Game Theory in Jurisprudence ebook

Preface

Game theory is a branch of mathematics devoted to the study of strategic interactions, i.e., interactions which involve more than one agent and in which the effects of each agent’s decision depend not only on her decision but also on the decisions of the other agents so that what each agent does depends on what she thinks the other agents will do. Game theory has been successfully applied in many areas of the natural sciences (especially evolutionary biology) and the social sciences (economics and sociology in particular). It has also proved to be a useful tool for legal scholars dealing with various branches of law (e.g., contract law or constitutional law), for the simple reason that strategic interactions constitute an important object of legal regulations and lie at the root of the legislative process. Moreover, game theory has offered insights not only regarding specific problems of various branches of law but also numerous legal-philosophical problems. Its knowledge, then, can undoubtedly help advance the research of both scholars interested in legal dogmatics and those more inclined towards legal philosophy (or jurisprudence – we treat these two terms – legal philosophy and jurisprudence – as equivalent). The main goal of this book is to present the basic solution concepts of game theory in this latter context – that of their various legal-philosophical applications. Herein also might lie its difference from other books on the applications of game theory in law (e.g., the excellent book Game Theory and the Law by C. Baird, R. H. Gertner, and R. C. Picker, and a collection of papers Game Theory and the Law edited by E. Rasmusen) which only deal with legal-philosophical problems marginally, being focused on specific legal problems of various branches of law. The reverse proportions are assumed in this book: it deals with the latter problems only marginally, being focused on the former. The ambitions of this book are modest: it is intended basically as an introduction to game theory for legal philosophers (this is the reason why it is focused primarily on the conceptual rather than mathematical aspects of game theory[1]). Nonetheless, we hope that, besides fulfilling this role, it can also offer some new insights regarding the applications of game theory in legal philosophy.

This book is divided into three parts: Part I is an introduction to game theory; Part II is devoted to the applications of game theory to the general issues of jurisprudence; Part III is devoted to the applications of game theory to more specific legal-philosophical issues that emerge in the context of concrete branches of the legal system. The Epilogue is a reflection on the relations between economic approach to law, the so called Law and Economics (postulating the analysis of law in terms of economic rationality and economic efficiency), whose game-theoretic approach to law is a special variety, and the projects of the naturalization of law. The book ends with Appendix in which the problem (only mentioned in Chapter 1) of the relationships between instrumental rationality and emotions is analyzed in detail.

PART ONE

INTRODUCTION TO GAME THEORY

Chapter 1

The basics of game theory

1. Twelve questions about game theory

In this chapter we shall present the basics of game theory to which we appeal in the following chapters devoted to the applications of game theory in jurisprudence. Our method of presentation will consist in asking twelve essential questions about game theory and in attempting to answer them in a concise manner.

2. What is game theory?

Game theory is a branch of rational choice theory, the other two branches being decision theory and social choice theory. What distinguishes game theory from the other branches of rational choice theory is that it deals with strategic situations, i.e., in situations in which there is more than one agent and each agent’s decision depends on what she expects the other agents to do because the outcomes of each agent’s decision depend on the decisions of those other agents. The twofold task of game theory is to provide theoretical models of these situations and to provide criteria of rational choice in them. In subsequent sections we shall present at length the manner in which game theory realizes this task. In this section we would like to devote some attention to the other branch of rational choice theory, viz., decision theory, because we shall appeal to its solution concepts in several parts of this book.

Decision theory studies decisions in parametric situations, which can be divided into two types: (1) (most frequent): situations in which there is only one person who “interacts” with ‘the world’ (e.g., an agent’s decision whether to take an umbrella or not when going outside will depend on her subjective probability distribution on variously defined ‘states of the world’, say, “it will rain”, “it will not rain”, or “it will rain lightly”, “it will rain heavily”, “it will not rain”); (2) (less frequent): situations in which there is a large number of persons, the outcomes of each person’s decision depend on the decisions of other persons, but various possible decisions of the others persons have to be viewed by each person as the ‘states of the world’, because for computational reasons it is impossible to form separate expectations regarding each other person’s decision (e.g., ‘the states of the world’ on which a person’s decision depends as to whether to buy or sell stocks are in fact determined by the decisions of others but each person makes the decision, for computational reasons, in a parametric rather than a strategic way). Thus, one can say that in situations (1) the states of the world are determined by the states of nature, whereas in situations (2) they are determined by a large number of human decisions. Now, the twofold task of decision theory (analogous to that of game theory) is to provide theoretical models of parametric situations and to provide criteria of rational choice in them. As for the former: depending on the relation between actions and outcomes, four kinds of a decision problem were distinguished, viz.: decision problem under certainty, if the relation is deterministic; decision problem under risk, if at least one action is correlated with a lottery, i.e., a set of possible outcomes, each of these outcomes being assigned objective probability; decision problem under uncertainty, if at least one action is correlated with a lottery whose outcomes cannot be assigned any probabilities (or at best can be assigned subjective probabilities); decision problem under ignorance, if, for at least one action, it is not known what outcomes it leads to. As for the latter: decision theory teaches us that an agent making a decision under certainty or under risk decides rationally iff she selects the action which maximizes her utility function. More accurately, under the conditions of certainty, a rational agent will choose the action determining the outcome that maximizes the value of her utility function. Under the conditions of risk, in which an agent’s actions are connected probabilistically with the outcomes, a rational agent will choose the action that maximizes the expected value of her utility function.[2] The expected value of a utility function for a given action is called the expected utility of this action. Thus, one can say that a rational agent making a decision under risk chooses the action with maximum expected utility. However, it should be noted that decision theorists have not worked out a unique criterion of rationality for choosing under certainty (we shall return to this issue in Chapter 5, Section 4) and did not work out any criterion for decision-making under ignorance.

Finally, social choice theory is a branch of rational choice theory dealing with the problem of social choice, i.e., the problem of aggregating individual preferences into a social preference. We shall not present this theory in more detail, as it will not play any role in our considerations (with a small exception in Chapter 4, Section 3, where we shall appeal to its basic concept – that of social preference).

3. What are the branches of game theory?

Game theory is divided into the classical and the non-classical. Classical game theory embraces non-cooperative game theory and cooperative game theory, whereas non-classical game theory is evolutionary game theory. Non-cooperative game theory analyzes non-cooperative games, i.e., games in which joint-action agreements between agents are not enforceable (binding), whereas cooperative game theory analyzes cooperative games, i.e., games in which joint-action agreements are enforceable (binding). Eric Rasmusen describes the differences between non-cooperative and cooperative game theory in the following way:

Cooperative game theory is axiomatic, frequently appealing to Pareto-optimality, fairness, and equity. Noncooperative game theory is economic in flavor, with solution concepts based on players maximizing their own utility functions subject to stated constraints. Or, from a different angle, cooperative game theory is a reduced-form theory, which focuses on properties of the outcome rather than on the strategies that achieve the outcome, a method which is appropriate if the modeling is too complicated (Rasmusen 2001b, p. 21).

Accordingly, the terms ‘cooperative’ and ‘non-cooperative’ refer not to the nature of the outcomes of the game, but to the way in which the player’s actions are implemented: collectively in the former case and individually in the latter. Thus, it is not the case (contrary to what the names of the games might suggest) that cooperation cannot be the result of non-cooperative games, and that cooperative games are free from competition between players. A more detailed discussion of cooperative game theory will be provided in Section 11.

A sub-branch of game theory is bargaining theory. Bargaining theory is aimed at solving the bargaining problem (i.e., the problem of distributing the surplus of goods between parties who contributed to bringing it about) by providing unique solutions to it. Bargaining theory can be constructed in two different ways: within cooperative game theory and within non-cooperative game theory. One of the most plausible (i.e., satisfying a set of plausible axioms) bargaining solutions provided within cooperative game theory is the Nash arbitration scheme, which prescribes the outcome that maximizes the product of the bargainer’s increments of utility in relation to their initial bargaining position. Bargaining theory will be presented in more detail in Section 12. To sum up, the ‘landscape’ of game theory can be presented in the following way:

4. What are the functions of game theory?

Game theory can fulfil three functions: descriptive, normative and clarificatory. The descriptive function embraces two more specific and interconnected functions, viz., explanation and prediction. Game theory can be interpreted as providing a model of human behaviour, i.e., as a tool for explaining and predicting human behaviour. By prediction we mean both what may be dubbed ‘prospective prediction’, i.e., predicting how people will behave in the future, and ‘retrospective prediction’ or ‘retrodiction’, i.e., figuring out how people may have behaved in the past. The normative function may be twofold: first, it consists in providing criteria of rationality (this is the direct and fundamental function of game theory); second, it may arguably consist in determining the content of other normative concepts than that of rationality (e.g., the concept of justice). It seems that, apart from the descriptive and normative function, one can also distinguish a clarificatory function of game theory, which consists in elucidating various concepts (e.g., the concept of convention). Of course, if the concept is a normative one (as is the case, for instance, with the concept of justice), the normative function is at the same time clarificatory. Likewise, a descriptive function may be clarificatory (in the sense that in order to precisely describe human behaviour one has to clarify at the same time the concepts indispensable for such description). Thus, while fulfilling a descriptive or normative function, game theory usually fulfils a clarificatory function at the same time.

5. What is a game?

A game (in a game-theoretic sense) is an interaction between two or more agents which is determined by the rules which specify: the list of players, the strategies available to each player, the sequence in which players make their moves, the payoffs of each player for all possible combinations of strategies pursued by the players. The above definition uses two concepts which need further clarification: strategies and payoffs. A strategy is a complete plan of action, i.e., a plan which specifies what the agent is supposed to do at each possible stage of the game. The payoffs of a player capture values, i.e., utility, the player assigns to the various outcomes of the game. They may reflect various – and not only selfish – motivations of players. For example, if a player cares about the realization of her opponent’s interests as much as she does about the realization of her own interests, then this ‘utilitarian’ motivation will be reflected in her utility function (which is a technical tool for presenting a player’s preferences over various outcomes of the game), and thereby in her payoffs. In the course of the analysis of particular problems by means of rational choice theory, one must be explicit about the assumptions concerning utility, especially those concerning the scale of utilities and the possibility of making interpersonal comparisons of utilities. The scale may be ordinal or cardinal, and the comparisons may be allowed or disallowed. A cardinal measure of preferences conveys information not only about the ordering of an agent’s preferences, but also about their strength. Thus, we have four possible combinations, which we rank from the weakest (1) to the strongest (4).[3]

Scale/ Interpersonal comparisons

Not Possible

Possible

Ordinal

1

3

Cardinal

2

4

Fig. 2. Assumptions about utility

In combination 3, interpersonal comparisons will be based on fuzzy calculations, while in combination 4 on strictly additive utilities. In general, game theorists try to make possibly weak assumptions concerning utility.[4]

[1] For readers who wish to broaden their knowledge of the mathematics of game theory we can recommend, e.g., Luce, Raiffa 1957, Gibbons 1992, Myerson 2002, Osborne, Rubinstein 1996, Rasmusen 2001a.

[2] This definition can be applied to conditions of uncertainty only if an agent at least implicitly assigns subjective probabilities to the outcomes of her actions; if she does not assign any probabilities to these outcomes, then this definition is useless; see Milnor 1964, Szaniawski 1971. French 1993.

[3] See, e.g. Lissowski 1986, pp. 156–158, or Hardin 1988, pp. 169–175.

[4] This does not apply, however, to cooperative game theory.