Pareto Optimal Allocation in Multi-agent Coalitional Games with Non-linear Payoffs

A fully connected network of multiple interacting agents modeled as a cooperative game to attain a common objective has found wide applications in the real world. Competitors frequently come together to work in coalitions that are mutually beneficial to them all, though the allocation of the mutual gains achieved is seldom easy. Shapley value is a popular way to compute payoffs in cooperative games where the agents are assumed to have deterministic, risk-neutral (linear) utilities. This paper explores a class of Multi-agent constant-sum cooperative games where the payoffs are random variables. We introduce a new model based on Borch's Theorem from the actuarial world of re-insurance, to obtain a Pareto optimal allocation for agents with risk-averse exponential utilities. This allocation problem seeks to maximize a linear sum of the expected utilities of a set of agents and the solution obtained at this optimal value naturally maximizes the social welfare of the grand coalition. The four main axioms of the Shapely Value, namely, nullity, additivity, symmetry and efficiency are satisfied by this solution. We show the correspondence of our solution to the Shapley value. As a result we can directly obtain the Shapley value from the allocation values obtained at the Pareto optimum as the individual utility achievements of the grand coalition.