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.
In this paper, we characterize two power indices introduced in [1] using two different modifications of the monotonicity property first stated by [2]. The sets of properties are easily comparable among them and with previous characterizations of other power indices.
: We propose a generalization of simple games to situations with coalitional externalities. The main novelty of our generalization is a monotonicity property that we define for games in partition function form. This property allows us to properly speak about minimal winning embedded coalitions. We propose and characterize two power indices based on these kind of coalitions. We provide methods based on the multilinear extension of the game to compute the indices. Finally, the new indices are used to study the distribution of power in the current Parliament of Andalusia.JEL Codes: C71.
We present parallel characterizations of two di erent values in the framework of restricted cooperation games. The restrictions are introduced as a nite sequence of partitions de ned on the player set, each of them being coarser than the previous one, hence forming a structure of di erent levels of a priori unions. On the one hand, we consider a value rst introduced in , 8), which extends the Shapley value to games with di erent levels of a priori unions. On the other hand, we introduce another solution for the same type of games, which extends the Banzhaf value in the same manner. We characterize these two values using logically comparable properties.
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