"[...] We saw that unacceptability and ties are a major source of intractability when computing Pareto optimal outcomes. In some cases, checking whether a given partition is Pareto optimal can be significantly harder than finding one.[...]" (Aziz et al., 2013)
So now I have a small question. Is it possible to conclude that all coalition games have a Pareto optimal solution, even though it is maybe extremely hard to find that solution, or is it possible that for some coalition games, a Pareto optimal solution may not exist?
Aziz, Haris, Felix Brandt, and Paul Harrenstein. "Pareto optimality in coalition formation." Games and Economic Behavior 82 (2013): 562-581.
I think that in all multi-period coalition games under certainty, the players select equilibrium strategies that generate outcomes on the production possibility and/or utility frontier that are by definition Pareto Optimal solutions. However, if a game is only played once, or there is uncertainty each period as to where the boundaries of the production possibility and/or utility frontier are relative to the strategy choices, a Pareto Optimal solution while ultimately pursued by the players, may never be achieved.
The answer depends on the model of the cooperative game you applied and what do you mean by "yield"? Cooperative game is presented in a classical kind as a characteristic function, which for each coalition put into correspondence a value of payoff. Pareto optimality means an allocation of payoff which cannot be improved by a grand coalition, i.e. the summation of individual values has to be not less when a value of grand coalition, normally equality takes place. Usually nobody discusses the way to find such distribution, but it is possible in an economic context. Moreover, a looking for a final allocation of total payoff just started from these individualized payoffs. As a result so-called C-core is specified as a set of all value allocations which can be dominated by no coalition, i.e. no coalition can suggest a feasible collection of payoffs improving position of every member of coalition. So, Pareto optimality is not matter in cooperative game context.
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