Random Order Approach to Shapley - value of Games andHaar measure 1

Imposing a group structure on the set of random orders, the paper reformulates and characterizes the random order value and more generally semi-value operators in a uni ed framework that encompasses games with nite and continuum of players and allows symmetry of the operators to be with respect to a subgroup of automorphisms. A set of orderings of players equipped with a group structure induced from the group structure of automorphisms together with a measure structure on it constitutes a group of random orders in the analysis. For nite games it is shown that given any xed group of random orders, the linear operator on the whole space of games that assigns to each game its expected marginal contribution is symmetric with respect to the associated group of automorphisms if and only if the randomness of the group of orders is generated by a right invariant Haar measure; as a corollary, the paper provides a group theoretic proof for the existence and uniqueness of random order Shapley value and semi-value operators that are symmetric with respect to the full group of automorphisms; the paper also shows that the random order semi-value operators constructed from a proper subgroup of orders coincide with the semi-value operators which are symmetric with respect to the full group of automorphisms on a linear subpace of games. Many of these results are also extended to games with continuum of players. Random Order Approach to Shapley-value of Games and Haar measure