Maskin Monotonicity Research Articles

Semi-random Impossibilities of Condorcet Criterion

The Condorcet criterion (CC) is a classical and well-accepted criterion for voting. Unfortunately, it is incompatible with many other desiderata including participation (PAR), half-way monotonicity (HM), Maskin monotonicity (MM), and strategy-proofness (SP). Such incompatibilities are often known as impossibility theorems, and are proved by worst-case analysis. Previous work has investigated the likelihood for these impossibilities to occur under certain models, which are often criticized of being unrealistic. We strengthen previous work by proving the first set of semi-random impossibilities for voting rules to satisfy CC and the more general, group versions of the four desiderata: for any sufficiently large number of voters n, any size of the group 1

Nash implementation on the basis of general priorities

We consider a school choice problem under general priorities with ties. Priorities in practice are usually complex since a school may rank students equally or care about an affirmative action policy. Thus, we do not specify a class of priorities already known, but abstractly treat all priorities such that a stable matching exists for all students' preference profiles. For those priorities, it is unknown whether stable matchings are achievable when students are strategic. We show that a stable correspondence is implementable in Nash equilibria. Then, we focus on the Pareto-frontier of stable matchings, which we call constrained efficient stable. We show that, under a reasonable class of priorities, a constrained efficient stable correspondence is Nash implementable if and only if it satisfies Maskin monotonicity. Finally, we identify a necessary and sufficient condition on priorities under which a constrained efficient stable correspondence is Nash implementable.

Maskin meets Abreu and Matsushima

The theory of full implementation has been criticized for using integer/modulo games, which admit no equilibrium (Jackson (1992)). To address the critique, we revisit the classical Nash implementation problem due to Maskin (1977, 1999) but allow for the use of lotteries and monetary transfers as in Abreu and Matsushima (1992, 1994). We unify the two well‐established but somewhat orthogonal approaches in full implementation theory. We show that Maskin monotonicity is a necessary and sufficient condition for (exact) mixed‐strategy Nash implementation by a finite mechanism. In contrast to previous papers, our approach possesses the following features: finite mechanisms (with no integer or modulo game) are used; mixed strategies are handled explicitly; neither undesirable outcomes nor transfers occur in equilibrium; the size of transfers can be made arbitrarily small; and our mechanism is robust to information perturbations.

Axiomatic foundations of the core for games in effectiveness form

We provide an axiomatic characterization of the core of games in effectiveness form. We point out that the core, whenever it applies to appropriate classes of these games, coincides with a wide variety of prominent stability concepts in social choice and game theory, such as the Condorcet winner, the Nash equilibrium, pairwise stability, and stable matchings, among others. Our characterization of the core invokes the axioms of weak nonemptiness, coalitional unanimity, and Maskin monotonicity together with a principle of independence of irrelevant states, and uses in its proof a holdover property echoing the conventional ancestor property. Taking special cases of this general characterization of the core, we derive new characterizations of the previously mentioned stability concepts.

Rationalizable implementation in finite mechanisms

We prove that the Maskin monotonicity⁎ condition (proposed by Bergemann et al. (2011)) fully characterizes exact rationalizable implementation in an environment with lotteries and transfers. Different from previous papers, our approach possesses many appealing features simultaneously, e.g., finite mechanisms with no integer game or modulo game are used; no transfers are made in any rationalizable profile; the message space is small; the implementation is robust to information perturbations in the sense of Oury and Tercieux (2012).

Influence in private-goods allocation

We reinterpret the ‘bossiness’ of a private-goods allocation rule (Satterthwaite and Sonnenschein, 1981) as the ability of an agent to ‘influence’ another’s welfare with no change to her own welfare. In applications where non-bossiness is not possible, we propose simple conditions on (1) which agents may have influence (acyclicity and preservation), and (2) the welfare consequences of influence (positivity and oppositeness). We apply these conditions to three well-known bossy rules: the ‘Vickrey rule’ in single-object auctions (Vickrey, 1961) (acyclic, positive), the ‘doctor-optimal stable rule’ in matching with contracts (Hatfield and Milgrom, 2005) (acyclic, positive, preserving) and ‘generalised absorbing top-trading cycles (GATTC) rules’ in housing markets with indifferences in preferences (Aziz and Keijzer, 2011) (acyclic, opposite, preserving). Under mild restrictions, we show how the nature of influence under a strategy-proof rule determines whether or not it satisfies weak group-strategy-proofness (requires acyclicity and either positivity or preservation), weak Maskin monotonicity (acyclicity and positivity) and Pareto-efficiency (acyclicity and oppositeness). In addition, we propose an influence-related generalisation of the efficiency-adjusted deferred acceptance mechanism in school choice (Kesten, 2010), and characterise influence for strategy-proof GATTC rules in housing markets.

Influence in Private-Goods Allocation

We reinterpret the `bossiness' of a private-goods allocation rule (Satterthwaite and Sonneschein, 1981) as the ability of an agent to `influence' another's welfare with no change to her own welfare. We propose simple conditions on (1) which agents may have influence (`acyclicity' and `preservation'), and (2) the welfare consequences of influence (`positivity' and `oppositeness'). We apply these conditions to three well-known bossy rules: the `Vickrey rule' in single-object auctions (Vickrey, 1961) (acyclic, positive), the `doctor-optimal stable rule' in matching with contracts (Hatfield and Milgrom, 2005) (acyclic, positive, preserving) and `generalised absorbing top-trading cycles (GATTC) rules' in housing markets with indifferences in preferences (Aziz and Keijzer, 2011) (acyclic, opposite, preserving). Under mild restrictions, we show how the nature of influence under a strategy-proof rule determines whether or not it satisfies `weak group-strategy-proofness' (requires acyclicity and either positivity or preservation), `weak Maskin monotonicity' (acyclicity and positivity) and `Pareto-efficiency' (acyclicity and oppositeness). In addition, we propose an influence-related generalisation of the`efficiency-adjusted deferred acceptance mechanism' in school choice (Kesten, 2010), and characterise influence for strategy-proof GATTC rules in housing markets.

Rationalizable Implementation of Correspondences

A new condition, which we call uniform monotonicity, is shown to be necessary and almost sufficient for rationalizable implementation of correspondences. Uniform monotonicity is much weaker than Maskin monotonicity and reduces to it in the case of functions. Maskin monotonicity, the key condition for Nash implementation, had also been shown to be necessary for rationalizable implementation of social choice functions. Our conclusion is that the conditions for rationalizable implementation are not only starkly different from but also, much weaker than those for Nash implementation when we consider social choice correspondences. Thus, dropping rational expectations significantly expands the class of rules that can be decentralized by communication-based economic institutions.

One-step-ahead implementation

In many situations, agents are involved in an allocation problem that is followed by another allocation problem whose optimal solution depends on how the former problem has been solved. In this paper, we take this dynamic structure of allocation problems as an institutional constraint. By assuming a finite number of allocation problems, one for each period/stage, and by assuming that all agents in society are involved in each allocation problem, a dynamic mechanism is a period-by-period process. This process generates at any period-t history a period-t mechanism with observable actions and simultaneous moves. We also assume that the objectives that a planner wants to achieve are summarized in a social choice function (SCF), which maps each state (of the world) into a period-by-period outcome process. In each period t, this process selects for each state a period-t socially optimal outcome conditional on the complete outcome history realized up to period t−1. Heuristically, the SCF is one-step-ahead implementable if there exists a dynamic mechanism such that for each state and each realized period-t history, each of its subgame perfect Nash equilibria generates a period-by-period outcome process that coincides with the period-by-period outcome process that the SCF generates at that state from period t onwards. We identify a necessary condition for SCFs to be one-step-ahead implemented, one-step-ahead Maskin monotonicity, and show that it is also sufficient under a variant of the condition of no veto-power when there are three or more agents. Finally, we provide an account of welfare implications of one-step-ahead implementability in the contexts of trading decisions and voting problems.

Temporary Implementation

In many situations, agents are involved in an allocation problem that is followed by another allocation problem whose optimal solution depends on how the former problem has been solved. In this paper, we take this dynamic structure of allocation problems as an institutional constraint. By assuming a finite number of allocation problems, one for each period/stage, and by assuming that all agents in society are involved in each allocation problem, a dynamic mechanism is a period-by-period process. This process generates at any period-t history a period-t mechanism with observable actions and simultaneous moves. We also assume that the objectives that a planner wants to achieve are summarized in a social choice function (SCF), which maps each state (of the world) into a period-by-period outcome process. In each period t, this process selects for each state a period-t socially optimal outcome conditional on the complete outcome history realized up to period t-1. Heuristically, the SCF is one-step-ahead implementable if there exists a dynamic mechanism such that for each state and each realized period-t history, each of its subgame perfect Nash equilibria generates a period-by-period outcome process that coincides with the period-by-period outcome process that the SCF generates at that state from period t onwards. We identify a necessary condition for SCFs to be one-step-ahead implemented, one-step-ahead Maskin monotonicity, and show that it is also sufficient under a variant of the condition of no veto-power when there are three or more agents. Finally, we provide an account of welfare implications of one-step-ahead implementability in the contexts of trading decisions and voting problems.

Nash implementation in production economies with unequal skills: a characterization

In production economies with unequal labor skills, where the planner is ignorant to the set of feasible allocations in advance of production, the paper firstly introduces a new axiom, Nonmanipulability of Irrelevant Skills (NIS), which together with Maskin Monotonicity constitute the necessary and sufficient conditions for Nash implementation. Secondly, the paper defines natural mechanisms, and then fully characterizes Nash implementation by natural mechanisms, using a slightly stronger variation of NIS and Supporting Price Independence. Following these characterizations, it is shown that there is a Maskin monotonic allocation rule which is not implementable when information about individual skills is absent. In contrast, many fair allocation rules, which are known to be non-implementable in the present literature, are implementable by the natural mechanisms. JEL Classification Numbers: C72, D51, D78, D82

Repeated implementation: A practical characterization

We characterize the social choice functions that are repeatedly implementable. The necessary and sufficient condition is formulated in terms of the equilibrium payoff set of an associated repeated game. It follows that the implementability of a function can be tested numerically by approximating the equilibrium payoff set. Additionally, with the help of our characterization, we demonstrate that an efficient function is implementable if and only if it satisfies a weaker version of Maskin monotonicity. As an application, we prove that utilitarian social choice functions are implementable by showing that continuation payoff promises effectively play the role of side-payments, which are needed for implementation in static setups.

Dividing bads under additive utilities

We compare the Egalitarian rule (aka Egalitarian Equivalent) and the Competitive rule (aka Comeptitive Equilibrium with Equal Incomes) to divide bads (chores). They are both welfarist: the competitive disutility profile(s) are the critical points of their Nash product on the set of efficient feasible profiles. The C rule is Envy Free, Maskin Monotonic, and has better incentives properties than the E rule. But, unlike the E rule, it can be wildly multivalued, admits no selection continuous in the utility and endowment parameters, and is harder to compute. Thus in the division of bads, unlike that of goods, no rule normatively dominates the other.

Treading a fine line: (Im)possibilities for Nash implementation with partially-honest individuals

This paper investigates the robustness of Dutta and Sen's (2012) Theorem 1 to weaker notions of truth-telling. It models individual i's honesty standard as a profile of (possibly non-empty) collections of ordered pairs of outcomes, one for each member of society, over which individual i feels truth-telling concerns. Individual i is honest provided that she states her true preferences as well as rankings (not necessarily complete) of outcomes that are consistent with the true preferences of individuals in her honesty standard. Under this notion of honesty, we offer a condition, called S(N)-partial-honesty monotonicity, which is necessary for Nash implementation when there are partially-honest agents. In an independent domain of preferences, we show that this condition is equivalent to Maskin monotonicity provided that honesty means stating the orderings of individuals (in a honesty standard) truthfully and individuals' honesty standards are non-connected.

Repeated Nash implementation

We study the repeated implementation of social choice functions in environments with complete information and changing preferences. We define dynamic monotonicity, a natural but nontrivial dynamic extension of Maskin monotonicity, and show that it is necessary and almost sufficient for repeated Nash implementation, regardless of whether the horizon is finite or infinite and whether the discount factor is “large” or “small.”

Nash implementing social choice rules with restricted ranges

We consider Nash implementation of social choice rules with restricted ranges, i.e., where some of the available alternatives are chosen at no preference profile. We show that the appropriate adaptation of Maskin monotonicity to this context depends on the range of the mechanisms: the wider is this range, the weaker is the monotonicity condition to be used. As a result, mechanisms employing outcome functions which allow for out-of-range alternatives at off-equilibrium messages can Nash implement social choice rules which fail to be Nash implementable by mechanisms whose ranges are restricted to the range of the social choice rule to be implemented. The Walrasian social choice correspondence is a particular instance of this. Moreover, social choice rules which are not Maskin monotonic can be “monotonized” by the addition of artificial out-of-range alternatives—a point we illustrate through Solomon’s Dilemma.

Feasible elimination procedures in social choice: An axiomatic characterization

Feasible elimination procedures (Peleg, 1978) play a central role in constructing social choice functions which have the following property: in the associated game form, for any preference profile there exists a strong Nash equilibrium resulting in the sincere outcome. In this paper we provide an axiomatic characterization of the social choice correspondences resulting from applying feasible elimination procedures. The axioms are anonymity, Maskin monotonicity, and independent blocking. We also show that these axioms are logically independent.

New axioms for deferred acceptance

We study the problem of allocating a set of indivisible objects to a set of agents based on agents’ preferences over objects and objects’ “choice functions” over agents, when monetary transfers are not allowed. Following Kojima and Manea (Econometrica 78(2):633–653, 2010) and Morrill (Int J Game Theory 42(1):19–28, 2013a), this paper provides four characterizations of the agent-proposing deferred acceptance allocation rule for all acceptant substitutable choice functions. It is the only rule satisfying any one of the following groups of axioms: (1) stability, rank monotonicity; (2) non-wastefulness, top best, weak consistency, rank monotonicity; (3) non-wastefulness, strong top best, weak Maskin monotonicity; (4) non-wastefulness, strong group rationality, rank monotonicity. These results suggested that two new axioms: rank monotonicity and weak consistency, deserve further attention. They also shed light on what distinguishes the agent-proposing deferred acceptance rule from the other rules.

A necessary and sufficient condition for weak Maskin monotonicity in an allocation problem with indivisible goods

We consider an allocation problem with indivisible goods, and provide a necessary and sufficient condition for weak Maskin monotonic allocation rules under non-wastefulness. The condition is based on robustness to group manipulation. Specifically, we introduce a new condition called the no improvement property of unmatched agents which means that unmatched agents cannot be strictly better off through any group manipulation. We show that a non-wasteful allocation rule satisfies weak Maskin monotonicity if and only if it satisfies the no improvement property of unmatched agents and weak group strategy-proofness. In addition, together with our result and that of Kojima and Manea (Econometrica 78:633–653, 2010), the deferred acceptance (DA) rules with acceptant substitutable priorities are characterized based on the conditions related to robustness to group manipulation.

Hedonic coalition formation games with variable populations: core characterizations and (im)possibilities

We consider hedonic coalition formation games with variable sets of agents and extend the properties competition sensitivity and resource sensitivity (introduced by Klaus, Games Econ Behav 72:172–186, 2011, for roommate markets) to hedonic coalition formation games. Then, we show that on the domain of solvable hedonic coalition formation games, the Core is characterized by coalitional unanimity and Maskin monotonicity (see also Takamiya, Maskin monotonic coalition formation rules respecting group rights. Niigata University, Mimeo, 2010, Theorem 1). Next, we characterize the Core for solvable hedonic coalition formation games by unanimity, Maskin monotonicity, and either competition sensitivity or resource sensitivity (Corollary 2). Finally, and in contrast to roommate markets, we show that on the domain of solvable hedonic coalition formation games, there exists a solution not equal to the Core that satisfies coalitional unanimity, consistency, competition sensitivity, and resource sensitivity (Example 2).