Sequential Solution Concepts in Cooperative Games with Generalized Characteristic Functions

Motivated by the fact that the worth of a coalition may depend on the order in which agents arrive, Nowak and Radzik (1994) (NR) introduced cooperative games with generalized characteristic functions. We study such temporal cooperative games (TCGs), where the worth function v is defined on sequences of agents π rather than sets S. This order sensitivity necessitates a re-examination of axioms for reward sharing. NR and subsequent work proposed several axioms; the resulting solution concepts are still inherently order-oblivious and closely tied to the Shapley value. In contrast, we focus on sequential solution concepts that explicitly depend on the realized order π. We study reward-sharing mechanisms satisfying incentive for optimal arrival(I4OA), which promotes orders maximizing total worth; online individual rationality (OIR), which ensures agents are not harmed by later arrivals; and sequential efficiency (SE), which requires that the worth of any sequence is fully distributed among its agents. These axioms are intrinsic to TCGs, and we characterize a class of reward-sharing mechanisms uniquely determined by them. The classical Shapley value does not directly extend to this setting. We therefore construct natural Shapley analogs in two worlds: a sequential world, where rewards are defined for each sequence–agent pair, and an extended world, where rewards are defined per agent, consistent with the NR framework. In both cases, the axioms of efficiency, additivity, and null player uniquely characterize the corresponding Shapley analogs. But these Shapley analogs are disjoint from the class of solutions satisfying the sequential axioms, even for convex and simple TCGs. Our results reveal a fundamental tension in temporal cooperative games: when order matters, solution concepts satisfying natural sequential incentives must differ structurally from Shapley-based allocations, motivating new axiomatic foundations for sequential reward sharing.