A model of strategic behaviour in repeated games

Abstract

This paper develops a general repeated game model over an arbitrary time domain. The paper considers the issue of formulating strategic behaviour in a general framework which includes continuous time behaviour. There are obvious requirements which a strategy must necessarily satisfy. These requirements are stated as axioms and within this axiomatic framework a class of strategies (called variable response strategies, VRS) is developed. In this class of strategy, a player is committed at any point in time to history independent behaviour for a positive length of time. However, this length of time of commitment depends on the way the history evolves locally. Thus, the speed of reaction to the evolution of the history is not bounded away from zero in the VRS formulation. Nevertheless, it is shown that these strategies generate unambiguous outcomes on any subgame and so are well defined. A characterization of the set of subgame perfect equilibria relative to variable response strategies is given. Following this discussion, we show that this class of strategy contains in a natural way a general class of differential game strategy. This result is of independent interest as it provides an alternative perspective on differential equations. The fact that VR strategies have a response time not bounded away from 0 is essential to this result.