Abstract
In this tutorial, we recall the main ingredients of the theory of dynamic games played over event trees and show step-by-step how to build a sustainable cooperative solution.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
To define a game in normal form, we need three elements: (a) a finite set of players M = {1, …, m}, (b) a strategy set S i of player i ∈ M, and (c) a payoff function \(\pi _{i}:\prod \limits _{i\in M}S_{i} \rightarrow \mathbb{R}\).
- 2.
For a detailed treatment in the context of this class of games, see Haurie et al. [22].
- 3.
We can easily extend our framework to the case where the players maximize a weighted sum of payoffs.
- 4.
The implicit assumption here is that players’ utilities (gains) are comparable and transferable; otherwise side payments do not make sense.
- 5.
The following example illustrates this statement. Consider a three-player cooperative game with characteristic function values given by
$$\displaystyle\begin{array}{rcl} v(\left \{1\right \})& =& v(\left \{2\right \}) = v(\left \{3\right \}) = 0, {}\\ v(\left \{1, 2\right \})& =& v(\left \{1, 3\right \}) = v(\left \{2, 3\right \}) = a,\quad v(\left \{1, 2, 3\right \}) = 1 {}\\ \end{array}$$where 0 < a ≤ 1. It is easy to verify that three cases can occur: (i) If 0 < a < 2∕3, then the core contains all imputations satisfying \(y_{j} \geq 0,\sum \limits _{j\in G}y_{j} \geq a\) and \(\sum \limits _{j\in M}y_{j} = 1.\) (ii) If a = 2∕3, then the core is a singleton, that is, the only imputation belonging to the core is \(\left (1/3, 1/3, 1/3\right )\). (iii) If a > 2∕3, then the core is empty.
References
Aumann, R.J.: The core of a cooperative game without side payments. Transactions of the American Mathematical Society 98, 539–552 (1961)
Avrachenkov, K., Cottatellucci, L., Maggi, L.: Cooperative Markov decision processes: Time consistency, greedy players satisfaction, and cooperation maintenance. International Journal of Game Theory 42, 239–262 (2013)
Breton, M., Sokri A., Zaccour, G.: Incentive equilibrium in an overlapping-generations environmental game. European Journal of Operational Research 185, 687–699 (2008)
Buratto, A., Zaccour, G.: Coordination of advertising strategies in a fashion licensing contract. Journal of Optimization Theory and Applications 142, 31–53 (2009)
Chander, P., Tulkens, H.: The core of an economy with multilateral environmental externalities. International Journal of Game Theory 23, 379–401 (1997)
Chiarella, C., Kemp, M.C., Long, N.V., Okuguchi, K.: On the economics of international fisheries. International Economic Review 25, 85–92 (1984)
Dockner, E., Jørgensen, S., Van Long, N., Sorger, G.: Differential Games in Economics and Management Science. Cambridge University Press, Cambridge (2000)
De Frutos, J., Martín-Herrán, G.: Does flexibility facilitate sustainability of cooperation over time? A case study from environmental economics. Journal of Optimization Theory and Applications 165, 657–677 (2015)
De Giovanni, P., Reddy, P.V., Zaccour, G.: Incentive strategies for an optimal recovery program in a closed-loop supply chain. European Journal of Operational Research 249, 605–617 (2016)
Dutta, P.K.: A folk theorem for stochastic games. Journal of Economic Theory 66, 1–32 (1995)
Ehtamo, H., Hämäläinen, R.P.: On affine incentives for dynamic decision problems. In: Basar, T. (ed.) Dynamic Games and Applications in Economics, pp. 47–63. Springer, Berlin (1986)
Ehtamo, H., Hämäläinen, R.P.: Incentive strategies and equilibria for dynamic games with delayed information. Journal of Optimization Theory and Applications 63, 355–370 (1989)
Ehtamo, H., Hämäläinen, R.P.: A cooperative incentive equilibrium for a resource management problem. Journal of Economic Dynamics and Control 17, 659–678 (1993)
Filar, J., Petrosjan, L.: Dynamic cooperative games. International Game Theory Review 2(1), 47–65 (2000)
Genc, T., Reynolds, S.S., Sen, S.: Dynamic oligopolistic games under uncertainty: A stochastic programming approach. Journal of Economic Dynamics & Control 31, 55–80 (2007)
Genc, T., Sen, S.: An analysis of capacity and price trajectories for the Ontario electricity market using dynamic Nash equilibrium under uncertainty. Energy Economics 30, 173–191 (2008)
Germain, M. Toint, P., Tulkens H., de Zeeuw A.: Transfers to sustain dynamic core-theoretic cooperation in international stock pollutant control. Journal of Economic Dynamics & Control 28(1), 79–99 (2003)
Gillies, D.B.: Some Theorems on N-Person Games, Ph.D. Thesis, Princeton University (1953)
Haurie, A.: A note on nonzero-sum differential games with bargaining solution. Journal of Optimization Theory and Applications 18, 31–39 (1976)
Haurie, A., Pohjola, M.: Efficient equilibria in a differential game of capitalism. Journal of Economic Dynamics and Control 11, 65–78 (1987)
Haurie, A., Krawczyk, J.B., Roche, M.: Monitoring cooperative equilibria in a stochastic differential game. Journal of Optimization Theory and Applications 81, 79–95 (1994)
Haurie, A., Krawczyk, J.B., Zaccour, G.: Games and Dynamic Games. Scientific World, Singapore (2012)
Haurie, A., Zaccour. G., Smeers, Y.: Stochastic equilibrium programming for dynamic oligopolistic markets. Journal of Optimization Theory and Applications 66(2), 243–253 (1990)
Haurie, A., Zaccour, G.: S-adapted equilibria in games played over event trees: An overview. Annals of the International Society of Dynamic Games 7, 367–400 (2005)
Jørgensen, S., Martín-Herrán, G., Zaccour, G.: Agreeability and time-consistency in linear-state differential games. Journal of Optimization Theory and Applications 119, 49–63 (2003)
Jørgensen, S., Martín-Herrán, G., Zaccour, G.: Sustainability of cooperation overtime in linear-quadratic differential game. International Game Theory Review 7, 395–406 (2005)
Jørgensen, S., Zaccour, G.: Time consistent side payments in a dynamic game of downstream pollution. Journal of Economic Dynamics and Control 25, 1973–1987 (2001)
Kaitala, V., Pohjola, M.: Economic development and agreeable redistribution in capitalism: Efficient game equilibria in a two-class neoclassical growth model. International Economic Review 31, 421–437 (1990)
Kaitala, V., Pohjola, M.: Sustainable international agreements on greenhouse warming: A game theory study. Annals of the International Society of Dynamic Games 2, 67–87 (1995)
Kanani Kuchesfehani, K., Zaccour, G.: S-adapted equilibria in games played over event trees with coupled constraints. Journal of Optimization Theory and Applications 166, 644–658 (2015)
Lehrer, E., Scarsini, M.: On the core of dynamic cooperative games. Dynamic Games and Applications 3, 359–373 (2013)
Martín-Herrán, G., Rincón-Zapatero, J.P.: Efficient Markov perfect Nash equilibria: Theory and application to dynamic fishery games. Journal of Economics Dynamics and Control 29, 1073–1096 (2005)
Martín-Herrán, G., Taboubi, S.: Shelf-space allocation and advertising decisions in the marketing channel: A differential game approach. International Game Theory Review 7(03), 313–330 (2005)
Martín-Herrán, G., Zaccour, G.: Credibility of incentive equilibrium strategies in linear-state differential games. Journal of Optimization Theory and Applications 126, 1–23. (2005)
Martín-Herrán, G., Zaccour, G.: Credible linear-incentive equilibrium strategies in linear-quadratic differential games. Annals of the International Society of Dynamic Games 10, 261–292 (2009)
Ordeshook, P.C.: Game theory and Political Theory. Cambridge University Press, Cambridge, UK (1986)
Osborne, M.J., Rubinstein, A.: A Course in Game Theory. MIT Press, Cambridge, MA (1994)
Parilina, E., Zaccour, G.: Node-consistent core for games played over event trees. Automatica 55, 304–311 (2015a)
Parilina, E., Zaccour, G.: Approximated cooperative equilibria for games played over event trees. Operations Research Letters 43, 507–513 (2015b)
Petrosjan, L.: Stable solutions of differential games with many participants. Viestnik of Leningrad University 19, 46–52 (1977)
Petrosjan, L.: Differential Games of Pursuit. World Scientific, Singapore, 270–282 (1993)
Petrosjan, L., Baranova, E.M., Shevkoplyas, E.V.: Multistage cooperative games with random duration. Proceedings of the Steklov Institute of Mathematics (Supplementary issues), suppl. 2, S126–S141 (2004)
Petrosjan, L., Danilov, N.N.: Stability of solutions in nonzero sum differential games with transferable payoffs. Journal of Leningrad University N1, 52–59 (in Russian) (1979)
Petrosjan, L., Danilov, N.N.: Cooperative Differential Games and Their Applications. Tomsk University Press, Tomsk (1982)
Petrosjan, L., Danilov, N.N.: Classification of dynamically stable solutions in cooperative differential games. Isvestia of High School 7, 24–35 (in Russian) (1986)
Petrosjan, L., Zaccour, G.: Time-consistent Shapley value of pollution cost reduction. Journal of Economic Dynamics and Control 27, 381–398 (2003)
Petrosjan, L., Zenkevich, N.A.: Game Theory. World Scientific, Singapore (1996)
Pineau, P.-O., Murto, P.: An oligopolistic investment model of the Finnish electricity market. Annals of Operations Research 121, 123–148 (2003)
Pineau, P.-O., Rasata, H., Zaccour, G.: A Dynamic Oligopolistic Electricity Market Model with Interdependent Segments, Energy Journal 32(4), 183–217 (2011a)
Pineau, P.-O., Rasata, H., Zaccour, G.: Impact of some parameters on investments in oligopolistic electricity markets. European Journal of Operational Research 213(1), 180–195 (2011b)
Predtetchinski, A.: The strong sequential core for stationary cooperative games. Games and Economic Behavior 61, 50–66 (2007)
Reddy, P. V., Shevkoplyas E., Zaccour, G.: Time-consistent Shapley value for games played over event trees. Automatica 49(6), 1521–1527 (2013)
Rosen, J.B.: Existence and uniqueness of equilibrium points for concave n-person games. Econometrica 33(3), 520–534 (1965)
Tolwinski, B., Haurie A., Leitmann, G.: Cooperative equilibria in differential games. Journal of Mathematical Analysis and Applications 119, 182–202 (1986)
Von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press, Princeton, NJ (1944)
Xu, N., Veinott Jr., A.: Sequential stochastic core of a cooperative stochastic programming game. Operations Research Letters 41, 430–435 (2013)
Yeung, D.W.K., Petrosjan, L.: Proportional time-consistent solutions in differential games. Proceedings of International Conference on Logic, Game Theory and Applications, Saint Petersburg, 254–256 (2001)
Yeung, D.W.K., Petrosjan, L.: Consistent solution of a cooperative stochastic differential game with nontransferable payoffs. Journal of Optimization Theory and Applications 124, 701–724 (2005a)
Yeung, D.W.K., Petrosjan, L.: Cooperative Stochastic Differential Games. Springer, New York, NY (2005b)
Yeung, D.W.K., Petrosjan, L.: Dynamically stable corporate joint ventures. Automatica 42, 365–370 (2006)
Yeung, D.W.K., Petrosjan, L., Yeung, P.M.: Subgame consistent solutions for a class of cooperative stochastic differential games with nontransferable payoffs. Annals of the International Society of Dynamic Games 9, 153–170 (2007)
Zaccour, G.: Théorie des jeux et marchés énergétiques: marché européen de gaz naturel et échanges d’électricité, Ph.D. Thesis, HEC Montréal (1987)
Zaccour, G.: Time consistency in cooperative differential games: A tutorial. INFOR 46(1), 81–92 (2008)
Acknowledgements
This paper draws from my previous work on the subject, in particular Reddy, Shevkoplyas and Zaccour [52] and Parilina and Zaccour [38]. I would like to thank my co-authors in these papers, as well as Alain Haurie and Leon Petrosjan for many stimulating discussions over the last two decades or so on dynamic games played over event trees and time consistency in dynamic games. Research supported by SSHRC, Canada.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer Science+Business Media LLC
About this chapter
Cite this chapter
Zaccour, G. (2017). Sustainability of Cooperation in Dynamic Games Played over Event Trees. In: Melnik, R., Makarov, R., Belair, J. (eds) Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science. Fields Institute Communications, vol 79. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-6969-2_14
Download citation
DOI: https://doi.org/10.1007/978-1-4939-6969-2_14
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4939-6968-5
Online ISBN: 978-1-4939-6969-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)