A new bargaining solution for finite offer spaces

Abstract

AbstractThe bargaining problem deals with the question of how far a negotiating agent should concede to its opponent. Classical solutions to this problem, such as the Nash bargaining solution (NBS), are based on the assumption that the set of possible negotiation outcomes forms a continuous space. Recently, however, we proposed a new solution to this problem for scenarios with finite offer spaces de Jonge and Zhang (Auton Agents Multi-Agent Syst 34(1):1–41, 2020). Our idea was to model the bargaining problem as a normal-form game, which we called the concession game, and then pick one of its Nash equilibria as the solution. Unfortunately, however, this game in general has multiple Nash equilibria and it was not clear which of them should be picked. In this paper we fill this gap by defining a new solution to the general problem of how to choose between multiple Nash equilibria, for arbitrary 2-player normal-form games. This solution is based on the assumption that the agent will play either ‘side’ of the game (i.e. as row-player or as column-player) equally often, or with equal probability. We then apply this to the concession game, which ties up the loose ends of our previous work and results in a proper, well-defined, solution to the bargaining problem. The striking conclusion, is that for rational and purely self-interested agents, in most cases the optimal strategy is to agree to the deal that maximizes the sum of the agents’ utilities and not the product of their utilities as the NBS prescribes.