Two n-person games may be said to be equivalent if they have the same Shapley value. In this paper, we attempt to simplify the determination of the equivalence of two games. We do this by recognizing that a pair of games are equivalent if and only if their difference lies in the null space of the Shapley value. Using the representation theory of the symmetric groups we construct a direct-sum decomposition of this null space into invariant subspaces. We then use this same theory to derive a characterization of a very general type of value, of which the Shapley value is one particular example.
We first show that, in general, a majority-voting game with vote-selling will not have any equilibria. We then evaluate the desirability of vote-selling, using a rudimentary 'theory of blocking trajectories.'
In this paper we attempt to decompose a game v into two different components, one lying in the null space of the Shapley value and the other in its orthogonal complement. We observe that the Shapley value of the former must be 0, so that the Shapley value of the latter coincides with the value of the original game. In this way we arrive at a new explicit formula for the Shapley value which, unlike the typical one, involves averages of player worths across coalition sizes. Central to our ideas is the game-theoretic contrast between the spaces in which each component lies.
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