Cooperative-competitive value

In Cooperation in Strategic Games Revisited (2013), Kalai and Kalai argued that in two-person strategic games with transferable utility, major variable-threat bargaining and arbitration solutions coincide. They presented the coco value as a broader theory built around that coincidence, together with a closed-form formula, an axiomatic characterization, and an extension to games with private signals.^[1]

For a two-player bimatrix game with payoff matrices A and B, the coco decomposition writes the original game as the sum of two parts: a team or common-interest game, in which both players receive ${\displaystyle (A+B)/2}$ at every action profile, and a zero-sum game, in which the players receive ${\displaystyle (A-B)/2}$ and ${\displaystyle (B-A)/2}$ respectively.^[1]

In the complete-information case, the coco value is obtained by taking the maxmax value of the team component and the minimax value of the zero-sum component, then combining them into a payoff pair.^[1] Kalai and Kalai described the concept as combining efficiency in the cooperative component with strategic advantage in the competitive component.^[1]

Kalai and Kalai also showed that the decomposition is straightforward to compute from the payoff matrices and that the coco value can be computed in polynomial time.^[1]

For two-player complete-information games, Kalai and Kalai characterized the coco value using axioms including efficiency, shift invariance, monotonicity in actions, payoff dominance, and invariance to redundant strategies.^[1][2] In their Bayesian extension they added monotonicity in information, under which strictly reducing a player's information cannot increase that player's value.^[1][2] Kalai and Kalai proved that these conditions uniquely determine the coco value.^[1]

Kohlberg and Neyman later wrote that the coco value coincides with the value of two-person strategic games in their broader framework for complete and incomplete information games.^[2] Andrés Salamanca compared the coco value with Roger Myerson's solution for Bayesian cooperative games with side payments, showing ex-ante utility equivalence under a verifiable-types assumption while also emphasizing differences in incentive compatibility and interim implementation.^[3]

In later research, Shiran Rachmilevitch showed that straightforward extension beyond two players is difficult: adding a mild dummy-player axiom to the two-player axioms makes them inconsistent for games with more than two players.^[5] Rachmilevitch also studied the relationship between the coco value and equilibrium payoffs in zero-sum and common-interest games, and later introduced an analogous coco decomposition in bargaining theory.^[4][6]

In multi-agent reinforcement learning, Sodomka, Hilliard, Littman and Greenwald introduced Coco-Q, which extends coco values to stochastic games with side payments.^[7] Later work on reward-transfer contracts in multi-agent systems referred to the coco value as a focal bargaining point for negotiating fair transfers in two-player games, and cooperative-AI work has cited it as a classical measure related to the costs of competition in a game.^[8][9]