Chapter 5: Infinite Dynamic Games with Discrete Strategy Space: A First Approach
5. Inﬁnite dynamic games with discrete strategy space: a ﬁrst approach INTRODUCTION 5.1 In Chapter 4 it was demonstrated that in ﬁnitely repeated simultaneous (sequential) move games with a single stage game Nash equilibrium (NE) (subgame-perfect equilibrium (SPE)), only this equilibrium will be played throughout the game. In negative externality games, like the PD game, this implies that cooperation cannot be established. In this section, we start again by considering a PD game and generalize the results subsequently. As pointed out in Section 4.3, we shall assume from now onwards simultaneous moves for simplicity, though in most cases an extension to cover the case of sequential moves is straightforward. In contrast to a ﬁnite setting, we cannot use backwards induction to solve the game. Though this may seem a disadvantage from an analytical point of view, it is this very fact which is responsible for obtaining more ‘optimistic’ results in an inﬁnite time horizon. Since the end of the game is not known with certainty, a player must always reckon with a costly and long-lasting punishment if s/he takes a free-ride. This might enforce compliance (Holler and Illing 1996, p. 22; Rasmusen 1995, p. 123; Taylor 1987). Generally, there are two interpretations of inﬁnite games: either, as the name suggests, the games last until perpetuity, or the end of the game is not known with certainty. The latter interpretation implies that there is a probability – though this might be very small – that the game continues (Gibbons 1992,...
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