Abstract
We study cooperative interval games. These are cooperative games where the value of a coalition is given by a closed real interval specifying a lower bound and an upper bound of the possible outcome. For interval cooperative games, several (interval) solution concepts have been introduced in the literature. We assume that each player has a different attitude towards uncertainty by means of the so-called Hurwicz coefficients. These coefficients specify the degree of optimism that each player has so that an interval becomes a specific payoff. We show that a classical cooperative game arises when applying the Hurwicz criterion to each interval game. On the other hand, the same Hurwicz criterion can also be applied to any interval solution of the interval cooperative game. Given this, we say that a solution concept is Hurwicz compatible if the two procedures provide the same final payoff allocation. When such compatibility is possible, we characterize the class of compatible solutions, which reduces to the egalitarian solution when symmetry is required. The Shapley value and the core solution cases are also discussed.
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Notes
For example, assume that players form an oligopoly that plans to create a cartel. The cartel can then anticipate their benefit as a monopoly. However, if two or more players are not present, the remaining players can not anticipate their exact benefit, as it would depend on whether the other players merge or not.
Other applications of Hurwicz coefficients in interval games appear in Lardon (2017) and Li (2016), who also deduce a (classical) cooperative game by using a selection via degrees of optimism. However, these degrees are coalition-dependent, not individual. Hence, they cannot be identified as Hurwicz coefficients in the same way we do here.
This is always possible to do since interval cooperative games generalize classical ones.
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Mallozzi, L., Vidal-Puga, J. Uncertainty in cooperative interval games: how Hurwicz criterion compatibility leads to egalitarianism. Ann Oper Res 301, 143–159 (2021). https://doi.org/10.1007/s10479-019-03379-9
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DOI: https://doi.org/10.1007/s10479-019-03379-9