Spectrum value for coalitional games

Abstract

Assuming a 'spectrum' or ordering on the players of a coalitional game, as in a political spectrum in a parliamentary situation, we consider a variation of the Shapley value in which coalitions may only be formed if they are connected with respect to the spectrum. This results in a naturally asymmetric power index in which positioning along the spectrum is critical. We present both a characterisation of this value by means of properties and combinatoric formulae for calculating it. In simple majority games, the greatest power accrues to 'moderate' players who are located neither at the extremes of the spectrum nor in its centre. In supermajority games, power increasingly accrues towards the extremes, and in unaninimity games all power is held by the players at the extreme of the spectrum.

The Effects of Excluding Coalitions

“…In this model, incompatible players are linked by an arc of the graph. As well as in the models by [3,5] , every coalition (especially the grand coalition) obtains a worth-the maximum that could achieved by the compatible players of the coalition. In the model by [4], loops or parallel arcs are not intended.…”

“…The ϕ E value enhances the approaches developed by [3] to model that players prefer some other players for cooperation. 1 Whereas the approach by [3,5] modifies the coalitional function of the game, the model by restricts the set of admissible permutations of players. For modeling the preferences of players, both models are insufficient.…”

“…The party from the middle admits cooperation with both parties. In the models by [3,5] the left and the right parties are connected via the middle party. Hence, all parties cooperate.…”

Values for cooperative games with restricted coalition formation

“…Proposition 7.1. Let (U, N , L) ∈ C S F N ,r =r cp , the following statements hold (1) If (N , L) is cycle-free, then r U (E) = rv(E) for any E ⊆ L.…”

“…However, the marginal contribution of player 4 in the order (3, 4, 1, 2, 5) decreases compared to its marginal contribution in the original order (1,2,3,4,5),…”