Uncertainty in cooperative interval games: how Hurwicz criterion compatibility leads to egalitarianism

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.

Shapley Mapping and Its Axiomatizations in n-Person Cooperative Interval Games

“…It should also be noted that α is endogenously determined depending on t ∈ w(N), rather than an exogenous parameter representing such factors as players' risk attitudes. Therefore, our α is not directly related to the Hurwicz criterion recently examined by Mallozzi and Vidal-Puga [29] in the context of interval game analyses. We do not impose any specific risk attitudes on players, and all of the results obtained here hold independent of those attitudes.…”

“…Example 3 (Alparslan Gök [25] and Palanci et al [30]). w(1) = [7,7], w(2) = [0, 0], w(3) = [0, 0], w(12) = [12,17], w(13) = [7,7], w(23) = [0, 0] and w(123) = [24,29]. As this game is size monotonic, the interval Shapley value exists: Φ(w) = ([27/2, 16], [13/2, 9], [4,4]).…”

“…The Shapley mapping, on the other hand, gives the following three-dimensional real-valued vectors for the realizations of the worth set of the grand coalition-24, 26.5 and 29 (the minimum, midpoint and maximum value of w(123), respectively): Comparing Shapley mapping σ * with interval Shapley mapping Φ, σ * (w) (24) and σ * (w) (29) are the same as the vectors consisting of the lower and upper bounds of each element in Φ(w), respectively. Similarly, each element of σ * (w) (26.5) is identical to the midpoint of the corresponding player's interval in Φ(w).…”

Some Properties of Interval Shapley Values: An Axiomatic Analysis

“…Meng et al [6] studied the Shapley value in interval fuzzy cooperative games based on Hukuhara's difference operator. Mallozzi and Vidal-Puga [7] examined interval games where players have different attitudes towards uncertainty represented by Hurwicz coefficients. Shino et al [8] examined the notion of the solution mapping as an alternative to the interval solution concept, proposed Shapley mapping as a specific form of the solution mapping, and showed its axiomatizations.…”