Sampling and Updating Higher Order Beliefs in Decision-theoretic Bargaining with Finite Interactive Epistemologies

In this paper we study the sequential strategic interactive setting of bilateral, two-stage, seller-offers bargaining under uncertainty. We model the epistemology of the problem in a finite interactive decision-theoretic framework and solve it for three types of agents of successively increasing (epistemological) sophistication (i.e. capacity to represent and reason with higher orders of beliefs). We relax typical common knowledge assumptions, which, if made, would be sufficient to imply the existence of a, possibly unique, game-theoretic equilibrium solution. We observe and characterize a systematic monotonic relationship between an agent’s beliefs and optimal behavior under a particular moment-based ordering of its beliefs. Based on this characterization, we present the spread-accumulatetechnique of sampling an agent’s higher order belief by generating “evenly dispersed” beliefs for which we (pre)compute offline solutions. Higher order prior belief identification is then approximated to arbitrary precision by identifying a (previously solved) belief “closest” to the true belief. These methods immediately suggest a mechanism for achieving a balance between efficiency and the quality of the approximation – either by generating a large number of offline solutions or by allowing the agent to search online for a “closer” belief in the vicinity of best current solution.

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