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

Chapter 2 is devoted to AI ethics, broadly defined. It provides an overview of ethical, responsible, safe, trustworthy, transparent and explainable, accurate, just and fair, accountable, sustainable, robust, accessible and inclusive AI. Just as the definition of AI itself is fraught with disagreement, words with a connotation of “good” AI have generated considerable controversy among academics, social movement activists, journalists, business leaders, and lawmakers. This chapter aims to represent the plurality of positions. Furthermore, the adjectives associated with getting AI right are mutually supportive, but tensions between desirable goals are mentioned as well.

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.

In this chapter, we provide an introduction to the countable-infinitary logic called L-omega-1-omega. There is no computability theory in this chapter. We prove all the main results that are about countable structures. This includes the development of Scott ranks, and the type omitting theorem. All of this is done with the authors perpective, so that it fits better with the computability notions to be developed later.

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.

This chapter explores different strands of the theory of two-player zero-sum games and equilibrium concepts for general multiplayer games. The conventional viewpoint is that equilibrium is an extension of the concept of value (and its associated optimal strategies) to non-zero-sum games, and the value is just a special case of an equilibrium payoff. However, it is argued that a number of important concepts apply only to one of these concepts.

This chapter generalizes entanglement to multipartite systems, focusing on robustness and scalability across qudits.

We had introduce the jump of a structure in Part I of this book series. We now iterate this jump through the transfinite. We show the first and second iterated-jump inversion theorems, and give some applications.

Overspill is a powerful tool for arguments at the boundary of the hyperarithmetic. We take the Harrison-linear ordering approach to overspill arugments. We used them to introduce structures of high-Scott rank.

This chapter introduces the three contributions in Part VI, “Individual Behavior in Strategic Interactions.” It frames the contributions in terms of two main issues that underlie models in behavioral game theory: (1) what motivates players (i.e., their goals or preferences) and (2) the mechanisms or procedures behind their choices.

Game theory has a long history in the political economy of trade policy. Beginning with work by Johnson in the 1950s, trade economists have used these tools to study strategic interactions between governments, interest groups representing industries or factors of production, political parties, and legislators representing different voting districts. Research has focused both on trade policies that have been set noncooperatively, sometimes in response to internal political pressures, and on the negotiation and features of cooperative trade agreements.