Deductive Reasoning in Extensive Games

Abstract

Wejustify the application to extensive games of a model of deductive reasoning based on three key features: 'caution', 'full belief of opponent rationality', and 'no extraneous restrictions on beliefs'. We apply the model to several examples, and show that it yields novel economic insights. The approach supports forward induction, without necessarily promoting backward induction. In many economic contexts decision makers interact and take actions that extend through time. A bargaining party makes an offer, which is observed by the adversary, and accepted, rejected or followed by a counter-offer. Firms competing in markets choose prices, levels of advertisement, or investments with the intent of thereby influencing the future behaviour of competitors. One could add many examples. The standard economic model for analysing such situations is that of an extensive game. This paper is concerned with the following question: what happens in an extensive game if players reason deductively by trying to figure out one another's moves? We have in Asheim and Dufwenberg (2002) (AD) proposed a model for deductive reasoning leading to the concept of 'fully permissible sets', which can be applied to many strategic situations. In this paper we argue that the model is appropriate for analysing extensive games and we apply it to several such games. 1. Motivation There is already a literature exploring the implications of deductive reasoning in extensive games, but the answers provided differ and the issue is controversial. Much of the excitement concerns whether or not deductive reasoning implies backward induction in games where that principle is applicable. We next discuss this issue, since it provides a useful backdrop against which to introduce and motivate our own approach. Consider the 3-stage 'Take-it-or-leave-it', introduced by Reny (1993) (a version of Rosenthal's (1981) centipede game), and shown in Figure 1 together with its pure strategy reduced strategic form (PRSF).' What would 2 do in F1 if called upon to play? Backward induction implies that 2 would choose d, which is consistent with the following idea: 2 chooses d because she 'figures out' that 1 would choose D at * During our work with this paper and its companion piece 'Admissibility and common belief', we have received helpful comments from many scholars, including Pierpaolo Battigalli and Larry Samuelson. We are very grateful to David de Meza and two referees for their constructive and insightful comments. Asheim gratefully acknowledges the hospitality of Harvard University and financial support from the Research Council of Norway. 1 We need not consider what players plan to do at decision nodes that their own strategy precludes them from reaching (Section 3.2).