium via PAOmechanisms is a strict superset of those that are non-bossy and implementable via OSPmechanisms.Our justification for the use of PAO mechanisms is non-standard, in the sense thatthe game-theoretical incentive properties of the PAO mechanisms are weaker than thealternatives - i.e., strategy-proof direct mechanisms and OSP mechanisms, but we testour conjecture via laboratory experiments.We test two allocation rules, the serial dictatorship (SD) and top trading cycles (TTC),and for each of them we construct three treatments: direct revelation implementation,PAO implementation, and OSP implementation. That is, for each one of these rules, weran each of the three different mechanisms for implementing them. We changed allocationrules within-subjects, while mechanisms implementing the rules between-subjects. SinceTTC is only OSP-implementable for certain “acyclic” priority structures (Troyan, 2019),we split the TTC treatments into cyclic and acyclic priority structures, with the formerhaving only direct and PAO implementations.
ICK-AN-OBJECT MECHANISMS 5
We find that, in fact, OSP implementations lead to higher truth-telling rates across theboard. When the OSP implementation exists (i.e., except for TTC with cyclic prioritystructures), OSP outperforms both PAO and direct implementations. When comparingthe direct implementation of TTC vs the PAO implementation of TTC, the experimentsshow that the PAO implementation leads to a higher proportion of subjects followingequilibrium strategies, and a higher average efficiency of allocations. When comparingdirect and PAO implementations for SD, we find no difference between them in proportionsof subjects following equilibrium strategies, but efficiency is significantly higher underPAO than under direct implementation. When we look more closely at the results for the OSP implementation of TTC, however,we see a big difference in the rates truthful behavior, depending on the nature of theobviously dominant strategy for a subject. When the obviously dominant strategy consistsof simply picking the most-preferred object, the rate of truthful behavior is 93%. Butwhen it requires some degree of “foresight,” in that it involves picking the “passing option”in terms of Pycia and Troyan (2019), it is only 56%. This result strongly supports thestrong OSP concept, and its myopic “picking” equilibrium, as being a better predictor ofbehavior than the OSP.To sum up, we show that the PAO environment has significant benefits over its directcounterpart in the general decision environment. When, however, there is the option ofusing OSP mechanism, our experiments suggest using them. Since many of the allocationrules used in real-life allocation problems are not OSP-implementable, but are PAO-implementable, we interpret our experimental results as a support in favor of the choiceof PAO mechanisms over their direct revelation counterparts.
Related literature.
In addition to the ones mentioned in the introduction, this paperis mainly related to two strands of the literature: the design of sequential allocationmechanisms and their theoretical properties, and the behavioral and experimental aspectsof market design.From the theoretical perspective, perhaps the closest paper to ours is Mackenzie andZhou (2020). They consider the family of menu mechanisms , which are also sequentialrevelation mechanisms in which participants are asked to choose from menus of theirpossible assignments. As in the case of PAO mechanisms, they focus on those in which anagent can never select an assignment twice. Differently from PAO mechanisms, however,the definition of menu mechanisms does not imply a restriction on the set of allocationrules that can be sequentialized, since an agent’s assignment does not have to be the lastchoice of an agent, and may in fact be an object that was not chosen. Despite consideringthis more general setup, they show that strategy-proof rules can be implemented in atruthful ex-post perfect equilibrium, a result similar to our Theorem 2. The experiments reported in B´o and Hakimov (2020) complement these with a comparison between thedirect revelation Gale-Shapley DA with the Iterative Deferred Acceptance mechanism, which is its PAOimplementation. The results are in line with the ones that we present here: the PAO implementation ofDA results in a higher proportion of truth-telling than its direct counterpart.
ICK-AN-OBJECT MECHANISMS 6
Another closely related paper is B¨orgers and Li (2019). As in our case, they are con-cerned about some notion of simplicity in mechanisms. They define the class of “strategi-cally simple” mechanisms, which are those in which an agent’s optimal strategy dependsonly on first-order beliefs about preferences and rationality. Similarly to the rules im-plemented in truthful equilibria in PAO mechanisms, these include all dominant strategymechanisms, but also extends to others. It is worth noting, however, that PAO mecha-nisms are not necessarily strategically simple.Other papers have also considered sequential versions of allocation rules, such as multi-unit auctions (Ausubel, 2004, 2006), stable (B´o and Hakimov, 2020; Kawase and Bando,2019; Haeringer and Iehle, 2019) and more general allocations (Schummer and Velez,2017). Moreover, there is a growing literature evaluating sequential mechanisms used inthe field (Gong and Liang, 2016; Grenet et al. , 2019; Veski et al. , 2017; Dur et al. , 2018).Other recent papers, such as Akbarpour and Li (2020) and Hakimov and Raghavan (2020)show that the use of sequential mechanisms can also be explained by their transparencyand credibility characteristics: the experience that participants have when interacting withthese mechanisms can convey information that helps them to be sure that the allocationis produced by following the rules.We also provide contributions to the literature documenting dominated behavior inmatching mechanisms. Shorrer and S´ov´ag´o (2018); Rees-Jones (2018); Hassidim et al. (2016) and Artemov et al. (2017) document dominated strategies being played in real-lifecentralized allocation processes. These are in line with laboratory experiments which alsoevaluate truthful behavior in strategy-proof mechanisms relative to dynamic mechanisms(Echenique et al. , 2016; B´o and Hakimov, 2020; Breitmoser and Schweighofer-Kodritsch,2019; Klijn et al. , 2019), and truthful behavior in TTC and SD (Chen and S¨onmez, 2002,2006; Pais and Pint´er, 2008; Guillen and Hakimov, 2017, 2018).2.
Model
Let A = { a , a , . . . , a n } be a finite set of agents and O = { o , o , . . . , o m } ∪ {∅} be aset of object types . Each agent a has strict preferences P a over the set O . Given P a we express the induced weak preference by R a . That is, oR a o (cid:48) if oP a o (cid:48) or o = o (cid:48) . We abusenotation and P a may represent its binary relation (ex: oP a o (cid:48) ) or a tuple of elements of O , for example, P a = ( o, o (cid:48) , ∅ , o (cid:48)(cid:48) ), which implies oP a o (cid:48) P a ∅ P a o (cid:48)(cid:48) . We also often treat tuplesof distinct elements of O as sets, if that does not create any ambiguity. For example, wemay say that γ = ( o , o , o ) and { o } ⊂ γ . Denote by P the set of all strict preferencesover O . An object o ∈ O is acceptable to agent a ∈ A if oP a ∅ . A preference profile is a list P = ( P a , P a , . . . , P a n ). We denote by P − a the set of all preferences in P exceptfor P a . A problem is a triple (cid:104) A, P, O (cid:105) . Let P = P n be the set of all preference profiles.An allocation is a function µ : A → O . For a given allocation µ , we say that agent a ’s allotment under µ is µ ( a ). Let M be the set of all allocations. A random allocation ICK-AN-OBJECT MECHANISMS 7 is a probability distribution over M . A rule is a function ϕ : P → M . Denote by ϕ a ( P ) = ϕ ( P ) ( a ). A rule ϕ is individually rational if, for any a , ϕ a ( P ) R a ∅ .A rule is strategy-proof if for every agent a and P (cid:48) ∈ P , ϕ a ( P a , P − a ) R a ϕ a ( P (cid:48) , P − a ).Define a choice history h as a sequence of tuples ((Ω , ω ) , (Ω , ω ) , . . . ), where forevery i , Ω i ⊆ O and ω i ∈ Ω i . That is, a choice history is a sequence of sets of object typesand elements of those sets. For example:(( { o , o , o , o , o } , o ) , ( { o , o , o } , o ))Since this will be used often in what follows, we denote by −→ h the last choice in h . Thatis, if h = ((Ω , ω ) , . . . , (Ω k , ω k )), −→ h = ω k . We say that h is a continuation history of h (cid:48) if all the tuples in h (cid:48) are also in h . We denote by H the set of all choice histories,including the empty choice history, represented by h ∅ . We say that a preference P a is consistent with the choice history h if for every (Ω i , ω i ) ∈ h and o ∈ Ω i , ω i R a o .We denote by P ( h ) the set of all preferences that are consistent with h , if any. We canalso say that a choice history is consistent with a preference using inverse reasoning, anddenote by h ( P i ) the set of all choice histories that are consistent with P i .Denote by collective history h A a list of n choice histories: ( h , h , . . . , h n ). We denoteby H A the set of all collective histories, by h A −∅ the collective history consisting of n emptychoice histories, and by h Ai the i-th element in h A . We also say that h A is a continuationcollective history of h A (cid:48) if each choice history in h A is a continuation history of itsrelated history in h A (cid:48) . A preference profile P = ( P a , P a , . . . , P a n ) is consistent withthe collective history h A if for every i , P a i is consistent with h Ai . We denote by P (cid:0) h A (cid:1) the set of preference profiles that are consistent with h A . We can also say thata collective history is consistent with a preference profile using inverse reasoning, anddenote by h A ( P ) the set of all collective histories that are consistent with the preferenceprofile P .We abuse notation as follows: ϕ (cid:0) h A (cid:1) = (cid:91) P ∈ P ( h A ) ϕ ( P )For a given collective history h A , therefore, ϕ (cid:0) h A (cid:1) is a (possibly empty) subset of M .The interpretation for it goes as follows. If we take the revealed preference approach tothe collective history h A , and consider for each agent the set of preferences that are con-sistent with the choices she makes in h A , ϕ (cid:0) h A (cid:1) tells you what allocations are consistentwith the application of the rule ϕ , given the information about the preference profile thatcan be deduced from the collective history h A . Note that this implies that a preference profile is consistent with a collective history if all histories areconsistent with some preference. In other words, it is necessary that all histories are rationalizable.
ICK-AN-OBJECT MECHANISMS 8
We also define the set of feasible matches after h A for a i , or µ ϕi (cid:0) h A (cid:1) as: µ ϕi (cid:0) h A (cid:1) = (cid:91) µ ∈ ϕ ( h A ) µ ( a i )Let Φ = (cid:0) O (cid:1) n . That is, Φ is the set of n-tuples with subsets of O .A menu function S : H A → Φ specifies, for each collective history, a list of menus tobe given to the agents. • S (cid:0) h A −∅ (cid:1) = ( φ , φ , . . . , φ n ), where for every i , φ i is a non-empty subset of O .These are the initial menus. • For any collective history h A , where h Ai = (cid:0) (Ω i , ω i ) , (Ω i , ω i ) , . . . , (cid:0) Ω ik i , ω ik i (cid:1)(cid:1) , S (cid:0) h A (cid:1) = ( φ , φ , . . . , φ n ), where φ i ⊆ Ω ik i \{ ω ik i } .Denote by S i ( · ) the i-th element in S ( · ) (or, when convenient, S a ( · ) to be the element in S ( · ) associated with agent a ). For a given menu function S , we define the pick-an-objectmechanism S as follows: • Period t = 1: For every agent a i , ask her to choose one item in S i (cid:0) h A −∅ (cid:1) , if themenu is non-empty. Let agent a i ’s choice, if any, be σ ti . Define h A − to be thecollective history such that for every agent a i who received a non-empty menu, h A − i = (( φ , σ i )). For those with an empty menu, let h A − i = (( ∅ )). • Period t >
1: Let ( φ t , φ t , . . . , φ tn ) = S (cid:0) h A − t (cid:1) . – If for all i , φ ti = ∅ , then the procedure stops, and outputs the allocation µ ,where for each i , µ ( a i ) is the last choice in h A − ti . – Otherwise, for every agent a i , ask her to choose one item in φ ti , if the menuis non-empty. Let agent a i ’s choice, if any, be σ ti . Define h A − ( t +1) as thecollective history such that for every agent a i who received a non-empty menu, h A − ( t +1) i = h A − ( t +1) i ⊕ ( φ ti , σ ti ), and for those with an empty menu, h A − ( t +1) i = h A − ti .Notice that since the menus that are given to each agent do not include her previouschoices and never include objects that were not present in previous menus, every col-lective history that results from any number of periods of a pick-an-object mechanism isconsistent with a non-empty set of preference profiles. When facing a pick-an-object mech-anism, one simple behavior that an agent may follow is what we call a straightforwardstrategy , which we define below. Definition 1.
An agent follows a straightforward strategy with respect to P ifwhenever presented with a menu I ⊆ O , she chooses the most-preferred element of I according to P . Notice that while this is a model of discrete object allocation, the notion of feasibility does not explicitlyconsider capacities. Feasibility is “encoded” in the allocation rules themselves: if an allocation is at theimage of the allocation function, then it is feasible. For each agent a i , k i represents the number of menus that she was given in h A . We use “ ⊕ ” to denote concatenation. That is, for example, (( φ, σ ) , ( φ (cid:48) , σ (cid:48) )) ⊕ ( φ (cid:48)(cid:48) , σ (cid:48)(cid:48) ) =(( φ, σ ) , ( φ (cid:48) , σ (cid:48) ) , ( φ (cid:48)(cid:48) , σ (cid:48)(cid:48) )). ICK-AN-OBJECT MECHANISMS 9 (a) Preferences and allocation P a o o o o o ∅ P a o o o o o ∅ P a o o o o o ∅ P a o o o ∅ o o (b) Continuation profiles P a o o o P a o o P a o o o o P a o o o ∅ Figure 1.
Monotonic discoverability: Example
Definition 2.
A pick-an-object mechanism S sequentializes the rule ϕ if, for any prefer-ence profile P , the pick-an-object mechanism S provides menus such that when each agent a i follows the straightforward strategy with respect to P a i , the outcome ϕ ( P ) is produced.We say that there exists a pick-an-object mechanism that sequentializes some rule ϕ ifthere exists a menu function S such that a pick-an-object mechanism S sequentializes ϕ .Let µ be an allocation. Let P = ( P a , P a , . . . , P a n ) be any element of P . We define thefunction L ( P, µ ) = { P (cid:48) ∈ P : ∀ a ∈ A and o, o (cid:48) ∈ O : oP a o (cid:48) P a µ ( a ) ⇐⇒ oP (cid:48) a o (cid:48) P (cid:48) a µ ( a ) } . Thatis, L ( P, µ ) contains all preference profiles that, for each agent a , agree with P a with respectto µ ( a ) and all objects preferred by a to µ ( a ), but may differ with respect to objects that µ ( a ) are preferred to, with respect to P a . We will say that L ( P, µ ) are the continuationprofiles of P at µ . Definition 3.
A rule ϕ satisfies monotonic discoverability if, for any allocation µ and preference profile P , either ϕ ( P ) = µ or there is an agent a ∗ ∈ A such that P (cid:48) ∈L ( P, µ ) = ⇒ µ ( a ∗ ) (cid:54) = ϕ a ∗ ( P (cid:48) ).For example, consider a rule ϕ which satisfies monotonic discoverability, a problemwhere A = { a , a , a , a } , O = { o , o , o , o , o , ∅} , and preferences are such as the onesin figure 1(a). Consider now the allocation µ indicated by the blue coloring in that figure.That is, µ ( a ) = o , µ ( a ) = o , etc. Notice that since the notion of allocation is notconnected to any concept of feasibility we can have that object type o is matched toboth a and a , regardless of any quantities or capacities that might be involved in ϕ .Suppose then that ϕ ( P ) (cid:54) = µ . Then, the fact that ϕ satisfies monotonic discoverabilityimplies that there is at least one agent a ∗ ∈ A who, for any continuation profile P ∗ at µ ,which comprise of all the completions of the gray areas in figure 1(b) that still constitutea preference profile, a ∗ will not be matched to the object µ ( a ∗ ) under ϕ .To understand the critical role that monotonic discoverability has when sequentializingrules, consider figure 2, which shows the same profile P . Suppose that we have a rule ϕ which satisfies monotonic discoverability, and start with the allocation µ that matcheseach agent with her most-preferred object type, highlighted in blue in figure 2, step 1. Wecan consider two cases. In the first, ϕ is such that for every preference profile in whichagent a i ’s top option is as in P , these agents are matched to their top choices. In thatcase, knowing those top choices already gives us enough information to determine this to ICK-AN-OBJECT MECHANISMS 10
Step 1 P a o o o o o ∅ P a o o o o o ∅ P a o o o o o ∅ P a o o o ∅ o o Step 2 P a o o o o o ∅ P a o o o o o ∅ P a o o o o o ∅ P a o o o ∅ o o Step 3 P a o o o o o ∅ P a o o o o o ∅ P a o o o o o ∅ P a o o o ∅ o o Step 4 P a o o o o o ∅ P a o o o o o ∅ P a o o o o o ∅ P a o o o ∅ o o Figure 2.
Monotonic discoverability and sequentializationbe the outcome that ϕ maps for any continuation profile. The second case is when thisis not true. That is, there are continuation profiles of P at µ for which ϕ does not mapthose profiles to µ . Let P be that profile. By monotonic discoverability, there is at leastone agent who, for every continuation profile of P at µ (and therefore also of P at µ )will not be matched to her match at µ . Without loss of generality, let a be such agent.Consider next the allocation µ , which is the same as µ except that the object mappedto a is the second one in her preference. That is shown in figure 2, step 2. Here, onceagain we have two cases. In the first, ϕ is such that for every continuation preferenceprofile of P at µ all agents are matched to their outcomes under µ , and another inwhich there is at least one, but potentially many, such profiles in which the outcomes aredifferent than µ . There is one thing we can say, though. Since the continuation profilesof P at µ are also continuation profiles of P at µ , for none of these cases is agent a matched to o . Suppose here, without loss of generality, that the second case is true,agents a and a are not matched to their outcomes under µ for any continuation profileof P at µ . Steps 3 and 4 represent a continuation of this argument, until at step 4, theallocation µ being evaluated is in fact the one that the rule ϕ maps to all continuationprofiles of P at µ . Notice that this process always converges to the allocation mappedby ϕ to the profile P , since we only move down the preference ordering of an agent whenwe have enough information to determine that ϕ rules out previously considered matchesto her.Following this process shows us two consequences of a rule satisfying monotonic discov-erability. The first is that while following this monotonic process of evaluating allocationsthat are at each step weakly worse (from the perspective of the underlying preference One simple example of this situation is when ϕ is a simple serial dictatorship, and no two agents havethe same object as their most-preferred one, as in P . ICK-AN-OBJECT MECHANISMS 11 profile), we end up at the allocation that ϕ maps to P . This fact is fundamental for apick-an-object mechanism to be able to match agents to their last choices without repe-tition.The second is that monotonic discoverability solves the informational trade-off thatis induced by a pick-an-object mechanism, that is, that in order to obtain more infor-mation about agents’ preferences, we must permanently reject the last choice she made.Monotonic discoverability guarantees that, as long as we follow the monotonic processover allocations that we just described, whenever we do not have enough information tosingle out an allocation (that is, there are still continuation profiles for which ϕ producesdifferent allocations), there is at least one agent whose last choice can be rejected andthen presented to a new menu.What monotonic discoverability eliminates, therefore, are situations of “ informationalgridlock ,” in which each agent may be matched to their last choice, but more informationabout their preferences is necessary to make a precise determination. In fact, this rela-tionship between monotonic discoverability and pick-an-object mechanism is as strong aspossible, as shown in Theorem 1 below. Theorem 1.
There exists a pick-an-object mechanism that sequentializes an individu-ally rational rule ϕ if and only if ϕ satisfies monotonic discoverability. Generalized Deferred Acceptance Procedures.
Many mechanisms used in match-ing are defined by algorithms which produce its outcomes, as opposed to axioms or objec-tive functions. Two classic examples are the Gale-Shapley deferred acceptance mechanism(DA) (Gale and Shapley, 1962) and Gale’s top trading Cycles (TTC) (Shapley and Scarf,1974). Their definitions describe step-by-step procedures which use agents’ preferencerankings (and often other information, such as priority orderings) and result in an allo-cation.Many of those procedures can be described as instances of the class of generalizedDA procedures . A generalized DA procedure is defined by an update function
Ψ :
M ×M → M , with the restriction that, if Ψ ( µ, µ (cid:48) ) = µ (cid:48)(cid:48) , for every a ∈ A it must be that µ (cid:48)(cid:48) ( a ) ∈ {∅ , µ ( a ) , µ (cid:48) ( a ) } . Given an update function Ψ, the procedure can be described bythe following algorithm: • Step
1: Let µ be an assignment in which, for all a ∈ A , µ ( a ) = ∅ , and µ ∗ be an assignment consisting of all agents matched to their top choice in P . Let,moreover, µ = Ψ ( µ , µ ∗ ). We say that every agent a ∈ A for which µ ∗ ( a ) (cid:54) = ∅ and µ ( a ) = ∅ was rejected by µ ∗ ( a ). • Step t >
1: Construct the assignment µ ∗ , in which every agent a who was rejectedat step t − P a , fromwhich she was not previously rejected. Moreover, let all other agents be matchedto ∅ in µ ∗ . Let, moreover, µ t = Ψ ( µ t − , µ ∗ ). If for every a ∈ A it is the case that µ t ( a ) ∈ { µ t − ( a ) , µ ∗ ( a ) } , stop the procedure and determine the assignment to be µ t . Otherwise, go to step t + 1. ICK-AN-OBJECT MECHANISMS 12
Generalized DA procedures, therefore, produce tentative allocations at each step, fol-lowing the agent’s preferences, until no agent has her choice rejected. It generalizes DAbecause the update function Ψ determines whether the tentative allocation of an agent toan object type becomes a rejection based on the entire tentative allocation and set of pro-posals. Note that it is different from DA, in which this is determined only by the tentativeallocation and proposals to the object type in question . If we use the college admissionsanalogy, in a generalized DA procedure, whether a student’s proposal is accepted or notmay depend not only on the college she applied to (and its tentatively matched students)but also on the entire tentative matching of students to colleges, and contemporaneousapplications.Most importantly for us, rules that are described by generalized DA procedures satisfymonotonic discoverability, as shown below.
Proposition 1. If ϕ is described by a generalized DA procedure, then ϕ satisfies mono-tonic discoverability. Proposition 1 implies that many mechanisms that are commonly used and referencedin the matching literature satisfy monotonic discoverability and can therefore be sequen-tialized by a pick-an-object mechanism.
Remark . The rules that satisfy monotonic discoverability include (i) DA, (ii) TTC, and(iii) Boston mechanism (Abdulkadiro˘glu and S¨onmez, 2003).Monotonic discoverability therefore gives a full characterization of the rules for which itis possible to use a pick-an-object mechanism to “sequentialize” the process of obtaininginformation about the participants’ preferences, and Proposition 1 gives us a family ofmechanisms that satisfy that condition. This does not, however, guarantee that it willbe in the participants’ own interest to truthfully reveal their preferences. When that isnot the case, then the outcomes produced by these pick-an-object mechanisms may differsubstantially, with respect to the agents’ true preferences, with that determined by therule being used. In other words, we also need to consider the implementation problemwhen using pick-an-object mechanisms.3.
Implementation in pick-an-object mechanisms
In this section we will consider the extensive-form game that is induced by pick-an-object mechanisms, and conditions on the rules that guarantee that it is in the partici-pants’ own interest to truthfully reveal their true preferences, by means of using straight-forward strategies. The equilibrium concept that we will use is that of ordinal perfectBayesian equilibrium (OPBE), introduced in B´o and Hakimov (2016). Loosely speak-ing, a strategy profile is an OPBE if at every information set, following the equilibriumstrategy first-order stochastically dominates any deviating strategy. It is, therefore, anordinal version of a perfect Bayesian Nash equilibrium, which will be formalized later inthis section.
ICK-AN-OBJECT MECHANISMS 13
Since every step in a pick-an-object mechanism is used to obtain (partial) informationabout the participants’ preferences over object types, straightforward strategies beingan OPBE implies that following them first-order stochastically dominates any deviatingstrategy regardless of past actions by the player being considered . That is, it implies thatproviding truthful information (in the form of choices based on true preferences) is thebest thing to do even when the information that was provided earlier is not truthful.This type of “truthful partial revelation” seems, at first sight, stronger than the usualincentive-compatibility constraints, in which all that is required is that reporting the truetype, and just that, is a best-response or a weakly dominant strategy.Our next result shows that strategy-proofness, a property that in its original definitionstates that reporting the true type is always a best-response in a direct revelation mech-anism, is in fact equivalent to a seemingly stronger property, which requires that truthfulpartial revelations is also always a best-response when an agent makes irreversible “mis-takes” in parts of its report.For any set I ⊆ O , let I ! denote the set of all permutations of the elements of I . Forany tuple γ of distinct elements of O (for example, γ = ( o , o , o )), let: P | γ ≡ (cid:91) λ ∈ ( O \ γ )! γ ⊕ λ That is, P | γ is the set of all preferences in which γ are the most-preferred object types,ordered as in the tuple γ itself. Let also P | I , where I ⊆ O , be the preference P restrictedto I (for example, if P = ( o , o , ∅ , o , o ), P | ( o ,o ) = ( o , o )). We can now define recursivedominance. Definition 4.
A rule ϕ satisfies recursive dominance if for every agent a , preference P a , preferences of other agents P − a , set I ⊆ O , permutation γ of I and P ∗ ∈ P | γ : ϕ a (cid:16) γ ⊕ P | O \ I , P − a (cid:17) R a ϕ a (cid:0) P ∗ , P − a (cid:1) A rule satisfies recursive dominance, therefore, if conditional on any ordering of a subsetof object types, submitting a truthful ordering of the remaining objects is always a best-response. One way to think about this is to consider, for example, a situation in which anagent a has to fill out a form with her preferences, which will be used to produce an allo-cation, but when she looks at the form she notices that the first k positions in the rankinghad been filled out by someone else. That is, suppose that O = { o , o , o , o , o , o , ∅} ,agent a ’s preferences are o P a o P a o P a o P a o P a o P a ∅ , but the form is partially filled outas below: o P o P o P (cid:3) P (cid:3) P (cid:3) P (cid:3) The usual properties considered when evaluating rules and mechanisms do not give anyguidance for the question of how agent a should fill those squares. But if we know that the For example, if I = { o , o , o } , then I ! = { ( o , o , o ) , ( o , o , o ) , ( o , o , o ) , ( o , o , o ) , ( o , o , o ) , ( o , o , o ) } . ICK-AN-OBJECT MECHANISMS 14 rule satisfies recursive dominance, then we know that, regardless of what other players aredoing, it is a best-response within this restricted domain of preferences imposedby the first three positions in the ranking above , to fill out the remaining positionsfollowing the true preferences P a . This would therefore result in the following ranking: o P o P o P o P o P o P ∅ Notice that the definition of recursive dominance implies strategy-proofness when I = ∅ .The result below shows, however, that strategy-proofness itself also implies recursivedominance. Proposition 2.
A rule is strategy-proof if and only if it satisfies recursive dominance.
Next, we formalize the description of the game, the participants’ information and beliefs.In order to do that, we will expand the definition of a pick-an-object mechanism to alsoinclude the information that it provides to the participants. We do this by defining, for agiven mechanism, the information structure that is associated with it.Let S be a menu function. We denote by H A S the set of all collective histories that canresult from the pick-an-object mechanism S . That is, H A S contains each collective historythat would result from each possible combinations of choices from all agents from themenus that are offered when using the pick-an-object mechanism S . Next, for each agent a ∈ A , let I a be agent a ’s information structure . I a = {I a , I a , . . . } is a partition ofthe set H A S , such that for every pair of collective histories h A , h A (cid:48) ∈ H A S :(1) If h A = (cid:0) h Aa , . . . , h Aa , . . . , h Aa n (cid:1) and h A (cid:48) = (cid:16) h Aa (cid:48) , . . . , h Aa (cid:48) , . . . , h Aa n (cid:48) (cid:17) and h Aa (cid:54) = h Aa (cid:48) ,then h A and h A (cid:48) must be in different elements of I a .(2) If S a (cid:0) h A (cid:1) (cid:54) = S a (cid:16) h A (cid:48) (cid:17) , then h A and h A (cid:48) must be in different elements of I a .In other words, if two collective histories are such that the agent a ’s choice histories aredifferent, or if they result in different menus to be given to a afterwards, these must be indifferent elements of the partition I a . The partition I a represents agent a ’s informationsets, and reflects the collective histories that an agent can differentiate based on whatshe can observe and by the information that is provided by the specific implementation ofthe pick-an-object mechanism . Collective histories in the same element of the partitioncannot be differentiated explicitly based on the information that the agent is given. Therestriction that we defined above is the minimal information that we assume the agentsmust have: they know the menus that were offered to them and the choices they made.Our results will be robust to finer partitions, including that of perfect information, inwhich each element of the partition contains only one collective history. We denote by I = ( I a , . . . , I a n ) the information structure of the game being considered. Givenan information structure I , a belief system θ is a collection of probability measures,one for each element of the partition in the information structures in I . We denote by θ a (cid:0) h A (cid:1) the probability associated with collective history h A in I a . ICK-AN-OBJECT MECHANISMS 15
Next, we define an agent a ’s strategy to be a function σ a : I a → O , where for every I ai ∈ I a , and h A ∈ I ai , σ a (cid:0) h A (cid:1) ∈ S a (cid:0) h A (cid:1) . That is, strategies map from information setsto objects in the menu offered to the agent. A collection with one strategy per agent isa strategy profile σ = ( σ a , . . . , σ a n ).Fix a belief system θ , and let a ∈ A be an agent and h A be a collective history in H A S .Let I ai be the set in agent a ’s information structure such that h A ∈ I ai . We define the outcome belief for a under θ , O θa (cid:0) h A , σ (cid:1) , as the distribution over assignments thatresult from following the pick-an-object mechanism S , starting from the collective history h A , in which agents follow the strategies in σ , given the distribution that θ puts over theelements of the set I ai .Let A and B be two random assignments. We denote by (cid:109) a the first-order stochasticdominance relation under P a . That is, A (cid:109) a B if for all o ∈ O , P r { A ( a ) = o (cid:48) | o (cid:48) R a o } ≥ P r { B ( a ) = o (cid:48) | o (cid:48) R a o } . We can now define an ordinal perfect Bayesian equilibrium (OPBE). Definition 5.
A strategy profile σ together with a belief system θ is an ordinal perfectBayesian equilibrium (OPBE) if for every a ∈ A , every h A ∈ H A S , and every strategy σ a (cid:48) for agent a : O θa (cid:0) h A , ( σ a , σ − a ) (cid:1) (cid:109) a O θa (cid:0) h A , ( σ a (cid:48) , σ − a ) (cid:1) Even though an OPBE is a refinement of the concept of a perfect Bayesian equilibrium, it may still suffer from the fact that an equilibrium may be supported by artificiallyconstructed belief systems. The following alternative equilibrium notion improves uponthat. Definition 6.
A strategy profile σ is a robust ordinal perfect Bayesian equilibrium if for every belief system θ , σ is an OPBE.We say that an allocation rule ϕ is pick-an-object implementable in some equilib-rium notion if there exists a pick-an-object mechanism S , which sequentializes ϕ , in whichstraightforward strategies constitute an equilibrium in that notion. Theorem 2.
A rule is pick-an-object implementable in robust ordinal perfect Bayesianequilibrium if and only if it is strategy-proof and satisfies monotonic discoverability.
Relation with Obviously Strategy-proof Mechanisms.
When introducing theconcept of obvious strategy-proofness (OSP), Li (2017) defined an OSP mechanism as onewhich has an equilibrium in obviously dominant strategies. A rule ϕ is OSP-implementableif there is a game and an obviously dominant strategy for each type of player in that game,such that the outcome produced by this strategy for each type profile is what is deter-mined by ϕ . Pycia and Troyan (2019) showed that, in an object allocation environmentsuch as the one that we use, every OSP-implementable rule can be implemented via a For simplicity, and without any consequence, we allow for the definition of strategies to make “choices”in collective histories in which an agent is not given a menu. If a strategy profile σ together with a belief system θ is an OPBE, then σ together with θ is a perfectBayesian equilibrium for any utility functions profile that represents the ordinal preferences in P . ICK-AN-OBJECT MECHANISMS 16 millipede game with a greedy strategy . Millipede games are sequential games where, ineach period some agent can either “pass” or “clinch” one of potentially multiple optionsin a menu, which corresponds to private allocations that they can guarantee (in our setup,therefore, she can clinch an object and leave with it). A greedy strategy consists of anagent choosing to “pass” as long as her most-preferred object can still be clinched from amenu in some continuation history, and clinching it whenever it is in a given menu. The first thing to note is that there is a direct relation between a greedy strategy ina millipede game and a straightforward strategy in a pick-an-object mechanism. In amillipede game, every time an agent interacts with the mechanism she is given a menu ofobjects that she can pick and keep for good and, potentially, also one option to “pass.”That is, as in pick-an-object mechanisms, in millipede games an agent is always matchedto the last object that she chose. In a millipede game, however, an agent can only choosean object once and it remains her final allocation, while in a pick-an-object mechanism, achosen object can be interpreted as her “tentative allocation,” and thus she can potentiallychoose multiple times. The authors show that the greedy strategy is obviously dominantin that game.As long as an agent is able to infer the set of clinchable objects in continuation histories,following the greedy strategy is very simple: the only object you choose from a menu isyour most preferred in the set of feasible objects. Otherwise, you simply pass. In a pick-an-object mechanism, agents who are presented with menus are always giventhe entire set of feasible objects. An agent following a straightforward strategy, therefore,chooses the most-preferred object among those that are feasible, and will only have tomake another choice once and if that object is no longer feasible. Notice that, as opposedto millipede games, in pick-an-object mechanisms agents do not have to infer the set offeasible objects: they are all given in the menus. Our next result relies on the commonproperty of non-bossiness, defined below.
Definition 7.
A rule ϕ is non-bossy if, for every a ∈ A , if ϕ a ( P a , P − a ) = ϕ a ( P (cid:48) a , P − a ),then ϕ ( P a , P − a ) = ϕ ( P (cid:48) a , P − a ).If a rule is non-bossy and OSP implementable, we obtain the following relation withPAO mechanisms: While each option in a menu given to a player in a millipede game corresponds to a private allocationfor that player, there might be multiple options in that menu associated with the same allocation. Whileevery option associated with an allocation results in the same outcome for the player making that choice,different choices among these might result in different outcomes for other players. Our results will consideronly situations in which that is not the case: different items in a menu correspond to different privateallocations for the agent making the choice. Therefore, we will ignore the possibility that they do not. Pycia and Troyan (2019) question, however, the simplicity of inferring the set of clinchable objects incontinuation histories, as it requires foresight from agents. They then introduce the concept of strongobvious strategy-proofness. In a strong OSP mechanism, the agents face a menu to choose from only once,and there is no passing option. The obviously dominant strategy in these games thus simply requireschoosing the best object from the single menu that is offered.
ICK-AN-OBJECT MECHANISMS 17
Theorem 3.
Every non-bossy OSP implementable rule is pick-an-object implementablein weakly dominant strategies.
Given the simplicity of these strategies, a question one may have is whether what drivesa high proportion of behavior in line with an obviously dominant strategy is, in fact, thesimplicity of the strategy itself, as opposed to more sophisticated arguments in terms ofthe kind of counterfactual reasoning that is eliminated in an OSP mechanism. If that isthe case, pick-an-object mechanisms could also lead agents to behave more in line withits theoretical prediction. 4.
Experiments
In this section, we present a series of experiments designed to test the performance ofPAO and OSP mechanisms for implementing different allocation rules, when compared tothe traditional direct revelation alternative. We chose two rules: top trading cycles (TTC)and serial dictatorship (SD). Ideally, we would have liked to compare implementing multi-ple allocation rules via PAO versus multiple alternative mechanisms. Our choice of usingSD and TTC is driven mainly by the importance of both rules for practical applications.Another contender could be DA, but the comparison of the PAO implementation of DAversus its direct counterpart was made in two recent papers B´o and Hakimov (2020) andKlijn et al. (2019). Both led to a similar conclusion, reporting the better performance ofthe PAO mechanism, with respect to the truth-telling rates and stability of the alloca-tions, when compared to the direct DA mechanism. Additionally, our choice of allocationrules was driven by the possibility of OSP implementation of both rules (for TTC, in thecase of acyclic priorities). This allows us to compare PAO versus OSP implementationsin a restricted setup.4.1.
Mechanisms.
In this subsection, we describe the six mechanisms that were used inthe experiments. For each allocation rule, namely TTC and SD, we use three differentmechanisms: direct, PAO, and OSP. The mechanisms correspond to treatments in theexperiments. Note that we describe the mechanisms in the same way that they weredescribed to the participants. We omit, for instance, the description of actions whensome or all objects are not acceptable, for simplicity, and to prevent confusion by thesubjects. The reason is that all objects lead to a positive payoff for the subjects in theexperiments. Moreover, in the experiments, subjects had to choose at least one object inall sequential mechanisms, and had to list all objects in the rank-order lists in the directmechanisms.
Direct TTC
Every participant submits her rank-order list of objects to a central authority. Thefollowing steps are executed by the central authority, without any further participationfrom the subjects.
ICK-AN-OBJECT MECHANISMS 18 • Step 1.1 All participants point to the object at the top of their submitted rank-ordered lists. Each object points to the participant with the highest priority atthat object. – The mechanism looks for cycles. There is at least one cycle. All participantsin the cycle are assigned the object they pointed to. • Step 1.2 The priorities of the objects are updated to account for assigned partici-pants. Submitted rank-ordered lists are updated to account for assigned objects.Steps 1–1.2 are repeated until all objects are assigned.Direct TTC is strategy-proof, and Pareto efficient (Abdulkadiro˘glu and S¨onmez, 2003).
PAO TTC
All participants are asked to pick one object from a menu with all objects. • Step 1.1. All participants point to the object they picked. All objects point to theparticipant with the highest priority at that object. – The mechanism looks for cycles. There is at least one cycle. All participantsin the cycle receive the object they pointed to. • Step 1.2 For the remaining participants, if their last picked object was alreadyassigned, the participant is asked to choose a new object from a menu of remainingones. Steps 1–1.2 are repeated until all the objects are assigned.In PAO-TTC, every participant following the straightforward strategy is a robust OPBE(see Theorem 1). If all participants follow straightforward strategies, the allocation isPareto efficient.
OSP TTC
Note that OSP TTC is defined only when the priorities of the objects are acyclic. Weuse the mechanism described in Troyan (2019). • Step 1.0. The mechanism first tentatively assigns each object to the participantwith the highest priority at that object (i.e., the participant tentatively owns theobject). • Step 1.1. One by one each participant who tentatively owns an object is askedwhether one of the objects that she owns is her favorite object. For each participantand object there are two possible answers: – If ”Yes,” the corresponding participant receives the object. Go to Step 1.2. – If ”No,” she is asked about the next object among the ones she tentativelyowns. If the participant answers “No” to all tentatively owned objects, thealgorithm moves to the next participant who owns at least one object. If allparticipants who tentatively own at least one object say ”No” to all ownedobjects, then each participant who tentatively owns at least one object isasked to pick one object among the objects she does not own.
ICK-AN-OBJECT MECHANISMS 19 ∗ All participants point to the object they picked. Each object points toits owner. The mechanism looks for cycles. There is at least one cycle.All participants in the cycle receive the object they picked. • Step 1.2 The priorities of the objects are updated to account for the participantswho left. Steps 1.1–1.2 are repeated until all objects are assigned.In OSP TTC, the truthful strategy is obviously dominant, leading to a Pareto efficientallocation.Note that in mechanisms implementing the SD allocation rule, instead of using ordinaltables priorities, we use priority scores. In the experiments, each subject knew her score,and knew that the higher the score, the higher her priority. Details are explained in theexperimental design section. Direct SD
Every participant submits her rank-order list of objects to a central authority. Thefollowing steps are executed by the central authority, without any further input from thesubjects. • Step 1. The participants with the highest priority score is assigned the top-rankedobject on her list. • Step 2. The participants with the second-highest priority score is assigned the topobject on her list, among the objects that remain. .... • Step N. The participant with Nth-highest priority score is assigned whatever objectremains.Direct SD is Pareto efficient and strategy-proof.
PAO SD • Step 1. All participants are asked to pick one object from a menu with all objects. • Step 2. The participant with the highest priority score is assigned the object shepicked. All other participants who chose this object are asked to pick a new objectfrom a menu containing the remaining objects. • Step 3. The participants with the second-highest priority score is assigned the lastobject she picked. All other participants with lower priority who chose this objectare asked to pick a new object from the menu containing the remaining objects..... • Step N+1. The participant with the lowest priority score is assigned the last objectshe picked.In PAO-SD, every player using the straightforward strategy is a robust OPBE (seeTheorem 1). If all participants follow straightforward strategies, the allocation is Paretoefficient. More specifically, the truthful strategy consists of only saying ”Yes” when the object is the most-preferred among the objects that are still available, and picking the most preferred object when asked topick from objects that she does not own.
ICK-AN-OBJECT MECHANISMS 20
OSP SD • Step 1. The participant with the highest priority is asked to pick an object andshe is assigned the object she picked. • Step 2. The participants with the second-highest priority is asked to pick an objectamong the remaining ones, and she is assigned the object she picked. .... • Step N. The participants with the lowest priority is assigned to the last objectthat remains.In OSP SD, straightforward strategies are strongly obviously dominant, leading to aPareto efficient allocation (Pycia and Troyan, 2019).4.2.
Experimental design.
In the experiment, there were eight objects and eight par-ticipants. In all treatments, participants received 22 euros if they were matched to theirmost-preferred object, 19 euros to their second most-preferred object, 16 euros to theirthird most-preferred object, and so on. Participants received 1 euro if they were matchedto their least-preferred object.Each session lasted for 21 rounds. At the end of the experiment, one round was ran-domly drawn to determine the participants’ payoffs.Each round represented a new market. The preferences used in each market were gen-erated following the designed market idea of Chen and S¨onmez (2006). For each market,agents’ ordinal preferences were generated from cardinal utilities by adding, for each ob-ject, a common and an idiosyncratic value. The common values for each object weredrawn from the uniform distribution with the range [0 , , We ran three treatments between-subjects: Direct, PAO, and OSP. The comparisonof Direct and PAO is our main focus. We additionally run OSP, which is only availablefor a restricted set of markets – namely those with acyclic priorities in the TTC rule –to disentangle the effect of the sequential pick-an-object environment from the effect ofthe simpler strategic environment of OSP mechanisms. We run three treatments within-subjects: TTC with cyclic priorities, TTC with acyclic priorities, and SD.Note that, with respect to the TTC rule, acyclic priorities is the most applicable setup,and therefore the main focus of our analysis for that rule. TTC with acyclic prioritiesis a simpler decision environment as the acyclic priorities make sure that, at any time ofthe mechanism, only two participants could be at the top of the priorities of all objects.
For rounds with acyclic priorities, priorities were drawn from the set of acyclic priorities. For eachround, a new draw of acyclic priorities was generated. TTC cycles, therefore, could involve only one or two agents.
ICK-AN-OBJECT MECHANISMS 21
Rounds Direct PAO OSP
Table 1.
Summary of treatmentsTTC with acyclic priorities is implementable through an OSP mechanism, thus allowingus to compare it to a PAO implementation of that rule. SD is arguably the simplestallocation rule, and also allows the implementation through the OSP mechanism, whichalso coincides with the environment of sequentially making choices from a menu, but unlikethe general PAO mechanism, does not require the simultaneous play of participants.For the first 14 rounds of the experiment, participants were matched using the TTCrule. During the first seven of these rounds, they faced markets with cyclic priorities intreatments Direct and PAO. Given the importance of this comparison, and to preventlearning effects from other treatments implemented within-subjects, we always run cyclicTTC in the first seven rounds. The same markets were used for Direct and PAO treat-ments. Since there is no OSP mechanism for TTC with cyclic priorities, for the OSPtreatment we generated acyclic priorities also for the first seven rounds. For rounds eightto 14, TTC with acyclic priorities was used to match participants to objects. The samemarkets were used in all three between-subjects treatments.In the first 14 rounds, participants observed the full priority tables of all objects. Theprovision of priority tables is necessary, as it allows participants to see when they areacyclic in the OSP TTC. Participants knew only their own preferences, however, andnot the preferences of other participants. They knew that other participants might havethe same or different preferences. We chose this informational environment in order tosimplify the processing of market information, as providing complete information wouldlead to a longer decision time every round.Finally, for the last seven rounds (rounds 15 to 21), the SD allocation rule was used. Inthe last seven rounds, instead of observing the priority table of objects, the participantswere assigned a priority score. They knew that the higher the score, the higher the priority.Participants were also informed that the priority scores would be drawn for each subjectand each round independently from a uniform distribution with the range [1 , andeach one of them was informed of their own draw. This choice was made to make sure thateven the participants with the lowest score have incentives to play a meaningful strategy.Otherwise, they would know that their choices were irrelevant for the allocation. As forpreferences, just like in the first 14 rounds, participants only knew their own preferencesbut were informed that other participants might have different preferences.Table 1 presents the summary of the experimental design. Each cell of the table rep-resents the mechanism that was used. Note that we explained to each subject two mech-anisms. To our knowledge, this is the first lab experiment which runs a within-subjects There were no ties. The design replicates the informational environment of Li (2017).
ICK-AN-OBJECT MECHANISMS 22 variation of the mechanisms. A probable reason for that is that mechanisms take too longto explain and are complicated for the subjects. In our case, however, all SD mechanismsare quite simple and straightforward to explain, thus we decided to run it within-subjectstogether with the TTC mechanisms. After each round, the participants learned the objectthey were matched to, but not the matches of the other participants.The experiment was run at the experimental economics lab at the Technical Universityof Berlin, from March to May 2019. We recruited student subjects from our pool withthe help of ORSEE (Greiner, 2015). The experiments were programmed in z-Tree (Fis-chbacher, 2007). For each of the three between-subjects treatments, independent sessionswere carried out. Each session consisted of either 16 or 24 participants that were splitinto two or three groups of eight for the entire session. We use fixed groups in order toincrease the number of independent observations and allow for maximum learning. Asevery round represents a new market and subjects play under incomplete informationabout the preferences of the other participants, subjects cannot identify the strategies ofthe players and their identities from previous rounds. Thus, we are not concerned aboutrepeated games caveats.In total, 15 sessions with 296 subjects were conducted. Thus, we have 96 subjectsand 12 independent observations in Direct and PAO treatments, and 108 subjects and 13independent observations for the OSP treatment. On average, the experiment lasted 110minutes, and the average earnings per subject were 27 euros, including a show-up fee ofnine euros.At the beginning of the experiment, printed instructions were given to the participants(see Appendix B). Participants were informed that the experiment was about the studyof decision-making. The instructions were identical for all participants of each treatment,explaining in detail the experimental setting. First, the mechanism implementing the TTCrule was explained, with an example. The participants were told that this mechanismwould be used to match them to objects for the first 14 rounds. Then, the mechanismimplementing the SD rule was explained, also with an example, and participants weretold that this mechanism would be used to match them to objects in the last sevenrounds. After round 14, participants were reminded of the switch of the mechanism.They were invited to re-read instructions for the second mechanism. Clarifying questionswere answered in private. Note that the switch between cyclic and acyclic priorities doesnot switch the mechanism, and it was not emphasized to subjects. They could, however,infer the difference from the priority tables.4.3.
Experimental Results.
The significance level of all our results is 5%, unless oth-erwise stated. We use signs > in the results between treatments to communicate signifi-cantly higher, and sign = to communicate the absence of statistically significant differencebetween treatments.4.3.1. Truthful reporting.
The focus of the paper is the comparison of the different pro-portions of equilibrium behavior induced by different mechanisms for the same allocation
ICK-AN-OBJECT MECHANISMS 23 rule. We consider the proportions of subjects following a dominant strategy of truthfulreporting in the direct mechanism, a straightforward strategy, which constitutes a robustOPBE in the PAO mechanisms, and the obviously dominant strategy in the OSP mech-anisms. To simplify the language of distinguishing between these strategies, we use theconcept of truthful strategy . A participant follows a truthful strategy under a direct mech-anism when she submits the truthful list of all eight objects. In PAO mechanisms andOSP-SD, a participant following the truthful strategy is equivalent to her following thestraightforward strategy. In OSP-TTC, participants following the truthful strategy re-quires truthfully answering the “yes-no” questions about the most-preferred object amongthe ones the participant has the highest priority, and in case of all “no,” truthful choiceof the favourite object among the other objects.
Result 1 (Behavior in line with the truthful strategy): (1) Under the TTC rule with cyclic priorities, the comparison of average proportionsof subjects behaving in line with the truthful strategy leads to the following results:PAO > Direct. (2) Under the TTC rule with acyclic priorities, the comparison of average proportionsof subjects behaving in line with the truthful strategy leads to the following results:OSP > PAO > Direct. (3) Under the SD rule, the comparison of average proportions of subjects behaving inline with the truthful strategy leads to the following results: OSP > PAO=Direct. Support:
Figure 3 presents the proportions of truthful strategies played by participants by treat-ments and rounds.First, under the TTC rule with cyclic priorities, the average proportion of truthfulstrategies under direct TTC is 20 percentage points lower than the proportion of truthfulstrategies under PAO TTC. The difference is significant. The significance of the differenceis robust to modifications of the definition of truthful strategy in the direct TTC. Notethat, in the setup of cyclic priorities, no mechanism that implements the TTC rule is OSP(Li, 2017). We observe that despite the strategy-proofness of direct TTC, the proportion Note that there are typically multiple undominated strategies, given the information available to thesubjects taking part in the experiments. We argue that the submission of the full truthful list (truncationsare not allowed by design), however, is the simplest strategy among the undominated. Nevertheless, asa robustness check we consider alternative definitions of truthful strategies throughout the section. The result is robust to two changes of the definition of truthful strategy in the direct TTC. First, ifinstead of requiring the full truthful list we count as truthful all truthful submissions until the guaranteedobject, the result remains. Second, if instead of requiring the full truthful list in the direct TTC we countas truthful all submissions with truthful ranking of objects until the assigned object, the result remains. If instead of requiring the full truthful list in the direct TTC we count as truthful all truthful submissionsuntil the guaranteed object, the result remains. If instead of requiring the full truthful list in the directTTC we count as truthful all submissions with the truthful ranking of objects until the assigned object,the difference between direct and PAO becomes not significant. If instead of requiring the full truthful list in the direct SD we count as truthful all truthful submissionsuntil the assigned object, the result remains. Note that under the informational conditions of the lastseven rounds, there’s no such thing as “guaranteed objects.”
ICK-AN-OBJECT MECHANISMS 24
Figure 3.
Truthful strategies by treatments and roundsDirect PAO OSP Direct=PAO Direct=OSP PAO=OSPp-value p-value p-valueTTC cyclic 35% 55% n/a 0.00 n/a n/aTTC acyclic 56% 67% 77% 0.00 0.00 0.00SD 72% 75% 98% 0.47 0.00 0.00
Table 2.
Proportions of the truthful strategiesof truthful strategies is just 35%, which is rather low. However, this rate is comparableto some other studies in the literature, that found similar rates of truthful reportingin TTC (for instance, 46% in Chen and S¨onmez (2006) and 41% Hakimov and Kesten(2018).) Sequentialization of the mechanism through the PAO environment leads toa significant increase in the proportion of equilibrium behavior. Note that this findingis similar to the finding of Klijn et al. (2019) and B´o and Hakimov (2020), who showthat the PAO implementation of the DA rule outperforms the direct mechanism. Thisis the main comparison of the experiment – in the absence of OSP alternatives, thePAO implementation can improve the proportions of equilibrium behavior relative to theimplementation through the direct mechanism. Two notable exceptions are Calsamiglia et al. (2010) and Pais and Pint´er (2008), who documentedhigher rates of truth-telling under TTC (62% and 85%). The high rate in Pais and Pint´er (2008) islikely driven by the fact that the rank-order list contained only three schools. Note, that in Hakimov andKesten (2018) in an environment with five schools, the rate is just 30%.
ICK-AN-OBJECT MECHANISMS 25
Second, in the case of TTC with acyclic priorities, the OSP TTC outperforms bothPAO and direct mechanisms. The difference is significant for both the test consideringonly rounds 8 to 14, or the test considering rounds 1 to 14 in OSP TTC versus rounds 8to 14 in direct and PAO TTC. As for the difference between Direct and PAO TTC, theproportion of truthful strategies is nine percentage points higher under PAO TTC, andthe difference is significant. The difference between Direct and PAO becomes smallerin TTC with acyclic priorities than in TTC with cyclic priorities. One can argue thatlearning is steeper under direct TTC. While this argument is hard to reject formally,suggestive evidence goes against this argument. More specifically, the coefficient for thevariable “round” in the probit regression of the dummy for the truthful strategy is notsignificant in either Direct nor PAO TTC. Also, there is a jump in the proportion oftruthful strategies between rounds 7 and 8 under both treatments, which suggest thatparticipants reacted to the switch of priorities from cyclic to acyclic.Third, for the SD rule, the use of the OSP mechanism results in almost universal (98%)truthful behavior, as evident from Table 2. The rate of truthful strategies in OSP SDis significantly higher than in the Direct and PAO treatments. There is no significantdifference between Direct and PAO. SD is a simple allocation rule, and thus the rates ofmanipulations are relatively low already under the Direct mechanism. Note that underPAO SD, participants might engage in multiple decisions, especially when they have alow priority: every time the chosen object is taken by someone with a higher priority, theagent is asked to pick another one.Next, we take a closer look at the determinants and the nature of deviations from thetruthful strategies.
Result 2 (Determinants of the truthful strategy): (1) Under the TTC rule with cyclic priorities, there is no significant correlation be-tween the propensity to play the truthful strategy and how high the agent is inthe priorities of objects in Direct, but there is significant correlation in PAO.(2) Under TTC rule with acyclic priorities, there is significant correlation betweenhow high the agent is in the priorities of objects and the propensity to play thetruthful strategy in all treatments. In OSP TTC, the rate of truthful behaviorin the passing strategies is much lower than in the clinching strategies, and thedifference is significant. (3) Under the SD rule, the manipulations in Direct and PAO mechanism are signifi-cantly more likely to appear if participants had a lower score. Note, however, that the significance of the difference is not robust to redefining truthful strategies indirect TTC as the truthful ranking of objects in the rank-order lists until the assigned object. In thiscase, the difference between Direct and PAO becomes not significant. We follow the insights from Pycia and Troyan (2019) and define “passing strategies” as OSP strategies,which involve saying “no” to all objects which the agent tentatively owns under OSP TTC. Thus, theagent has to pass the decision to the next agent, hoping that a better object will appear in his choice set.“Clinching strategies” are OSP strategies that require only saying “yes” to the most-preferred object inOSP TTC. Thus, whenever an agent is asked to act, she tentatively owns the most-preferred object.
ICK-AN-OBJECT MECHANISMS 26 (1) (2) (3) (4) (5) (6)Direct TTC PAO TTC Direct TCC PAO TTC Direct PAOcyclic cyclic acyclic acyclic SD SDRank at best obj. -.015 -.049*** -.088*** -.117***(.010) (.008) (.014) (.015)Rank at 2nd-best obj. .003 .026*** .007 .011(.011) (.008) (.012) (.007)Av. rank at rem. obj. .026 .039** .053** .053***(.028) (.015) (.023) (.016)Score in SD .006*** .004***(.001) (.001)Observations 385 623 371 532 462 231No. of clusters 12 12 12 12 12 12log(likelihood) -265.56 -413.00 -238.66 -308.94 -281.28 -155.20Marginal effects of Probit regressions of the truthful strategy. The sample excludes participantswho either always played the truthful strategy or never played the truthful strategy within atreatment. Rank at best obj. is the rank of the participant in the priority order of her favoriteobjects (goes from 1 to 8). Rank at 2nd-best obj. is the rank of the participant in the priorityorder of her second favorite objects (goes from 1 to 8). Av. rank at rem. obj. is the averagerank of the participant in priorities of all remaining objects. p < .
05, *** p < .
01. Standarderrors are clustered at the level of matching groups and are presented in parentheses.
Table 3.
Marginal effects of the probit model of truthful strategies on thepriority ranks in TTC and the score in SD for Direct and PAO treatments
Support:
Table 3 shows the marginal effects of probit regression models for the dummy of playingthe truthful strategy, depending on the priority of the participants.In all treatments, except Direct TTC with cyclic priorities, the priorities of the objectscan partially explain deviations from the truthful strategy. The higher the priority in themost-preferred object (i.e., the lower the rank), the more likely it is that the participantwill play the truthful strategy. The rank in other objects has the opposite average effect:the lower the priority in other objects, the more likely it is that the participants will playtruthful strategy. This result suggests that subjects attempt to list the object where theyhave relatively high priority at the top of the rank-order list or escape ranking highly theobjects where they have very low priority. Note that this is a typical deviation also inSD, where participants with a low score misrepresent their preferences more often thanparticipants with a high score. Interestingly, the priorities of objects are significant inexplaining deviations from truthful strategies in the Direct treatment only under TTCwith acyclic priorities, and not under cyclic priorities.The difference between the determinants of truthful strategies under PAO and DirectTTC under cyclic priorities is intriguing. In PAO, priorities are significant in determiningsubjects’ strategies already in cyclic TTC, while in Direct TTC, they are significant onlyin the case of acyclic priorities, thus when more objects “agree” on priorities. There aretwo distinct ways of how one can try to “manipulate” their reports under TTC:
ICK-AN-OBJECT MECHANISMS 27 • Cycles manipulations: misrepresentations of preferences such that by pointing toa different object, one participates in a cycle earlier. The manipulation should bebased on the subject’s beliefs about the preferences of other subjects. Note, thatthis manipulation is relevant only when the participant has trading power at someobject, i.e., he is ranked at the top of priority list of an object. • Clinching manipulations: misrepresentations of preferences such that by pointingto a different object, one clinches an object earlier (participants in the self-cycle).Thus, in order to make clinching manipulations, participants should consider thepriorities of objects, and rank high the objects where they have a higher priority.Notice that none of these manipulations can have the stated desirable effects. Theyrepresent (incorrect) rationalizations for the departure from truthful strategies.Under cyclic TTC, in PAO the significance of priorities in model (2) of table 3 em-phasizes attempts of the clinching manipulations. Note that in PAO, subjects effectivelyobserve cycles that happen in previous steps of the mechanism, by observing the avail-able objects and the objects that are gone from the menus. This might emphasize theimportance of priorities, i.e., the order in which the objects point at the subjects. Also,PAO makes more evident that cycles manipulations are not effective because if one is atthe top of an object, the object never disappears from the choice set, independent of theassignments of other objects (and thus other cycles). Thus, the lower frequency of at-tempts at cycles manipulations might explain the difference in the proportion of truthfulstrategies between PAO and Direct TTC.Note, that once priorities in TTC are switched from cyclic to acyclic, the relevance ofcycles manipulations is much smaller relative to one’s position in priorities, as priorities areclose to common, and they primarily influence the chances of getting the objects. That iswhy the increase in truthful strategies is higher in Direct TTC than in PAO TTC. Indeed,if cycles manipulations are more likely to be present in Direct TTC than in PAO TTCin cyclic priorities environment, once they become less relevant one can expect a higherincrease in truthful strategies in Direct TTC than in PAO TTC. This also explains lowertreatment differences between PAO and Direct in the acyclic environment. Finally, inSD, there is no difference between PAO and direct treatments, which is in line with theargument above, as cycles manipulations do not exist in SD.The presence of skipping or clinching types of manipulations in SD resembles similarstrategies in DA. A recent paper by Artemov et al. (2017) argues that the manipulationsin SD by participants with low grades are likely to be driven by the multiplicity of equilib-rium strategies, and participants making mistakes, rather than by the strategic motivesof participants. Indeed, if subjects have a strong belief that preferences are correlated,and thus that it is impossible to get the most-preferred objects if the score is low, theskipping of top choices might be inconsequential for the allocation. Thus, PAO SD doesnot improve on the Direct mechanism, as a truthful strategy may contain multiple de-cisions for low-score players, and many deviations from the truthful strategy are indeed
ICK-AN-OBJECT MECHANISMS 28
TTC acyclicN % of truthfulClinching actions 836 93%Passing actions 620 56%
Table 4.
Truthful strategies by clinching and passing in OSP TTCnon-consequential. If anything, in PAO SD, a participant has more reasons to performdeviations from truthful strategy if he believes they are irrelevant, as it can save time.OSP SD, thus, reduces these deviations, as the truthful strategy is a unique equilibrium,and every participant makes only one decision. As for the positive effect of OSP TTC,the truthful strategy is also a unique equilibrium for many participants. We take a closerlook at the determinants of the truthful strategies under OSP TTC next.In line with the theory, obvious strategy-proofness leads to a higher rate of truthfulstrategies both under TTC and SD. Note that, in general, OSP-TTC is not stronglyobviously strategy-proof, and the obviously dominant strategy might contain so-called“passing” actions that require forward-looking from participants. In line with the def-inition of Pycia and Troyan (2019), we can categorize possible paths of the obviouslydominant strategies of the participants into two groups:(1)
Clinching actions – paths of the obviously dominant strategy that contain onlyclinching actions. If by the time the OSP TTC mechanism interviewed a partic-ipant (thus, she was at the top of the priority of at least one object), she was atthe top of the priority of her favorite remaining object. This strategy is stronglyobviously strategy-proof, as it does not require passing to the other player, andthus does not require a foresight from the participant.(2)
Passing actions – paths of the obviously dominant strategy that contain atleast one passing action. If by the time the OSP TTC mechanism interviewed aparticipant (thus, she was at the top of the priority of at least one object), shewas not at the top of the priority of her favorite remaining object, and, thus, OSPrequires to “pass” the turn of interview to another participant. This path of theobviously dominant strategy does require a foresight from the participant. In thecontext of OSP TTC, these strategies require saying “no” to all objects and thenpicking the favorite object among the ones where the participant does not havethe top priority.Table 4 presents the proportion of truthful strategies in OSP TTC, depending on thepath of the OSP strategy. The rate of truthfulness in the passing actions is much lowerthan in the clinching actions. In fact, the rate is not significantly different from truthfulrates under direct TTC (p=0.94), and is significantly lower than under PAO ( p < . ICK-AN-OBJECT MECHANISMS 29
This result supports the concept of strong obvious strategy-proofness by Pycia andTroyan (2019). Indeed, when the market is such that the preferences and priorities of theobject are strongly negatively correlated (the agents prefer objects that rank them thelowest), the obvious strategy-proofness of OSP TTC might not result in the high rates ofoptimal behavior, as most paths of the OSP strategy will contain passing actions.Summing up the subsection on individual strategies, experimental results support useof the PAO mechanisms in the complex environment, where an OSP mechanism is notavailable. Once the environment is simple enough to allow for the presence of the OSPmechanisms, they should be used. The benefit of the OSP mechanisms comes mostlythrough the presence of the paths in OSP which contain only clinching actions.4.3.2.
Efficiency.
While the focus of our experiment is on the subjects’ individual behav-ior, in this subsection we look at the efficiency of the reached allocations. On one side,the matching game is often a zero sum game, and difference in efficiency is unlikely toappear. Thus, as in many previous experiments, we do not expect to find large differ-ences in efficiency between treatments. The differences might appear only due to Paretodominated allocations. Nevertheless, we would like to have a measure of the relevance ofthe manipulations to the allocations, as especially in the case of SD, many misrepresen-tations by participants with lower scores might be irrelevant for the allocation. Here, weadopt two approaches. First, similar to other market design experiments we compare theaverage ranks of assigned objects in the true preferences of participants. Second, similarto Li (2017) we consider the binary variable of reaching the allocation that would havebeen reached if everyone followed the truthful strategy.
Result 3 (Efficiency): (1) Under the TTC rule with cyclic priorities: • the comparison of average ranks of objects reached in the resulting alloca-tion in the true preferences of participants leads to the following results:Direct > PAO. • the comparison of shares of predicted allocations to the following results:PAO=Direct.(2) Under the TTC rule with acyclic priorities: • the comparison of average ranks of objects reached in the resulting alloca-tion in the true preferences of participants leads to the following results:Direct > PAO, OSP = PAO (round 8-14), OSP = Direct (round 8-14). • the comparison of shares of predicted allocations to the following results:Direct=PAO > OSP.(3) Under the SD rule : • the comparison of average ranks of objects reached in the resulting alloca-tion in the true preferences of participants leads to the following results:Direct > OSP=PAO. Consider the theoretical argument of (Liu and Pycia, 2016) for further details of this discussion.
ICK-AN-OBJECT MECHANISMS 30
Figure 4.
Average rank of assigned objects by treatments • the comparison of shares of predicted allocations to the following results:OSP > PAO=Direct.
Support:
Figure 4 shows the average rank of the assigned objects under the true preferencesof participants by rounds. Thus, the higher the rank, the worse the assignment forparticipants. Table 5 shows the average ranks of assigned objects by treatments. UnderTTC with cyclic priorities (first seven rounds), the average rank of the assigned objectsis significantly higher under Direct than under PAO. The difference is, on average, 0.29of a rank. While it might appear small, one has to take into account that the differencesin the efficiency are hard to get between the mechanisms, as often a worse assignment ofone participant leads to a better assignment of the other. Note that we do not present theresults for the OSP treatment for the first seven rounds, as the comparison would not bemeaningful, as the participants played under different priorities, and thus the equilibriumallocations are different.Under TTC with acyclic priorities, there is a small but statistically significant differ-ence between PAO and Direct. Again, the average assignment is significantly better forparticipants under the PAO treatment. There is no significant difference between OSPand other treatments. Thus, a higher rate of truthful strategies in OSP does not result inbetter average allocation of objects. We will consider the reasons for these findings later
ICK-AN-OBJECT MECHANISMS 31
Direct PAO OSP Direc=PAO Direct=OSP PAO=OSPp-value p-value p-valueTTC cyclic 2.87 2.58 n/a 0.00 n/a n/aTTC acyclic 2.43 2.19 2.34 0.00 0.33 0.08SD 2.49 2.35 2.37 0.02 0.02 0.66 dummyfor the corresponding treatment in the OLS regression of the rank of assigned object inthe true preferences of participants on the treatment dummy, with the sample restrictedto the treatments involved in test. The standard errors of all regressions are clustered
Table 5.
Average rank of assigned objects in the true preferences of theparticipants Direct PAO OSP Direc=PAO Direct=OSP PAO=OSPp-value p-value p-valueTTC cyclic 5.9% 5.9% n/a 1.00 n/a n/aTTC acyclic 22.6% 26.2% 12.6% 0.72 0.05 0.01SD 38.1% 50% 83.% 0.16 0.00 0.00
Table 6.
Proportions of equilibrium allocations reached.in this section when analyzing the average cost of the deviation from truthful behaviorby treatments.Finally, under the SD rule, the average ranks in the Direct mechanism are significantlyhigher than under OSP and PAO. Despite a similar rate of truthful strategies, the con-sequence of the deviations from truthful strategies is different between Direct and PAOtreatments. On the other hand, despite the big difference in truthful rates between PAOand OSP, the average rank of assigned objects do not differ significantly.Given the fact that it might be the case that the equilibrium allocation has a lowersum of ranks than the allocation when a participant deviates from the truthful strategy,it is essential to look at different criteria. One measure of the success of the mechanism iswhether the desired allocation is reached. This approach is used in Li (2017) to estimatethe performance of direct versus OSP SD. We adopt this approach for all treatments.Note, however, that in Li (2017), the market consists of four participants, while in ourcase, it consists of eight participants. In the larger market, we can expect a lower rateof the equilibrium allocations reached, as it is enough for one participant in the group todeviate from truthful strategy in the consequential way to distort the whole allocation.Table 6 presents the proportion of equilibrium allocations by treatments. Under TTCwith cyclic priorities, only 5.9% of allocations in Direct and PAO were equilibrium allo-cations. This low rate is not surprising, given the rate of the truthful strategies manip-ulations, and the fact that even one consequential manipulation distorts the allocations.Under TTC with acyclic priorities, we observe that the OSP treatment has the lowestrate of equilibrium allocations. Despite the higher rate of truthful strategies, as everydeviation from the truthful strategy is consequential for the allocation, it leads to only12.6% of equilibrium allocations on average, which is significantly lower than in Direct
ICK-AN-OBJECT MECHANISMS 32 and PAO treatments. Finally, under SD, OSP outperforms Direct and PAO mechanisms.Thus, we replicate the finding by Li (2017).While the comparison of the rate of equilibrium allocations is useful for a smallermarket, or under very low deviations from truthful strategies, we think the criterion isnot very informative about the consequences of deviations in the case of a high rate ofdeviations from truthful strategies. Another approach would be to analyze the differencein the consequence of deviation from truthful strategies from an individual perspective bytreatments.
Result 4 (Cost of deviation from truthful strategy): (1) Under the TTC rule with cyclic priorities the average cost of deviation from thetruthful strategy under Direct is 3.97 euros, under PAO 3.61 euros, with no sig-nificant difference between treatments.(2) Under the TTC rule with acyclic priorities the average cost of deviation from thetruthful strategy in Direct is 2.43 euros, in PAO 1.98 euros, and in OSP 3.36 euroswith no significant difference between treatments.(3) Under the SD rule the average cost of deviation from the truthful strategy inDirect is 2.53 euros, in PAO 1.55 euros, and in OSP 6.01 euros with all differencesbetween treatments being statistically significant.
Support:
Table 7 presents the results of the OLS estimation of the effect of misreporting on thepayoff of the subjects.Each regression includes 56 dummies for each combination of ID and round, to accountfor the “role-specific” fixed effects, as the roles (a combination of preferences and pri-orities/scores) vary the prospects of earning high payoffs. Thus, the coefficient for thenon-truthful dummy presents the average differences between subjects who play truth-fully relative to subjects who play non-truthfully in the Direct mechanism, controlling forthe role of the subjects. Under TTC with cyclic priorities (Model (1) of Table 7), thedeviation from the truthful strategy, on average, leads to a loss of 3.97 euros in Direct(note that the maximum payoff for the allocation is 22 euros), while the deviations are 38cents less costly in PAO, though the difference is not significant. Under TTC with acyclicpriorities (Model (2) of Table 7), the deviation from the truthful strategy, on average,leads to a loss of 2.43 euros in Direct, while the deviations are 45 cents less costly in PAO,and 93 cents more costly in OSP, though the differences are not significant. Finally, underSD, (Model (3) of Table 7), the deviation from the truthful strategy, on average, leads toa loss of 2.53 euros in Direct, while the deviations are 98 cents less costly in PAO, and3.48 euros more costly in OSP, with all differences being significant. The difference canbe explained by the fact that skipping in PAO is less consequential, due to intermediateupdates on which objects are left for allocation, which is not the case in Direct. Thehighest cost of deviations in OSP can be explained by the fact that all deviations from
ICK-AN-OBJECT MECHANISMS 33 (1) (2) (3)payoff payoff payoffTTC cyclic TTC acyclic SDNon-truthful strategy -3.97*** -2.43*** -2.53***(.29) (.29) (.24)Non-truthful in PAO .38 .45 .98**(.39) (.38) (.36)Non-truthful in OSP -.93 -3.48***(.60) (.76)Dummies for each participant ID in each round yes yes yesObservations 1344 2800 2072No. of clusters 24 37 37R .287 .370 .759log(likelihood) -4063.06 -7796.13 -5136.51Notes: OLS regression. * p < .
10, ** p < .
05, *** p < .
01. Standard errors are clustered atthe level of matching groups and are presented in parentheses. Non-truthful is a dummy fornot playing the truthful strategy. Non-truthful in PAO is the interaction of the Non-truthfuldummy and the dummy for PAO treatment. Non-truthful in OSP is the interaction of theNon-truthful dummy and the dummy for OSP treatment.
Table 7.
OLS regression of payoff on the dummy for non-truthful strategy the truthful strategy are payoff-relevant, as it is a unique equilibrium strategy in OSP,unlike in the other treatments. 5.
Conclusion
Recent empirical evidence raises concern about the practical success of direct matchingmechanisms with a dominant strategy of truthful reporting. Recent work by Li (2017) sheda new light on the design of market mechanisms, emphasizing the importance of simplerand thus potentially more successful solutions for practice. The hope was, however,not long-lasting, as many desirable allocation rules cannot be implemented via obviousstrategy-proof mechanisms.Our paper takes a different stand on potential solutions to the perceived complexity ofdirect mechanisms. When a rule cannot be implemented via OSP mechanisms but belongsto an extensive family of rules, which include many commonly considered in practice andin the literature, we suggest using PAO mechanisms. Similarly to OSP mechanisms, thesecan also implement those rules with an attractive and simple equilibrium strategy.Our experimental evidence, together with recent evidence by B´o and Hakimov (2020);Klijn et al. (2019), show that improvement over direct mechanisms in allocations and thepercentage of people following an equilibrium is possible for allocation rules for which OSPimplementation is not available. These results might appear puzzling, and may invitefurther research on the understanding of the relative strength of different equilibriumconcepts in predicting behavior, especially of inexperienced participants.
ICK-AN-OBJECT MECHANISMS 34
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Appendix A. Proofs
Theorem 1.
There exists a pick-an-object mechanism that sequentializes an individu-ally rational rule ϕ if and only if ϕ satisfies monotonic discoverability.Proof. First we assume that ϕ satisfies monotonic discoverability and construct a pick-an-object function S such that its mechanism sequentializes ϕ .Define a pick-an-object function S such that for every i and h A ∈ H A : S i (cid:0) h A (cid:1) = ∅ if (cid:12)(cid:12) ϕ (cid:0) h A (cid:1)(cid:12)(cid:12) = 1 or −→ h Ai ∈ µ ϕi (cid:0) h A (cid:1) µ ϕi (cid:0) h A (cid:1) otherwiseTo show that S sequentializes ϕ , we need to show that: (i) if the collective history h A is such that (cid:12)(cid:12) ϕ (cid:0) h A (cid:1)(cid:12)(cid:12) >
1, at least one student must be given a non-empty menu, that is,there must be at least one i such that S i (cid:0) h A (cid:1) (cid:54) = ∅ ; (ii) When an empty menu is returnedfor all agents, their last choices must be the allocation that ϕ determines what should beproduced by any preference profile consistent with the collective history.First, (i). Suppose not. Then (cid:12)(cid:12) ϕ (cid:0) h A (cid:1)(cid:12)(cid:12) > i , S i (cid:0) h A (cid:1) = ∅ . By the definitionof S , this implies that for all i , −→ h Ai ∈ µ ϕi (cid:0) h A (cid:1) . That is, the last choice of each student isa feasible match after h A .Let −→ µ be the allocation that matches each agent with her last choice in h A , that is,for every i , −→ µ ( a i ) = −→ h Ai , and P A be any preference profile consistent with h A . Mono-tonic discoverability implies that either (a) for every preference profile P (cid:48) consistent withcontinuations of h A , ϕ ( P (cid:48) ) = −→ µ , which is a contradiction with (cid:12)(cid:12) ϕ (cid:0) h A (cid:1)(cid:12)(cid:12) >
1, or (b) thatthere is at least one agent a i ∗ such that for all P ∈ L (cid:0) P A , −→ µ (cid:1) , −→ h Ai ∗ (cid:54) = µ i ∗ ( P ), which isagain a contradiction with −→ h Ai ∗ ∈ µ ϕi ∗ (cid:0) h A (cid:1) .Now, to (ii). Since an empty menu is given to all students, then by the definition of S ,either (a) (cid:12)(cid:12) ϕ (cid:0) h A (cid:1)(cid:12)(cid:12) = 1 or (b) for every i , −→ h Ai ∈ µ ϕi (cid:0) h A (cid:1) . Consider first (a). By definitionof the notation, (cid:12)(cid:12) ϕ (cid:0) h A (cid:1)(cid:12)(cid:12) = 1 implies that for all preference profiles consistent with h A ,the rule ϕ determines the same allocation. Suppose, however, that ϕ (cid:0) h A (cid:1) (cid:54) = −→ µ , and let a i be an agent for whom ϕ i (cid:0) h A (cid:1) (cid:54) = −→ µ ( a i ). Clearly, ϕ i (cid:0) h A (cid:1) cannot be any choice madebefore −→ µ ( a i ) in h A , since by design of the pick-an-object mechanism it is only rejected ifit is not a feasible match anymore. So it can either be an object which was present in amenu previous to the one where −→ h Ai was chosen but not in some future menu, or in themenu given when −→ h Ai was chosen. The first option contradicts the way in which menusare constructed: menus contain all feasible matches conditional on the collective history.So if at some point the object type was not feasible anymore, it cannot be that agent’sassignment under ϕ . For the second, this implies that the allocation was determined by ϕ to match agent a i to some object type o ∗ that was not chosen. Notice, however, that since ϕ is individually rational, a collective history cannot point to a single allocation unlessthe agents’ preferences consider these objects acceptable. That is, the collective historymust include agents choosing, at some point, these objects from a menu that includes the ICK-AN-OBJECT MECHANISMS 38 option “ ∅ ”. Therefore, it cannot be the case that ϕ is individually rational, (cid:12)(cid:12) ϕ (cid:0) h A (cid:1)(cid:12)(cid:12) = 1,and an ϕ (cid:0) h A (cid:1) matches some agent to an object that she did not choose from a menu.Now, case (b). Here, all the last choices of all agents are feasible in ϕ (cid:0) h A (cid:1) . That is,there is no agent who will not be matched to their last choice for a preference profile thatis consistent with ϕ (cid:0) h A (cid:1) . But then monotonic discoverability implies that ϕ (cid:0) h A (cid:1) = −→ µ ,which is what we wanted to show.Next, we will show that if a rule ϕ does not satisfy monotonic discoverability, then itcannot be sequentialized by some pick-an-object function S . Suppose not. Then there isa rule ϕ ∗ that does not satisfy monotonic discoverability and a pick-an-object mechanism S that sequentializes it. Since ϕ ∗ does not satisfy monotonic discoverability, there exists apreference profile P ∗ and an allocation µ ∗ such that (i) ϕ ∗ ( P ∗ ) (cid:54) = µ ∗ and (ii) for each agent a ∈ A , there is at least one preference profile P ∗ ,a ∈ L ( P ∗ , µ ∗ ) where ϕ ∗ a ( P ∗ ,a ) = µ ∗ ( a ).The first thing to note next is that (ii) implies that when all agents follow straight-forward strategies with respect to P ∗ , there must be for each agent a a period in whichshe chooses µ ∗ ( a ), and that this must happen before she chooses other objects that sheis matched to by ϕ ∗ for any profile in L ( P ∗ , µ ∗ ). That is, since when following P ∗ “up to µ ∗ ” there is some continuation in which a is matched to µ ∗ ( a ), then a must first choosethat object from a given menu.The next observation is that, after the period in which a chooses µ ∗ ( a ), the determina-tion of whether an agent a will be matched to µ ∗ ( a ) or some other object below µ ∗ ( a ) inher preference cannot depend on that agent’s preferences among objects below µ ∗ ( a ) in herpreference, since in order to obtain information about that part of agent a ’s preferences requires rejecting µ ∗ ( a ) as a potential allocation for a . This implies that whether a willbe matched to µ ∗ ( a ) or not depends on information about the other agents’ preferences.More than that, (ii) specifically implies that the conclusion that a will not be matched to µ ∗ ( a ) cannot be reached before some other agent b makes choices after having her choiceof µ ∗ ( b ) rejected. This, therefore, has the following implications: • Every agent a ∗ following the straightforward strategy with respect to P ∗ will, insome period, choose her allocation under µ ∗ , • In any periods that follow, in which the other agents did not yet choose theirallocation under µ ∗ , agent a ∗ ’s allocation may or not be determined to be µ ∗ ( a ∗ ),but will not have her choice rejected, since there is still some continuation in which µ ∗ ( a ∗ ) will be her allocation, and rejections are final.The two implications above result in the following dynamic when agents follow straight-forward strategies with respect to P ∗ : agents make choices over menus until, at somepoint, they choose their allocation under µ ∗ . After a certain number of periods, therefore,we reach a point in which all agents’ last choices are their allocations in µ ∗ . By (i), µ ∗ should not be the allocation, but by (ii), for each agent, more information about thepreference profile is necessary to point out the correct allocation to be produced. Thisrequires rejecting at least one of the agents’ choices, but by (ii) for every agent there is a ICK-AN-OBJECT MECHANISMS 39 continuation in which she is matched to her assignment under µ ∗ . So no more informationcan be obtained when using any pick-an-object function, leading to a contradiction. (cid:3) Proposition 1. If ϕ is described by a generalized DA procedure, then ϕ satisfies mono-tonic discoverability.Proof. In light of Theorem 1, it suffices to show that there is a pick-an-object mechanismthat sequentializes the rule that is specified by the generalized DA procedure. Let Ψ ∗ bethe update function used to describe the rule ϕ . We construct the menu function S ∗ asfollows: S ∗ (cid:0) h A −∅ (cid:1) = ( O, O, O, . . . )The value of S ∗ for other collective histories are determined, recursively, as follows. Let h A be a collective history for which the value of S ∗ (cid:0) h A (cid:1) has already been determined as S ∗ (cid:0) h A (cid:1) = ( φ , φ , . . . , φ n ).For each choice profile ( o , o , . . . , o n ), where for each i ∈ A , o i ∈ φ i if φ i (cid:54) = ∅ and o i = ♦ otherwise, perform the following:(1) Construct the assignments µ and µ as follows: • For every a i ∈ A : – If o i = ♦ and h A (cid:54) = h A −∅ , let µ ( a i ) = −→ h Ai . – Otherwise, let µ ( a i ) = ∅ . • For every a i ∈ A , let µ ( a i ) = ∅ if o i = ♦ , and µ ( a i ) = o i otherwise.(2) For every a i ∈ A , define the choice history h i to be: • h i = h Ai ⊕ ( φ i , o i ) if o i (cid:54) = ♦ , • h i = h Ai otherwise.(3) Let µ = Ψ ∗ ( µ , µ ). • If for every a ∈ A it is the case that µ t (3) ∈ { µ ( a ) , µ ( a ) } , then S ∗ ( h , h , . . . , h n ) =( ∅ , ∅ , . . . , ∅ ). • Otherwise, S ∗ ( h , h , . . . , h n ) = ( φ (cid:48) , φ (cid:48) , . . . , φ (cid:48) n ), where for each a i ∈ A : – φ (cid:48) i = ∅ if µ ( a i ) ∈ { µ ( a i ) , µ ( a i ) } , – φ (cid:48) i = O \ (cid:83) (Ω ,ω ) ∈ h Ai ω otherwise.Notice first that S ∗ is a menu function: initial menus are the entire set O , and for everycollective history the menus given are precisely the last menu an agent was given with herlast choice removed. Since S ∗ is defined recursively for each collective history that canbe generated by choices from the menus that can be offered, every possible path of thepick-an-object mechanism is well defined. Consider now the pick-an-object mechanism S ∗ and the agents in A following straightforward strategies with respect to P . Here the symbol ♦ is used as a placeholder for the agents who are not presented with a menu to choosefrom. ICK-AN-OBJECT MECHANISMS 40
What follows is an exact reproduction of the steps of the generalized DA procedureunder the preference profile P . Since the menus always include all elements of O minus theagents’ past choices, agents will make choices from the top until their last choice followingtheir preference, as in the generalized DA. Agents who have their last choices rejected andare given new menus, for any collective history, are determined by the function Ψ ∗ , sothat the way in which the sequences of choices from menus determine whether an agentis tentatively matched or not is given by that function. And finally, whenever the lastchoice of agents should be determined as the outcome, S ∗ returns a list of empty menus.We can conclude, therefore, that the pick-an-object mechanism S ∗ sequentializes therule ϕ . (cid:3) Proposition 2.
A rule is strategy-proof if and only if it satisfies recursive dominance.Proof.
Suppose not. Then there is a strategy-proof rule ϕ , an agent a with preference P a ,and other agents with preferences P − a , where: ϕ a (cid:0) P ∗ , P − a (cid:1) P a ϕ a (cid:0) P (cid:48) , P − a (cid:1) ( a P ∗ = γ ⊕ λ , P (cid:48) = γ ⊕ P | O \ γ , and λ (cid:54) = P | O \ γ .First, note that if ϕ a ( P (cid:48) , P − a ) ∈ λ , then ϕ a ( P ∗ , P − a ) ∈ λ . Otherwise, an agentwhose true preference is P (cid:48) would obtain a more preferred allocation by reporting thepreference P ∗ , a violation of strategy-proofness of ϕ . Similarly, if ϕ a ( P (cid:48) , P − a ) ∈ γ , then ϕ a ( P ∗ , P − a ) ∈ γ . Otherwise, an agent whose true preference is P ∗ would obtain a morepreferred allocation by reporting the preference P (cid:48) . We have, therefore, two cases: (c1) ϕ a ( P (cid:48) , P − a ) ∈ γ and ϕ a ( P ∗ , P − a ) ∈ γ , and (c2) ϕ a ( P (cid:48) , P − a ) ∈ λ and ϕ a ( P ∗ , P − a ) ∈ λ .For ease of notation, let ϕ a ( P (cid:48) , P − a ) = o and ϕ a ( P (cid:48) , P − a ) = o . By assumption ( a o (cid:54) = o .Consider case ( c P (cid:48) and P ∗ compare objects o and o , there are fourpossibilities: ( i ): o P (cid:48) o and o P ∗ o , ( ii ): o P (cid:48) o and o P ∗ o , ( iii ): o P (cid:48) o and o P ∗ o ,and ( iv ): o P (cid:48) o and o P ∗ o . Since in case ( c
1) both o and o are in γ , P (cid:48) and P ∗ compare both objects in the same way, so neither ( ii ) or ( iii ) can be true. Next, if ( i ) istrue, then an agent with preferences P ∗ would obtain a better allocation when reporting P (cid:48) instead, and if ( iv ) is true, then an agent with preferences P (cid:48) would obtain a betterallocation when reporting P ∗ instead, both contradictions with strategy-proofness of ϕ .We conclude, therefore, that case ( c
1) is false.Next, consider case ( c P (cid:48) and P ∗ compare objects o and o . Using the same argument as above, possibilities ( i ) and ( iv )contradict the strategy-proofness of ϕ , and are therefore false. Now consider possibility( ii ). By the definition of P (cid:48) , since o and o are in λ , P (cid:48) compares o and o in the sameway as P a . But here o P (cid:48) o , and by ( a o P a o , a contradiction. The last possibility is( iii ). Here, o P (cid:48) o , which does not contradict ( a o P ∗ o , which implies that anagent with true preferences P ∗ would be better off by reporting P (cid:48) , a contradiction withthe strategy-proofness of ϕ , implying that ( iii ) is also false and finishing the proof. (cid:3) ICK-AN-OBJECT MECHANISMS 41
Theorem 2.
A rule is pick-an-object implementable in a robust ordinal perfect Bayesianequilibrium if and only if it is strategy-proof and satisfies monotonic discoverability.Proof.
Let ϕ be a rule that is strategy-proof and satisfies monotonic discoverability. ByTheorem 1, the mechanism S ϕ , described in the proof of that theorem, sequentializes ϕ .That is, given some preference profile P , if every agent follows the straightforwardstrategy, the pick-an-object mechanism S ϕ produces the outcome ϕ ( P ).We will use the following claim: Claim.
Let S ϕ be a pick-an-object mechanism, and h A be any collective history in H A S ϕ .Then, there is a list of tuples ( γ , γ , . . . , γ n ) , each with distinct elements of O , such thatthe set of outcomes produced by S ϕ in any continuation history of h A is given by: (cid:91) P ∈ P | γ ×···× P | γn ϕ ( P ) Proof.
Given the definition of S ϕ , agents’ choices are used as revealed preference, and theallocation that is produced is one that ϕ indicates being the unique allocation for thepreference profiles consistent with these choices. By assumption, all agents followed somearbitrary strategy up to the collective history h A .When considering any agent a ∈ A , after the menus that were given to her and herchoices, present in collective history h A , there are potentially multiple preferences overthe objects in O \ µ ϕa (cid:0) h A (cid:1) that are consistent with the choices made by a . One thing wecan say, however, is that none of the objects in µ ϕa (cid:0) h A (cid:1) were chosen, since all of themmay still become an outcome for a in a continuation collective history, by the definitionof µ ϕa itself. We can partition O into three sets O a , O a , O a : O a being the set of objectsthat a chose from a menu in some period in h A , O a = µ ϕa (cid:0) h A (cid:1) , and O a having all theother objects.Clearly, any preference P a in which: • (i) o ∈ O a and o (cid:48) ∈ O a implies oP a o (cid:48) and • (ii) o ∈ O a , o (cid:48) ∈ O a , and o was chosen by a in h A before o (cid:48) implies oP a o (cid:48) is consistent with the collective history h A . Moreover, any continuation history of h A is consistent with preferences with the following structure: o a , P a o a , P a · · · P a o a ,k P a { O a } P a { O a } where (cid:0) o a , , o a , , . . . o a ,k (cid:1) is the set O a ordered by the period in which the object waschosen by a from a menu in , and { O a } and { O a } represent any permutation of theelements in these sets. Since, for each agent a , all menus and choices that will take place Note that preferences in which the objects in O a are intertwined between the objects in O a and O a are also consistent with h A . But since the pick-an-object mechanism being considered produces anoutcome without eliciting more information, we can safely conclude that the allocation rule yields thesame allocation for any preference consistent with those choices, in particular those preferences in whichall objects in O a are preferred to all objects in O a , which in turn are all preferred to the objects in O a . ICK-AN-OBJECT MECHANISMS 42 in any continuation collective history involve only objects in O a , continuation historieswill be consistent with any particular ordering of the objects in O a .For every a and O a defined above, let o a , , o a , , . . . , o a ,(cid:96) be a permutation of the elementsof O a . The reasoning above allows us to conclude, therefore, that conditional on h A , theoutcome that will be produced by the pick-an-object mechanisms in question is ϕ ( P ∗ ),where for each agent a ∈ A its preference in the profile P ∗ is the following: P ∗ a = γ a ⊕ λ a where: γ a = (cid:0) o a , , o a , , . . . , o a ,k , o a , , o a , , . . . , o a ,(cid:96) (cid:1) and λ a is a permutation of the elements of O a consistent with the choices made by a , after the collective history h A , over subsets of O a . Since S ϕ sequentializes ϕ , anycombination of values of ( λ a ) a ∈ A above which lead to different outcomes under ϕ arecontinuation collective histories of h A in H A S ϕ . Therefore, for every P ∈ P | γ × · · · × P | γ n and µ = ϕ ( P ), there is a collection of tuples ( λ a ) a ∈ A such that the preference profile P ∗ constructed above is such that µ = ϕ ( P ) = ϕ ( P ∗ ), finishing the proof. (cid:3) Suppose now, for contradiction, that ϕ is not implementable in robust OPBE. Then,there is a belief system θ for which a strategy profile in which all agents follow straight-forward strategies is not an OPBE. That is, there is an agent a and a strategy σ (cid:48) a , whichis not straightforward after h A , for which O θa (cid:0) h A , ( σ a , σ − a ) (cid:1) does not first-order stochas-tically dominate O θa (cid:0) h A , ( σ (cid:48) a , σ − a ) (cid:1) , where ( σ a , σ − a ) is the strategy profile in which allagents follow strategies that are straightforward in any continuation of h A . That is, if A = O θa (cid:0) h A , ( σ a , σ − a ) (cid:1) and A (cid:48) = O θa (cid:0) h A , ( σ (cid:48) a , σ − a ) (cid:1) , there is an object o ∈ O for which: P r { A = o (cid:48) | o (cid:48) R a o } < P r { A (cid:48) = o (cid:48) | o (cid:48) R a o } That is, the probability of obtaining an object at least as good as o is strictly higherunder A (cid:48) than under A . Since these random outcomes are produced by a distributionover deterministic outcomes, this in turn implies that there is at least one collectivehistory h A ∗ ∈ I ai , where h A ∈ I ai and θ a (cid:0) h A ∗ (cid:1) >
0, in which the deterministic outcomeof following the strategy profile ( σ (cid:48) a , σ − a ) after h A ∗ is strictly preferred by a over thedeterministic outcome that a obtains under the profile ( σ a , σ − a ) after h A ∗ . Let θ ∗ be abelief system in which θ a (cid:0) h A ∗ (cid:1) = 1, o = O θ ∗ a (cid:0) h A ∗ , ( σ a , σ − a ) (cid:1) and o (cid:48) = O θ ∗ a (cid:0) h A ∗ , ( σ (cid:48) a , σ − a ) (cid:1) (notice that since θ ∗ is degenerate at the information set I ai , outcomes are deterministicfor any given strategy profile). Since o (cid:48) P a o , the outcome of following the strategy profile( σ (cid:48) a , σ − a ) is different from following ( σ a , σ − a ), implying that under the first profile, agent a makes at least one choice from a menu that is not straightforward. That is, agent a chose a less preferred object with respect to P a than another that was in the menu. That is, h A ∗ and h A are in the same information set. ICK-AN-OBJECT MECHANISMS 43
As by the claim above, the outcome of following the profile ( σ a , σ − a ) after h A ∗ is ϕ (cid:0) P ∗ a , P ∗− a (cid:1) , whereas following the profile ( σ (cid:48) a , σ − a ) after h A ∗ yields ϕ (cid:0) P ∗∗ a , P ∗− a (cid:1) , where P ∗ a and P ∗∗ a differ only in how they rank the objects in O \ γ a , all of them being at the tailof a preference ranking that is the same for all the remaining objects. Since P ∗ a ranks theobjects in O \ γ a with respect to P a and P ∗∗ a does not, agent a obtaining a more preferredobject under ϕ (cid:0) P ∗∗ a , P ∗− a (cid:1) contradicts the recursive dominance of ϕ .Finally, suppose that there is a rule ϕ ∗ which is not strategy-proof but is pick-an-objectimplementable in a robust OPBE. Let the pick-an-object mechanism S ϕ ∗ implement ϕ ∗ in a robust OPBE. By theorem 1, ϕ ∗ satisfies monotonic discoverability. Since ϕ ∗ is notstrategy-proof, then there is a preference profile P , an agent a ∈ A and a preference P (cid:48) a (cid:54) = P a for which: ϕ a ( P (cid:48) a , P − a ) P a ϕ a ( P a , P − a )For any agent a , let σ a denote the straightforward strategy with respect to P a , and σ the corresponding strategy profile. Since S ϕ ∗ sequentializes ϕ ∗ , when all agents follow σ , the outcome produced by S ϕ ∗ is ϕ a ( P a , P − a ). Moreover, the outcome ϕ a ( P (cid:48) a , P − a )is produced when all agents but a follow straightforward strategies, and a follows thedeviation strategy σ (cid:48) a , “straightforward as if her preference was P (cid:48) a ”. Clearly, since theoutcome is different, when following the deviation strategy σ (cid:48) a , agent a makes at leastone choice that is not straightforward. Consider next the earliest collective history h A in which the choices under the profiles σ and ( σ (cid:48) a , σ − a ) differ, and a belief system θ forwhich θ a (cid:0) h A (cid:1) = 1. By assumption, by following σ (cid:48) a instead of σ a after h A , agent a isstrictly better off. Which is a contradiction with σ a first-order stochastically dominatingany other strategy, including σ (cid:48) a . (cid:3) Theorem 3.
Every non-bossy OSP implementable rule is pick-an-object implementablein weakly dominant strategies.Proof.