Capability-Weighted Group Utility Maximizer for Network Coalitional Games under Uncertainty

In this paper we study network games where agents with different skills come together to cooperate and yet competitively pursue individual goals. We propose a multi-agent based utilitarian approach to model the payoff allocation problem for a class of such games where the capabilities of the agents and the payoffs are not known with certainty. The primary objective is to maximize a linear sum of the expected utilities of risk-averse agents; and we consider constant risk-aversion with exponential utility functions. We pose the problem as a stochastic cooperative game which is solved in two phases. In the first phase we apply a learning mechanism on this 'social' network of fully connected agents to arrive at a consensus on the capability of every agent in the coalition thus resolving uncertainty in capabilities. Agents initially start with a social influence matrix reflecting the influence agents have on each other and prior subjective beliefs of the capabilities of the others and these beliefs evolve through a process of interaction. We use a variant of the DeGroot algorithm to show that over time learning results in a dynamic update of the beliefs and the social influence matrix leading to a consensus. We provide theoretical convergence proofs for the algorithm. The second phase involves optimizing a capability-weighted sum of the expected utilities of the agents to achieve a group Pareto optimal solution. In this paper we propose a new framework called the Capability Weighted Group Utility Maximizer developed around Borch's theorem borrowed from the actuarial world of insurance to obtain a fair distribution of the stochastic payoffs once a consensus is reached on the capabilities of the agents in the coalition.