The Nakamura numbers for computable simple games

The Nakamura number of a simple game plays a critical role in preference aggregation (or multi-criterion ranking): the number of alternatives that the players can always deal with rationally is less than this number. We comprehensively study the restrictions that various properties for a simple game impose on its Nakamura number. We find that a computable game has a finite Nakamura number greater than three only if it is proper, nonstrong, and nonweak, regardless of whether it is monotonic or whether it has a finite carrier. The lack of strongness often results in alternatives that cannot be strictly ranked. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users. We would like to thank an anonymous referee for useful suggestions. The discussion in footnote 3 and Remark 4, among other things, would not have been possible without his/her suggestion.     

A note on the core and von Neumann-Morgenstern solutions of simple games

Following the results of Nakamura (1979) and Muto (1984), we derive, for a given proper voting game G, the upper bound on the size of the space of alternatives, which guarantees that the core constitutes a von Neumann-Morgenstern solution for any profile of voter's preferences. In particular, we show that if the space of alternatives consists of more than two elements, then, in general, the core is not a von Neumann-Morgenstern solution.We wish to thank Bezalel Peleg for pointing us out that one of the results in the previous version of this paper has already been proved by Muto (1984) and two anonymous referees for their useful remarks.     

Stable sets and cores for multi-criteria simple games and for their extensions

In this article we deal with multi-criteria simple games which constitute an extension of the basic framework of voting systems and related social-choice situations. We generalize two closely related solution concepts for these games, the stable sets and the core. A relationship between stable sets and minimal winning coalitions is established. Two generalizations of the core for multi-criteria simple games are characterized in terms of veto players. In addition, extended multi-criteria simple games obtained from different aggregation operations, such as union, intersection, marginalization, and composition, are introduced. It is shown that a voting system can be established when these operations are applied to complex voting systems.     

On the structure of minimal winning coalitions in simple voting games

According to Coleman’s index of collective power, a decision rule that generates a larger number of winning coalitions imparts the collectivity a higher a priori power to act. By the virtue of the monotonicity conditions, a decision rule is totally characterized by the set of minimal winning coalitions. In this paper, we investigate the structure of the families of minimal winning coalitions corresponding to maximal and proper simple voting games (SVG). We show that if the proper and maximal SVG is swap robust and all the minimal winning coalitions are of the same size, then the SVG is a specific (up to an isomorphism) system. We also provide examples of proper SVGs to show that the number of winning coalitions is not monotone with respect to the intuitively appealing system parameters like the number of blockers, number of non-dummies or the size of the minimal blocking set.     

Non-collegial simple games and the nowhere denseness of the set of preference profiles having a core

In recent papers, somewhat conflicting results on the generic emptiness of the core have been proven. Rubinstein (1979) has shown that the core is generically empty in the Kannai topology even without a restriction on the dimensionality of the alternative space. Schofield (1980) finds the core generically empty in the Whitney-C topology, but requires a dimensionality condition. This note explores the apparent conflict by proving a generalization of Rubinstein's result for the Whitney-C ⁰ topology, and showing that the result does not extend to the Whitney-C ¹ topology without a dimensionality condition.I would like to thank Kim Border and Richard McKelvey for their helpful discussion and comments. This research was supported in part by the National Science Foundation under Grant no. SES 78-2478781     

Language,games and language games

How do social values come about and gain legitimacy? Starting from the premise that discourses of social analysis affect the ways in which social norms develop and proliferate, this article models the evolution of professional codes and dialects using Wittgenstein’s idea of a language game. A language game is formalized as a repeated game of tacit coordination played among participants with informational asymmetries. The informational asymmetries model the different meanings that people assign to the same word used in a conversation. A language is formalized as a code that emerges as a result of repeated interactions in a language game. The paper argues that certain codes—such as those based on the real number system—lead to more reliable strategies in language games. The result is used to argue that professional dialects based on axiomatizable codes—such as physics, mathematics and economics—are less likely to experience fragmentation into intra-disciplinary ‘sects,’ camps and incommensurable paradigms than are professional dialects that are not based on an axiomatizable code—such as sociology, psychology, organization studies, and strategic management studies. The idea of a language game is extended to explore ways in which certain disciplines can establish cognitive jurisdiction over particular phenomena, starting from a particular set of codes, and thereby claim ‘cognitive monopolies.’ A rudimentary theory of the market for ideas is advanced.     

Cooperative games and cooperative organizations

It is well known that game theory has two major branches, cooperative and noncooperative game theory. Noncooperative game theory is the better known and more influential of the two. A difference is that cooperative game theory admits of binding agreements to choose a joint strategy in the mutual interest of those who agree. Cooperative organizations, too, are seen as being in the mutual interest of the members, but there has been little contact between the two bodies of thought. This paper surveys cooperative game theory and explores the extent to which cooperative game theory may help us to understand (and perhaps extend) cooperative organizations. In particular, reciprocity motives are introduced into the cooperative game analysis, and this may provide a link between cooperative game theory and cooperative organizations.     

Stability of voting games

This paper generalizes the result of Le Breton and Salles (1990) about stable set (far-sighted core of order 1) for voting games to far-sighted core of arbitrary order. Let m be the number of alternatives, n be the number of voters and G(n,k) be a proper symmetric simple game in which the size of a winning coalition is greater or equal to k. It is shown that the far-sighted core of order d for G(n,k) is nonempty for all preference profiles and for all n and k with n/(n–k)=v ¹ iff m(d+1)(v–1).This paper is part of my dissertation. I am grateful to my thesis advisor Leonid Hurwicz for his guidance and encouragement. I would like to thank Edward Green, Lu Hong, James Jordan, Andrew McLennan, Herve Moulin and Marcel Richter for their very helpful suggestions. Especially a referee and Maurice Salles made many good comments. Of course, any errors that remain are the sole responsibility of the author.     

Chivalry and solidarity in ultimatum games

We report the results of ultimatum game experiments designed to test for differences in the behavior of women and men. Women's proposals are on average more generous than men's, regardless of the sex of the partner, and women respondents are more likely to accept an offer of a given amount. A given offer is more likely to be accepted if it comes from a woman; we term this result chivalry . Women paired with women almost never fail to reach an agreement; we term this result solidarity . Age, earnings, and race also significantly affect proposals and the rates of rejection.     

Logical abilities and behavior in strategic-form games

This paper investigates the impact of cognitive abilities and preferences on behavior in games. We first measure subjects’ logical abilities and social preferences and then evaluate their influence on behavior in strategic-form games. For this purpose, we estimate how logical abilities and preferences relate to the probability of a subject being of a specific behavioral type. We find that better logical abilities increase the likelihood of sophisticated behavior. This shift is stronger for subjects who are classified as “selfish”. However, not even the most logically able, selfish subjects are identified as Nash types.     

Gamson’s law and hedonic games

This note adds one celebrated coalition formation game due to Gamson (Am Soc Rev 26:373–382 1961a, Am Soc Rev 26:565–573, 1961b) in the list of applications of the theory of hedonic games explored by Banerjee et al. (Soc Choice Welf 18:135–153, 2001) and Bogomolnaia and Jackson (Games Econ Behav 38: 204–230, 2002). We apply their results to study the original Gamson game and offer extensions both to a multi-dimensional characteristics space and to an infinite number of players.     

Scoring rule voting games and dominance solvability

This article studies the dominance solvability (by iterated deletion of weakly dominated strategies) of general scoring rule voting games when there are three alternatives. The scoring rules we study include Plurality rule, Approval voting, Borda rule, and Relative Utilitarianism. We provide sufficient conditions for dominance solvability of general scoring rule voting games. The sufficient conditions that we provide for dominance solvability are in terms of one statistic of the game: sufficient agreement on the best alternative or on the worst alternative. We also show that the solutions coincide with the set of Condorcet Winners whenever the sufficient conditions for dominance solvability are satisfied. Approval Voting performs the best in terms of our criteria.     

Intentions and ethical behavior in trust games

This paper explores if knowledge of one's partner's intention affects cooperative behavior. Results of a trust game experiment show that Player 1's elicited intentions are consistent over an imperfect information treatment, when Player 2 is not aware of her partner's intention, and a perfect information treatment, when she knows it. Evidence highlights that people's intentions on one side of the games shape their beliefs as to how their partner wants them to act when the roles are reversed. Moreover, in the perfect information treatment, participants act in a manner consistent with their own intentions, suggesting that ethical considerations permeate behavior.     

Undominated strategies and coordination in normalform games

This paper studies the question whether there are nondictatorial procedures for collective decision making which ensure that collective decisions are Pareto-efficient if all agents choose strategies that are not weakly dominated. It is shown that contrary to what one might expect the answer is not entirely negative.This paper is a shortened and revised version of Börgers (1989a). I would like to thank John Moore and Ariel Rubinstein for their comments on a very early version of this research. I have also benefited from the reactions of seminar participants in London, Visegard, Paris, and Basel. I am grateful to a referee and to an associate editor for helpful suggestions.     

On the core of voting games

The purpose of this paper is to analyse a special class of cooperative games called voting games in the cases where the set of alternatives is finite or a convex and compact subset of an Euclidean space. In a first part we provide a complete classification of these games according to the non-emptiness of the core. Then in a second part we prove that the set of continuous preference profiles having a core is small from a topological point of view. The contribution of the paper is mainly a simplification of the existing proofs and the introduction of a suitable topology to formulate the second question.     

Communication networks and their role in cooperative games

We consider cooperative games subordinated to communication networks, which indicate those players able to communicate directly as well as the strengths of their relationships. We develop an index of a player's ability to communicate with others and we also create a new game whose solution reflects this ability in conjunction with the economic capabilities described by the original game.     

Depth of reasoning in strategic form games

The depth of human reasoning in interactive decision-making situations can be assessed experimentally in dominance solvable non-cooperative games. We first review the studies employing two sorts of games—the Beauty Contest game and the Investment game. Studies employing these games show that: (i) people typically engage in relatively low levels of strategic reasoning; and (ii) there are individual differences in the depth of reasoning. There are other lines of research employing other types of games and definitions of the depth of reasoning that support the two conclusions above. We argue that these studies underestimate depth of reasoning because it is also necessary to apply a mental model about other players’ reasoning ability in order to play well. Implications of those findings to the theory-of-mind or Machiavellian Intelligence research are discussed.