Determinacy of equilibrium in outcome game forms

We show the generic finiteness of the number of probability distributions on outcomes induced by Nash equilibria for two-person game forms such that either (i) one of the players has no more than two strategies or (ii) both of the players have three strategies, and (iii) for outcome game forms with three players, each with at most two strategies. Finally, we exhibit an example of a game form with three outcomes and three players for which the Nash equilibria of the associated game induce a continuum of payoffs for an open non-empty set of utility profiles.     

Spatial pillage game

A pillage game is a coalitional game as a model of Hobbesian anarchy. The spatial pillage game introduces a spatial feature into the pillage game. Players are located in regions and can travel from one region to another. The players can form a coalition and combine their power only within their destination regions, which limits the exertion of the power of each coalition. Under this spatial restriction, a coalition can pillage less powerful coalitions without any cost. The feasibility of pillages between coalitions determines the dominance relation that defines stable states in which powers among the players are endogenously balanced. With the spatial restriction, the set of stable states changes. However, if the players have forecasting ability, then the set of stable states does not change with the spatial restriction. Core, stable set, and farsighted core are adopted as alternative solution concepts.     

Coalitional games with veto players: Consistency, monotonicity and Nash outcomes

Following Dagan et al. [Dagan, N., Volij, O., Serrano, R. (1997). A non-cooperative view on consistent bankruptcy rules, Games Econ. Behav. 18, 55–72], we construct an extensive form game for veto-balanced TU games in which a veto player is the proposer and the other players are responders. The set of Nash outcomes of this extensive form game is described, and compared to solutions of TU games such as the nucleolus, kernel and egalitarian core. We find necessary and sufficient conditions under which the nucleolus of the game is a Nash outcome.     

Cores of effectivity functions and implementation theory

In a committee where cooperative voting occurs, effectivity functions describe the blocking power of coalitions. It is a binary relation that says for each coalition T and each subset of outcomes B whether or not T can force the final outcome within B. The corresponding cooperative stability notion generalizes the familiar core of a simple game. We study those effectivity functions yielding a non-empty core for all preference profiles, of which additive effectivity functions are an example. This proves to be closely related to implementation by means of the strong equilibrium concept.     

Preference revelation games and strong cores of allocation problems with indivisibilities

This note considers the equilibrium outcomes of the preference revelation games in the general model of indivisible good allocation introduced by [Sönmez, T., 1999. Strategy-proofness and essentially single-valued cores. Econometrica 67, 677–689]. We adopt the concepts of coalitional equilibria and cores which are both defined in terms of the weak deviation or blocking by a prescribed class of admissible coalitions. We prove that if the solution which induces preference revelation games is individually rational and Pareto optimal and the class of admissible coalitions is “monotonic,” then the set of coalitional equilibrium outcomes coincides with the core. And we point out that the preceding analysis in the context of marriage problems [Gale, D., Shapley, L., 1962. College admissions and the stability of marriage. American Mathematical Monthly 69, 9–15] is hardly extended to the general model.     

An indifference trap of voting

In this paper strategic situations of voting and abstentions are analysed in a three-candidate voting model where voters are indifferent to various alternatives and candidates are interested in winning the election and seeing their most preferred alternative being realized. A specific decision problem is analysed, described by an Indifference Trap Game, with respect to, e.g., the corresponding Nash equilibria, perfect equilibria, and maximin solution. A second-best outcome is contained in the choice set of all three solutions while the Nash equilibrium concept is compatible with the Pareto efficient outcomes of the game.An alternative scenario where candidates suffer from incumbency but voters still are indifferent to some alternatives also supports the second-best outcome. Again, various solutions concepts are applied. We conclude that indifferent voters imply an eminent coordination problem for the candidates in the given voting game which, in general, ends up in inefficient outcomes. The inherent complexity of the decision situation cannot be sufficiently reduced to single out one and only one outcome.     

Theorems on the core of an economy with infinitely many commodities and consumers

It is known that the classical theorems of Grodal [Grodal, B., 1972. A second remark on the core of an atomless economy. Econometrica 40, 581–583] and Schmeidler [Schmeidler, D., 1972. A remark on the core of an atomless economy. Econometrica 40, 579–580] on the veto power of small coalitions in finite dimensional, atomless economies can be extended (with some minor modifications) to include the case of countably many commodities. This paper presents a further extension of these results to include the case of uncountably many commodities. We also extend Vind’s [Vind, K., 1972. A third remark on the core of an atomless economy. Econometrica 40, 585–586] classical theorem on the veto power of big coalitions in finite dimensional, atomless economies to include the case of an arbitrary number of commodities. In another result, we show that in the coalitional economy defined by an atomless individualistic model, core–Walras equivalence holds even if the commodity space is non-separable. The above-mentioned results are also valid for a differential information economy with a finite state space. We also extend Kannai’s [Kannai, Y., 1970. Continuity properties of the core of a market. Econometrica 38, 791–815] theorem on the continuity of the core of a finite dimensional, large economy to include the case of an arbitrary number of commodities. All of our results are applications of a lemma, that we prove here, about the set of aggregate alternatives available to a coalition. Throughout the paper, the commodity space is assumed to be an ordered Banach space which has an interior point in its positive cone.     

The core of the inductive limit of a direct system of economies with a communication structure

This paper studies direct systems of finite, that is with finitely many agents, communication structures, finite (balanced and superadditive) NTU games and finite economies. The inductive limit of such a system is again a communication structure, an NTU game or an economy, this time possibly with infinitely many agents. As a matter of fact, each infinite communication structure, each infinite NTU game and each infinite economy, is the inductive limit of a direct system of finite communication structures, finite NTU games and finite economies. A communication game is an NTU game with a communication structure on the set of players. To each economy, there corresponds a balanced and superadditive NTU game. To each economy with a communication structure on the set of agents, there corresponds a communication game. In the paper it is proved that the core of the inductive limit of a direct system of communication games is not empty and in fact the intersection of the cores of the finite communication games of the direct system. It follows that each infinite economy (with or without a communication structure on the set of agents) has a nonempty core. A direct system of economies is a generalisation of the Debreu and Scarf [Debreu, G., Scarf, H. A limit theorem on the core of an economy, International Economic Review 4, pp. 235–246.] example of `replica economies'. The proof of the nonemptiness of the core of the inductive limit of a direct system of economies is along the lines of the proof by Debreu and Scarf. As by-product it is shown that an NTU game is totally balanced if and only if all its finite subgames are balanced.     

国际环境合作的经济学分析

本文利用现代经济学理论与方法，构筑了一个分析国际环境合作问题的经济学框架。分析的中心是建立一个静态模型来寻求作为一个联盟形态的国际环境合作的内部和外部稳定性条件，主要结论是稳定的国际环境合作是由模型给定的参数内生地决定的。与当前大多数研究的一个重要不同是，我们的模型没有采用通常的减少污染排放这一分析参数，而是选用了更为直接的污染排放量这一指标。在运用联盟理论对两类情形即不合作和完全合作进行模型化处理的基础上，本文导出了联盟成员国与非成员国福利水平和污染排放量的若干命题，并建立起国际环境合作稳定性的条件。     

Strategic complementarities and nested potential games

This paper reports that every finite game of strategic complementarities is a nested pseudo-potential game defined by Uno [Uno, H., 2007. Nested potential games. Economics Bulletin 3(17), 1–8] if the action set of each player is one-dimensional, except possibly for one player.     

Nash equilibria in a model of multiproduct price competition: an assignment problem

We study the market interaction of a finite number of single-product firms and a representative buyer, where the buyer consumes bundles of these goods. The buyers’ value function determines their willingness to pay for subsets of goods. We show that Nash-equilibrium outcomes are solutions of the linear relaxation of an integer programming assignment problem and that they always exist. The Nash-equilibrium price set is characterized by the Pareto frontier of the associated dual problem’s projection on the firms’ price vectors. We identify the Nash-equilibrium prices for monotonic buyers’ value functions and, more importantly, we show that some central solution concepts in cooperative game theory are (subgame perfect) equilibrium prices of our strategic game.     

Measuring the set of blocking coalitions in infinite dimensional economies

The classical core-Walras equivalence is investigated via the measurement of the set of blocking coalitions. In the framework of continuum economies with an infinite dimensional space of commodities, we find the measure of the set of coalitions that block a non-competitive Pareto optimal allocation. Then, from the relation between coalitions of a continuum economy with a finite number of types and fuzzy coalitions of an economy with finitely many agents, the previous results are translated as results on the measure of blocking fuzzy coalitions. Both results imply classically formulated core-Walras equivalence theorems. Finally, the measure of blocking "social communication structures" is determined.     

The Theory of Socially Embedded Games: Applications and Extensions to Open and Closed Games

An earlier article, drawing on the mathematical theory of rules and rule complexes, extends and generalizes game theory (GGT). The theory has been used to conceptualize and analyze diverse social relationships, roles, and games as particular types of rule complexes.For instance, a social role, as a major basis of a parent's action in a game, consists of at least four key components – which are mathematical objects – in the determination of action: value complex, model of reality (including beliefs and knowledge bases), a repertoire of acts, routines, programs, and strategies, and modality, a role-specific algorithm for determining or generating action in game settings. This article applies and extends GGT in analyses of a market bargaining game (a type of open game) and of the classical game of prisoners' dilemma (a type of closed game). The applications show the concrete effects of social embeddedness on game structuring, game interaction patterns and outcomes, and social equilibria.     

Coalition formation in general NTU games

A general nontransferable utility (NTU) game is interpreted as a collection of pure bargaining games that can be played by individual coalitions. The threatpoints or claims points respectively, in these pure bargaining games reflect the players' opportunities outside a given coalition. We develop a solution concept for general NTU games that is consistent in the sense that the players' outside opportunities are determined by the solution to a suitably defined reduced game. For any general NTU game the solution predicts which coalitions are formed and how the payoffs are distributed among the players.     

Representation of transferable utility games by coalition production economies

We prove that, by the method of construction of a coalition production economy due to Sun et al. [Sun, N., Trockel, W., Yang, Z., 2008. Competitive outcomes and endogenous coalition formation in an n$n$-person game. Journal of Mathematical Economics 44, 853–860], every transferable utility (TU) game can be generated by a coalition production economy. Namely, for every TU game, we can construct a coalition production economy that generates the given game. We briefly discuss the relationship between the core of a given TU game and the set of Walrasian payoff vectors for the induced coalition production economy.     

Repeated implementation and complexity considerations

A game form is commitment-free if single actions of players do not have physical consequences, i.e., affect the continuation game. Such game can be thought to represent inifinite interaction with complete patience. A choice rule can be Nash implemented via a commitment-free mechanism if and only if it coincides with the feasible set of a normal form game. However, when players are complexity averse (in the lexicographic sense), then any Nash implementable choice rule becomes available. I am very grateful to Matt Jackson and a referee for their comments. I also thank Klaus Kultti and Hannu Salonen for useful conversations.     

Generic impossibility of Arrow’s impossibility theorem

Our major objective is to show that if there are infinitely many alternatives and society can be decomposed into the sum of two non-negligible coalitions, the set of all Arrovian collective choice rules that satisfy the ultrafilter property (its set of decisive coalitions form an ultrafilter), is nowhere dense in the set of all Arrovian collective choice rules. This result entails that (1) almost all Arrovian collective choice rules are non-dictatorial and (2) in any neighborhood of a dictatorial Arrovian collective choice rule, a nondictatorial Arrovian collective choice rule can be found, i.e., the property of admitting a dictator is not a locally stable property for Arrovian collective choice rules in the sense that the property does not withstand a small perturbation.     

A theory of coalition formation in committees

Let G = (N,W) be a strong weighted majority game and let A be a set of alternatives. Denote by L the set of linear orders on A. A social choice function F:L^N → A is a representation of G if the simple game G¹(F) associated with F equals G. A coalition S is determining in G if it satisfies the following condition. Let F be a representation of G and let R^N ? L^N. Then, if a simple majority of the members of S consider an alternative x to be their best choice, then S can ‘enforce’ x to be a Nash equilibrium payoff in the resulting non-cooperative voting game g(F,R^N). In this paper we generalize the above notion of a determining coalition to committees (i.e., proper and monotonic simple games), and give a complete characterization of the set of determining coalitions of a committee. Furthermore, we discuss our notion of a determining coalition in the light of some real-life data on formation of coalitions in town councils in Israel.     

On pure-strategy Nash equilibria in price–quantity games

This paper examines the existence and characteristics of pure-strategy Nash equilibria in oligopoly models in which firms simultaneously set prices and quantities. Existence of a pure-strategy equilibrium is proved for a class of price–quantity games. If the demand function is continuous, then the equilibrium outcome is similar to that of a price-only model. With discontinuous demand and limited spillover, there are rationing equilibria in which combined production falls short of market demand. Moreover, there might again be an equilibrium reflecting the outcome of a price game. Competition in price and quantity thus yields Bertrand outcomes under a variety of market conditions.     

Network structure,games, and agent dynamics

Consider a group of agents embedded in a network, repeatedly playing a game with their neighbors. Each agent acts locally but through the links of the network local decisions percolate to the entire population. Past research shows that such a system converges either to an absorbing state (a fixed distribution of actions that once attained does not change) or to an absorbing set (a set of action distributions that may cycle in finite populations or behave chaotically in unbounded populations). In many network games, however, it is uncertain which situation emerges. In this paper I identify two fundamental network characteristics, boundary consistency and neighborhood overlap, that determine the outcome of all symmetric, binary-choice, network games. In quasi-consistent networks these games converge to an absorbing state regardless of the initial distribution of actions, and the degree to which neighborhoods overlap impacts the number and composition of those absorbing states.