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This chapter focuses on the existence of Nash equilibria. The existence problem is first introduced with pure-strategy reaction curves for Noisy and Silent Duel. A proof of John Nash's existence theorem is then sketched using the Kakutani fixed-point theorem. The latter may be deduced from Brouwer's more famous fixed-point theorem. A sketch of a proof of Brouwer's theorem is offered using the fact that Hex cannot end in a draw. Games often have many Nash equilibria, which creates an equilibrium selection problem. Various possible approaches to the equilibrium selection problem are reviewed, including the evolutionary approach and Thomas Schelling's notion of a focal point.

An agent in a game of strategy often has a preference ranking of options without a stable top. Strategic reasoning leads the players to a profile of strategies that are jointly self-supporting. Such a strategic equilibrium is not always a Nash equilibrium, but every Nash equilibrium is a strategic equilibrium. In a game with multiple strategic equilibria, principles of comprehensive rationality select an equilibrium that is Pareto optimal among strategic equilibria. The strategies forming the equilibrium are jointly rational and so constitute a solution of the game. The chapter’s appendix reviews Skryms’s account of joint deliberational equilibrium.

This chapter examines the sociobiology of human cooperation. Given the tendency of people to copy the successful and the fact that natural selection favors the more fit, the chapter asks how our altruistic preferences overcame the cultural and biological evolutionary handicaps entailed by the reduced payoffs that they elicited. To answer this question, two major biological explanations of cooperation are discussed: inclusive fitness in either a kin-based or a multi-level selection model, and reciprocal altruism and its indirect reciprocity and costly signaling variants. The chapter explores a model of inclusive fitness based on group differentiation and competition, clarifying what is meant by multi-level selection and how it works. It also discusses models that address equilibrium selection, the link between standing strategy and indirect reciprocity, and positive assortment. Finally, it assesses the mechanisms and motives underlying helping behavior.

This chapter offers an approach to applying game theory to David Hume's ideas on moral philosophy. It begins by explaining how the Humean approach allows a different interpretation of experimental results in games like the Ultimatum Game compared to that taken for granted in classical economics. It then briefly reviews ideas from orthodox social choice theory as a preliminary to arguing that making interpersonal comparisons of utility is necessary for a full analysis of fairness norms. Harsanyi's theory of interpersonal comparison is then introduced. Harsanyi's utilitarian theory and Rawls' egalitarian theory are discussed. The chapter ends with an account of an experiment that is intended to illustrate how fairness norms may evolve over time.

This chapter looks at the development of a formal model that specifies the relationships between leaders and members. Organizational leaders can develop reputations for effectiveness; this reputation makes possible leadership rents, which expand cooperation and compensate the leader when information acquisition and communication are costly. Members agree to contribute to the leader's compensation up to a point. The form these rents take, whether monetary or political, affects the scope of union activities. Since these outcomes are only a few of many possible, the chapter introduces the concept of organizational governance institutions that embody the union's solution to the equilibrium selection problem. As in all formal models, particularly those intended as part of an analytic narrative, the study lays out the paths not taken as well as the one followed.

This chapter lays the foundations for a revised and more inclusive view of the government. It argues that the law must be perceived differently to gain a better understanding on reality while building models of economics, politics, and society. It is shown that the law has similarities with equilibrium-selection norms discussed in the previous chapter.

This chapter considers the role that compensating payments can play in promoting international cooperation. The theory developed here explains that such payments cannot increase cooperation when countries are alike, but that they can have a huge effect when countries are very different. Side payments can therefore only really help when the conditions allow them to play this role. The theory is again applied to the Montreal Protocol.