Subgame Perfect Equilibria in Continuous-Time Repeated Games

This paper considers subgame perfect equilibria of continuous-time repeated games with perfect monitoring when immediate reactions to deviations are allowed. The set of subgame perfect equilibrium payoffs is shown to be a fixed-point of a set-valued operator introduced in the paper. For a large class of discrete time games the closure of this set corresponds to the limit payoffs of  when the discount factors converge to one. It is shown that in the continuous-time setup pure strategies are sufficient for obtaining all equilibrium payoffs supported by the players' minimax values. Moreover, the equilibrium payoff set is convex and satisfies monotone comparative statics when the ratios of players' discount rates increase.</p

Rangaistusten laskeminen puhtailla strategioilla toistetuissa peleissä

A repeated game is a decision making situation where players interact over and over again. They are used to model long-term relationships, cooperation and competition in a rational manner. In repeated interaction a basic solution concept is subgame-perfect equilibrium. It a special case of Nash equilibrium and requires players to play Nash equilibrium in all possible situation during the game. Often many subgame-perfect equilibria exist and computing those areas complex as in single-shot games, where players meet and choose an action only once [Borgs et al., 2010]. Worst equilibria allows a computationally efficient way to check, if an arbitrary solution is an equilibrium solution of a game [Abreu, 1986].Berg and Kitti [2011] have introduced a method for computing all the equilibrium solutions of a game, assuming that the worst equilibrium payoffs are known. This is because rational players can only play an equilibrium solution. The worst equilibrium is the strongest punishment, which the players can use for threaten each other and force others to play another equilibrium. It is an open question, whether it is possible to solve these punishment strategies in practice. This work examines computing threat points and strategies first with unlimited and then with bounded rationality meaning bounded computing capacity. An algorithm for optimal punishments with pure strategies is introduced. It is suitable for an arbitrary number of players and strategies and it accepts also unequal discount factors. In the end the performance of the algorithm is analyzed and limits to finding punishment strategies with given computing capacity is discussed.Toistetulla pelillä tarkoitetaan päätöksentekotilannetta, jossa samat pelaajat kohtaavat toisensa yhä uudelleen. Toistettuja pelejä käytetään, jotta voidaan mallintaa rationaalisella tavalla pitkäaikaisia vuorovaikutussuhteita ja niissä tapahtuvaa kilpailua ja yhteistyötä. Toistetussa vuorovaikutuksessa vakiintunut ratkaisukäsite on osapelitäydellinen tasapaino, joka vaatii pelaajaa toimimaan Nashin tasapainoperiaatteen mukaisesti kaikissa tilanteissa. Usein toistetussa pelissä tällaisia tasapainoja on useita ja niiden laskeminen on yhtä vaikeaa kuin kertapelissäkin, jossa pelaajat kohtaavat päätöksentekotilanteen vain kerran [Borgs et al., 2010]. On todistettu, että pelaajien kannalta huonoimpien tasapainoratkaisujen tunteminen mahdollistaa muiden tasapainoehdokkaiden tarkistamisen laskennallisesti tehokkaasti [Abreu, 1986]. Lisäksi Berg ja Kitti [2011] ovat kehittäneet laskentamenetelmän, jolla pelin tasapainoratkaisut voidaan löytää, mikäli huonoimmat tasapainoratkaisut tunnetaan. Tämä perustuu siihen, että rationaaliset pelaajat voivat päätyä ainoastaan tasapainoratkaisuun ja huonoin tasapainoratkaisu on voimakkain uhka, jota pelaajat voivat käyttää painostuskeinona pysyäkseen muissa tasapainoratkaisuissa. On kuitenkin avoin kysymys, pystytäänkö uhkausstrategioita käytännössä ratkaisemaan. Tässä työssä tarkastellaan uhkausstrategioiden laskemista ensin rajoittamattoman ja sitten rajoitetutun rationaalisuuden, eli käytännössä rajoitetun laskentakapasiteetin näkökulmasta. Työssä esitellään algoritmi, jolla voidaan laskea voimakkaimmat puhtailla strategioilla aikaansaadut uhat annetulle kertapelille ja tarkastellaan sitä, millaisia uhkausstrategioita voidaan löytää annetun laskentakapasiteetin rajoissa

Equilibrium restoration in a class of tolerant strategies

This study shows that in a two-player infinitely repeated game where one is impatient, Pareto-superior subgame perfect equilibrium can still be achieved. An impatient player in this paper is depicted as someone who can truly destroy the possibility of attaining any feasible and individually rational outcome that is supported in equilibrium in repeated games, as asserted by the Folk Theorem. In this scenario, the main ingredient for the restoration of equilibrium is to introduce the notion of tolerant trigger strategy. Consequently, the use of the typical trigger strategy is abandoned since it ceases to be efficient as it only brings automatically the game to its punishment path, therefore eliminating the possibility of extracting other feasible equilibria. I provide a simple characterization of perfect equilibrium payoffs under this scenario and show that cooperative outcome can be approximated

Tolerance, Cooperation, and Equilibrium Restoration in Repeated Games

This study shows that in a two-player infinitely repeated game where one is patient and the other is impatient, Pareto-superior subgame perfect equilibrium can be achieved. An impatient player in this paper is depicted as someone who can truly destroy the possibility of attaining any feasible and individually rational outcome that is supported in equilibrium in repeated games, as asserted by the Folk Theorem. In this scenario, the main ingredient for the restoration of equilibrium is to introduce the notion of tolerant trigger strategy. Consequently, the use of the typical trigger strategy is abandoned since it ceases to be efficient as it only brings automatically the game to its punishment path, therefore eliminating the possibility of extracting other feasible equilibria. I provide a simple characterization of perfect equilibrium payoffs under this scenario and show that cooperative outcome can be approximated

A Theory of Partnership Dynamics: Learning, Specific Investment, and Dissolution

This paper explores the benefits and drawbacks of potential partnership dissolution through an infinite-period, dynamic game-theoretic model of learning and endogenous dissolution. As partners learn about the quality of their partnership relative to their outside opportunities, the rents associated with the partnership change, effecting a related change in the strangth of incentives to provide effort. The paper develops an incentive-constrained dynamic programming algorithm for the computation of optimal symmetric equilibria of dynamic games with known worst punishments (such as dissolution here). The scheme is much simpler than the more general set-valued approach pioneered by Abreu, Pearce, and Stacchetti in that it only requires the computation of one value function at each iteration. The algorithm is then used to show that rather mild supermodularity conditions lead to effort levels in the optimal equilibria which rise in the expected quality of the partnership.Center for Research on Economic and Social Theory, Department of Economics, University of Michiganhttp://deepblue.lib.umich.edu/bitstream/2027.42/100933/1/ECON380.pd