Approximating Mixed Nash Equilibria using Smooth Fictitious Play in Simultaneous Auctions (Short Paper) Enrico H. Gerding ∗ Zinovi Rabinovich ∗ Andrew Byde ∗,† Edith Elkind ∗ Nicholas R. Jennings ∗ {eg,zr,ab06v,ee,nrj}@ecs.soton.ac.uk ∗ School of Electronics and Computer Science, University of Southampton, UK † Hewlett-Packard Laboratories, Bristol, UK. ABSTRACT We investigate equilibrium strategies for bidding agents that partic- ipate 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 dis- cretising the bid space, we are able to use smooth fictitious play to compute approximate solutions. Specifically, we find that the re- sults do indeed converge to ǫ-Nash mixed equilibria and, therefore, we are able to locate equilibrium strategies in such complex games where no known solutions previously existed. 1. INTRODUCTION The rapid increase of online auctions such as eBay, QXL, and Ya- hoo! has spawned considerable research in the field of auctions and automated bidding agents. In such auctions we increasingly observe different sellers offering similar or even identical goods and services at the same time. In eBay alone, for example, the Nintendo Wii game console has nearly 2000 listings at the time of writing, of which over 1500 are proper auctions. In addition to the web, such auctions are also considered a key approach to achieve effective allocation of tasks and resources within a number of research areas of multi-agent systems, including Grid computing and multi-robot coordination. Against this background, it is impor- tant to develop intelligent agents that are able to bid effectively in such auctions. In particular, this paper considers bidding strategies when multiple auctions selling substitutable goods are held simul- taneously. Whereas most of the previous research in this domain focuses on best-response or heuristic strategies, here we extend this research by considering equilibrium outcomes when several agents optimise their utility by participating in multiple auctions. To this end, we compute the equilibrium using fictitious play,a game-theoretic learning algorithm that optimises behaviour based on the opponents’ history of play. This algorithm has the property Cite as: Approximating Mixed Nash Equilibria using Smooth Fictitious Play in Simultaneous Auctions (Short Paper), Gerding, Rabinovich, Byde, Elkind, and Jennings, Proc. of 7th Int. Conf. on Autonomous Agents and Multiagent Systems (AAMAS 2008), Padgham, Parkes, Müller and Parsons (eds.), May, 12-16., 2008, Estoril, Portugal, pp. XXX-XXX. Copyright c  2008, International Foundation for Autonomous Agents and Multiagent Systems (www.ifaamas.org). All rights reserved. that, if the strategies converge, in the limit these strategies are a Nash equilibrium solution [4, Ch 2, Prop 2.1]. Specifically, here we apply an approach called smoothed or cautious fictitious play which is able to converge to approximate, also called ǫ-Nash mixed, equilibria. Formally, an equilibrium is ǫ-Nash if any single agent cannot gain more than ǫ by deviating from it. To date, much of the existing research dealing with multiple si- multaneous auctions typically assumes that bidders choose one of them and then bid optimally in that auction. Previous research has shown, however, that if all opponents follow this strategy and bid in a single auction, and given that the auctions do not have reserve prices, it is a best response to bid in all available auctions [5]. Here we extend this work by investigating equilibrium strategies where all the agents participate in multiple auctions. Finding an equi- librium outcome in this setting is a challenging problem, however, since no closed-form solution exists even for the best response case, and finding the equilibrium by brute-force search is computation- ally intractable (especially when considering mixed strategies). As a result, this setting has received very little attention in the literature and the work that exists operates in very limited cases. In this case, the seminal paper by Engelbrecht-Wiggans and We- ber [2] provides one of the starting points for the game-theoretic analysis of markets where buyers have substitutable goods. They derive a mixed Nash equilibrium for the special case where the number of buyers is large. Moreover, they assume that bidders have the same valuations and not all bidders can bid in all auctions. Our analysis, on the other hand, does not make these assumptions. Following this, [6] studied the case of simultaneous auctions with complementary goods. The setting provided in [6] is further ex- tended to the case of common values in [8]. However, neither of these works extend easily to the case of substitutable goods which we consider. This case is studied in [9], but the scenario consid- ered is restricted to three sellers and two bidders and with each bid- der having the same value (and thereby knowing the value of other bidders). The space of symmetric mixed equilibrium strategies is derived for this special case, but these results do not generalise to settings with more bidders and sellers, and, most importantly, to settings where bidders have different valuations. In more detail, this work advances the state-of-the-art in the fol- lowing ways. First, we derive equations for the bidder’s expected utility in the case when all bidders use mixed strategies. In this way we can compute the equilibrium using smooth fictitious play without actually simulating the auctions. These equations cannot be easily computed for very large inputs, however, and therefore we limit the strategy space by assuming that bidders bid at most