This chapter discusses empirical evidence for a range of decision procedures underlying individuals’ choices in games. These include (1) quantal response equilibria. These are stochastic choice models where a player does not best reply to their beliefs, but instead chooses actions with a higher expected payoff with a higher probability. It also includes (2) cognitive hierarchy models where, roughly speaking, players possess different, discrete levels of cognition. A level-0 player chooses some fixed anchor strategy. And, recursively, having fixed the strategies of players of level 0,...,k-1, a level-k player best responds to a belief that all other players are of a lower level. Lastly, it includes (3) a variety of learning models.

This chapter is the introduction to Part I, “The early history of game theory from Borel”. Starting from a 1921 paper by Émile Borel, it surveys the historical development of game theory during its first few decades.

Game theory in biology emerged as a theoretical and modeling approach to the evolutionary study of behavior and other phenotypes. It was inspired by the prominence of game theory in economics. This chapter gives a perspective on this influence of economic thinking on biology. Which ideas were taken from economics and the social sciences to biology, how were they modified, which biological phenomena were they applied to, and what are the successes, challenges, and the possible future of the field? It focuses on two of the earliest applications of game theory in biology, namely sex-ratio theory and animal contests.

This chapter reviews literature on game-theoretic analysis of voting. Both cooperative and noncooperative concepts are used to answer questions, such as, How do candidates or parties propose alternatives to voters in strategic interactions? Why do voters vote? What are the implications of asymmetric information for candidates’ and voters’ incentives? Do prevoting deliberations improve information sharing? If so, through what type of rules? Sophisticated voters may act strategically, and therefore it matters whether one’s choices are pivotal. In the presence of private information, the mechanism design approach is highly appropriate, as voters’ incentives can be heavily influenced by the institutional settings that determine how votes are transformed to election outcomes. The analysis of information aggregation in large-scale elections brings important insights to our understanding of representative democracy. Due to the nonexistence of a core and the cyclical structure of pairwise comparison, there may be a fundamental difficulty in the preference aggregation by majoritarian democracy in large-scale elections. The chapter concludes with questions for future research: How does the limitation of preference/information aggregation in large-scale elections affect the stability of representative democracy? What determines the robustness of democratic norms? What is the role of the media in the presence of information asymmetry, particularly in ideological battles where information filtering can play an exacerbating role?

One relatively recent development related to the growth of practical market design is the need to deal with big strategy sets that may involve parts of the economic environment beyond the boundaries of the individual marketplaces. A related matter is that in naturally occurring environments, strategies may be discovered in the course of play. Discovering new strategies is very much like inventing new technology, or new game theory: all of these things can change the game in important ways. These issues blur the borders of what historically were regarded as separate domains of game theory, namely the theories of cooperative and non-cooperative games.

The purpose of this chapter is to review the key contributions of game theory to the field of cultural evolution, focusing particularly on interfaces between cultural evolution and economics. Because many readers may not be familiar with the interdisciplinary field of cultural evolution, it begins with a brief orientation to this field as a scientific enterprise and then highlights the important ways that game theory has been deployed in both theoretical and empirical research within the field, noting spillovers and interactions with economics.

This chapter discusses both motivations and choice mechanisms that underly how people make strategic choices. It lists multiple areas where our understanding could benefit from closer study. About the early work by Tversky and Kahneman on framing (i.e., the dependence of human choice behavior on different presentations of what to rational agents should be irrelevant factors), it concludes that one must make a choice between normative adequacy and descriptive accuracy. Concerning recent work on reciprocity, it argues that players’ reactions to, for instance, kind acts may lead to volatile behavior in settings with noise, whereas reciprocity toward perceived kind types can be more forgiving and result in more stable reciprocal relations.

A large share of individuals deviates from self-interested behavior in many paradigmatic games, but in many other strategic situations almost all individuals behave in a self-interested manner. Models with heterogeneous social preferences provide a unifying understanding for these seemingly contradictory facts by focusing on the interaction between agents with other-regarding and selfish preferences. This focus explains why and when selfish agents behave as if they were other-regarding, as well as to why and when other-regarding agents behave as if they were selfish. This focus also helps understand (1) the importance of seemingly irrelevant institutional details, (2) the role of contractual incompleteness for the behavioral relevance of social preferences, (3) the role of social preferences for the prevalence of contractual incompleteness, and (4) why social preferences are an important component in explaining key characteristics of the employment relation. More recent evidence suggests that the empirical distribution of social preferences can be parsimoniously characterized by a small number of preference types which also have out-of-sample predictive power for important behaviors such as the demand for politically enforced redistribution.

Under what conditions do the behaviors of players, who play a game repeatedly, converge to a Nash equilibrium? If one assumes that the players’ behavior is a discrete-time or continuous-time rule whereby the current mixed strategy profile is mapped to the next, this becomes a problem in the theory of dynamical systems. We apply this theory, and in particular the concepts of chain recurrence, attractors, and the Conley index, to prove a general impossibility result: There exist games for which any dynamics will fail to converge, from certain initial conditions, to the set of Nash equilibria. The games which help prove this impossibility result are degenerate, but we conjecture that an analogous result holds, under complexity assumptions, for nondegenerate games. We also prove a stronger result for approximate Nash equilibria: For a set of games of positive measure, there are no game dynamics that converge to the set of approximate Nash equilibria for some substantial approximation bound. These impossibility results also apply to dynamics with memory. We argue that these results further weaken the appeal of the Nash equilibrium as the solution concept of choice in game theory, and discuss alternatives suggested by the dynamics point of view.

This chapter introduces the three contributions that constitute Part III, “Population Dynamics, Learning, and Biology.” These contributions discuss biology and population dynamics in game theory. The chapters concern models of strategy adaptation: process models. Such models have refined our understanding of Nash and other equilibrium concepts, and the evolution of population shares through time is itself an object of interest from the perspectives of biological reproduction and of learning human (and artificial) agents.

This chapter tells how von Neumann and Morgenstern were brought together to write the “Theory of Games and Economic Behavior.” It discusses von Neumann’s early involvement in games before his emigration, Morgenstern’s curious career in interwar Vienna, their unlikely collaboration as exiles at Princeton during World War II, and the effect of war and the Cold War on the reception of their research.

This chapter gives an extensive overview of techniques and algorithms for representing and solving large imperfect-information extensive-form games and reports on recent breakthroughs that have been achieved for the game of poker. These breakthroughs were made possible by advances in three key areas: (1) game abstraction (i.e., the systematic construction of significantly smaller extensive-form games that are strategically similar to the original game), (2) equilibrium-finding algorithms, and (3) solving subgames during game play in much finer abstractions than would be possible in advance. A new proposal put forward is to reason about games whose rules are modeled via a programming language.

This chapter points to a fundamental difficulty associated with the formal study of dynamic adjustment processes toward a Nash equilibrium in the context of social and economic problems (i.e., for human players). This difficulty has created an unfortunate dichotomy of researchers and has hindered progress in this area of research. It suggests, with a couple of examples, that a promising way to overcome this problem is to strengthen the empirical side of research on adjustment dynamics.

This chapter introduces the six contributions in Part IV, “Computer Science.” The main focus is on topics in algorithmic game theory, algorithmic mechanism design, and computational social choice.

This chapter describes the early contributions of Émile Borel to the theory of games. It also discusses early reactions and controversies.

It took some thirty years before the game theoretic ideas of Émile Borel became known to a wider audience, with the publication of the seminal book by John von Neumann and Oskar Morgenstern. Similarly, it took thirty years for the evolutionary approach of Brown, von Neumann, and Nash to be taken up by a wider community. By now, a substantial set of potential updating mechanisms has been modeled and analyzed via game dynamics. Large as it is, it is yet unlikely to capture the full range of adaptive behavior used by human players. A closer relation between the dynamics of nonequilibrium play and empirical data on adaptation and learning is sorely needed. This is a topic where psychology and economics can fruitfully join hands.

This chapter introduces the three contributions that constitute Part VII, “Political Science,” about game theoretic models in political science, armed conflict, and trade policy.

This chapter summarizes three key contributions of Borel’s 1921 paper: (1) the strategic normalization of games in extensive form, (2) the introduction of randomized strategies, and (3) expected payoff maximization. It also discusses the impact Borel had on other early contributors to game theory, notably von Neumann, Nash, and Schelling.

This chapter shows how the theory of symmetric two-player zero-sum games, which was initiated by Borel in 1921, can be used for randomly selecting an alternative based on quantified pairwise comparisons between alternatives. It points out desirable properties satisfied by the equilibrium distribution and gives examples where these distributions arise as the limit of simple dynamic processes that have been studied across various disciplines, such as population biology, quantum physics, and machine learning.