National security games

Issues that arise in using game theory to model national security problems are discussed, including positing nation-states as players, assuming that their decision makers act rationally and possess complete information, and modeling certain conflicts as two-person games. A generic two-person game called the Conflict Game, which captures strategic features of such variable-sum games as Chicken and Prisoners' Dilemma, is then analyzed. Unlike these classical games, however, the Conflict Game is a two-stage game in which each player can threaten to retaliate — and carry out this threat in the second stage — if its opponent chose noncooperation in the first stage.Conditions for the existence of different pure-strategy Nash equilibria, or stable outcomes, are found, and these results are extended to situations in which the players can select mixed strategies (i.e., make probabilistic threats or choices). Although the Conflict Game sheds light on the rational foundations underlying arms races, nuclear deterrence, and other strategic situations, more detailed assumptions are required to tie this generic game to specific conflicts.Steven J. Brams gratefully acknowledges the financial support of the National Science Foundation under Grant No. SES85-20154, the Sloan Foundation, and the Guggenheim Foundation.D. Marc Kilgour gratefully acknowledges the financial support of the Natural Sciences and Engineering Research Council of Canada under Grant No. A8974.     

Game Theory in Philosophy

Game theory is the mathematical study of strategy and conflict. It has wide applications in economics, political science, sociology, and, to some extent, in philosophy. Where rational choice theory or decision theory is concerned with individual agents facing games against nature, game theory deals with games in which all players have preference orderings over the possible outcomes of the game. This paper gives an informal introduction to the theory and a survey of applications in diverse branches of philosophy. No criticism is reviewed. Game theory is shown at work in discussions about epistemological dependence (prisoner’s dilemma), liberalism and efficiency (Nash equilibrium), Hume’s concept of convention (correlated equilibrium), morality and rationality (bargaining games), and distributive justice and egalitarianism (evolutionary game theory). A guide to the literature provides hints at applications in collective intentionality, epistemology, ethics, history of philosophy, logic, philosophy of language, and political philosophy.     

Collective rationality in interactive decisions: evidence for team reasoning

Decision theory and game theory rest on a fundamental assumption that players seek to maximize their individual utilities, but in some interactive decisions it seems intuitively reasonable to aim to maximize the utility of the group of players as a whole. Such team reasoning requires collective preferences and a distinctive mode of reasoning from preferences to decisions. Findings from two experiments provide evidence for collective preferences and team reasoning. In lifelike vignettes (Experiment 1) and abstract games (Experiment 2) with certain structural properties, most players preferred team-reasoning strategies to strategies supporting unique Nash equilibria, although individually rational players should choose equilibrium strategies. These findings suggest that team reasoning predicts strategy choices more powerfully than orthodox game theory in some games.     

Mutual support in games: Some properties of Berge equilibria

The Berge equilibrium concept formalizes mutual support among players motivated by the altruistic social value orientation in games. We prove some basic results for Berge equilibria and their relations to Nash equilibria, and we provide a straightforward method for finding Berge equilibria in n-player games. We explore some specific examples, and we explain how the Berge equilibrium provides a compelling model of cooperation in social dilemmas. We show that the Berge equilibrium also explains coordination in some common interest games and is partially successful in explaining the payoff dominance phenomenon, and we comment that the theory of team reasoning provides alternative solutions to these problems.     

Psychologically rational choice: selection between alternatives in a multiple-equilibrium game

Choice is modeled by game theory through analyses of the structure of a game situation. However, at least some choices, such as those in games that have more than one rational solution, are difficult to address under standard game theory. We investigated choice in a simple multiple-equilibrium game, Wolf’s Dilemma, and found that choice depends on both structural components of the game, such as the number of opponents, and non-structural components, such as judgments about opponents’ likely choices. Significant effects of trial-to-trial sequence were identified. We conclude that game theoretic methods cannot deal with multiple and equilibrium games in the absence of a psychological understanding of choice, and that psychological measures are required to predict performance in such games.     

The Incompleteness of Theories of Games

We first state a few previously obtained results that lead to general undecidability and incompleteness theorems in axiomatized theories that range from the theory of finite sets to classical elementary analysis. Out of those results we prove several incompleteness theorems for axiomatic versions of the theory of noncooperative games with Nash equilibria; in particular, we show the existence of finite games whose equilibria cannot be proven to be computable.     

Three ancient problems solved by using the game theory logic based on the Shapley value

The ancient problems of bankruptcy, contested garment, and rights arbitration have generated many studies, debates, and controversy. The objective of this paper is to show that the Shapley value from game theory, measuring the power of each player in a game, may be consistently applied for getting the general one-step solution of all these three problems viewed as n-person games. The decision making is based on the same tool, namely the game theory logic based on the use of the Shapley value, but the specific games involved are slightly different in each problem. The kind of claims of the players, the relationship between the given claims and the given resources available, and the particular way of calculating the generalized characteristic function of the game determine the specific type of game which has to be solved in each of the three ancient problems mentioned. The iterative use of the Shapley value may also justify the well-known Aumann–Maschler step-by-step procedure for solving the bankruptcy problem.

     

On coalition formation in simple games: A mathematical analysis of Caplow's and Gamson's theories

In this paper, we propose a theory of coalition formation in simple games. The process of coalition formation is modeled as an abstract game. Two solutions of abstract games, the core and the dynamic solution, are used as the predictions of our model. Two classical theories of coalitions in sociology due to Caplow and Gamson are reformulated in a more general and mathematical setting. These theories are then analyzed using the techniques of our theory.     

Classical Game Theory, Socialization and the Rationalization of Conventions

The paper begins by providing a game-theoretic reconstruction of Gilbert’s (1989) philosophical critique of Lewis (1969) on the role of salience in selecting conventions. Gilbert’s insight is reformulated thus: Nash equilibrium is insufficiently powerful as a solution concept to rationalize conventions for unboundedly rational agents if conventions are solutions to the kinds of games Lewis supposes. Both refinements to NE and appeals to bounded rationality can plug this gap, but lack generality. As Binmore (this issue) argues, evolutive game theory readily explains the origin of conventional behavior, but that is not Lewis’s project. Gilbert’s critique is generalized by reference to Bacharach’s (2006) work on team reasoning in games. The paper then argues that although Lewis’s account of the rationalization of conventions is shown by the reconstruction of Gilbert’s critique to be incomplete, Gilbert is wrong to conclude that classical (‘eductive’) game theory lacks the resources to explain conformity to conventions among people. A game-theoretic account of the dynamics of socialization, based on Ross’s (2005, 2006) idea of ‘game determination’, rationalizes choices of conventional strategies in overlapping generations contexts, provided agents are products of evolutionary selection and know that other players are also such products.     

Hard and Soft Preparation Sets in Boolean Games

A fundamental problem in game theory is the possibility of reaching equilibrium outcomes with undesirable properties, e.g., inefficiency. The economics literature abounds with models that attempt to modify games in order to avoid such undesirable properties, for example through the use of subsidies and taxation, or by allowing players to undergo a bargaining phase before their decision. In this paper, we consider the effect of such transformations in Boolean games with costs, where players control propositional variables that they can set to true or false, and are primarily motivated to seek the satisfaction of some goal formula, while secondarily motivated to minimise the costs of their actions. We adopt (pure) preparation sets (prep sets) as our basic solution concept. A preparation set is a set of outcomes that contains for every player at least one best response to every outcome in the set. Prep sets are well-suited to the analysis of Boolean games, because we can naturally represent prep sets as propositional formulas, which in turn allows us to refer to prep formulas. The preference structure of Boolean games with costs makes it possible to distinguish between hard and soft prep sets. The hard prep sets of a game are sets of valuations that would be prep sets in that game no matter what the cost function of the game was. The properties defined by hard prep sets typically relate to goal-seeking behaviour, and as such these properties cannot be eliminated from games by, for example, taxation or subsidies. In contrast, soft prep sets can be eliminated by an appropriate system of incentives. Besides considering what can happen in a game by unrestricted manipulation of players’ cost function, we also investigate several mechanisms that allow groups of players to form coalitions and eliminate undesirable outcomes from the game, even when taxes or subsidies are not a possibility.     

Social norms and the traditional deterrence game

Bicchieri (The grammar of society: The nature and dynamics of norms, 2006, xi) presents a formal analysis of norms that answers the questions of “when, how, and to what degree” norms affect human behavior in the play of games. The purpose of this paper is to apply a variation of the Bicchieri norms analysis to generate a model of norms-based play of the traditional deterrence game (Zagare and Kilgour, Int Stud Q 37:1–27, 1993; Morrow, Game theory for political scientists, 1994), the paradigmatic model of conflict initiation in International Relations. The deterrence game is modeled here as a sequential decision problem. As such, our analysis is an adaptation of Bicchieri’s game-theoretic formalization of norms to what we will call the norms account of the game. We find that the standard account of the traditional deterrence game is a special case of the norms account of the game. We also show that the adaptation of Bicchieri’s analysis of social norms yields new and interesting claims regarding when, how, and to what degree norms operate as a constraint on risk-related behavior in the traditional deterrence game. Moreover, we discuss how the results of the model provide testable propositions of relevance to the role of norms in international interactions.     

Game relativity: how context influences strategic decision making

Existing models of strategic decision making typically assume that only the attributes of the currently played game need be considered when reaching a decision. The results presented in this article demonstrate that the so-called "co-operativeness" of the previously played prisoner's dilemma games influence choices and predictions in the current prisoner's dilemma game, which suggests that games are not considered independently. These effects involved reinforcement-based assimilation to the previous choices and also a perceptual contrast of the present game with preceding games, depending on the range and the rank of their co-operativeness. A. Parducci's (1965) range frequency theory and H. Helson's (1964) adaptation level theory are plausible theories of relative judgment of magnitude information, which could provide an account of these context effects.     

How Braess’ paradox solves Newcomb's problem*

Newcomb's problem is regularly described as a problem arising from equally defensible yet contradictory models of rationality. Braess’ paradox is regularly described as nothing more than the existence of non‐intuitive (but ultimately non‐contradictory) equilibrium points within physical networks of various kinds. Yet it can be shown that Newcomb's problem is structurally identical to Braess’ paradox. Both are instances of a well‐known result in game theory, namely that equilibria of non‐cooperative games are generally Pareto‐inefficient. Newcomb's problem is simply a limiting case in which the number of players equals one. Braess’ paradox is another limiting case in which the ‘players’ need not be assumed to be discrete individuals. The result is that Newcomb's problem is no more difficult to solve than (the easy to solve) Braess’ paradox.     

The Motivational Pull of Video Games: A Self-Determination Theory Approach

Four studies apply self-determination theory (SDT; Ryan & Deci, 2000) in investigating motivation for computer game play, and the effects of game play on well-being. Studies 1–3 examine individuals playing 1, 2 and 4 games, respectively and show that perceived in-game autonomy and competence are associated with game enjoyment, preferences, and changes in well-being pre- to post-play. Competence and autonomy perceptions are also related to the intuitive nature of game controls, and the sense of presence or immersion in participants’ game play experiences. Study 4 surveys an on-line community with experience in multi-player games. Results show that SDT’s theorized needs for autonomy, competence, and relatedness independently predict enjoyment and future game play. The SDT model is also compared with Yee’s (2005) motivation taxonomy of game play motivations. Results are discussed in terms of the relatively unexplored landscape of human motivation within virtual worlds.     

Dynamic probability of reinforcement for cooperation: Random game termination in the centipede game

Experimental games have previously been used to study principles of human interaction. Many such games are characterized by iterated or repeated designs that model dynamic relationships, including reciprocal cooperation. To enable the study of infinite game repetitions and to avoid endgame effects of lower cooperation toward the final game round, investigators have introduced random termination rules. This study extends previous research that has focused narrowly on repeated Prisoner's Dilemma games by conducting a controlled experiment of two‐player, random termination Centipede games involving probabilistic reinforcement and characterized by the longest decision sequences reported in the empirical literature to date (24 decision nodes). Specifically, we assessed mean exit points and cooperation rates, and compared the effects of four different termination rules: no random game termination, random game termination with constant termination probability, random game termination with increasing termination probability, and random game termination with decreasing termination probability. We found that although mean exit points were lower for games with shorter expected game lengths, the subjects' cooperativeness was significantly reduced only in the most extreme condition with decreasing computer termination probability and an expected game length of two decision nodes.     

施特劳斯政治哲学的精神与中国传统之重建

该文认为政治就是以公共权力分配为核心内容的游戏;政治学与政治哲学的根本区别在于政治学研究权力游戏规则的运行规律,而政治哲学则研究权力游戏规则的合法性;古典西方政治哲学所形成的是一种正义传统,而它正与中国古典政治哲学的天道传统相呼应。中国近代所形成的革命文化破坏了我们自己的政治哲学传统,重建我们自己的政治哲学和政治文化传统,是当代中国文化建设的核心任务之一。     

Reflections on Ethics and Game Theory

Applications of game theory to moral philosophy are impededby foundational issues and troublesome examples. In the first part of this paper,questions are raised about the appropriate game-theoretical frameworks for applications to moralphilosophy and about the proper interpretations of the theoretical devices employed inthese frameworks. In the second part, five examples that should be of particular interest to thoseinterested in the connections between ethics and game theory are delineated and discussed. Thefirst example comprises games in which there is an outcome unanimously preferred to the``solution' of the game, appropriately defined. The second comprises games whose solution callsfor different players to employ different strategies. The third comprises games whosesolution calls for players to adopt mixed strategies. The fourth comprises games whose solutionrequires players to cycle among a variety of strategies. The fifth comprises games whose solutionrequires players to discriminate in morally inappropriate ways.