Axiomatizing Rationality

Games and Economic Behavior

We provide a sound and complete axiomatization for a class of logics appropriate for reasoning about the rationality of players in games, and show that essentially the same axiomatization applies to a very wide class of decision rules. We also consider games in which players may be uncertain as to what decision rules their opponents are using, and define in this context a new solution concept, D-rationalizability.

2004

Game Logic with Preferences (GLP), is a logic that makes it possible to reason about how information or assumptions about the preferences of other players can be used by agents in order to realize their own preferences. We extend the work done on this logic by looking at the satisfiability problem for this logic. We introduce an axiom system and show the soundness of this system.

We consider strategic-form games with ordinal payoffs and provide a syntactic analysis of common belief/knowledge of rationality, which we define axiomatically. Two axioms are considered. The first says that a player is irrational if she chooses a particular strategy while believing that another strategy is better. We show that common belief of this weak notion of rationality characterizes the iterated deletion of pure strategies that are strictly dominated by pure strategies. The second axiom says that a player is irrational if she chooses a particular strategy while believing that a different strategy is at least as good and she considers it possible that this alternative strategy is actually better than the chosen one. We show that common knowledge of this stronger notion of rationality characterizes the restriction to pure strategies of the iterated deletion procedure introduced by Stalnaker (1994). Frame characterization results are also provided.

2008

We consider strategic-form games with ordinal payoffs and provide a syntactic analysis of common belief/knowledge of rationality, which we define axiomatically. Two axioms are considered. The first says that a player is irrational if she chooses a particular strategy while believing that another strategy is better. We show that common belief of this weak notion of rationality characterizes the iterated deletion of pure strategies that are strictly dominated by pure strategies. The second axiom says that a player is irrational if she chooses a particular strategy while believing that a different strategy is at least as good and she considers it possible that this alternative strategy is actually better than the chosen one. We show that common knowledge of this stronger notion of rationality characterizes the restriction to pure strategies of the iterated deletion procedure introduced by Stalnaker (1994). Frame characterization results are also provided.

“The logic of best action from a deontic perspective” and “Obligation, free choice and the logic of weakest permission” argued that, in games, obligations and permissions should be viewed, respectively, as giving necessary and sufficient conditions for rationality. This gives rise to a specific deontic logic where, for instance, O and P are not dual notions and P becomes a “free choice” permission operator. Similar deontic logics have been proposed in the literature, as early as van Benthem’s Minimal deontic logics, and more recently in “On deontic action logics based on boolean algebra”. In this paper we study the relation between these deontic logics for rational agency in games. We compare their deductive power, provide translation results, and emphasize the different views they take on what players ought to, or may do.

1998

Backward induction is a form of logical reasoning, based on the technique of mathematical induction, that is applicable to certain sequential games. I shall illustrate it with a simple version of the ancient game of nim. There are 20 matches on the table, and two players take turns in removing either one or two matches, the winner being the player who takes the last one, leaving the other player unable to move. Ernst Zermelo (1912) proved the first major theorem of game theory to the effect that, in games of this type, either one player or the other must have a guaranteed winning strategy. In this case, which player wins, and how?

Linguistics and Philosophy, 2014

I am very grateful to Franke (2014) for giving me the opportunity of clarifying my views. I also appreciate his epistemic justification of the solution concept I employed in my article (2013). I have argued that iterated admissibility (or IA) can easily explain scalar implicatures. In order to respond to Franke's objections I need to embed that account in a larger strategy, to the effect that forward induction (FI) is the most appropriate solution concept in game-theoretic pragmatics. Forward induction first appeared in the '80s in the work of Elon Kohlberg and Jean-François Mertens. It is not a sharply defined notion, it rather embodies some intuitive principles that have been formalised in several different ways. Its intuitive significance can be expressed as the principle that, when a player observes an unexpected action on the part of some opponent, whenever this is possible, she should revise her beliefs in a way compatible with the assumption that the opponent is acting rationally. It covers several formal criteria such as the intuitive criterion (Cho and Kreps 1987), never a weak best response and stability (Kohlberg and Mertens 1986). Also IA is a possible formalisation of FI, even if it is one of the oldest solution concepts in game theory, dating back to the '50s. The connection between FI and IA hinges on the fact that a reasonable way of rationalising an opponent's action is to presume that he did not choose a dominated strategy. Like its very name suggests, IA is an iterative procedure, but so are many FI-based criteria, e.g. never a weak best response and some varieties of the intuitive criterion. According to the dominant view in game theory, the acceptable solutions must be Nash equilibria. It is well known, though, that many Nash equilibria are implausible, S. Pavan (B)

2014

The theories of Nash noncooperative solutions and of rationalizability intend to describe the same target problem of ex ante individual decision making, but they are distinctively different. We consider what their essential difference is by giving a unified approach and parallel derivations of their resulting outcomes. Our results show that the only difference lies in the use of quantifiers for each player’s predictions about the other’s possible decisions; the universal quantifier for the former and the existential quantifier for the latter. Based on this unified approach, we discuss the statuses of those theories from the three points of views: Johansen’s postulates, the free-will postulate vs. complete determinism, and prediction/decision criteria. One conclusion we reach is that the Nash theory is coherent with the free-will postulate, but we would meet various difficulties with the rationalizability theory.

Bulletin of Economic Research, 2001

This special issue of Knowledge, Rationality and Action contains a selection of papers presented at the sixth conference on "Logic and the Foundations of the Theory of Games and Decisions" (LOFT6), which took place in Leipzig, in July 2004. The LOFT conferences have been a regular biannual event since 1994. 1 The first conference was hosted by the Centre International de Recherches Mathematiques in Marseille (France), the next four took place at the International Center for Economic Research in Torino (Italy) and the most recent one was hosted by the Leipzig Graduate School of Management in Leipzig (Germany). The LOFT conferences are interdisciplinary events that bring together researchers from a variety of fields: computer science, economics, game theory, linguistics, logic, mathematical psychology, philosophy and statistics. In its original conception, LOFT had as its central theme the application of logic, in particular modal epistemic logic, to foundational issues in the theory of games and individual decision-making. Epistemic considerations have been central to game theory for a long time. The expression interactive epistemology has been used in the game-theory literature to refer to the analysis of what individuals involved in a strategic interaction know about facts concerning the external world as well as facts concerning each other's knowledge and beliefs. What is relatively new is the realization that the tools and methodology that were used in game theory are closely related to those already used in other fields, notably computer science and philosophy. Modal logic turned out to be the common language that made it possible to bring together different professional communities. The insights gained and the methodology employed in one field can benefit researchers in a different field. Indeed, new and active areas of research have sprung from the interdisciplinary exposure provided by the LOFT conferences. 2 Over time the scope of the LOFT conferences has broadened to encompass other tools, besides modal logic, that can be used to shed light on the general issues of rationality and agency. Topics that have fallen within the LOFT umbrella include epistemic and [1]

International Game Theory Review, 2007

Game-theoretic solution concepts describe sets of strategy profiles that are optimal for all players in some plausible sense. Such sets are often found by recursive algorithms like iterated removal of strictly dominated strategies in strategic games, or backward induction in extensive games. Standard logical analyses of solution sets use assumptions about players in fixed epistemic models for a given game, such as mutual knowledge of rationality. In this paper, we propose a different perspective, analyzing solution algorithms as processes of learning which change game models. Thus, strategic equilibrium gets linked to fixed-points of operations of repeated announcement of suitable epistemic statements. This dynamic stance provides a new look at the current interface of games, logic, and computation.