Lexicographic composition of simple games

Certain voting bodies can be modeled as a simple game where a coalition's winning depends on whether it wins, blocks or loses in two smaller simple games. There are essentially five such ways to combine two proper games into a proper game. The most decisive is the lexicographic rule, where a coalition must either win in G₁, or block in G₁ and win in G₂. When two isomorphic games are combined lexicographically, a given role for a player confers equal or more power when held in the first game than the second, if power is assessed by any semi-value. A game is lexicographically separable when the players of the two components partition the whole set. Games with veto players are not separable, and games of two or more players with identical roles are separable only if decisive. Some separable games are egalitarian in that they give players identical roles.     

Axiomatizations of two types of Shapley values for games on union closed systems

A situation in which a finite set of players can obtain certain payoffs by cooperation can be described by a cooperative game with transferable utility, or simply a TU-game. A (single-valued) solution for TU-games assigns a payoff distribution to every TU-game. A well-known solution is the Shapley value. In the literature various models of games with restricted cooperation can be found. So, instead of allowing all subsets of the player set N to form, it is assumed that the set of feasible coalitions is a subset of the power set of N. In this paper, we consider such sets of feasible coalitions that are closed under union, i.e. for any two feasible coalitions also their union is feasible. We consider and axiomatize two solutions or rules for these games that generalize the Shapley value: one is obtained as the conjunctive permission value using a corresponding superior graph, the other is defined as the Shapley value of a modified game similar as the Myerson value for games with limited communication.     

Group play in games and the role of consent in network formation

We study games played between groups of players, where a given group decides which strategy it will play through a vote by its members. When groups consist of two voting players, our games can also be interpreted as network-formation games. In experiments on Stag Hunt games, we find a stark contrast between how groups and individuals play, with payoffs playing a primary role in equilibrium selection when individuals play, but the structure of the voting rule playing the primary role when groups play. We develop a new solution concept, robust-belief equilibrium, which explains the data that we observe. We provide results showing that this solution concept has application beyond the particular games in our experiments.     

On the stability of coalition structures

We resolve a seeming conflict between a non-existence result on solutions to coalition formation in hedonic games [Barberà, S., Gerber, A., 2007. A note on the impossibility of a satisfactory concept of stability for coalition formation games. Economics Letters 95, 85–90] and the universal existence of stable coalition structures in TU games under the χ-value [Casajus, A., 2008. Outside options, component efficiency, and stability, Games and Economic Behavior (forthcoming). doi: 10.1016/j.geb.2007.04.003].     

An admissible set occurring in various bargaining situations

Given a set of alternatives S and a binary relation M on S the admissible set of the pair (S, M) is defined to be the set of maximal elements with respect to the transitive closure of M. It is shown that existing solutions in game theory and mathematical economics are special cases of this concept (they are admissible sets of a natural S and M). These include the core of an n-person cooperative game, Nash equilibria of a noncooperative game, and the max-min solution of a two-person zero sum game. The competitive equilibrium prices of a finite exchange economy are always contained in its admissible set. Special general properties of the admissible set are discussed. These include existence, stability, and a stochastic dynamic process which leads to outcomes in the admissible set with high probability.     

Stable governments and the semistrict core

We consider the class of proper monotonic simple games and study coalition formation when an exogenous weight vector and a solution concept are combined to guide the distribution of power within winning coalitions. These distributions induce players' preferences over coalitions in a hedonic game. We formalize the notion of semistrict core stability, which is stronger than the standard core concept but weaker than the strict core notion and derive two characterization results for the semistrict core, depending on conditions we impose on the solution concept. A bounded power condition, which connects exogenous weights and the solution, turns out to be crucial. It generalizes a condition termed “absence of the paradox of smaller coalitions” that was previously used to derive core existence results.     

Regularity of pure strategy equilibrium points in a class of bargaining games

Summary. We develop an index theory for the Stationary Subgame Perfect (SSP) equilibrium set in a class of n-player sequential bargaining games with probabilistic recognition rules. For games with oligarchic voting rules (a class that includes unanimity rule), we establish conditions on individual utilities that ensure that for almost all discount factors, the number of SSP equilibria is odd and the equilibrium correspondence lower-hemicontinuous. For games with general, monotonic voting rules, we show generic (in discount factors) determinacy of SSP equilibria under the restriction that the agreement space is of dimension one. For non-oligarchic voting rules and agreement spaces of higher finite dimension, we establish generic determinacy for the subset of SSP equilibria in pure strategies. The analysis also extends to the case of fixed delay costs. Lastly, we provide a sufficient condition for uniqueness of SSP equilibrium in oligarchic games.Received: 13 May 2004, Revised: 1 March 2005, JEL Classification Numbers: C62, C72, C78.I thank John Duggan and participants of the 2003 annual meeting of the American Political Science Association, Philadelphia, PA, the Political Economy Seminar at Northwestern University, and the Economic Theory seminar at the University of Rochester for helpful comments.     

Cooperative vs non-cooperative truels: little agreement, but does that matter?

It is well-known that non-cooperative and cooperative game theory may yield different solutions to games. These differences are particularly dramatic in the case of truels, or three-person duels, in which the players may fire sequentially or simultaneously, over one round or n rounds. Representative solution concepts (Nash and subgame-perfect equilibrium; two notions of core) are compared, and little agreement is found among them. Although it might be desirable to subsume these different solutions within a common framework, such unification seems unlikely since they are grounded in fundamentally different notions of stability.     

Stability by deterrence in cooperative non-sidepayment n-person games

This paper reports several experimental tests of the deterrence set, a solution concept for n-person games proposed by Laffond and Moulin (1977, 1981). This solution concept is distinctive because it specifies equilibria attained through the use of threats, where threats are conceptualized as costly not only to the target but also to the user. The laboratory tests were conducted in the context of 3- and 4-person cooperative non-sidepayment matrix games. In the first experimental test, the deterrence set was juxtaposed against the imputation set. Results indicate that the deterrence set has greater predictive accuracy than the imputation set. In a series of further tests, the deterrence set was juxtaposed against both the imputation set and the von Neumann-Morgenstern stable set solution. Results again show that the deterrence set is more accurate than the imputation set, but neither the deterrence set nor the stable set is reliably more accurate than the other. Overall, these results indicate that deterrence is a viable basis for stability in cooperative non-sidepayment games.     

Stability when mobility is restricted by the existing coalition structure

In many social and economic situations the optimal solution requires the formation of coalitions that partition the set of players. When the individual player is small relative to the size of the existing coalitions, it seems realistic to assume that the prevailing coalition structure dictates the set of possible blocking coalitions. Specifically, it is assumed that an individual does not consider forming any coalition, but rather joining an already existing one. Two solution concepts for these games are investigated: structural equilibrium and stable payoffs, which are derived from the application of ψ-stability to the core and to the bargaining set, respectively. To this end an extension of the bargaining set to games without side payments is offered. Both solution concepts are shown to exist for some coalition structure. However, while structural equilibrium may fail to exist for any non trivial coalition structure, for every coalition structure there exists a stable payoff.     

Endogenous transfers in the Prisoner's Dilemma game: An experimental test of cooperation and coordination

We test a two-stage compensation mechanism for promoting cooperation in Prisoner's Dilemma games. Players first simultaneously choose binding non-negative amounts to pay their counterparts for cooperating, and then play the induced game knowing these amounts. In our games, all payment pairs consistent with mutual cooperation in subgame-perfect equilibrium transform these games into coordination games, with both mutual cooperation and mutual defection as Nash equilibria in the second stage. When endogenous transfer payments are not permitted, cooperation is much less likely. Mutual cooperation is most likely when the (sufficient) payments are identical, and it is also substantially more likely with payment pairs that bring the mutual-cooperation payoffs closer together. Both the Fehr–Schmidt and Charness–Rabin models predict that transfers that make final payoffs closer are preferred; however, they do not explain why equal transfers are particularly effective. Transfers are also effective in sustaining cooperation even when they are imposed and not chosen.     

Competitive outcomes and the inner core of NTU market games

We consider the inner core as a solution concept for cooperative games with non-transferable utility (NTU) and its relationship to payoffs of competitive equilibria of markets that are induced by NTU games. An NTU game is an NTU market game if there exists a market such that the set of utility allocations a coalition can achieve in the market coincides with the set of utility allocations the coalition can achieve in the game. In this paper, we introduce a new construction of a market based on a closed subset of the inner core which satisfies a strict positive separability. We show that the constructed market represents the NTU game and, further, has the given closed set as the set of payoff vectors of competitive equilibria. It turns out that this market is not uniquely determined, and thus, we obtain a class of markets. Our results generalize those relating to competitive outcomes of NTU market games in the literature.     

The social cost problem,rights, and the (non)empty core

We revisit the “Coase theorem” through the lens of a cooperative game model which takes into account the assignment of rights among agents involved in a problem of social cost. We consider the case where one polluter interacts with many potential victims. Given an assignment or a mapping of rights, we represent a social cost problem by a cooperative game. A solution consists in a payoff vector. We introduce three properties for a mapping of rights. First, core compatibility indicates that the core of the associated cooperative games is nonempty. Second, Kaldor‐Hicks core compatibility indicates that there is a payoff vector in the core where victims are fully compensated for the damage once the negotiations are completed. Third, no veto power for a victim says that no victim has the power to veto an agreement signed by the rest of the society. We then demonstrate two main results. First, core compatibility is satisfied if and only if the rights are assigned either to the polluter or to the entire set of victims. Second, there is no mapping of rights satisfying Kaldor‐Hicks core compatibility and no veto power for a victim.     

Anonymous yes–no voting with abstention and multiple levels of approval

Symmetric (3,2) simple games serve as models for anonymous voting systems in which each voter may vote “yes,” abstain, or vote “no,” the outcome is “yes” or “no,” and all voters play interchangeable roles. The extension to symmetric (j,2) simple games, in which each voter chooses from among j ordered levels of approval, also models some natural decision rules, such as pass–fail grading systems. Each such game is determined by the set of (anonymous) minimal winning profiles. This makes it possible to count the possible systems, and the counts suggest some interesting patterns. In the (3,2) case, the approach yields a version of May's Theorem, classifying all possible anonymous voting rules with abstention in terms of quota functions. In contrast to the situation for ordinary simple games these results reveal that the class of simple games with 3 or more levels of approval remains large and varied, even after the imposition of symmetry.     

On the Generic Finiteness of Equilibrium Outcomes in Plurality Games

In this paper we prove that for generic (noncooperative) voting games under plurality rule the set of equilibria that induce a mixed distribution over the outcomes (i.e., with two or more candidates elected with positive probability) is finite and, furthermore, each of these equilibria is regular. From that we deduce the finiteness of the set of equilibrium distributions over outcomes. Furthermore we offer an example (S. Govindan and A. McLennan, 1997, “On the Generic Finiteness of Equilibrium Outcome Distributions in Game Forms,” mimeo) that shows the impossibility of extending such results to a general framework, even just to voting games. Journal of Economic Literature Classification Numbers: C72, D72.     

On core stability, vital coalitions, and extendability

If a TU game is extendable, then its core is a stable set. However, there are many TU games with a stable core that are not extendable. A coalition is vital if there exists some core element x such that none of the proper subcoalitions is effective for x. It is exact if it is effective for some core element. If all coalitions that are vital and exact are extendable, then the game has a stable core. It is shown that the contrary is also valid for matching games, for simple flow games, and for minimum coloring games.     

Alpha- and beta-characteristic functions in cooperative non-sidepayment games: An experimental test

This article investigates theories of payoff allocation in cooperative non-sidepayment games, that is, games where players can form coalitions and coordinate strategies but cannot redistribute payoffs via sidepayments. A labotatory experiment was conducted with 4-person, 3-strategy non-sidepayment games. The goal of this experiment was to ascertain which characteristic function, V_α(S) or V_β(S), provides the more accurate basis for payoff predictions from solution concepts. Predictions from each of three solution concepts (core, stable set, imputation set) were computed on the basis of both V_α(S) and V_β(S), making a total of six theories under test. The results showed two basic findings. First, for each of the three solutions, the prediction computed on the basis of V_α(S) was more accurate than the prediction computed from V_α(S). Second, the β-core was the most accurate of the six theories tested. Overall, these results support the view that V_β(S) is superior to V_α(S) as a basis for payoff predictions in cooperative non-sidepayment games.     

Coalition-Proof Nash Equilibria II. Applications

In Bernheim, Peleg, and Whinston (“Coalition-Proof Equilibria. I. Concepts,” J. Econ. Theory 42 (1987), 1–12), we proposed the notion of Coalition-Proof Nash equilibrium and Perfectly Coalition-Proof Nash equilibrium as solution concepts for strategic environments in which players can freely discuss their strategies, but cannot make binding commitments. This paper undertakes applications to several economic problems, including the behavior of Cournot oligopolists, oligopolistic entry deterrence, cooperation in finite horizon games, and social choice rule implementation.     

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.     

Market Games in Large Economies with a Finite Number of Types

We study market games derived from an exchange economy with a continuum of agents, each having one of finitely many possible types. The type of agent determines his initial endowment and utility function. It is shown that, unlike the well-known Shapley–Shubik theorem on market games (Shapley and Shubik in J Econ Theory 1:9–25, 1969), there might be a (fuzzy) game in which each of its sub-games has a non-empty core and, nevertheless, it is not a market game. It turns out that, in order to be a market game, a game needs also to be homogeneous. We also study investment games – which are fuzzy games obtained from an economy with a finite number of agents cooperating in one or more joint projects. It is argued that the usual definition of the core is inappropriate for such a model. We therefore introduce and analyze the new notion of comprehensive core. This solution concept seems to be more suitable for such a scenario. We finally refer to the notion of feasibility of an allocation in games with a large number of players. Some of the results in this paper appear in a previous draft distributed by the name “Cooperative investment games or Population games”. An anonymous referee of Economic Theory is acknowledged for his/her comments