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Approximating mixed Nash equilibria using smooth fictitious play in simultaneous auctions

We investigate equilibrium strategies for bidding agents that participate in multiple, simultaneous second-price auctions with perfect substitutes. For this setting, previous research has shown that it is a best response for a bidder to participate in as many such auctions as there are available, provided that other bidders only participate in a single auction. In contrast, in this paper we consider equilibrium behaviour where all bidders participate in multiple auctions. For this new setting we consider mixed-strategy Nash equilibria where bidders can bid high in one auction and low in all others. By discretising the bid space, we are able to use smooth fictitious play to compute approximate solutions. Specifically, we find that the results do indeed converge to e-Nash mixed equilibria and, therefore, we are able to locate equilibrium strategies in such complex games where no known solutions previously existed.