Abstract
This paper analyzes the core of cooperative games generated by asymmetric aggregative normal-form games, i.e., games where the payoff of each player depends on his strategy and the sum of the strategies of all players. We assume that each coalition calculates its worth presuming that the outside players stand alone and select individually best strategies (Hart and Kurz Econometrica 51:1047–1064, 1983; Chander and Tulkens Int J Game Theory 26:379–401, 1997). We show that under some mild monotonicity assumptions on payoffs, the resulting cooperative game is balanced and has a non-empty core (which is the \(\gamma \)-core). Our paper thus offers an existence result for a core notion which is frequently encountered in the theory and applications of cooperative games with externalities.
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Notes
We should point that the roots of this approach go back to Von Neumann and Morgenstern (1944), in the context of three-player games. We thank an anonymous referee for pointing out this.
Although all players in \(\Gamma \) stand alone, we continue to categorize them in accordance with \(\Gamma ^S\) to facilitate the comparisons that will follow.
Our notation implies that when we take the derivative of \(u_k(x_k,x)\) over the first argument \(x_k,\) we treat x as fixed. We hope this will not confuse the reader.
The function \(\beta (x_k,x)\) is simply the marginal payoff function of player k when he stands alone, i.e., when \(T\setminus \{k\}=\emptyset \) (in which case \(\beta _k(x_k,x)\) reduces to \(\alpha _k(x_k,x)\)) or when he does not stand alone, i.e., when \(T\setminus \{k\}\ne \emptyset \).
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Stamatopoulos, G. On the \(\gamma \)-core of asymmetric aggregative games. Theory Decis 88, 493–504 (2020). https://doi.org/10.1007/s11238-019-09733-4
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DOI: https://doi.org/10.1007/s11238-019-09733-4