Perceptron versus automaton in the finitely repeated prisoner’s dilemma

We study the finitely repeated prisoner’s dilemma in which the players are restricted to choosing strategies which are implementable by a machine with a bound on its complexity. One player has to use a finite automaton while the other player has to use a finite perceptron. Some examples illustrate that the sets of strategies which are induced by these two types of machines are different and not ordered by set inclusion. Repeated game payoffs are evaluated according to the limit of means. The main result establishes that a cooperation at almost all stages of the game is an equilibrium outcome if the complexity of the machines the players may use is limited enough and if the length T of the repeated game is sufficiently large. This result persists when more than T states are allowed in the player’s automaton. We further consider a variant of the model in which the two players are restricted to choosing strategies which are implementable by perceptrons and prove that the players can cooperate at most of the stages provided that the complexity of their perceptrons is sufficiently reduced.     

Stubbornness,Power, and Equilibrium Selection in Repeated Games with Multiple Equilibria

Axelord’s [(1970), Conflict of Interest, Markham Publishers, Chicago] index of conflict in 2 × 2 games with two pure strategy equilibria has the property that a reduction in the cost of holding out corresponds to an increase in conflict. This article takes the opposite view, arguing that if losing becomes less costly, a player is less likely to gamble to win, which means that conflict will be less frequent. This approach leads to a new power index and a new measure of stubbornness, both anchored in strategic reasoning. The win probability defined as power constitutes an equilibrium refinement which differs from Harsanyi and Selten’s [(1988), A General Theory of Equilibrium Selection in Games, MIT Press, Cambridge] refinement. In contrast, Axelrod’s approach focuses on preferences regarding divergences from imaginary outmost rewards that cannot be obtained jointly. The player who is less powerful in an asymmetric one-shot game becomes more powerful in the repeated game, provided he or she values the future sufficiently more than the opponent. This contrasts with the view that repetition induces cooperation, but conforms with the expectation that a more patient player receives a larger share of the pie.      

K–Player Additive Extension of Two-Player games with an Application to the Borda Electoral Competition Game

In this note we introduce the notion of K–player additive extension of a symmetric two-player game and prove a result relating the equilibria in mixed strategies in the two games. Then we apply the result to the Borda electoral competition game.     

A choice for ‘me’ or for ‘us’? Using we-reasoning to predict cooperation and coordination in games

Cooperation is the foundation of human social life, but it sometimes requires individuals to choose against their individual self-interest. How then is cooperation sustained? How do we decide when instead to follow our own goals? I develop a model that builds on Bacharach (in: Gold, Sugden (eds) Beyond individual choice: teams and frames in game theory, 2006) ??circumspect we-reasoning?? to address these questions. The model produces a threshold cost/benefit ratio to describe when we-reasoning players should choose cooperatively. After assumptions regarding player types and beliefs, we predict how the extent of cooperation varies across games. Results from two experiments offer strong support to the models and predictions herein.     

A differential approach to the repeated prisoner's dilemma

The repeated prisoner's dilemma game is converted into a differential game by assuming that the players, instead of making decisions individually for each repetition of the prisoner's dilemma game, make decisions on the ratio of cooperative and noncooperative games that they wish to play over the next few moves, and that the actual plays are then determined using this ratio and a randomizing procedure. Although it sounds like a significant departure, this assumption is probably not too different from reality.Since each player can always obtain by his own action at least the payoff which he would receive from the noncooperative-noncooperative pair of strategies, that part of the differential game which is dominated by such a dual noncooperative strategy for either player is an unlikely outcome. This dominated area can be readily computed for any game, including those with more than two players.Formal testing with empirical data was impossible because of uncertainty about the proper null hypothesis. Nevertheless, experimental results reported by Rapoport are consistent with the theory.The authors are, respectively, Graduate Research Asistant in Economics and Professor of Economics and Public Choice at Virginia Polytechnic Institute and State University.     

Two-Speed Evolution of Strategies and Preferences In Symmetric Games

Agents in a large population are randomly matched to play a certain game, payoffs in which represent fitness. Agents may have preferences that are different from fitness. They learn strategies according to their preferences, and evolution changes the preference distribution in the population according to fitness. When agents know the preferences of the opponent in a match, only efficient symmetric strategy profiles of the fitness game can be stable. When agents do not know the preferences of the opponent, only Nash equilibria of the fitness game can be stable. For 2 × 2 symmetric games I characterize preferences that are stable.Jel Codes: C72, A13     

Rules, Social Ontology and Collective Identity

Mainstream game theory explains cooperation as the outcome of the interaction of agents who permanently pursue their individual goals. Amartya Sen argues instead that cooperation can only be understood by positing a type of rule-following behaviour that can be (and often is) out of phase with the pursuit of individual goals, due to the existence of a collective identity. However, Sen does not clarify the ontological preconditions for the type of social behaviour he describes. I will argue that Sen's account of collective identity can be best interpreted in the light of John Searle's notion of collective intentionality, while Sen's explanation of rule-following behavior and agency is best understood using the critical realist transformational model of social activity.     

Implementing egalitarianism in a class of Nash demand games

We add a stage to Nash’s demand game by allowing the greedier player to revise his demand if the demands are not jointly feasible. If he decides to stick to his initial demand, then the game ends and no one receives anything. If he decides to revise it down to \(1-x\), where x is his initial demand, the revised demand is implemented with certainty. The implementation probability changes linearly between these two extreme cases. We derive a condition on the feasible set under which the two-stage game has a unique subgame perfect equilibrium. In this equilibrium, there is first-stage agreement on the egalitarian demands. We also study two n-player versions of the game. In either version, if the underlying bargaining problem is “divide-the-dollar,” then equal division is sustainable in a subgame perfect equilibrium if and only if the number of players is at most four.     

Endogenizing the order of moves in matrix games

Players often have flexibility in when they move and thus whether a game is played simultaneously or sequentially may be endogenously determined. For 2 × 2 games, we analyze this using an extended game. In a stage prior to actual play, players choose in which of two periods to move. A player moving at the first opportunity knows when his opponent will move. A player moving at the second turn learns the first mover's action. If both select the same turn, they play a simultaneous move subgame.If both players have dominant strategies in the basic game, equilibrium payoffs in the basic and extended games are identical. If only one player has a dominant strategy or if the unique equilibrium in the basic game is in mixed strategies, then the extended game equilibrium payoffs differ if and only if some pair of pure strategies Pareto dominates the basic game simultaneous play payoffs. If so, sequential play attains the Pareto dominating payoffs. The mixed strategy equilibrium occurs only when it is not Pareto dominated by some pair of pure strategies.In an alternative extended game, players cannot observe delay by opponents at the first turn. Results for 2×2 games are essentially the same as with observable delay, differing only when only one player has a dominant strategy.     

The relationship between children's social concerns and their social interaction strategies: differences between rejected and accepted boys

In this study involving 55 fourth and fifth grade boys, children's concerns in their peer interactions, their social interaction strategies, and the relationship between their concerns and strategies were examined. Compared to peer accepted boys, submissive rejected boys cared less about sustaining interactions with peers and aggressive rejected boys cared less about peers' feelings. Aggressive rejected boys, and rejected boys who were neither highly aggressive nor highly submissive, also suggested more aversive strategies for handling conflictual interpersonal situations than did accepted boys. Of particular interest is that this was true even when their concerns in those situations were similar to those of accepted boys. The implications of these findings for children's social competence are discussed.     

Indefinitely repeated games: A response to Carroll

In a recent volume of this journal John Carroll argued that there exist only uncooperative equilibria in indefinitely repeated prisoner's dilemma games. We show that this claim depends on modeling such games as finitely but indefinitely repeated games, which reduce simply to finitely repeated games. We propose an alternative general model of probabilistically indefinitely repeated games, and discuss the appropriateness of each of these models of indefinitely repeated games.     

The Emergence of Reactive Strategies in Simulated Heterogeneous Populations

The computer simulation study explores the impact of the duration of social impact on the generation and stabilization of cooperative strategies. Rather than seeding the simulations with a finite set of strategies, a continuous distribution of strategies is being defined. Members of heterogeneous populations were characterized by a pair of probabilistic reactive strategies: the probability to respond to cooperation by cooperation and the probability to respond to defection by cooperation. This generalized reactive strategy yields the standard TFT mechanism, the All-Cooperate, All-Defect and Bully strategies as special cases. Pairs of strategies interacted through a Prisoner's Dilemma game and exerted social influence on all other members. Manipulating: (i) the initial distribution of populations' strategies, and (ii) the duration of social influence, we monitored the conditions leading to the emergence and stabilization of cooperative strategies. Results show that: (1) The duration of interactions between pairs of strategies constitutes a crucial factor for the emergence and stabilization of cooperative strategies, (2) Unless sufficient learning intervals are provided, initializing the simulations with cooperative populations does not guarantee that cooperation will sustain.     

惩罚机制对阻断腐败循环的演化博弈分析

腐败循环是一种典型的社会困境问题,其根源在于反腐败组织无法打破腐败利益交易双方在长期博弈中形成的稳定均衡态势。通过演化博弈分析发现,惩罚机制可以起到促进反腐败组织协调高效运作的作用,有助于打破腐败交易双方的长期合作关系,从而为最终阻断腐败循环提供可能性。     

The Trial and Crucifixion of Jesus: A Modest Proposal

I model an attempt by radical parties to topple a modus vivendi between a ruling government and a moderate opposition group. Cooperation between the regime and the moderate opposition is possible if each player prefers mutual cooperation to mutual confrontation. If each player also prefers mutual confrontation to cooperating while the other defects then radical parties have a chance at breaking up this accord. Radical parties can succeed in bringing the government and opposition to mutual confrontation if they can agree on power-sharing arrangements after regime change. This paper also resolves central questions surrounding the trial and crucifixion of Jesus Christ. I use an institutional approach to infer player preferences from historical and biblical sources and then use game theory to model the interactions between participants in these events. In so doing, I clarify aspects of the Gospel narrative that have puzzled readers for the past 2000 years.     

A Banzhaf share function for cooperative games in coalition structure

A cooperative game with transferable utility–or simply a TU-game– describes a situation in which players can obtain certain payoffs by cooperation. A value function for these games assigns to every TU-game a distribution of payoffs over the players. Well-known solutions for TU-games are the Shapley and the Banzhaf value. An alternative type of solution is the concept of share function, which assigns to every player in a TU-game its share in the worth of the grand coalition. In this paper we consider TU-games in which the players are organized into a coalition structure being a finite partition of the set of players. The Shapley value has been generalized by Owen to TU-games in coalition structure. We redefine this value function as a share function and show that this solution satisfies the multiplication property that the share of a player in some coalition is equal to the product of the Shapley share of the coalition in a game between the coalitions and the Shapley share of the player in a game between the players within the coalition. Analogously we introduce a Banzhaf coalition structure share function. Application of these share functions to simple majority games show some appealing properties.     

Why do groups cooperate more than individuals to reduce risks?

Previous research has discovered a curious phenomenon: groups cooperate less than individuals in a deterministic prisoner’s dilemma game, but cooperate more than individuals when uncertainty is introduced into the game. We conducted two studies to examine three possible processes that might drive groups to be more cooperative than individuals in reducing risks: group risk concern, group cooperation expectation, and pressure to conform to social norms. We found that ex post guilt aversion and ex-post blame avoidance cause group members to be more risk concerned than individuals under uncertainty. These concerns drive groups to choose the cooperation (and risk-reduction) strategy more frequently than individuals. Groups also have higher cooperation expectations for their corresponding groups than individuals have for their corresponding individuals. We found no evidence of pressure to conform to social norms driving groups to be more cooperative than individuals.     

Using Turn Taking to Mitigate Coordination and Conflict Problems in the Repeated Battle of the Sexes Game

The Battle of the Sexes game, which captures both coordination and conflict problems, has been applied to a wide range of situations. We show that, by reducing distributional conflict and enhancing coordination, (eventual) turn taking supported by a “turn taking with independent randomizations” strategy allows players to engage in intertemporal sharing of the gain from cooperation. Using this insight, we decompose the benefit from turn taking into conflict-mitigating and coordination-enhancing components. Our analysis suggests that an equilibrium measure of the “degree of intertemporal conflict” provides an intuitive way to understand the sources of welfare gain from turn taking in the repeated Battle of the Sexes game. We find that when this equilibrium measure is lower, players behave less aggressively and the welfare gain from turn taking is higher.      

The Independent Localisations of Interaction and Learning in the Repeated Prisoner's Dilemma

The results of a series of computer simulations demonstrate how the introduction of separate spatial dimensions for agent interaction and learning respectively affects the possibility of cooperation evolving in the repeated prisoner's dilemma played by populations of boundedly-rational agents. In particular, the localisation of learning promotes the emergence of cooperative behaviour, while the localisation of interaction has an ambiguous effect on it.     

Won't You Be My Neighbor? Norms of Cooperation,Public Broadcasting,and the Collective Action Problem*

Objectives. The logic of collective action suggests that public broadcasting stations should not receive the empirically observed level of member support they do. Why do people contribute to public television when they can view it without contributing? Methods. The hypothesis tested is that “norms of cooperation” govern the behavior of individuals in collective action situations. This article tests the hypothesis with an original survey of public television viewers in three large communities. Results. The survey data provide support for the “norms of cooperation” hypothesis. The higher the level of characteristics of an individual that measure cooperation, the more likely the individual is to give to public broadcasting, all other factors being equal. Conclusions. Norms of cooperation—an important part of social capital—help overcome the logic of collective action where it concerns public television contributions.     

论社会治理原理与原则

社会治理时代正向我们走来,引起人们的极大关注。有什么样的社会就会有什么样的社会统治、社会管理、社会治理。在后工业化与全球治理的背景下,我国多质态的社会统治、社会管理、社会治理还将历时态与共时态地长期并存,同时呈现以政党治理、国家治理、政府治理三位一体的“独家管理”为主、社会自理或自治、合作或共同管理、相互管理或相互治理的多种模式。社会管理或治理主客体的二重性,往往使得原生性主体缺位,派生性或衍生性主体越位,形成社会管理或社会治理主客体之间的不对称、不对等的互动互补趋势。这种趋势必然要求在社会管理和社会治理过程中遵循共同但有区别的原则,以实现对社会的有效管理和治理。