# Learning Strategic Value and Cooperation in Multi-Player Stochastic Games through Side Payments Source: [Learning Strategic Value and Cooperation in Multi-Player Stochastic Games through Side Payments]() Collection index: [llms.txt]() Data updated: 2026-05-13T05:00:12.693Z Authors: Yixin Chen, Jeffrey Richley, Darleen Perez-Lavin, Jessica Singh Syal, Solmaz Kia, Alan Kuhnle Published: 2023-03-09T14:57:15.000Z ## Abstract We study general-sum, multi-player stochastic games with transferable utility, motivated by settings where agents can use side payments to make cooperation individually rational. Building on the Harsanyi--Shapley (HS) value for normal-form games, we introduce two HS-based value notions for stochastic games: HS-S, defined by aggregating dynamic coalition-versus-complement threat powers, and Coco-S, defined as fixed points of a statewise HS Bellman operator. We extend HS-style axioms to the stochastic setting and show that HS-S is the unique mapping satisfying them. We prove that HS-S and Coco-S coincide in all two-player stochastic games, but can disagree when $n\>2$, via an explicit three-player counterexample. We prove existence and uniqueness of Coco-S fixed points for all two-player games and for three-player two-state games via topological degree theory, and provide an axiomatic characterization of Coco-S through a new \\emph{Markov Consistency} axiom that distinguishes it from HS-S. Finally, we give sampling-based estimators with finite-sample guarantees and empirically compare the induced values, policies, and side payments on multi-player grid-game benchmarks. ## Recorded benchmark results A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage. ## Original sources and implementation [Original paper]() [PDF]()