Deliberation in Equilibrium: Bargaining in Computationally Complex Problems

We develop a normative theory of interaction— negotiation in particular—among self-interested computationally limited agents where computational actions are game-theoretically treated as part of an agent’s strategy. We focus on a 2-agent setting where each agent has an intractable individual problem, and there is a potential gain from pooling the problems, giving rise to an intractable joint problem. At any time, an agent can compute to improve its solution to its problem, its opponent’s problem, or the joint problem. At a deadline the agents then decide whether to implement the joint solution, and if so, how to divide its value (or cost). We present a fully normative model for controlling anytime algorithms where each agent has statistical performance profiles which are optimally conditioned on the problem instance as well as on the path of results of the algorithm run so far. Using this model, we analyze the perfect Bayesian equilibria of the games which differ based on whether the performance profiles are deterministic or stochastic, whether the deadline is known or not, and whether the proposer is known in advance. Finally, we present algorithms for finding the equilibria.