On the use (and abuse) of Logic in Game Theory

Abstract

A quick glance at the opening paragraphs in many of the classic logic textbooks reveals a common view: Logical methods highlight the reasoning patterns of a single (idealized) agent engaged in some form of mathematical thinking.1 However, this traditional view of the “subject matter” of logic is expanding. There is a growing literature using phrases such as “rational interaction” or “information flow” to describe its subject matter while still employing traditional logical methods. The clearest example of this can be found in the work of Johan van Benthem and others on logical dynamics (van Benthem, 1996; van Ditmarsch et al., 2007; Pacuit, 2013b); Rohit Parikh and others on social software (Parikh, 2002; van Eijck and Vergrugge, 2009); Samson Abramsky and others on game semantics for linear logic (Abramsky, 2007), and Mike Wooldridge, Valentin Goranko and others on logics for multiagent systems (Wooldridge, 2000; Goranko and Jamroga, 2004; van der Hoek and Wooldridge, 2003). There are many issues driving this shift in thinking about what logic is about (see van Benthem, 2005, for a discussion). One important reason for this shift is the close connection between logic and game theory. For centuries, logicians have used game-theoretic notions in their analyses. (See Hodges (2013); Gradel (2011) for an overview of applications of game theory in logic.) In the past 20 years or so, the influence has started to run in the other direction. There is now a very active research area focused on adapting existing logical systems and developing new ones to reason about different aspects of a game situation. Full coverage of this fascinating research area would require much more space than I have in this short article. So, I will not attempt a comprehensive overview of the use of logic in game theory2 and, instead, will focus on just a few key topics. The first topic is the question of motivation: Why should game theorists learn some logic, and, vice versa, why should logicians learn some game theory? There is a very simple reason why logicians have become interested in game theory: The mathematical models that game theorists use to describe a game situation are natural models for many existing logical languages (e.g., epistemic, doxastic and preference logics, as well as conditional logics, temporal logics and logics of action). Thus, game theory provides a fertile testing ground for a variety of logical systems. The following