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Gabriel Frahm, Rational Choice and Strategic Conflict: The Subjectivistic Approach to Game and Decision Theory

Andrés Perea
p. 403-408
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Gabriel Frahm, Rational Choice and Strategic Conflict: The Subjectivistic Approach to Game and Decision Theory, Berlin: De Gruyter, 2019, 340 pages, ISBN 978-311059515-4

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1By reading the literature, one often gets the impression that decision theory and game theory represent two separate worlds. This not only applies to classical texts such as the game theory books by von Neumann and Morgenstern and Luce and Raiffa, but also to more recent treatments of game theory. Many people will find this puzzling, as game theory may naturally be viewed as an instance of decision theory where every player faces his own, personal decision problem in which the uncertainty is about the possible choices of his opponents, and thereby about the possible outcomes in the game.

2This book is a delightful attempt at presenting decision theory and game theory in a unified manner, and in my view it has brilliantly succeeded in its mission. It takes a bottom-up approach to game theory, by constructing it from purely decision theoretic principles, which enables the author to provide a seamless transition between decision theory and game theory. By modelling every player as a decision maker with his own, personal set of states (representing the events about which he is uncertain), it transforms a game into a collection of one-person decision problems—one for every player. This is what the author calls the subjectivistic approach to game theory. As natural as this approach may seem, it clearly separates this monograph from most other books on game theory.

3Another key distinguishing feature in the book is its focus on the interdependence between choices and states—a phenomenon to which the author refers as counterfactual reasoning. That is, the decision maker may change the course of events by the choice he makes, and this effect has to be taken into account ex-ante when comparing the different choices to one other. This phenomenon is rarely discussed in books on decision theory and game theory, but it is at the center stage of this monograph. It leads to a new class of games, which the author calls coherent games, in which a player knows that his action has an influence on the actions of his opponents, and to a new solution concept for such games, termed rational solution. This concept is at the heart of the game theoretic analysis in the book. It is quite contrary to most other game theory books that typically revolve around the notion of Nash equilibrium and its refinements.

4Towards the end, the book compares the subjectivistic approach to other ways of looking at game theory, such as the traditional approach (as presented in standard game theory textbooks), the epistemic approach and the theory of moves by Steven Brams. This detailed discussion, which spans over three chapters, clearly positions the subjectivistic approach within some of the existing approaches to game theory, and provides a critical discussion of the assumptions made in each of these frameworks. Not many books contain such an extensive, and critical, comparison with other approaches. In the remainder of this review, I will zoom in on some of the distinguishing features discussed immediately above.

A Subjectivistic Approach

5The book starts with a detailed account of one-person decision theory, focusing on Savage’s axiomatic foundation of expected utility. It emphasizes that all ingredients in Savage’s framework, the set of choices, the set of states of the world, and the set of consequences, may be perceived as personal, and hence may be different across decision makers. Also the probabilities that a decision maker assigns to the various states of the world are subjective. This is important, as a game will later be viewed as a collection of one-person decision problems in which every player faces his own, personal set of states of the world, his own set of possible consequences and assigns his own, subjective probabilities to the various states. This, in fact, is the essence of the subjectivistic approach in this book.

6Not only does Savage’s framework justify the use of utilities in a game matrix or tree, it can also be viewed as a theory of personal probability. Indeed, Savage’s postulates on the decision maker’s preferences over acts induce a unique subjective probability distribution over the set of states. Quite deservedly, the author zooms in on this peculiar aspect of Savage’s theory, as it justifies the use of personal probability in games later on.

Counterfactual Reasoning

7What distinguishes this book from most other treatments of decision theory is the key role of counterfactual reasoning. By this, the author denotes the phenomenon that the choice by the decision maker may have an influence on the course of events taking place in the world. Hence, if the decision maker compares his choice to another choice, he must reason about the effect that the other choice would have on the events that are happening in the world.

8Most accounts of decision theory (including Savage’s) assume, in contrast, that the decision maker cannot influence (or, better said, believes that he cannot influence) the course of events. Counterfactual reasoning, however, enables the author (after others) to explain several well-known paradoxes, such as Newcomb’s paradox and Ellsberg’s paradox, by staying within the realm of expected utility. In Ellsberg’s paradox, for instance, the frequently observed behavior can be explained if we assume that the person, by placing a particular bet, believes to influence the configuration of balls in the urn. Similarly, in Newcomb’s paradox, the person may choose the single box if he believes that his choice may influence the content of the opaque box.

9At the same time, counterfactual reasoning can be shown to be implicitly present in well-known decision theoretic criteria such as Wald’s maxmin rule and Savage’s minimax regret rule, and von Neumann’s maxmin criterion for two-person zero-sum games. In short, counterfactual reasoning seems to be able to explain well-known phenomena in novel ways, and may thus be worthwhile to explore in depth. This is precisely what this book does: It puts counterfactual reasoning at center stage, which leads to a novel and fresh approach to both decision theory and game theory.

From Decision Theory to Game Theory

10This book stands out by its very smooth, and natural, transition from decision theory to game theory. A game is modelled as a collection of one-person decision problems—one for every player—in which each player faces his own particular set of states of the world. Quite naturally, the states of the world about which a particular player is uncertain correspond to the combinations of actions that can be chosen by his opponents. If the player satisfies the Savage postulates, as is assumed in this book, his preferences can be summarized by a subjective probability distribution over the opponents’ action combinations, together with a utility function.

11This setting can be expanded as to allow for counterfactual reasoning. In the context of a game, counterfactual reasoning would mean that the player expects his own action to affect the opponents’ actions. In the book, a scenario describes how the opponents’ actions would be influenced by the action of the player at hand. In the spirit of Savage, the player would then hold a subjective probability distribution over the possible scenarios, rather than the possible opponents’ actions. To make this compatible with the Savage approach, one would have to refine the state space as to include the interdependence between the player’s own actions and the opponents’ actions. The standard decision theoretic approach, in which counterfactual reasoning is excluded, can be retrieved as a special case where the player assigns probability zero to all scenarios where different actions by himself induce different opponents’ actions.

12This approach does not only apply to one-shot games, but also to dynamic games. The author models a dynamic game as a collection of one-person decision trees—one for every player—in which the chance moves either correspond to real moves of nature, or to the opponents’ actions chosen at the various points in time.

13Summing up, the book views a game, be it static or dynamic, with or without counterfactual reasoning, as a collection of one-person decision problems, in which every player faces his own, personal set of states of the world, and forms his own, personal probability distribution over these states. This is the basis for the “subjectivistic” approach to games proposed by this book.

14I personally find this approach very appealing and natural, and it makes the step from one-person decision theory to game theory much smaller than usual. It is refreshing to see that games can be modelled, and analyzed, from a purely decision theoretic point of view.

The Rational Solution

15The key game theoretic concept in the book is not Nash equilibrium, but the rational solution. It is a novel concept especially designed for so-called coherent or quasi-coherent games, in which every player engages in counterfactual reasoning. That is, each player knows or, at least, believes that his own action may affect the actions chosen by his opponents. That is, every player has a clear idea of how his own actions influence the actions of the others.

16The rational solution represents a situation where no player can improve the outcome by choosing a different action, given how he thinks the alternative action would influence the opponents’ actions. The key difference with Nash equilibrium is thus that the latter does not take into account counterfactual reasoning, whereas the rational solution does.

17By an intuitive and constructive proof, it is shown that every coherent game with compact action sets and continuous payoff functions allows for a rational solution. In some of these games, Nash equilibria may fail to exist. Moreover, in some situations the rational solution is able to explain phenomena that Nash equilibria cannot. For instance, cooperation in the Prisoner’s Dilemma can be achieved in a rational solution if the player believes that switching to the non-cooperative action induces the other to switch alike. It is to be expected that the rational solution will also shed a new light on some applications of game theory to economics and other disciplines.

18Conceptually, the rational solution may be viewed as a very basic solution that is exclusively based on decision theoretic principles. It only requires every player to choose optimally given his own counterfactual reasoning.

Comparison to Other Approaches

19The last three chapters discuss in detail the differences between the subjectivistic approach advocated by this book and some alternative treatments of game theory in the literature. An important difference with the traditional game theoretic approach is that the latter often seems to take an objectivistic, rather than a subjectivistic, stance. In the concept of correlated equilibrium, for instance, the common prior assumption indicates that the players’ subjective probability distributions come from a common source (thus live on a common state space), which seems to be in conflict with the subjectivistic view. The book also provides a critical discussion of Nash equilibrium, questioning the assumptions that players randomize and hold correct beliefs about each other’s actions.

20It is argued by the author that the epistemic approach to game theory is closer to the subjectivistic approach than the traditional one. Indeed, both take a one-person perspective in which the game is analyzed from the subjective viewpoint of a single player. The most important difference, according to the book, lies in the common belief assumption, which is central to the epistemic approach but not to the subjectivistic approach. Common belief in rationality, which states that a player does not only choose rationally, but also believes that others choose rationally, believes that others believe that others choose rationally, and so on, lies at the heart of epistemic game theory. The rational solution, on the other hand, does not require higher-order belief in rationality, and the author argues that in many real-life situations the conditions of common belief in rationality may not be met.

21The theory of moves by Steven Brams, which is discussed in the last chapter, plays a special role in this book as, together with the rational solution, it seems to be the only systematic attempt at incorporating counterfactual reasoning in the analysis of a game. In fact, as the author states, the theory of moves was a principal motivation for him to study counterfactual reasoning in games. The book provides a rigorous comparison between the rational solution and Brams’ so-called “non-myopic” equilibrium. Conceptually, the main difference between the two is that Brams’ theory of moves is based on the assumption that players have complete (but not necessarily perfect) information and common knowledge. In contrast, coherent games are based only on the assumption of perfect (but not necessarily complete) information, which means that the players know the strategy of each other. Complete information and common knowledge play no role at all in coherent games. Moreover, the moves and countermoves in Brams’ model are interpreted as events that can really happen, whereas in the author’s framework these countermoves remain hypothetical events that merely take place in the player’s mind. Nevertheless, the theory of moves can be viewed as a refinement of the author’s approach, as every non-myopic equilibrium in Brams’ model corresponds to a rational solution if the game at hand is considered coherent. To this extent, Frahm’s approach is a generalization of Brams’.

22This book is a wonderful, systematic attempt at placing game theory within the framework of one-person decision theory, instead of treating both as separate entities. The seamless transition from decision theory to game theory that is delivered by this book will, without any doubt, be an eye-opener to many readers. What I have also appreciated is that the book, in its quest for a decision theoretic foundation of game theory, does not settle for compromises. Rather than trying to accommodate existing concepts, like Nash equilibrium, correlated equilibrium and rationalizability, into the framework, the author looks for a game theory concept that can be based on decision theoretic principles alone. This finally culminates in the subjectivistic approach to game theory, and the associated rational solution, which offers a novel approach to game theory as a whole. The high degree of originality, together with the beautiful writing style and the many insightful examples, make the book a pleasure to read. When taken together, all of these aspects make the book one of a kind, and a must-read for anyone who is interested in the foundations of decision theory, the foundations of game theory, and the intriguing connections between the two.

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Andrés Perea, « Gabriel Frahm, Rational Choice and Strategic Conflict: The Subjectivistic Approach to Game and Decision Theory »Œconomia, 10-2 | 2020, 403-408.

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Andrés Perea, « Gabriel Frahm, Rational Choice and Strategic Conflict: The Subjectivistic Approach to Game and Decision Theory »Œconomia [En ligne], 10-2 | 2020, mis en ligne le 01 juin 2020, consulté le 20 avril 2021. URL : http://journals.openedition.org/oeconomia/8322 ; DOI : https://doi.org/10.4000/oeconomia.8322

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