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Temporal Cooperative Games

Ashwin Goyal, Drashthi Doshi, Swaprava Nath

TL;DR

This paper introduces temporal cooperative games (TCGs) where worth depends on the arrival sequence of agents, not just the participating set, and proposes three temporal properties—incentive for optimal arrival ($I4OA$), online individual rationality ($OIR$), and sequential efficiency ($SE$)—as foundations for fair, real-time reward sharing. It constructs SeqShare, a class of mechanisms that uniquely satisfy these properties under a basis-consistency condition, and defines MargSol and Ext-Shap as Shapley-inspired solutions in sequential and extended spaces, proven to be uniquely characterized by efficiency, additivity, and null-player-type properties. A key and surprising result is the fundamental tension between temporal properties and Shapley-inspired axioms: no mechanism can simultaneously satisfy both sets, and this incompatibility persists even for restricted classes like convex and simple TCGs. The work thus highlights a core trade-off in sequential environments and motivates future research into new fair and efficient solution concepts tailored to arrival-order dependent worths.

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.

Temporal Cooperative Games

TL;DR

This paper introduces temporal cooperative games (TCGs) where worth depends on the arrival sequence of agents, not just the participating set, and proposes three temporal properties—incentive for optimal arrival (), online individual rationality (), and sequential efficiency ()—as foundations for fair, real-time reward sharing. It constructs SeqShare, a class of mechanisms that uniquely satisfy these properties under a basis-consistency condition, and defines MargSol and Ext-Shap as Shapley-inspired solutions in sequential and extended spaces, proven to be uniquely characterized by efficiency, additivity, and null-player-type properties. A key and surprising result is the fundamental tension between temporal properties and Shapley-inspired axioms: no mechanism can simultaneously satisfy both sets, and this incompatibility persists even for restricted classes like convex and simple TCGs. The work thus highlights a core trade-off in sequential environments and motivates future research into new fair and efficient solution concepts tailored to arrival-order dependent worths.

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 becomes a function of the sequence of agents rather than just the set . 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.

Paper Structure

This paper contains 19 sections, 10 theorems, 36 equations, 7 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Related Work
  4. Preliminaries
  5. The SeqShare Class of Mechanisms
  6. Remark.
  7. Remark.
  8. Properties of SeqShare
  9. Properties Inspired by Shapley in TCGs
  10. Solutions Satisfying Shapley-inspired Properties
  11. Uniqueness of MargSol
  12. Uniqueness of Ext-Shap
  13. Remark.
  14. Why Symmetry Is Not Necessary?
  15. Remarks.
  16. ...and 4 more sections

Key Result

Lemma 1

For a TCG $\langle N,v \rangle$, if a solution concept $\phi$ satisfies I4OA and SE then there must be a basis solution.

Figures (7)

  • Figure 1: A game that satisfies basis conditions.
  • Figure 2: (a) The first player arrives and all the value generated is assigned to player P1. (b) The second player P2 arrives and the marginal value generated by the arrival of this player is distributed among P1 and P2 in a way that ensures the total reward of each agent is less than the upper bound. The total reward for any agent is the sum of the reward in the previous round and the share of marginal increase assigned in the current round. (c) The third player P3 arrives and the marginal value generated by P3 is distributed among P1, P2 and P3 keeping total rewards lower than the upper bound. (d) Finally, P4 arrives and the marginal value generated by this player can only be divided among P1,P2 and P4 as P3 has already reached its maximum possible reward.
  • Figure 3: SeqShare is not compatible with sanchez1997values
  • Figure 4: A generic TCG serving as counterexample for multiple settings for different values of $a$ and $b$, e.g., no solution in SeqShare can be reduced to Ext-Shap.
  • Figure 5: MargSol violates I4OA
  • ...and 2 more figures

Theorems & Definitions (35)

  • Definition 1: Online Individual Rationality
  • Definition 2: Sequential Efficiency (SE)
  • Definition 3: Incentive for Optimal Arrival (I4OA)
  • Definition 4: Basis conditions
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Definition 5
  • Definition 6: Sequential Additivity (SA)
  • ...and 25 more