On October 2, 2024, Springer published a significant book by researcher Pierre Dehez titled “Game Theory for the Social Sciences: Conflict, Bargaining, Cooperation, and Power”. This book presents the fundamental concepts of game theory and explains its uses and applications in social sciences. Starting from non-cooperative games and progressing to cooperative games, it explores a wide range of topics, including the analysis of zero-sum games, voting rules and their practical implications, and other applications of game theory. Uniquely, it emphasizes cooperative concepts and their normative applications, providing a fresh perspective.
The goal of game theory is to propose solutions to decision-making problems involving multiple individuals who may have conflicting interests. According to the traditional view, non-cooperative games provide a detailed model of strategic interaction between individuals, while cooperative games describe intuitively appropriate characteristics for group decisions.
The book offers a coherent set of classical game theory tools, from equilibrium concepts to bargaining solutions and power indicators, all explained with examples and practical applications.
“In economics, and generally in our society, the behavior of the entire system results from the direct interaction between individuals who, unlike actors in the typical framework, consider the people they are interacting with. This changes the perspective through which we analyze social sciences and, particularly, economics. Game theory allows us to understand this interaction, which can be cooperative or non-cooperative, where the term strategic behavior is appropriate.”
The book is divided into five chapters. The first chapter provides a general introduction to the concepts, history, and current state of game theory. The second chapter focuses on non-cooperative games, which are games where players are not bound by pre-existing agreements. If pre-existing agreements are possible, they are included within the strategies available to the players. This non-cooperative framework does not exclude cooperation between players. Except in the case of zero-sum games, games often involve a mixture of conflict and convergence of interests. This is precisely what makes their study interesting.
The chapter focuses on games involving two players, which is not a real limitation, as these games contain most of the components needed to analyze the conceptual difficulties of non-cooperative games. It is also limited to games with complete information: each player is assumed to know all the game’s data, including the preferences of their opponent. It also addresses individual decision-making under uncertainty in two-player non-cooperative games, using symbolic examples in two representations: the extensive form, where players move sequentially, and the strategic form, where each player, when making a strategic choice, ignores the decision made by the other. Though focused on non-cooperative situations, it does not exclude cooperation. The introduction of mediators or correlated strategies represents steps toward cooperation. However, binding agreements are excluded. Behind the intervention of a mediator or the use of correlated strategies, there is no formal contract that players agree upon and sign. Players can avoid relying on the mediator or deviate from the proposal they receive from a random mechanism.
The third chapter addresses bargaining and consensus issues. The bargaining problem faced by two players is defined by a set of outcomes, with one of them designated as the disagreement outcome, also known as the status quo. While their preferences are in conflict, they know that reaching an agreement is desirable as long as there are better outcomes than the status quo. As a result, they have an interest in bargaining. Bargaining either leads to a consensus on a particular outcome, or it fails, and they revert to the disagreement outcome. The status quo acts as a threat. If the two parties eventually reach a consensus, this consensus must result from a compromise: one party may gain more at the expense of the other.
We do not describe here an actual negotiation process nor attempt to predict the outcome of the negotiation. Instead, we adopt a normative viewpoint. The idea here is to propose a rule based on the combination of a set of desirable properties (axioms) that can be applied to any bargaining problem to produce a solution. In this way, the solution to the bargaining problem is directly related to the importance of the axioms underlying the rule. This is the approach explained by Nash, marking the first step toward cooperation.
The fourth chapter deals with cooperation and the distribution of the benefits of cooperation among any number of players, assuming that benefits are transferable and agreements enforceable. When two players face each other, the issue of cooperation is simple: they either cooperate or they do not. However, when there are three or more players, they can form coalitions. This greatly enriches the analysis. The concept of coalition is, in fact, central to the analysis of games involving more than two players. If a group of players decides to form a coalition, it means they separate from the “grand coalition” of all players to cooperate among themselves. Nevertheless, the question at hand concerns cooperation among all players and the distribution of the resulting “social” outcome among them. Thus, the role of coalitions remains hypothetical. Beyond the maximum gain that players can generate entirely through cooperation, we need to know the gain that each coalition of players can achieve by cooperating among its members, independent of the actions of players outside the coalition. When a coalition forms, what matters is the outcome it can achieve, without specifying the organizational details.
Cooperative game theory is concerned with issues such as cooperation, coalition formation, organizational structure, commitment, trust, compromise, threat, and enforceability, and even the entire legal/social/cultural environment. It places less emphasis on tactical optimization, detailed spelling out of rules, and numerical computation of outcomes and rewards. The possibility of binding and enforceable agreements is a distinguishing feature between cooperative and non-cooperative games. As Aumann puts it, “A game is called cooperative if there is a mechanism available, such as a court, to enforce agreements. In a cooperative game, any possible outcome can be achieved if the players agree on the appropriate agreement.”
Cooperative games, also called coalition games, are divided into two types depending on whether transfers between players are possible. Here, we assume that transfers are possible, assuming that benefits are specified in terms of some commodity money. These are games with transferable utility, also known as side-payment games.
One of the peculiarities of cooperative games is the existence of multiple solution concepts. This arises from the fact that the study of cooperative games naturally invites normative reflection. In this chapter, we will analyze three concepts: the core, the Shapley value, and the nucleolus. The core defines allocations of the social outcome that are immune to deviations by player coalitions. It may be empty or contain many distributions. The Shapley value and the nucleolus are allocation rules that define a unique allocation of the social outcome. These rules are based on desirable properties, axioms, just like Nash’s solution to the bargaining problem.
The fifth chapter examines voting and power issues. If political science, as an empirical discipline, is the study of the formation and sharing of power, then the issue of power is a central focus in the analysis of political processes. But what do we mean by power? Can it be measured? This chapter analyzes collective decisions within committees. A committee is a group of decision-makers who make decisions according to established rules. These rules are translated into a list of winning coalitions, groups of decision-makers who, according to the rules, are able to make a decision. The set of decision-makers and the list of winning coalitions define the voting game. At the start of the decision-making process, a proposal is put on the table, and the question is which subgroups of decision-makers, if they agree, can together impose a decision, i.e., accept or reject the proposal.
Not all decision-makers necessarily have the same weight in the decision-making process. Some committees may have different numbers of votes, some may enjoy veto power, and others may have special constraints, such as minority protection. Examples of such committees include shareholder general meetings, parliaments (where decision-makers are defined by parties), and international organizations such as the European Council or the United Nations Security Council.
The chapter discusses in detail the problem of measuring the relative power of each committee member, beyond the apparent power granted to them by the rules. We aim to measure each decision-maker’s ability to influence the voting outcome in light of the prevailing rules.
The book introduces many key concepts, but four stand out as principal concepts: Nash equilibrium, Nash bargaining solution, the core, and the Shapley value.
What sets this book apart from many other game theory books is its clear and accessible writing style, avoiding complex mathematics. It provides many different and interesting examples, from fair division problems and bargaining to voting systems. It is an essential study of a frequently referenced but often misunderstood subject, serving as a necessary reference for students of political science, economics, law, and related social sciences.
