A reformulation of Aumann-Shapley random order values of non- atomic games using invariant measures

In this paper the random order approach to values of non-atomic games is reformulated by generating random orders from a fixed subgroup of automorphisms, $\Theta$ that admits an invariant probability measurable group structure. The resulting $\Theta$-symmetric random order value operator is unique and satisfies all the axioms of a $\Theta$-symmetric axiomatic value operator. It is shown that for the uncountably large invariant probability measurable group $\left(\breve\Theta,\breve{\cal B},\breve\Gamma\right)$ of Lebesgue measure preserving automorphisms constructed in Raut [1996], $\breve\Theta$-symmetric random order value exists for most games in BV and it coincides with the fully symmetric Aumann-Shapley axiomatic value on pNA. Thus by restricting the set of admissible orders suitably the paper provides a possibility result to the Aumann-Shapley Impossibility Principle for the random order approach to values of non-atomic games.