Top

Theory and Decision

Published in:

22-06-2017

Stability and cooperative solution in stochastic games

Authors: Elena M. Parilina, Alessandro Tampieri

Published in: Theory and Decision | Issue 4/2018

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

This paper analyses the principles of stable cooperation for stochastic games. Starting from the non-cooperative version of a discounted, non zero-sum stochastic game, we build its cooperative form and find the cooperative solution. We then analyse the conditions under which this solution is stable. Principles of stability include subgame consistency, strategic stability and irrational behaviour proof of the cooperative solution. We finally discuss the existence of a stable cooperative solution, and consider a type of stochastic games for which the cooperative solution is found and the principles of stable cooperation are checked.

previous article Prosocial consequences of third-party anger

next article Consistency of determined risk attitudes and probability weightings across different elicitation methods

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Available only for authorised users

From now on, we use the notation \(\eta _{i}\) if player i uses the stationary strategy in the game. When a player i uses a behaviour strategy (not necessarily stationary), we use the notation \(\varphi _{i}\).

Without loss of generality, we may find the maximum in Eq. (3) over the set of pure actions of coalition N.

The existence of the minmax value of two-player discounted stochastic game is proved by Shapley (1953a).

In Eq. (6), the maximum in \(\min _{\hat{\eta }_{N\backslash S}}\max _{\eta _{S}}\sum _{i\in S}E_{i}^{\omega }(\eta _{S},\eta _{N\backslash S})\) is found over the set of pure strategies of coalition S, while the minimum in \(\max _{\hat{\eta }_{S}}\min _{\eta _{N\backslash S}}\sum _{i\in S}E_{i}^{\omega }(\eta _{S},\eta _{N\backslash S})\) is found over the set of pure strategies of coalition \(N\backslash S\).

The property of superadditivity is not needed and is often omitted in cooperative game theory, because in real life there are a lot of motivations to consider both profitable and non-profitable coalitions. As Aumann and Dreze (1974, p. 233) note, there are arguments for superadditivity that are quite persuasive, but, as they also note, superadditivity is quite problematic in some economic applications.

We suppose that the allocation mechanism gives a unique imputation for each subgame. But the principle of subgame consistency may be extended in the case when the imputation is a set.

See Petrosjan and Danilov (1979) and Baranova and Petrosjan (2006).

If the solution is the Shapley value, the nucleolus or another single-valued solution.

Ehtamo and Hamalainen (1989) model this discrete-time effect in differential games.

The strict definition of the behaviour strategy is given in the Appendix in the proof of Proposition 3.

Things change for subgame perfectness. In this case, we need to prove that Eq. (13) holds for all possible histories and all stages. Therefore, we need to determine the strategy of a player even if more than one player deviates. Strategy (44), p. 31, defines the behaviour of the player given any history.

In the case of multiple Nash equilibria, one of them should be chosen for the realisation of the punishment. Notice that this can be implemented because players use correlated strategies.

Note that it is possible to formulate an analogous condition for repeated games.

Notice that the actions of the players from coalition \(N\backslash z\) are correlated.

Amir, R. (2003). Stochastic games in economics: The lattice-theoretic approach. In A. Neyman & S. Sorin (Eds.), Stochastic games and applications (pp. 443–453)., NATO Science Series C, Mathematical and Physical Sciences Netherlands.: Springer.CrossRef

Aumann, R. J. (1974). Subjectivity and correlation in randomized strategies. Journal of Mathematical Economics, 1, 67–96.CrossRef

Aumann, R. J., & Dreze, J. H. (1974). Cooperative games with coalition structure. International Journal of Game Theory, 3, 217–237.CrossRef

Aumann, R. J., & Peleg, B. (1960). Von Neumann–Morgenstern solutions to cooperative games without side payments. Bulletin of the American Mathematical Society, 66, 173–179.CrossRef

Baranova, E. M., & Petrosjan, L. A. (2006). Cooperative stochastic games in stationary strategies. Game Theory and Applications, 11, 7–17.

Dutta, P. (1995). A folk theorem for stochastic games. Journal of Economic Theory, 66, 1–32.CrossRef

Ehtamo, H., & Hamalainen, R. P. (1989). Incentive strategies and equilibria for dynamic games with delayed information. Journal of Optimization Theory and Applications, 63, 355–369.CrossRef

Fink, A. M. (1964). Equilibrium in a stochastic n-person game. Journal of Science of the Hiroshima University, A–I 28, 89–93.

Herings, P. J., & Peeters, R. J. A. P. (2004). Stationary equlibria in stochastic games: Structure, selection, and computation. Journal of Economic Theory, 118, 32–60.

Kohlberg, E., & Neyman, A. (2015). The cooperative solution of stochastic games. Harvard Business School. Working Paper, No. 15-071, March 2015.

Parilina, E. M. (2015). Stable cooperation in stochastic games. Automation and Remote Control, 76, 1111–1122.CrossRef

Parilina, E., & Zaccour, G. (2015). Node-consistent core for games played over event trees. Automatica, 53, 304–311.CrossRef

Petrosjan, L. A. (1977). Stability of the solutions in differential games with several players. In Vestnik Leningrad. University MMatematika Mekhanika Astronomiya, vol. 19 (pp. 46–52). (in Russian).

Petrosjan, L. A. (1993). Strongly time consistent differential optimality principles Vestnik Sankt-Peterburgskogo Universiteta. Ser 1. Matematika Mekhanika Astronomiya, 4, 35–40.

Petrosjan, L. A. (1998). Semicooperative games Vestnik Sankt-Peterburgskogo Universiteta. Ser 1. Matematika Mekhanika Astronomiya, 2, 62–69. (in Russian).

Petrosjan, L. A. (2006). Cooperative Stochastic Games. Advances in dynamic games, Annals of the International Society of Dynamic Games, 8, 139–146.CrossRef

Petrosjan, L. A., & Danilov, N. N. (1979). Stability of the solutions in nonantagonistic differential games with transferable payoffs. Vestnik Leningrad. Univ. Mat. Mekh. Astronom., 1, 52–59. (in Russian).

Petrosjan, L. A., & Zenkevich, N. A. (2015). Conditions for sustainable cooperation. Automation and Remote Control, 76(10), 1894–1904.CrossRef

Schmeidler, D. (1969). The nucleolus of a characteristic function game. SIAM Journal of Applied Mathematics, 17(6), 1163–1170.CrossRef

Shapley, L. S. (1953a). Stochastic games. Proceedings of National Academy of Sciences of the USA, 39, 1095–1100.CrossRef

Shapley, L. S. (1953b). A value for n-person games. Contributions to the Theory of Games-II. In H. W. Kuhn & A. W. Tucker (Eds.), Annals of Mathematical Studies (Vol. 28, pp. 307–317). Princeton: Princeton University Press.

Takahashi, M. (1964). Stochastic games with infinitely many strategies. Journal of Science of the Hiroshima University, A–I 28, 95–99.

von Neumann, J., & Morgenstern, O. (1944). Theory of Games and Economic Behavior. Princeton: Princeton University Press.

Xu, N., & Veinott, A, Jr. (2013). Sequential stochastic core of a cooperative stochastic programming game. Operations Research Letters, 41, 430–435.CrossRef

Yeung, D. W. K. (2006). An irrational-behavior-proof condition in cooperative differential games. International Game Theory Review, 8, 739–744.CrossRef

Title: Stability and cooperative solution in stochastic games
Authors: Elena M. Parilina
Alessandro Tampieri
Publication date: 22-06-2017
Publisher: Springer US
Published in: Theory and Decision / Issue 4/2018
Print ISSN: 0040-5833
Electronic ISSN: 1573-7187
DOI: https://doi.org/10.1007/s11238-017-9619-7

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Other articles of this Issue 4/2018

The probability of majority inversion in a two-stage voting system with three states

Timing effect in bargaining and ex ante efficiency of the relative utilitarian solution

Communication, leadership and coordination failure

Consistency of determined risk attitudes and probability weightings across different elicitation methods

Prosocial consequences of third-party anger

This or that? Sequential rationalization of indecisive choice behavior

Premium Partner