Temporal Cooperative Games

Classical cooperative game theory assumes that the worth of a coalition depends only on the set of agents involved, but in practice, it may also depend on the order in which agents arrive. Motivated by such scenarios, we introduce temporal cooperative games (TCG), where the worth $v$ becomes a function of the sequence of agents $π$ rather than just the set $S$. This shift calls for rethinking the underlying axioms. A key property in this temporal framework is the incentive for optimal arrival (I4OA), which encourages agents to join in the order maximizing total worth. Alongside, we define two additional properties: online individual rationality (OIR), incentivizing earlier agents to invite more participants, and sequential efficiency (SE), ensuring that the total worth of any sequence is fully distributed among its agents. We identify a class of reward-sharing mechanisms uniquely characterized by these three properties. The classical Shapley value does not directly apply here, so we construct its natural analogs in two variants: the sequential world, where rewards are defined for each sequence-player pair, and the extended world, where rewards are defined for each player alone. Properties of efficiency, additivity, and null player uniquely determine these Shapley analogs in both worlds. Importantly, the Shapley analogs are disjoint from mechanisms satisfying I4OA, OIR, and SE, and this conflict persists even for restricted classes such as convex and simple TCGs. Our findings thus uncover a fundamental tension: when players arrive sequentially, reward-sharing mechanisms satisfying desirable temporal properties must inherently differ from Shapley-inspired ones, opening new questions for defining fair and efficient solution concepts in TCGs.