Monotonic and implementable solutions in generalized matching problems

We consider one-to-one matching problems when preferences are weak orders and search for (Maskin-)monotonic and implementable solutions. We show that for any coalition structure , the strong -core is monotonic if and only if it is the individually rational solution. We introduce a solution, which we call the “union strong -core”, and show that this solution is the minimal way to expand the strong -core recovering monotonicity. We also show that the union strong -core is implementable.     

Ordinal efficiency and dominated sets of assignments

Using lotteries is a common tool for allocating indivisible goods. Since obtaining preferences over lotteries is often difficult, real-life mechanisms usually rely on ordinal preferences over deterministic outcomes. Bogomolnaia and Moulin (J. Econom. Theory 19 (2002) 623) show that the outcome of an ex post efficient mechanism may be stochastically dominated. They define a random assignment to be ordinally efficient if and only if it is not stochastically dominated. In this paper we investigate the relation between ex post efficiency and ordinal efficiency. We introduce a new notion of domination defined over sets of assignments and show that a lottery induces an ordinally efficient random assignment if and only if each subset of the full support of the lottery is undominated.     

Stochastic stability for roommate markets

We show that for any roommate market the set of stochastically stable matchings coincides with the set of absorbing matchings. This implies that whenever the core is non-empty (e.g., for marriage markets), a matching is in the core if and only if it is stochastically stable, i.e., stochastic stability is a characteristic of the core. Several solution concepts have been proposed to extend the core to all roommate markets (including those with an empty core). An important implication of our results is that the set of absorbing matchings is the only solution concept that is core consistent and shares the stochastic stability characteristic with the core.     

New axioms for top trading cycles

The school choice problem is of great importance both in theory and practice. This paper studies the (student-optimal) top trading cycles mechanism (TTCM) in an axiomatic way. We introduce two new axioms: MBG (mutual best group)-quota-rationality and MBG-robust efficiency. While stability implies MBG-quota rationality, MBG-robust efficiency is weaker than robust efficiency, which is stronger than the combination of efficiency and group strategy proofness. The TTCM is characterized by MBG-quota rationality and MBG-robust efficiency. Our results construct a new basis to compare the TTCM with the other school choice mechanisms, in terms of both fairness and incentive-related axioms.     

Market structure and matching with contracts

Ostrovsky (2008) [9] develops a theory of stability for a model of matching in exogenously given networks. For this model a generalization of pairwise stability, chain stability, can always be satisfied as long as agents' preferences satisfy same side substitutability and cross side complementarity. Given this preference domain I analyze the interplay between properties of the network structure and (cooperative) solution concepts. The main structural condition is an acyclicity notion that rules out the implementation of trading cycles. It is shown that this condition and the restriction that no pair of agents can sign more than one contract with each other are jointly necessary and sufficient for (i) the equivalence of group and chain stability, (ii) the core stability of chain stable networks, (iii) the efficiency of chain stable networks, (iv) the existence of a group stable network, and (v) the existence of an efficient and individually stable network. These equivalences also provide a rationale for chain stability in the unrestricted model. The (more restrictive) conditions under which chain stability coincides with the core are also characterized.     

Sorting versus screening: Search frictions and competing mechanisms

In a market where sellers compete by posting trading mechanisms, we allow for a general search technology and show that its features crucially affect the equilibrium mechanism. Price posting prevails when meetings are rival, i.e., when a meeting by one buyer reduces another buyer's meeting probability. Under price posting buyers reveal their type by sorting ex-ante. Only if the meeting technology is sufficiently non-rival, price posting is not an equilibrium. Multiple buyer types then visit the same sellers who screen ex-post through auctions.     

Von Neumann-Morgenstern stable sets, discounting, and Nash bargaining

We establish a link between von Neumann-Morgenstern stable set and the Nash solution in a general n-player utility set. The stable set-solution is defined with respect to a dominance relation: payoff vector u dominates v if one player prefers u even with one period delay. We show that a stable set exists and, if the utility set has a smooth surface, any stable set converges to the Nash bargaining solution when the length of the period goes to zero.     

Efficient assignment respecting priorities

A widespread practice in assignment of heterogeneous indivisible objects is to prioritize some recipients over others depending on the type of the object. Leading examples include assignment of public school seats, and allocation of houses, courses, or offices. Each object comes with a coarse priority ranking over recipients. Respecting such priorities constrains the set of feasible assignments, and therefore might lead to inefficiency, highlighting a tension between respecting priorities and Pareto efficiency. Via an easily verifiable criterion, we fully characterize priority structures under which the constrained efficient assignments do not suffer from such welfare loss, and the constrained efficient rule (CER) is indeed efficient. We also identify the priority structures for which the CER is singleton-valued and group strategy-proof.     

Stable schedule matching under revealed preference

Baiou and Balinski (Math. Oper. Res., 27 (2002) 485) studied schedule matching where one determines the partnerships that form and how much time they spend together, under the assumption that each agent has a ranking on all potential partners. Here we study schedule matching under more general preferences that extend the substitutable preferences in Roth (Econometrica 52 (1984) 47) by an extension of the revealed preference approach in Alkan (Econom. Theory 19 (2002) 737). We give a generalization of the Gale-Shapley algorithm and show that some familiar properties of ordinary stable matchings continue to hold. Our main result is that, when preferences satisfy an additional property called size monotonicity, stable matchings are a lattice under the joint preferences of all agents on each side and have other interesting structural properties.     

Why do popular mechanisms lack efficiency in random environments?

We consider the problem of randomly assigning n indivisible objects to n agents. Recent research introduced a promising mechanism, the probabilistic serial that has superior efficiency properties than the most common real-life mechanism random priority. On the other hand, mechanisms based on Gale's celebrated top trading cycles method have long dominated the indivisible goods literature (with the exception of the present context) thanks to their outstanding efficiency features. We present an equivalence result between the three kinds of mechanisms, that may help better understand why efficiency differences among popular mechanisms might arise in random environments. This result also suggests that the probabilistic serial and the random priority mechanisms can be viewed as two top trading cycles based mechanisms that essentially differ in the initial conditions of the market before trading starts.     

Simple Markovian equilibria in dynamic spatial legislative bargaining

The paper proves, by construction, the existence of Markovian equilibria in a dynamic spatial legislative bargaining model. Players bargain over policies in an infinite horizon. In each period, a sequential protocol of proposal-making and voting, with random proposer recognitions and a simple majority, produces a policy that becomes the next period's status-quo; the status-quo is endogenous. The construction relies on simple strategies determined by strategic bliss points computed by the algorithm we provide. A strategic bliss point, the dynamic utility ideal, is a moderate policy relative to a bliss point, the static utility ideal. Moderation is strategic and germane to the dynamic environment; players moderate in order to constrain the future proposals of opponents. Moderation is a strategic substitute; when a player's opponents do moderate, she does not, and when they do not moderate, she does. We provide conditions under which the simple strategies induced by the strategic bliss points computed by the algorithm deliver a Stationary Markov Perfect equilibrium, and we prove its existence in generic games with impatient players and in symmetric games. Because the algorithm constructs all equilibria in simple strategies, we provide their general characterization, and we show their generic uniqueness.     

Collective choice under dichotomous preferences

Agents partition deterministic outcomes into good or bad. A mechanism selects a lottery over outcomes (time-shares). The probability of a good outcome is the canonical utility. The utilitarian mechanism averages over outcomes with largest “approval”. It is efficient, strategyproof, anonymous and neutral.We reach an impossibility if, in addition, each agent's utility is at least , where n is the number of agents; or is at least the fraction of good to feasible outcomes. We conjecture that no ex ante efficient and strategyproof mechanism guarantees a strictly positive utility to all agents, and prove a weaker statement.     

The probabilistic serial mechanism with private endowments

A random assignment is ordinally efficient if it is not stochastically dominated with respect to individual preferences over sure objects. When there are no private endowments, the set of ordinally efficient random assignments is characterized by the eating algorithm (Bogomolnaia and Moulin, 2001). When there are private endowments, the main requirement is individual rationality; however, the eating algorithm fails to deliver this property. Our contribution is the natural generalization of the eating algorithm for this general class of problems. The family of this generalized eating algorithm characterizes the set of individually rational and ordinally efficient random assignments. A special solution in this family, the individually rational probabilistic serial (${\mathrm{PS}}^{\mathbf{IR}}$), also achieves a new fairness axiom, no justified-envy. However, it is not immune to strategic manipulation. We show that individual rationality, no justified-envy and strategy-proofness are incompatible.     

Dynamic signaling and market breakdown

We consider the effects a public revelation of information (e.g. rating, grade) has on trading in a dynamic signaling model. Competing buyers offer prices to a privately informed seller who can reject them and delay trade. Delay is costly and the seller has no commitment to its duration. The external public information allows for signaling in equilibrium. More interestingly, we characterize the dynamics of trade and prices. If signals are noisy, no trade takes place just before the revelation of external information. If signals are fully revealing, then trade occurs even close to revelation, however, transaction prices are discontinuous.     

Hope springs eternal: learning and the stability of cooperation in short horizon repeated games

This paper considers learning rates in finitely repeated prisoners’ dilemmas. If players think their opponents might be relatively cooperative (e.g., tit-for-tat or grim types), they will cooperate in finitely repeated prisoners’ dilemmas (see Kreps et al., J. Econom. Theory 27 (1982) 245). However, if there are actually no cooperative types, players will eventually learn this and cooperation will break down. This paper shows that this learning is extremely slow, so it will take an extremely long time for cooperation to break down.Thus, suppose the world is either “good” or “bad.” The probability of a grim type is δ>0 if the world is good, and zero if the world is bad. Successive generations pair up to play finitely repeated prisoners’ dilemmas. Players observe play in previous generations and use Bayes’ rule to update their prior, π, that the world is good. We show that, if the world is really bad, then π falls per generation on average. Thus, if δ is small, there is less cooperation if the world is good, but cooperation may become more stable. For a representative 19 period repeated prisoners’ dilemma, beliefs fall one percentage point on average after a thousand generations.To derive these learning rates, we must refine existing results on the sensitivity of repeated games to Kreps et al. (1982) type perturbations. Specifically, we show cooperation is possible in perturbed prisoners’ dilemmas repeated O(log(1/δ)) times. This improves significantly on the O(1/δ) results in previous work. The paper thus provides two new reasons why cooperation tends to be stable, even in short horizon repeated games.     

Bargaining and inequity aversion: On the efficiency of the double auction

In a bargaining setting with asymmetrically informed, inequity-averse parties, a fully efficient mechanism (i.e., the double auction) exists if and only if compassion is strong. Less compassionate parties do not trade in the double auction in the limit of strong envy.     

THE SECOND‐PRICE AUCTION SOLVES KING SOLOMON'S DILEMMA*

The planner wants to give k identical, indivisible objects to the top k valuation agents at zero costs. Each agent knows her own valuation of the object and whether it is among the top k. Modify the (k+ 1)st‐price sealed‐bid auction by introducing a small participation fee and the option not to participate in it. This simple mechanism implements the desired outcome in iteratively undominated strategies. Moreover, no pair of agents can profitably deviate from the equilibrium by coordinating their strategies or bribing each other.     

Unmediated communication in games with complete and incomplete information

We study the effects of adding unmediated communication to static, finite games of complete and incomplete information. We characterize S^U(G), the set of outcomes of a game G, that are induced by sequential equilibria of cheap talk extensions. A cheap talk extension of G is an extensive-form game in which players communicate before playing G. A reliable mediator is not available and players exchange private or public messages that do not affect directly their payoffs. We first show that if G is a game of complete information with five or more players and rational parameters, then S^U(G) coincides with the set of correlated equilibria of G. Next, we demonstrate that if G is a game of incomplete information with at least five players, rational parameters and full support (i.e., all profiles of types have positive probability), then S^U(G) is equal to the set of communication equilibria of G.     

A three-player dynamic majoritarian bargaining game

We analyze an infinitely repeated divide-the-dollar bargaining game with an endogenous reversion point. In each period a new dollar is divided among three legislators according to the proposal of a randomly recognized member—if a majority prefer so—or according to previous period's allocation otherwise. Although current existence theorems for Markovian equilibria do not apply for this dynamic game, we fully characterize a Markov equilibrium. The equilibrium is such that irrespective of the discount factor or the initial division of the dollar, the proposer eventually extracts the whole dollar in all periods. We also show that proposal strategies are weakly continuous in the status quo that equilibrium expected utility is not quasi-concave, and the correspondence of voters’ acceptance set (the set of allocations weakly preferred over the status quo) fails lower hemicontinuity.     

Indeterminacy and directed search

The directed search approach assumes each seller posts a fixed price and, ex post, randomly allocates the good should more than one buyer desire the good. This paper assumes sellers can post prices which are contingent on ex post realized demand; e.g. an advertisement might state the Bertrand price should there be more than one buyer, which corresponds to an auction outcome. Competition in fixed prices and ex post rationing describes equilibrium behavior. There is also real market indeterminacy: a continuum of equilibria exists which are not payoff equivalent. Sellers prefer the equilibrium in auctions.