# Igre s ponavljanjem

Petrunić, Ivan (2015) Igre s ponavljanjem. Diploma thesis, Faculty of Science > Department of Mathematics.

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## Abstract

Insights from game theory are used today in economics, sociology, psychology, political science and in many other areas. In this thesis we study repeated games, which could be used to model situations where a game is played repeatedly. In repeated games, we can observe phenomena which are not present in one-stage games. We analyze three variants of the repeated game model. In the first variant, the game lasts a finite number of stages T, and each player wants to maximize his average payoff. In the second variant, the game lasts an infinite number of stages, and each player wants to maximize his long-run average payoff. In the third variant, the game lasts an infinite number of stages, and each player wants to maximize his discounted payoff. Since the number of strategies available to each player increases very quickly as the number of stages increases, we have not tried to find all the equilibria of repeated games. Instead, our aim was to characterize the sets of equilibrium payoffs. For T-stage repeated games, we have shown that playing the base game equilibrium at every stage leads to an equilibrium of the repeated game. But we have also shown that a repeated game has other equilibria as well. We have seen that, at every equilibrium of the repeated game, the payoff to each player is at least equal to his minmax value. Finally, we have proved the Folk Theorem, which characterizes the set of equilibrium payoffs of the T-stage repeated game. For infinitely repeated games, we have shown that they lead to equilibrium payoffs which cannot be obtained as equilibrium payoffs of finite games whose lengths increase to infinity. Using the expectation of the limit of average payoffs, we have defined an equilibrium of infinitely repeated games; we have then stated the Folk Theorem for such games. Finally, for discounted games, we have taken into account the time value of money. We have seen that we can model long repeated games using discounted games where the discount factor approaches 1. We have observed that the analogue of the Folk Theorem for infinitely repeated games does not hold for discounted games. Finally, using additional technical conditions, we have stated the Folk Theorem for discounted games.

Item Type: Thesis (Diploma thesis) Čaklović, Lavoslav 2015 41 NATURAL SCIENCES > Mathematics Faculty of Science > Department of Mathematics Iva Prah 23 Oct 2015 11:40 23 Oct 2015 11:40 http://digre.pmf.unizg.hr/id/eprint/4182

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