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On Equilibria of N-seller and N-buyer Bargaining Games

A group of players which contain n sellers and n buyers bargain over the partitions of n pies. A seller(/buyer) has to reach an agreement with a buyer (/seller) on the division of a pie. The players bargain in a system like the stock market: each seller(buyer) can either offer a selling(buying) price to all buyers(sellers) or accept a price offered by another buyer(seller). The offered prices are known to all. Once a player accepts a price offered by another one, the division of a pie between them is determined. Each player has a constant discounting factor and the discounting factors of all players are common knowledge. In this article, we prove that the equilibrium of this bargaining problem is a unanimous division rate, which is equivalent to Nash bargaining equilibrium of a two-player bargaining game in which the discounting factors of two players are the average of n buyers and the average of n sellers respectively. This result shows the relevance between bargaining equilibrium and general equilibrium of markets.