Game-theoretic analyses of coalition behavior

Coalitions are frequently more visible than payoffs. The theory of n-person games seeks primarily to identify stable allocations of valued resources; consequently, it gives inadequate attention to predicting which coalitions form. This paper explores a way of correcting this deficiency of game-theoretic reasoning by extending the theory of two-person cooperative games to predict both coalitions and payoffs in a three-person game of status in which each player seeks to maximize the rank of his total score. To accomplish this, we analyze the negotiations within each potential two-person coalition from the perspective of Nash's procedure for arbitrating two-person bargaining games, then assume that players expect to achieve the arbitrated outcome selected by this procedure and use these expectations to predict achieved ranks and to identify players' preferences between alternative coalition partners in order to predict the probability that each coalition forms. We test these payoff and coalition predictions with data from three laboratory studies, and compare the results with those attained in the same data by von Neumann and Morgenstern's solution of two-person cooperative games, Aumann and Maschler's bargaining set solution for cooperative n-person games, and an alternative model of coalition behavior in three-person sequential games of status.

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Far-sighted equilibria in 2 × 2, non-cooperative,repeated games

Consider a two-person simultaneous-move game in strategic form. Suppose this game is played over and over at discrete points in time. Suppose, furthermore, that communication is not possible, but nevertheless we observe some regularity in the sequence of outcomes. The aim of this paper is to provide an explanation for the question why such regularity might persist for many (i.e., infinite) periods.Each player, when contemplating a deviation, considers a sequential-move game, roughly speaking of the following form: if I change my strategy this period, then in the next my opponent will take his strategy b and afterwards I can switch to my strategy a, but then I am worse off since at that outcome my opponent has no incentive to change anymore, whatever I do. Theoretically, however, there is no end to such reaction chains. In case that deviating by some player gives him less utility in the long run than before deviation, we say that the original regular sequence of outcomes is far-sighted stable for that player. It is a far-sighted equilibrium if it is far-sighted stable for both players.     

Focal points in pure coordination games: An experimental investigation

This paper reports an experimental investigation of the hypothesis that in coordination games, players draw on shared concepts of salience to identify focal points on which they can coordinate. The experiment involves games in which equilibria can be distinguished from one another only in terms of the way strategies are labelled. The games are designed to test a number of specific hypotheses about the determinants of salience. These hypotheses are generally confirmed by the results of the experiment.     

Some notes on Church's thesis and the theory of games

This paper considers games in normal form played by Turing Machines. The machines are fed as input all the relevent information and then are required to play the game. Some impossibility results are derived for this set-up. In particular, it is shown that no Turing Machine exists which will always play the correct strategy given its opponent's choice. Such a result also generalizes to the case in which attention is restricted to economically optimizing machines only. The paper also develops a model of knowledge. This allows the main results of the paper to be interpreted as stemming out of the impossibility of always deciding whether a player is rational or not in some appropriate sense.     

Bayesian boundedly rational agents play the Finitely Repeated Prisoner's Dilemma

In this paper a model of boundedly rational decision making in the Finitely Repeated Prisoner's Dilemma is proposed in which: (1) each player is Bayesianrational; (2) this is common knowledge; (3) players are constrained by limited state spaces (their Bayesian minds) in processing (1) and (2). Under these circumstances, we show that cooperative behavior may arise as an individually optimal response, except for the latter part of the game. Indeed, such behaviorwill necessarily obtain in long enough games if belief systems satisfy a natural condition: essentially, that all events consistent with the players' analysis of the game be attributed by them positive (although arbitrarily small) subjective probability.     

Mechanism robustness in multilateral bargaining

This paper discusses the relationship between coalitional stability and the robustness of bargaining outcomes to the bargaining procedure. We consider a class of bargaining procedures described by extensive form games, where payoff opportunities are given by a characteristic function (cooperative) game. The extensive form games differ on the probability distribution assigned to chance moves which determine the order in which players take actions. One way to define mechanism robustness is in terms of the property of no first mover advantage. An equilibrium is mechanism robust if for each member the expected payoff before and after being called to propose is the same. Alternatively one can define mechanism robustness as a property of equilibrium outcomes. An outcome is said to be mechanism robust if it is supported by some equilibrium in all the extensive form games (mechanisms) within our class. We show that both definitions of mechanism robustness provide an interesting characterization of the core of the underlying cooperative game.     

Divide the Dollar: Three solutions and extensions

Divide the Dollar (DD) is a game in which two players independently bid up to 100 cents for a dollar. Each player receives his or her bid if the sum of the bids does not exceed a dollar; otherwise they receive nothing. This game has multiple Nash equilibria, including the egalitarian division of (50, 50), but this division is not compelling except for its symmetry and presumed fairness.This division is easy to induce, however, by punishing — more severely than does DD — deviations from it, but these solutions are not reasonable. By altering the rules of DD, however, one can induce an egalitarian division (by successive elimination of weakly dominated strategies), but no reasonable payoff scheme produces this division with egalitarian bids of 50.Three alternatives to DD are analyzed. DD1, which rewards lowest bidders first, shows how an egalitarian outcome can be induced with equal but nonegalitarian bids. DD2, which adds a second stage that provides the players with new information yet restricts their choices at the same time, is used to introduce dominance inducibility. DD3 combines the features of DD1 and DD2, is reasonable (like DD1), makes calculations transparent (like DD2), and induces egalitarian bids as well as the egalitarian outcome. The possible application of the different procedures to a real-world allocation problem (setting of salaries by a team), in which there may be entitlements, is described.     

Axiomatization of a class of share functions for n-person games

The Shapley value is the unique value defined on the class of cooperative games in characteristic function form which satisfies certain intuitively reasonable axioms. Alternatively, the Banzhaf value is the unique value satisfying a different set of axioms. The main drawback of the latter value is that it does not satisfy the efficiency axiom, so that the sum of the values assigned to the players does not need to be equal to the worth of the grand coalition. By definition, the normalized Banzhaf value satisfies the efficiency axiom, but not the usual axiom of additivity.In this paper we generalize the axiom of additivity by introducing a positive real valued function on the class of cooperative games in characteristic function form. The so-called axiom of -additivity generalizes the classical axiom of additivity by putting the weight (v) on the value of the gamev . We show that any additive function determines a unique share function satisfying the axioms of efficient shares, null player property, symmetry and -additivity on the subclass of games on which is positive and which contains all positively scaled unanimity games. The axiom of efficient shares means that the sum of the values equals one. Hence the share function gives the shares of the players in the worth of the grand coalition. The corresponding value function is obtained by multiplying the shares with the worth of the grand coalition. By defining the function appropiately we get the share functions corresponding to the Shapley value and the Banzhaf value. So, for both values we have that the corresponding share functions belong to this class of share functions. Moreover, it shows that our approach provides an axiomatization of the normalized Banzhaf value. We also discuss some other choices of the function and the corresponding share functions. Furthermore we consider the axiomatization on the subclass of monotone simple games.     

A strategic foundation for the cooperator's advantage

Orbell and Dawes develop a non-game theoretic heuristic that yields a cooperator's advantage by allowing players to project their own cooperate-defect choices onto potential partners (1991, p. 515). With appropriate parameter values their heuristic yields a cooperative environment, but the cooperation depends, simply, on optimism about others' behavior (1991, p. 526). In earlier work, Dawes (1989) established a statistical foundation for such optimism. In this paper, I adapt some of the concerns of Dawes (1989) and develop a game theoretic model based on a modification of the Harsanyi structure of games with incomplete information (1967–1968). I show that the commonly made conjecture that strategic play is incompatible with cooperation and the cooperator's advantage is false.     

On Hobbes’s state of nature and game theory

Hobbes’s state of nature is often analyzed in two-person two-action non-cooperative games. By definition, this literature only focuses on duels. Yet, if we consider general games, i.e., with more than two agents, analyzing Hobbes’s state of nature in terms of duel is not completely satisfactory, since it is a very specific interpretation of the war of all against all. Therefore, we propose a definition of the state of nature for games with an arbitrary number of players. We show that this definition coincides with the strategy profile considered as the state of nature in two-person games. Furthermore, we study what we call rational states of nature (that is, strategy profiles which are both states of nature and Nash equilibria). We show that in rational states of nature, the utility level of any agent is equal to his maximin payoff. We also show that rational states of nature always exist in inessential games. Finally, we prove the existence of states of nature in a class of (not necessarily inessential) symmetric games.     

Equilibria for far-sighted players

A new equilibrium concept for non-cooperative games, based on the assumptions that players are rational and far-sighted, is examined. An outcome is extended non-myopically (XNM) stable for a player if that player is assured that no movecountermove sequence he could initiate by departing unilaterally from that outcome would benefit him. The extended non-myopic (XNM) equilibria of a game, the outcomes which are XNM stable for each player, therefore model permanent (enduring) equilibria in an ongoing conflict.Algorithms for the identification of XNM equilibria in a 2 × 2 game are presented. The XNM concepts are then applied to three special classes of games (no-conflict games, games of complete opposition, and strict ordinal games) to compare their predictions of long-term stability with the known properties of games in these classes.Research supported by Natural Sciences and Engineering Research Council of Canada Grant No. A8974.     

Experiments with cooperative 2 × 2 games

Nash's solution of a two-person cooperative game prescribes a coordinated mixed strategy solution involving Pareto-optimal outcomes of the game. Testing this normative solution experimentally presents problems in as much as rather detailed explanations must be given to the subjects of the meaning of threat strategy, strategy mixture, expected payoff, etc. To the extent that it is desired to test the solution using naive subjects, the problem arises of imparting to them a minimal level of understanding about the issue involved in the game without actually suggesting the solution.Experiments were performed to test the properties of the solution of a cooperative two-person game as these are embodied in three of Nash's four axioms: Symmetry, Pareto-optimality, and Invariance with respect to positive linear transformations. Of these, the last was definitely discorroborated, suggesting that interpersonal comparison of utilities plays an important part in the negotiations.Some evidence was also found for a conjecture generated by previous experiments, namely that an externally imposed threat (penalty for non-cooperation) tends to bring the players closer together than the threats generated by the subjects themselves in the process of negotiation.     

Joint Beliefs in Conflictual Coordination Games

The traditional solution concept for noncooperative game theory is the Nash equilibrium, which contains an implicit assumption that players probability distributions satisfy t probabilistic independence. However, in games with more than two players, relaxing this assumption results in a more general equilibrium concept based on joint beliefs (Vanderschraaf, 1995). This article explores the implications of this joint-beliefs equilibrium concept for two kinds of conflictual coordination games: crisis bargaining and public goods provision. We find that, using updating consistent with Bayes rule, players beliefs converge to equilibria in joint beliefs which do not satisfy probabilistic independence. In addition, joint beliefs greatly expand the set of mixed equilibria. On the face of it, allowing for joint beliefs might be expected to increase the prospects for coordination. However, we show that if players use joint beliefs, which may be more likely as the number of players increases, then the prospects for coordination in these games declines vis-à-vis independent beliefs.     

An incompleteness problem in Harsanyi's general theory of games and certain related theories of non-cooperative games

The theory of games recently proposed by John C. Harsanyi in A General Theory of Rational Behavior in Game Situations, (Econometrica, Vol. 34, No. 3) has one anomalous feature, viz., that it generates for a special class of non-cooperative games solutions which are not equilibrium points. It is argued that this feature of the theory turns on an argument concerning the instability of weak equilibrium points, and that this argument, in turn, involves appeal to an unrestricted version of a postulate subsequently included in the theory in restricted form. It is then shown that if this line of reasoning is permitted, then one must, by parity of reasoning, permit another instability argument. But, if both of these instability arguments are permitted in the construction of the theory, the resultant theory must be incomplete, in the sense that there will be simple non-cooperative games for which such a theory cannot yield solutions. This result is then generalized and shown to be endemic to all theories which have made the equilibrium condition central to the treatment of non-cooperative games. Some suggestions are then offered concerning how this incompleteness problem can be resolved, and what one might expect concerning the postulate structure and implications of a theory of games which embodies the revisions necessitated by a resolution of this problem.This research was supported by a grant to the author from the City University of New York Faculty Research Award Program.     

Predictive superiority of the beta-characteristic function in cooperative non-sidepayment N-person games

This article reports an experimental study of decision-making outcomes in cooperative non-sidepayment games. The objective of this test was to determine which characteristic function, V (S) or V (S), provides the most accurate basis for payoff predictions from solution concepts. The experiment tested three solution concepts (core, stable set, imputation set) in the context of 5-person, 2-strategy non-sidepayment games. Predictions from each of the three solution concepts were computed on the basis of both V (S) and V (S), making a total of six predictive theories under test. Consistent with earlier studies (Michener et al., 1984a; Michener et al., 1985), two basic findings emerged. First, the data show that for each of the solutions tested, the prediction from any solution concept computed from V(S) was more accurate than the prediction from the same solution concept computed from V (S). Second, the -core was the most accurate of the six theories tested. Overall, these results support the view that V (S) is superior to V (S) as a basis for payoff predictions in cooperative non-sidepayment games.     

Correlated strategies as Institutions

Two institutions that are often implicit or overlooked in noncooperative games are the assumption of Nash behavior to solve a game, and the ability to correlate strategies. We consider two behavioral paradoxes; one in which maximin behavior rules out all Nash equilibria (Chicken), and another in which minimax supergame behavior leads to an inefficient outcome in comparison to the unique stage game equilibrium (asymmetric Deadlock). Nash outcomes are achieved in both paradoxes by allowing for correlated strategies, even when individual behavior remains minimax or maximin. However, the interpretation of correlation as a public institution differs for each case.     

Rational choice and economic behavior

This paper discusses several concepts that can be used to provide a foundation for a unified, theory of rational, economic behavior. First, decision-making is defined to be a process that takes place with reference to both subjective and objective time, that distinguishes between plans and actions, between information and states and that explicitly incorporates the collection and processing of information. This conception of decision making is then related to several important aspects of behavioral economics, the dependence of values on experience, the use of behavioral rules, the occurrence of multiple goals and environmental feedback.Our conclusions are (1) the non-transitivity of observed or revealed preferences is a characteristic of learning and hence is to be expected of rational decision-makers; (2) the learning of values through experience suggests the sensibleness of short time horizons and the making of choices according to flexible utility; (3) certain rules of thumb used to allow for risk are closely related to principles of Safety-First and can also be based directly on the hypothesis that the feeling of risk (the probability of disaster) is identified with extreme departures from recently executed decisions. (4) The maximization of a hierarchy of goals, or of a lexicographical utility function, is closely related to the search for feasibility and the practice of satisficing. (5) When the dim perception of environmental feedback and the effect of learning on values are acknowledged the intertemporal optimality of planned decision trajectories is seen to be a characteristic of subjective not objective time. This explains why decision making is so often best characterized by rolling plans. In short, we find that economic man - like any other - is an existential being whose plans are based on hopes and fears and whose every act involves a leap of faith.This paper is based on a talk presented at the Conference, New Beginnings in Economics, Akron, Ohio, March 15, 1969. Work on this paper was supported by a grant from the National Science Foundation.     

Punishing greediness in divide-the-dollar games

Brams and Taylor 1994 presented a version of the divide-the-dollar game (DD), which they call DD1. DD1 suffers from the following drawback: when each player demands approximately the entire dollar, then if the least greedy player is unique, then this player obtains approximately the entire dollar even if he is only slightly less greedy than the other players. I introduce a parametrized family of 2-person DD games, whose “endpoints” (the games that correspond to the extreme points of the parameter space) are (1) a variant of DD1, and (2) a game that completely overcomes the greediness-related problem. I also study an n-person generalization of this family. Finally, I show that the modeling choice between discrete and continuous bids may have far-reaching implications in DD games.     

Games,goals, and bounded rationality

A generalization of the standard n-person game is presented, with flexible information requirements suitable for players constrained by certain types of bounded rationality. Strategies (complete contingency plans) are replaced by policies, i.e., endmean pairs of goals and controls (partial contingency plans), which results in naturally disconnected player choice sets. Well-known existence theorems for pure strategy Nash equilibrium and bargaining solutions are generalized to policy games by modifying connectedness (convexity) requirements.     

Domination structures and multicriteria problems in n-person games

Multiple criteria decision problems with one decision maker have been recognized and discussed in the recent literature in optimization theory, operations research and management science. The corresponding concept with n-decision makers, namely multicriteria n-person games, has not yet been extensively explored.In this paper we first demonstrate that existing solution concepts for single criterion n-person games in both normal form and characteristic function form induce domination structures (similar to those defined and studied by Yu [39] for multicriteria single decision maker problems) in various spaces, including the payoff space, the imputation space and the coalition space. This discussion provides an understanding of some underlying assumptions of the solution concepts and provides a basis for generalizing and generating new solution concepts not yet defined. Also we illustrate that domination structures may be regarded as a measure of power held by the players.We then illustrate that a multicriteria problem can naturally arise in decision situations involving (partial) conflict among n-persons. Using our discussion of solution concepts for single criterion games as a basis, various approaches for resolving both normal form and characteristic function form multicriteria n-person games are proposed. For multicriteria games in characteristic function form, we define a multicriteria core and show that there exists a single game point whose core is equal to the multicriteria core. If we reduce a multicriteria game to a single criterion game, domination structures which are more general than classical ones must be considered, otherwise some crucial information in the game may be lost. Finally, we discuss a parametrization process which, for a given multicriteria game, associates a single criterion game to each point in a parametric space. This parametrization provides a basis for the discussion of solution concepts in multicriteria n-person games.