Temporal Cooperative Games

2025-10-13

AI Summary

Classical cooperative game theory assumes coalition values depend solely on member sets, ignoring the order of player arrivals—a limitation that undermines temporal realism in dynamic collaboration settings. Method: We propose “temporal cooperative games,” introducing arrival order as a fundamental variable and reconstructing core axioms to define temporal properties: Incentive for Optimal Arrival (I4OA), Online Individual Rationality (OIR), and Sequential Efficiency (SE). We integrate time-series modeling into value allocation and axiomatically construct Shapley-like values in both sequential and extended domains. Contribution/Results: We prove that no mechanism can simultaneously satisfy I4OA, OIR, and SE—this incompatibility persists even in convex and simple games. Leveraging efficiency, additivity, and null-player axioms, we uniquely characterize our sequential Shapley-like values. This establishes the first reward allocation framework that jointly ensures temporal consistency and fairness, bridging a critical gap between cooperative game theory and sequential decision-making.

Technology Category

• Game Theory and Economic Paradigms: Cooperative Game Theory
• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Sequential Decision Making

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

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.

Problem

• Extends cooperative games to account for agent arrival sequences
• Identifies mechanisms satisfying temporal incentive and efficiency properties
• Reveals incompatibility between Shapley value analogs and temporal properties

Innovation

• Introduces temporal cooperative games with sequence-dependent worth
• Defines three key properties: I4OA, OIR, and SE
• Constructs Shapley analogs for sequential and extended worlds