Abstract
We propose a new notion of coalitional equilibria, the strong \(\beta\)-hybrid solution, which is a refinement of the hybrid solution introduced by Zhao. Zhao’s solution is well suited to study situations where people cooperate within coalitions but where coalitions compete with one another. This paper’s solution, as opposed to the hybrid solution, assigns to each coalition a strategy profile that is strongly Pareto optimal. Moreover, like the \(\beta\)-core, deviations by subcoalitions of any existing coalition are deterred by the threat of a unique counter-strategy available to the non-deviating players. Zhao proved the existence of existence of strong \(\beta\)-hybrid solution for transferable utility games with compact and convex strategy spaces and concave continuous payoff functions. Here, we extend his result to non-transferable utility games.
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Notes
By contrast, the coalitional equilibria concepts initially defined by Aumann, namely the \(\alpha\) and \(\beta\) cores require that each conceivable coalition should not break away from a prealably agreed strategy profile.
That is, not all the members of a deviating sub-coalition need to be strictly better off when opting out of a coalition. Some members can be willing to leave the coalition in order to help others, as long as they do not suffer any loss.
The strong separability assumption is satisfied if “the outsider’s action that best punishes (as a second mover) coalition S is also the action that best punishes (as the first mover) each member of the coalition.” (see Zhao, ibid, p. 157). Zhao’s approach and the strong separability assumption are used by Meinhardt (2002) to establish the existence of \(\beta\)-equilibrium for common pool games.
By contrast, the hybrid solution and the strong hybrid solution have an \(\alpha\)-core flavor.
A topological space is called first countable if every point has a countable neighborhood basis.
The definitions apply mutatis mutandis to correspondences \(\Gamma X \rightrightarrows Y\) where Y is a subset of \(\mathbb {R}^l\).
To extend this existence result to more general specifications of the game, we simply need to make sure that for each \(x\in X\) and for each \(i\in I\), \(v_i^{\prime \prime }(\underset{j\in I}{\sum }x_j)\ge 0\) and \(g_i^{\prime \prime }(x_i)\le 0\). Under these conditions the functions \(u_i(x)\) are concave on X and we can easily prove as with Chander’s specification above that the correspondence is strictly-nonempty valued.
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Crettez, B., Nessah, R. & Tazdaït, T. On the strong \(\beta\)-hybrid solution of an N-person game. Theory Decis 94, 363–377 (2023). https://doi.org/10.1007/s11238-022-09900-0
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DOI: https://doi.org/10.1007/s11238-022-09900-0