5.2.Infinitely Repeated Games.1.0(博弈论讲-Arizona State University).pdfVIP

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5.2.Infinitely Repeated Games.1.0(博弈论讲-Arizona State University).pdf

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5.2.Infinitely Repeated Games.1.0(博弈论讲-Arizona State University)

Infinitely Repeated Games with Discounting Page 1 jim@ Jim Ratliff /gametheory Infinitely Repeated Games with Discountingù Introduction ______________________________________________________________________1 Discounting the future______________________________________________________________2 Interpreting the discount factor______________________________________________________3 The average discounted payoff ______________________________________________________4 Restricting strategies to subgames ____________________________________________________7 Appendix: Discounting Payoffs______________________________________________________10 In a hurry? ____________________________________________________________________10 The infinite summation of the discount factors ________________________________________10 An infinite summation starting late__________________________________________________11 The finite summation ____________________________________________________________12 Endless possibilities ______________________________________________________________13 Introduction We’ll now discuss repeated games which are “infinitely repeated.” This need not mean that the game never ends, however. We will see that this framework is appropriate for modeling situations in which the game eventually ends (with probability one) but the players are uncertain about exactly when the last period is (and they always believe there’s some chance the game will continue to the next period). We’ll call the stage game G and interpret it to be a simultaneous-move matrix game which remains exactly the same through time. As usual we let the player set be I={1,…,n}. Each player has a pure action space Ai.1 The space of action profiles is A=X i˙IùAi. Each player has a von Neumann- Morgenstern utility function defined over the outcomes of G, gi:ùA§?. The stage game repeats each period, starting at t=0. Although each stage game is a simultaneous- move game, so that each player acts in ignorance of