Cooperative Games with Externalities and Probabilistic Coalitional Beliefs
Cooperative game theory studies situations where groups or coalitions of players act collectively by signing binding agreements. The starting point of the theory is to determine the worth each coalition can achieve when its members act independently of the players outside the coalition. In games with orthogonal coalitions, i.e., coalitions that do not affect one another, this task is quite straightforward, as it suffices to study the actions of the insiders only. However, when orthogonality is absent, or in other words, when there are inter-coalitional externalities, the specification of the worth of a coalition requires the studying of the actions of the players in all coalitions. Therefore, when a number of players contemplate to form a coalition in an environment with externalities they need to have a theory, or a conjecture, about the actions of the players outside the proposed coalition. Clearly, different conjectures lead to different specifications of the worth or value of the coalition, which in turn affects the outcome of the game. In particular, these conjectures determine the core of the cooperative game. The core is the set of all outcomes (allocations of the value that the entire society of players generates) that no coalition has incentive to block and act on its own. Non-emptiness of the core means that cooperation among all players in the game is feasible. The literature on cooperative games with externalities has proposed a number of such coalitional conjectures, each giving rise to a specific core notion. According to Î± and Î²-conjectures (Aumann 1959), the members of a coalition compute their worth assuming that the outside players select their strategies so as to minimize the payoff of the coalition; the Î± and Î²-core are then defined with respect to the resulting coalitional payoffs. According to Î³-conjectures (Chander&Tulkens 1997), it is assumed that the outsiders select individual best strategies, i.e., they form only singleton coalitions; the Î³-core is then accordingly defined. The same approach can be followed under the additional assumption that each coalition assumes for itself the role of Stackelberg leader (Currarini&Marini 2003). The r-theory (Huang&Sjostrom 2003; Koczy 2007) proposes that the members of a coalition compute their worth by looking recursively on the sub-games played among the outsiders; the r-core arises when the solution concept employed in these sub-games is the core itself. Economists often restore to cooperative games with externalities to model various economic environments. Applications include the use of Î± and Î²-core in oligopolistic markets (Zhao 1999; Norde et al. 2002; Lardon 2010); the use of Î³-core in economies with production externalities (Chander&Tulkens 1997; Chander 2007; Helm 2012), in oligopolies (Rajan 1989; Lardon 2010; Lardon 2012) or in extensive-form games (Chander&Wooders 2012); the use of sequential Î³-core for cooperative games with strategic complements (Currarini&Marini 2003) or for economies with enviromental externalities (Marini 2013), etc. The main focus of these papers is to find conditions under which the core is non-empty. The current paper focuses too on cooperative games with externalities but takes a different route. It assumes that when a group of players S contemplate to break off from the rest of the society, they are uncertain about the partition that the players outside S will form. Hence, they assign various probability distributions on the set of all possible partitions. These probabilistic beliefs do not necessarily reflect the behavior of the outsiders, i.e., beliefs need not be consistent with actual choices. Given the beliefs, no natter how they form, one can compute the expected worth of S and define the core of the resulting cooperative game. The task that arises then is to find conditions on the data of the game (i.e., payoff functions and probability distributions) that guarantee the non-emptiness of the core, or, in other words, guarantee that the cooperation of all players in the game is feasible. The motivation of our paper is twofold. First, we are intersted in generalizing (some of) the existing approaches on the definition of the core. For example, the Î³-core notion is a special case of our approach that arises when each coalition assigns probability one to the event that the outsiders will form only singleton coalitions. Secondly, our paper could be read as a work on bounded rationality in its relation to cooperative games. The assignment of a (non-equilibrium) probability distribution on the set of parititions of the outsiders may reflect the cognitive inability of the members of a coalition to accurately deduce the outsiders' equilibrium partition. In this sense, probabilistic beliefs act as a rule of thumb. This approach is particularly relevant for games with a large number of players, where the number of different partitions can be very large. We apply the above framework to cooperative games generated by (symmetric) aggregative normal form games, i.e., games where the payoff of a player depends on his strategy and on the sum of the strategies of all players. Many economic models have an aggregative structure, such as oligopoly models, rent-seeking games, contest games, etc. We focus, in particular, on aggregative games that satisfy the following properties: (a) each player's payoff function has a bilinear form (this gives us the family of linear aggregative games introduced by Martimort&Stole 2010); (b) each player's payoff and marginal payoff decrease in the aggregate value of all players' strategies. The bilinear form assumption, in particular, is used as it allows us to simplify considerably the objective function of each coalition.
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- Aymeric Lardon, 2012. "The γ-core in Cournot oligopoly TU-games with capacity constraints," Theory and Decision, Springer, vol. 72(3), pages 387-411, March.
- Aymeric Lardon, 2012. "The γ-core in Cournot oligopoly TU-games with capacity constraints," Post-Print halshs-01071838, HAL.
- Aymeric Lardon, 2012. "The γ -core in Cournot oligopoly TU-games with capacity constraints," Post-Print halshs-00667669, HAL.
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