Partially symmetric values

We consider spaces of differentiable nonatomic and mixed vector measure games, pNA and pM, with finitely or countably many types of players. Type-symmetric values on these spaces of games are investigated (all Aumann and Shapley conditions except symmetry are assumed, the latter being replaced by a weaker assumption of covariance under automorphisms of the space of players that preserve each type). We show that if the types are uncountable, then type-symmetric values are random path values. In particular, the symmetric values on pM are characterized as mixtures of values defined in Hart (1973).