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Sequential Solution Concepts in Cooperative Games with Generalized Characteristic Functions

NeurIPS 2026 · Multi-agent systems and games › Games and equilibria · approach: Cooperation and social dilemmas · 0 citations (OpenAlex)

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Studies temporal cooperative games (generalized characteristic functions over sequences), proposes sequential solution concepts (SeqShare class under I4OA, OIR, SE) and Shapley analogs (MargSol, Ext-Shap), and shows the two families are disjoint even for convex and simple games.

Problem

Reward sharing in cooperative games where a coalition's worth depends on the order in which agents arrive; existing generalized-characteristic-function solutions are order-oblivious and tied to the Shapley value.

Approach

Defines order-sensitive (sequential) axioms I4OA, OIR and SE, characterizes the SeqShare class satisfying them via basis conditions, builds sequential Shapley analogs (MargSol, via efficiency, additivity, null player) and their averaged extended version (Ext-Shap), and proves these families are disjoint.

Key results

• SeqShare class satisfies OIR, I4OA and SE iff the game satisfies basis conditions; otherwise no solution satisfies I4OA and SE.
• MargSol is the unique sequential solution satisfying SE, SA and SNP; Ext-Shap is the unique extended solution satisfying EE, EA and ENP.
• No SeqShare solution reduces to Ext-Shap even for convex and simple TCG examples (e.g., Ext-Shap_1 = 8/6 vs phi_1(pi*) = 1 in one example).

Surprising findings

• Sequential additivity, null player and efficiency already imply symmetry, so symmetry is not needed as an axiom, unlike the classical Shapley characterization.
• MargSol violates I4OA, and the sequential order-sensitive and Shapley-like properties are incompatible even in convex and simple games.

Limitations

• Results are purely theoretical and small illustrative examples; no empirical evaluation.
• The SeqShare class is non-unique, and a unique solution requires additional properties not identified here.
• The paper itself notes gaps such as an unfinished or placeholder proof and undefined references in the text.

Experimental setup

Baselines: NR value SB value Shapley value

Builds on: Nowak and Radzik generalized characteristic function Shapley value Shapley axiomatization

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