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.

Chapter 3 presents the other side of the coin, namely AI risks and harms. Automated decision systems, chatbots, recommender systems, and other AI-powered software and platforms have been found to cause potential risks or actual harms to affected persons and communities. Such risks and harms include bias and discrimination, surveillance, inaccurate, incorrect and unreliable output, disinformation, misinformation or manipulation, harm to life, livelihood and wellbeing, privacy violations, decline in product and service quality, political polarization, online radicalization and algorithmic censorship, and job replacement. Some of these harms, such as bias and discrimination, have already been experienced frequently, while others, like job replacement, point to future risks. It is also worth noting that AI risks and harms often aggravate existing social and political problems. For example, political polarization and radicalization, while exacerbated by algorithmic curation, appear to have origins in societal divisions. Finally, AI is criticized for causing system-level harm in the form of environmental degradation, exploitation of labor, and market concentration.

This chapter examines bipartite systems, focusing on residual uncertainty and the role of conditional entropy.

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.

Under the hood of the game metatheorem are the iterated true-stage systems. They proved a way of approximating Simga-alpha information in a way that is combinatorially clean. As an application we prove the tree-of-structures theorem, that requires the full power of an iterated true-stage system and cannot be proved with the game metatheorem.

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.