# Endogenous interval games in oligopolies and the cores

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## Abstract

In this article we study interval games in oligopolies following the \(\gamma \)-approach. First, we analyze their non-cooperative foundation and show that each coalition is associated with an endogenous real interval. Second, the Hurwicz criterion turns out to be a key concept to provide a necessary and sufficient condition for the non-emptiness of each of the induced core solution concepts: the interval and the standard \(\gamma \)-cores. The first condition permits to ascertain that even for linear and symmetric industries the interval \(\gamma \)-core is empty. Moreover, by means of the approximation technique of quadratic Bézier curves we prove that the second condition always holds, hence the standard \(\gamma \)-core is non-empty, under natural properties of profit and cost functions.

## Keywords

Interval game Oligopoly \(\gamma \)-Cores Hurwicz criterion Quadratic Bézier curve## JEL Classification

C71 D43## References

- Abreu, D. (1988). On the theory of infinitely repeated games with discounting.
*Econometrica*,*56*, 383–396.CrossRefGoogle Scholar - Alparslan-Gök, S., Branzei, O., Branzei, R., & Tijs, S. (2011). Set-valued solution concepts using interval-type payoffs for interval games.
*Journal of Mathematical Economics*,*47*, 621–626.CrossRefGoogle Scholar - Alparslan-Gök, S., Miquel, S., & Tijs, S. (2009). Cooperation under interval uncertainty.
*Mathematical Methods of Operations Research*,*69*, 99–109.CrossRefGoogle Scholar - Aumann, R. (1959). Acceptable points in general cooperative n-person games. In Iuce Tucker (Ed.),
*Contributions to the theory of games IV. Annals of Mathematics Studies*(Vol. 40). Princeton: Princeton University Press.Google Scholar - Bézier, P. (1976). Numerical definition of experimental surfaces.
*Neue Technik*,*18*(7–8), 487–489 and 491–493.Google Scholar - Bondareva, O. N. (1963). Some applications of linear programming methods to the theory of cooperative games.
*Problemi Kibernetiki*,*10*, 119–139.Google Scholar - Chander, P., & Tulkens, H. (1997). The core of an economy with multilateral environmental externalities.
*International Journal of Game Theory*,*26*, 379–401.CrossRefGoogle Scholar - Debreu, G. (1976). Smooth preferences: A corrigendum.
*Econometrica*,*44*, 831–832.CrossRefGoogle Scholar - Driessen, T. S., & Meinhardt, H. I. (2005). Convexity of oligopoly games without transferable technologies.
*Mathematical Social Sciences*,*50*, 102–126.CrossRefGoogle Scholar - Friedman, J. W. (1971). A noncooperative equilibrium for supergames.
*Review of Economic Studies*,*38*, 1–12.CrossRefGoogle Scholar - Han, W., Sun, H., & Xu, G. (2012). A new approach of cooperative interval games: The interval core and Shapley value revisited.
*Operations Research Letters*,*40*, 462–468.CrossRefGoogle Scholar - Hurwicz, L. (1951). Optimality criteria for decision making under ignorance. Discussion Paper 370, Cowles Commission.Google Scholar
- Katzner, D. W. (1968). A note on the differentiability of consumer demand functions.
*Econometrica*,*36*(2), 415–418.CrossRefGoogle Scholar - Lardon, A. (2012). The \(\gamma \)-core of cournot oligopoly games with capacity constraints.
*Theory and Decision*,*72*, 387–411.CrossRefGoogle Scholar - Lekeas, P. V., & Stamatopoulos, G. (2014). Cooperative oligopoly games with boundedly rational firms.
*Annals of Operations Research*,*223*(1), 255–272.CrossRefGoogle Scholar - Monteiro, P. K., Páscoa, M. R., & da Costa Werlang, S. R. (1996). On the differentiability of the consumer demand function.
*Journal of Mathematical Economics*,*25*, 247–261.CrossRefGoogle Scholar - Moore, R. (1979).
*Methods and applications of interval analysis*. Philadelphia: SIAM.CrossRefGoogle Scholar - Myerson, R. B. (1978). Threat equilibria and fair settlements in cooperative games.
*Mathematics of Operations Research*,*3*(4), 265–274.CrossRefGoogle Scholar - Norde, H., Pham Do, K. H., & Tijs, S. (2002). Oligopoly games with and without transferable technologies.
*Mathematical Social Sciences*,*43*, 187–207.CrossRefGoogle Scholar - Okuguchi, K., & Szidarovszky, F. (1990).
*The theory of oligopoly with multi-product firms*. Berlin: Springer.CrossRefGoogle Scholar - Rader, T. (1979). Nice demand functions II.
*Journal of Mathematical Economics*,*6*, 253–262.CrossRefGoogle Scholar - Shapley, L. S. (1967). On balanced sets and cores.
*Naval Research Logistics Quaterly*,*14*, 453–460.CrossRefGoogle Scholar - Zhao, J. (1999). A \(\beta \)-core existence result and its application to oligopoly markets.
*Games and Economic Behavior*,*27*, 153–168.CrossRefGoogle Scholar