About Coalitional Equilibrium

In this paper, consider the n-person normal form game under uncertainty. Many coalition related concepts of equilibrium or solutions have been introduced for n-person normal form games. The main initial motivation for the inception of this direction of research is to overcome one of the drawbacks of Nash equilibrium (NE), namely, NE is not immune against coalition deviation. A coalition may improve the payoff of all its members by collectively deviating from NE. In this article, the Coalitional Equilibrium (CE), is introduced for normal form games under uncertainty. Regarding the undetermined parameters, it is assumed that the players know their range of variation only; no probability information is available (for known or unknown reasons). In the process of modelling game phenomena, considering uncertainty leads to more adequate results and decisions, which is supported by the numerous publications related to this domain (a google search on the topic “mathematical modelling under uncertainty”, returns more than one million links to related works). The uncertainty appears because of incomplete information about the players’ strategy sets, the strategies being selected by each player and the related payoffs.
A question arises: How a player can, at the same time, consider the game’s strategic and cooperation aspects, and the presence of uncertainty when selecting his/her strategy?
In this paper, the following approach to formalize the cooperation aspect of the game is adopted. It is assumed that the cooperation character of the game consists in the fact that any nonempty subset of players has the possibility to form a coalition through communication and coordination by agreeing to select a bundle of strategies to achieve the best possible payoff for all its members. This assumption means that the interests of all possible coalitions are considered. Further, it is also assumed that the game is without side payments or non-transferable utility. The concept of coalitional equilibrium (CE) is introduced for the described game. This concept is based on the synthesis of the notions of individual rationality and collective rationality in normal form games without side payments, and a proposed coalitional rationality. Sufficient conditions for the existence of CE in pure strategies are established via the saddle point of the Germeier convolution function of the players’ payoff functions. Finally, following the approach of Borel, von Neumann and Nash, a theorem of existence of CE in mixed strategies is proved under common minimal mathematical conditions for normal form games (compactness of players’ strategy sets, uncertainty set and continuity of payoff functions).

Keywords: Normal form game without side payments, uncertainty, guarantee, mixed strategies, Germeier convolution, saddle point, equilibrium