A model of strategic behaviour in repeated games
The paper considers the problem of formulating strategic behaviour in the context of continuous time repeated games. There are well-known difficulties involved in defining continuous time strategies. These difficulties arise from the fact that time is not well ordered and lead to coherency problems: strategy desciptions which appear unambiguous may be consistent with multiple outcomes and hence are incoherent. The paper begins by observing that there are obvious restrictions 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 is developed. In this class of strategy, a player is committed at any point in time to history independent behavior for a positive length of time. However, the length of the commitment depends on the way the history evolves locally. The speed of reaction to the evolution of the history is not bounded away from zero in the variable response formulation. 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, it is shown that the class of variable response strategies contains in a natural way a general class of differential game strategy. The set of variable response strategies is dense in the set of functional differential equations (defined with the same time domain and with the action space as state space). The fact that variable response strategies have a response time not bounded away from 0 is essential to this result.
scientific article; zbMATH DE number 57177