Outperforming the Competition in Multi-Unit Sealed Bid Auctions

Outperforing the Copetition in Multi-Unit Sealed Bid Auctions ABSTRACT Ioannis A. Vetsikas School of Electronics and Coputer Science University of Southapton Southapton SO17 1BJ, UK iv@ecs.soton.ac.uk In this paper, we exaine the behavior of bidding agents that are in direct copetition with the other participants in an auction setting. Thus the agents are not siply trying to axiize their own utility, rather they wish to axiize a weighted difference of their own gain to that of their copetitors. By so doing, this work significantly extends the existing state-of-the-art results on single unit auctions, by generalizing to the ulti-unit case. Specifically, our ain result is the derivation of syetric Bayes-Nash equilibria for these agents in both th and +1 th price sealed bid auctions. Subsequently, we use these equilibria to exaine the profits of different agents and show that aiing to beat the copetition is ore effective than pure self interest in any copetitive setting. Finally, we exaine how the auctioneer s revenue is affected and find that the weight that agents place in iniizing the opponents profit deterines whether the th or the +1 th price auction yields a higher revenue. Categories and Subject Descriptors I.2.11 [ARTIFICIAL INTELLIGENCE]: Multiagent systes; I.2.11 [ARTIFICIAL INTELLIGENCE]: Intelligent Agents General Ters Theory, Econoics, Experientation Keywords gae theory, bidding strategies, equilibriu analysis, revenue, siulation 1. INTRODUCTION Gae theory is widely used in ulti-agent systes, as a way to odel and predict the interactions between rational agents. Auctions have also becoe quite popular, especially Perission to ake digital or hard copies of all or part of this work for personal or classroo use is granted without fee provided that copies are not ade or distributed for profit or coercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific perission and/or a fee. AAMAS 7 May 14 18 27, Honolulu, Hawai i, USA. Copyright 27 IFAAMAS. Nicholas R. Jennings School of Electronics and Coputer Science University of Southapton Southapton SO17 1BJ, UK nrj@ecs.soton.ac.uk in the last decade, particularly with ebay and other siilar copanies bringing auctions to the Internet and illions of online bidders. Now, in ost of the cases exained, either by gae theory or by experiental analyses on auction participant behaviour, a key assuption is that bidders are rational and self-interested i.e. they care only about axiizing their profit in the auction, and not about the profit of other bidders. However, there are observations and scenarios that cannot convincingly be explained by these assuptions alone. These usually involve direct copetition between copanies or individual agents, where fewer participants ean higher profits for those involved, or cases in which ranking and/or relative profit is iportant. In such cases, the agents will try to axiize the difference of their utility to that of any other copetitor. In ore detail, gae theorists have studied a nuber of exaples where self-interest, at least in the short ter, is not as iportant as retaining a hold on the arket, especially in scenarios that involve onopolies or oligopolies [6]. In addition, there are exaples of real-world scenarios, where yopic self-interest alone cannot explain the observed behaviour. On ebay auctions, for exaple, bidders soeties get into bidding wars, which cause those auctions to close uch higher than other auctions selling the exact sae coodity; this could be attributed partially to a love for winning and a total utility that takes into account the fact itself that the bidder lost. In addition, in any European countries, where obile phone spectru licenses were sold in various auction settings, it was observed that any teleco copanies bid higher than the estiated value of the corresponding licenses, in order to deny the fro their copetition and to discourage saller copetitors fro enetring the arket. Furtherore, in a copetition setting, like the Trading Agent Copetition TAC, it has been observed that participants tend to bid quite high for coodities that are in short deand and which are needed in order to ake profit; while part of this bidding behaviour can be attributed to the valuation they have of these ites, part of this is also due to the desire to outperfor the opposition. In fact, in [1], the authors suggest this copetition factor as the ain reason why the ost successful strategies bid ore aggressively than the assuption of self-interested agents alone would predict. Against this background, this work considers that agents not only wish to axiize their own profit, but at the sae tie they also wish to iniize the profits of their opponents. Specifically, our assuption for the odel used by agents in such settings is that they wish to axiize a 72 978-81-94262-7-5 RPS c 27 IFAAMAS

weighted su of their profit relatively to the profit of their copetitors. The weight of this su,, is the spite or copetition coefficient, that denotes the degree of copetition in the setting under exaination. It can vary fro self-interested agents, when =, to copletely alicious ones, which only care about iniizing the gain of their opponents, when = 1. Now, there is also a sall, but growing, literature of papers that consider cases either equivalent or very siilar to this one. For soe of the ost relevant work see [1, 2, 4, 5, 7, 9]. This paper significantly extends the work presented in these papers and, in particular, in [2]. To be ore precise, while these papers exaine the strategies and the equilibria for single unit auctions, i.e. second price in ost cases and/or first price auctions, we exaine the case of ulti-unit auctions. This is an iportant extension, because in theoretical analyses, as well as in real-world scenarios, it is usually the case that ore than one instance of a particular coodity is available. This paper is organized as follows. In section 2, we forally present the auction setting we exaine. Then we provide, for the first tie, syetric Bayes-Nash equilibria for both th and +1 th price sealed bid auctions in sections 3 and 4. We further prove that allowing the bidders to bid for ultiple ites, which they don t need, in order to deprive the fro the copetition, does not change the equilibriu strategies. 1 These equilibria, along with the corresponding proofs, are the priary contributions of this paper. Fro this analysis, we observe a nuber of differences with the single ite case that was exained in related work. For exaple, for large, the equilibriu strategy for the th price auction is to bid ore than the true valuation, and we also deonstrate that the equilibriu strategy for the +1 th price auction is independent of the nuber of ites sold. In section 5, we exaine how the auctioneer s revenue is affected. In particular, we show that revenue equivalence does not hold, and, in fact, the revenue of the +1 th price auction is only higher than that for the equivalent th price auction when is sall; which is not observed in single unit auctions. In section 6, we exaine the profit that various strategies yield when pitted against various other strategies, like itself or strategies that do not take the copetition into account as if =. Fro this, we deduce that the ost successful strategies in a copetition setting are those derived for a coefficient that is just right to beat the opposition. 2. PROBLEM STATEMENT In this section we forally describe the auction setting to be analyzed and define the objective function that the agents wish to axiize. We also give the notation that we use. In particular, we will copute and analyze the syetric 2 Bayes-Nash equilibria for sealed bid auctions where 1 identical ites are being sold. The two ost coon auction settings in this context are the th and +1 th price auctions, in which the top bidders win one ite 1 This eans that the coputed equilibria are false-naeproof, i.e. that no agent ay profit by subitting bids under an assued identity. 2 This eans that all agents use the sae bidding strategy. This is a coon assuption ade in gae theory, in order to restrict the space of strategies that we exaine. It is likely that in addition to the syetric equilibria we copute there are also asyetric ones. each at a price equal to the th and +1 th highest bid respectively. Specifically, we assue that N bidders where N participate in the auction and these agents have a private valuation utility u i for acquiring any one of the traded ites; these valuations are assued to be i.i.d. fro a distribution with cuulative distribution function cdf F u, which is the sae for all bidders. Furtherore, let and u h be the bounds for the possible values of rando variables u i as defined by F u: 3 = ax{u F u =} u h = in{u F u =1} Finally, we will consider bidding strategies gu that are strictly increasing and differentiable over [,u h], that is the area where F u is defined. These are the assuptions coonly ade when coputing equilibria in the auction theory literature [3]. In this work, we define the objective function i.e. the total utility that each agent tries to axiize in the sae way as in [2]: Definition 1. The objective function that each agent wishes to axiize is given by: U i =1 ũ i where [, 1] is a paraeter called the spite coefficient 4, ũ i is the gain of agent i i.e. ũ i =, if it does not win any ites, and ũ i = u i p i,ifitdoesandp i is the total payent the agent ust ake to the auctioneer. We would like to point out that fro the point of view of an agent copeting against other agents, the following definition of the objective function ight look ore natural: 5 U i = ũ i γ This is because the agent cares about axiizing its own profit inus the weighted profit of its opponents, where the weight γ. For exaple, if we have a copetition like TAC, then the agent wishes to axiize the difference of its profit to the expected profit of any opponent, and hence we would use this objective function with γ = 1.Another N 1 paraeter that is also interesting in a copetition setting, is when γ = 1, because then the agent axiizes his gain against the su of the gains of every opponent. However, this objective function is equivalent to the objective function j i j i ũ j 1 presented in definition 1, when γ =, so definition 1 is ore general, and this is the reason why we use it. Moreover, axiizing the difference of the agent s profit to that of any opponent, and to that of all opponents, respectively, can easily be represented using coefficient = 1,and = 1, N 2 in the objective fuction. 3 For exaple, if F u is the unifor distribution U[, 1], then = and u h =1. IfF u is such that u i can take values in [, +, then = and u h = +. We use these tight bounds to define the boundary conditions for the equilibria we copute. 4 We also call this the copetition coefficient, since it deterines how uch the agents take their copetitors into account in the total utility. 5 In fact this is the total utility function used in [7]. ũ j The Sixth Intl. Joint Conf. on Autonoous Agents and Multi-Agent Systes AAMAS 7 73

Now that we have defined the general for of the function that each agent tries to axiize, each agent i will do so by subitting a particular bid v i for the ite it needs. Here the optial bids for the agent are those given fro the equilibriu strategy u, assuing that other agents use the sae strategy. Thus the bid axiizing the objective function should be v i = u i. To copute the probability distributions of the order statistics of the opponents bids and valuations, we use the inforation provided by the following lea: Lea 1. The cdf of the k th order statistic, denoted U k, of the N 1 valuations utilities u i of the opponents of any given agent, is given by: Prob[U k x] =Φ kx while the cdf of the k th order statistic, denoted B k,ofthe N 1 bids of these opponents, when each of the bids v i = u i according to the equilibriu stategy u, is given by: Prob[B k x] =Φ kg 1 x where CN 1,i is the total nuber of possible cobinations of i ites chosen fro N 1, and k 1 Φ kx = CN 1,i F x N 1 i 1 F x i 1 i= Proof. The probability distribution fro which each utility u i is drawn has cdf F u; the probability distribution fro which the bids v i are drawn has cdf Gx =F g 1 x. We also know that the k th order statistic of a set of N 1 i.i.d. rando variables drawn fro a distribution with cdf Hx is k 1 i= CN 1,i HxN 1 i 1 Hx i see [8]. Using this equation for G = F and G = H, respectively, we get the two equations of this lea. We will use these forulae extensively in the coputations of the Bayes-Nash equilibria in the next sections. 3. M-TH PRICE AUCTIONS In this section we exaine the syetric Bayes-Nash equilibriu that exists in the case of an th price auction. Specifically, each agent bids v i for a unit of the ite on sale, and we initially assue that it is only allowed to bid for one ite and no ore. Theore 1. In the case of an th price sealed bid auction with N participating bidders, in which each agent i is interested in purchasing one unit of the good for sale with inherent utility valuation for that ite equal to u i, u i is i.i.d. drawn fro F u, andan-coefficient for outperforing its copetition, the following bidding strategy constitutes a syentric Bayes-Nash equilibriu: u =u F u N 1 if <1, if =1,and u =u +F u N 1 if >1. u F z N 1 dz 2 u =u 3 u F z N 1 dz 4 Proof. See the appendix, section A.1. It is interesting to notice that for sall values of the spite coefficient naely < 1 the equilibriu strategy is to bid less than the agent s true valuation, for = 1 to bid truthfully, and for higher values of > 1 tobidore than the true valuation. This is quite different fro the first price auction result exained in [2], because in that case, we do not observe the behavior of bidding ore than the true valuation. However, in an auction where ultiple ites are sold, the price paid by all winners is the sae and equal to the lowest winning bid. This eans that quite often the price paid is significantly lower than one s bid, especially as gets bigger, and this leads to agents bidding higher than they would in an identical auction where fewer ites are sold. In addition, using sall values of eans that the agents priarily care about axiizing their own profit, so they will bid a lot less aggressively, than in the case where they care ostly about iniizing the profit of their opponents large. These two facts when taken together explain why in ulti-unit auctions there are cases for large when agents should bid ore than their true valuation for an ite. Corollary 1. In the case that F u is the cdf of a unifor distribution U[, 1], the equilibriu strategy is: β u β > 1+β u β = u = β 1+β u + 5 u β β< β 1 1+β 1 lnu u β = 1 where β = N. 1 We use this equilibriu strategy derived for F u being the unifor distribution U[, 1], in our analysis of the auctioneer s revenue in section 5 and the siulations we present in section 6. Naturally when there are >1 ites for sale, it ight be beneficial to attept to purchase ultiple ites, which the agent does not use 6, in order to deny the fro its opponents. Now, even if the rules prohibit agents fro bidding for ultiple ites, they can always subit bids under a false identity. Moreover, there are cases when this akes sense, so we will assue that it will happen. As an exaple consider the following. Assue that = 1 and the opponents bid erroneously according to g u instead of g 1u. The opponents who win an ite ake a profit, since g u <u, so the payent is less than their valuations. By bidding for a second ite and winning it, the agent denies its copetitors a gain equal to U 1 p p is the payent, which is a positive value, and forces the to pay a higher new payent p >pfor the rest of the 2 ites that they win. However this case arises due to the fact that the opponents do not bid according to the correct equilibriu strategy. In the case that they do bid as per the equilibriu strategy, a bidder i does not gain by bidding for a second ite, because the profit fro its opponents losing an ite and increasing their payents is offset on expectation fro the fact that the opponents ake a payent for one less ite, while bidder i pays for one ore itself. We exaine this case in the next theore. 6 We are assuing free disposal, eaning that an agent is allowed to throw away ites that it does not wish to use without gain or loss of utility. 74 The Sixth Intl. Joint Conf. on Autonoous Agents and Multi-Agent Systes AAMAS 7

Theore 2. In the case of the th price sealed bid auction described by theore 1, when each bidder is allowed to place an unrestricted nuber of bids for ultiple ites, bidding for exactly one ite according to the function u given in theore 1, constitutes a syetric Bayes-Nash equilibriu. Proof. See the appendix, section A.2. The two theores presented in this section prove that the strategy u that we coputed is indeed a Bayes-Nash equilibriu in an th price auction, even when the agents are allowed to place ultiple bids, for ites that they don t need. 4. M+1-TH PRICE AUCTIONS In this section we exaine the syetric Bayes-Nash equilibriu that exists in the case of an +1 th price auction. Again we initally assue that each agent bids v i for a unit of the ite on sale, and that each agent is allowed to bid for only one ite and no ore. Theore 3. In the case of an +1 th price sealed bid auction with N participating bidders, each agent i interested in purchasing one unit of the good for sale with inherent utility valuation for that ite equal to u i, u i is i.i.d. drawn fro F u, andan-coefficient for outperforing its copetition, the following bidding strategy constitutes a syetric Bayes-Nash equilibriu: u =u +1 F u 1 Proof. See the appendix, section A.3. u 1 F z 1 dz 6 Fro this, it is interesting to notice that the agent always bids ore than its true valuation when > and also that the equilibriu strategy does not depend on the nuber of agents N or the nuber of ites for sale, but only on the copetition coefficient. While these observations are consistent with those for the second price auction result exained in [2] in that case the authors showed that the strategy did not depend on the nuber of bidders N, we further show that the nuber of ites sold, alsodoesnot atter for the equilibriu strategy. Corollary 2. In the case that F u is the cdf of a unifor distribution U[, 1], the equilibriu strategy is: u = u + 7 1+ We use this equilibriu strategy derived for F u being the unifor distribution U[, 1], in our analysis of the auctioneer s revenue in section 5 and the siulations we present in section 6. As before, since there are >1 ites for sale, one ight think that it would be beneficial to attept to purchase ultiple ites, which the agent does not use, in order to deny the fro its opponents. We exaine this case in the next theore. Theore 4. In the case of the +1 th price sealed bid auction described by theore 3, when each bidder is allowed to place an unrestricted nuber of bids for ultiple ites, bidding for exactly one ite according to the function u given in theore 3, constitutes a syetric Bayes- Nash equilibriu. Expected Revenue 3.5 3 2.5 2 1.5 1 6 8 1 12 14 16 18 2 Bidders Participating N ER =2 I ER.4 =2 II ER.4 =2 ER =3 I ER.4 II ER.4 =3 =3 ER =4 Figure 1: Exaination of the expected revenue ER I and ER II forn = 6,...,2 participating bidders. Graphed for =the revenue is the sae for both auctions and =.4. Expected Revenue 3 2.5 2 1.5 1 =4 =3 =2 I ER.4 II ER.4 =4 =4.2.4.6.8 1 coefficient Figure 2: Exaination, for various values of, ofthe expected revenue ER I for th price auctions solid lines and ER II for +1 th price auctions dotted lines. Fro botto to top the pairs of lines are for =2, =3and =4ites sold respectively. The nuber of agents participating is N =2. Proof. SKETCH Due to space liitations, we only present a brief sketch of the proof for this case. In a siilar way to the proof of theore 2, we can show that the total utility, when bidder i bids for two ites placing bids v i and ṽ i, is a su of two parts. The first depending on v i, which is axiized when v i = u i, eaning that the top bid should follow the equilibriu strategy derived in theore 3. The second depending on ṽ i, which is axiized when the bidder does not really bid any aount that will allow it to win, i.e. ṽ i + 1+. Of course, given that, in the +1 th price auction, the equilibriu strategy does not depend on the nuber of opponents, nor the ites sold, it only really akes sense to bid for worthless ites if the opponents ake a serious istake in the strategy that they use. For exaple, if =1,and all the opponents bid less than their true valuation, then, in that case, it does ake sense to bid for a second ite. The Sixth Intl. Joint Conf. on Autonoous Agents and Multi-Agent Systes AAMAS 7 75

.25.2.15.7.6.5 Expected Profit.1.5.5.1 3x 2x +1xg g 2x +1xg 1x +2xg g 1x +2xg 2x +1x 2x +1x 1x +2x 1x +2x.2.4.6.8 1 coefficient Expected Profit.4.3.2.1.1 6x 5x +1xg g 5x +1xg 1x +5xg g 1x +5xg 5x +1x 5x +1x 1x +5x 1x +5x.2.2.4.6.8 1 coefficient N=3 bidders participating N=6 bidders participating Figure 3: Exaination of the expected profit in an th price auction =2withN participating bidders. For various coefficients the perforance of agents using the u strategy is copared against agents using the g u and u strategies. 5. REVENUE ANALYSIS In this section we exaine the auctioneer s revenue for both the th and +1 th price auctions. In particular, the expected revenue ER I in an th price auction with N bidders, whose valuations are drawn fro F u, when they all bid according to function g I u is: ER I = gω I Ψ ω dω 8 while the expected revenue ER II for the equivalent +1 th price auction, when the bidders bid according to function g II u is: ER II = g II ω Ψ +1ω dω 9 where Ψ kx = k 1 i= CN,i F x N i 1 F x i. Using these equations we exaine the expected revenue, when the nuber N of participating bidders changes figure 1 7, and when the copetition coefficient a is varied figure 2. For = both expected revenues are equal, as we expected given that the Revenue Equivalence Theore applies in that case. In [2], it is shown that the presence of the coefficient, when and 1, leads to the second price auction yielding ore profit for the auctioneer than the equivalent first price auction, ER I ER II, and also that as increases, so does the expected revenue. Fro our experients, we concur that as increases, so does the expected revenue. However, it is no longer the case that ER I ER II. Based on our observations, we know that for close to, it is ER I ER II, and for close to 1, it is ER I ER II. Given this, we believe that the following holds: 7 It should be noted that the different cases in figure 1 and this is also the case for the different experients in figures 3 and 4 are denoted by different sybols,,+,* in order to ake it easy to distinguish the. The expected gain in figures 1 and 2 is coputed accurately using nuerical integration. Clai 1. For the sae coefficient, when the bidders use the syetric Bayes-Nash equilibria strategies, the th and +1 th price auctions yield the sae expected revenue, when =or = 1, while, for << 1,the +1th price auction yields ore revenue, and for > 1,theth price auction yields ore revenue. In fact, we expect the proof for the case when [, 1 ]to be very siilar to the proof of the theore for the single unit case when the expected revenue of the first price auction is no ore than that of the second price auction, presented in [2]. To prove the clai when > 1, will require using the fact that the two revenues are equal for = 1 and the equilibriu strategies that we coputed, and is left for future work. 6. SIMULATIONS In this section we present the results of siulations that were conducted to epirically verify the validity of the coputed equilibria and to analyze the profits derived by various agents when pitted against other agents. In figure 3 we present the expected gain 8 for a siulation of an th price auction with = 2 ites being sold, for N participating bidders N =3, 6 for varying values of the coefficient. In each case we perfored five siulations: i strategy u against itself, ii strategy u against strategy g u which is a self-interested agent, and iii strategy u against strategy u which is an agent axiizing the relative difference between its own profit to that of any copetitor. For the latter two cases, we conducted two siulations, one in which N 1 of the agents used strategy u and one in which only one agent used strategy u. We also did the sae siulations for the case of an 8 In figures 3 and 4, the expected gain is coputed by discretizing the space of possible valuations for all agents soe billions of cases to exaine and then coputing a near accurate estiated error is well under.1% estiate of the expected gain as the average gain fro all these cases. 76 The Sixth Intl. Joint Conf. on Autonoous Agents and Multi-Agent Systes AAMAS 7

Expected Profit.15.1.5 4x 3x +1xg g 3x +1xg 1x +3xg g 1x +3xg 3x +1x 3x +1x 1x +3x 1x +3x.2.4.6.8 1 coefficient Figure 4: Exaination of the expected profit in an +1 th price auction =2withN =4participating bidders. For various coefficients the perforance of agents using the u strategy is copared against agents using the g u and u strategies. +1 th price auction, with N = 4 participating bidders and the results are presented in figure 4. There are a nuber of observations that can be ade fro these siulations. When ore aggressive agents 9 participate in the auction, the profits for all agents are decreased. Furtherore, we verify that each strategy u does constitute a best response to itself, when the objective function is equal to the weighted difference of the profits with weight equal to. Even for = 1, when the profit is lowest, the agent does what is best, because the opponents profit is also lowest. However, we further observe that in cases where N 1 of these aggressive agents using strategy u participate, their profit is alost the sae as when all of the use u, while the opponent which is using a less aggressive strategy u <, obtains a uch higher profit. Our observations are different in the case that there is only one aggressive agent. In this case, the aggressive agents beat the agents using g u quite often, although this depends on the relative nuber of bidders N to the nuber of ites sold, with higher N or lower reducing this benefit. Moreover, they even anage to achieve a score which is reasonably close to that of the agents using strategy u. These latter agents using strategy u always obtain a higher profit than their opponents, which is not entirely surprising given that they try to axiize the profit difference between theselves and any opponent. They also tend to get relatively good profit, no atter what agents they copete against, which indicates that their perforance is relatively robust under ost if not all possible opposition. This observation and the fact that, in direct copetition, which is the ost coon case in real world scenarios, they 9 These are agents using strategy u, where the coefficient takes high values i.e. close to 1, and thus they place bids at higher prices. always outperfor their opponents, suggest that we should consider using this strategy unless there is a strong reason to believe that the coefficient has soe different value. 7. CONCLUSIONS We exained the behavior of bidding agents that wish to axiize a weighted difference of their own gain to that of their copetitors. We provided, for the first tie, the syetric Bayes-Nash equilibria that exist for the standard ulti-unit auctions, naely both the th and +1 th price sealed bid auctions, thus extending the existing state-of-theart results, which were analyzing only single unit auctions. We also observed a nuber of differences between the single and ulti-unit cases; the ost notable of which is that for > 1, the equilibriu strategy for the th price auction becoes to bid ore than the true valuation. We further proved that allowing the bidders to bid for ultiple ites, that they don t need, in order to deprive the fro the copetition, does not change the equilibriu strategies. Then we exained how the auctioneer s revenue is affected. We showed that revenue equivalence does not hold, and in fact the expected revenue of the +1 th price auction is higher than that of the equivalent th price auction for certain values of, while for others the opposite is true. Finally, we conducted siulations to epirically verify the validity of our coputed strategies. Indeed we observed that the agent using a = 1 i.e. trying to axiize the profit difference N between itself and any opponent outperfors its copetitors, when conducting a coparison of the relative profits. There are a nuber of unresolved issues in this paper. In particular we plan to exaine the revenue analysis and prove the clai that we ade in section 5. We are also currently working towards generating equilibria for ultideand auctions when = and then will plan to extend these results to any value of. 8. ACKNOWLEDGMENTS We would like to thank Alex Rogers and Radjeep Dash at the University of Southapton for coents on this work. This research was undertaken as part of the ALADDIN Autonoous Learning Agents for Decentralised Data and Inforation Systes project and is jointly funded by a BAE Systes and EPSRC Engineering and Physical Research Council strategic partnership EP/C54851/1. 9. 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APPENDIX A. PROOFS OF MAIN THEOREMS We use the following lea to siplify the differential equations in the proofs: Lea 2. For all N,, suchthatn the following equations hold: Φ u i Φu i Φ =N F u i 1u i F u i Φ u i Φ+1u i Φ = F u i u i 1 F u i Proof. Coputing the derivative of Φ u i using equation 1 leaves only one ter at the end and then it s easy to prove the first equation. Then to prove the second one, observe that Φu i Φ 1 u i Φ +1 u i Φ u i = N F u i 1 F u i. A.1 Proof of Theore 1 We assue that all bidders except i follow the strategy u. Bidder i bids v i, the bid that axiizes his objective function on expectation. Let C be the su on expectation of the top 1 highest opponent valuations. Since in all cases that we will exaine, whether bidder i wins or not, we know that the opponents with the top 1 valuations will win an ite, we know that they will gain this aount C fro doing so. Now: 1 1 C = E U j = EU j = 1 ω Φ jω dω is a constant and does not depend on the bid v i. We need to consider the following three cases: i When B >v i, bidder i does not win any ite and the closing price is B. Therefore bidder i s gain is and the opponents ake a gain fro gaining an extra ite the th, in addition to the 1 ites that they always win this was counted in the coputation of constant C, but they also ust ake total payents of B. The total additional 1 expected utility for bidder i in this case is hence: ΔU 1 = v i ω g 1 ω d dω Φg 1 ω dω ii When B 1 >v i B,bidderiwins an ite and the closing price is v i. Therefore bidder i s gain is u i v i and the opponents pay 1 v i for the ites that they win. The total additional expected utility for bidder i is: ΔU 2 = 1 u i v i+ 1 v i Φg 1 v i Φ 1g 1 v i iii When v i B 1, bidder i wins an ite and the closing price is B 1. Therefore bidder i s gain is u i B 1 and the opponents ust pay 1 B 1 for the ites that they purchase. The total additional expected utility for bidder i in this case is: vi ΔU 3 = 1 ui ω+ 1 ω d Φ 1g 1 ω dω dω 1 We ean additional to the fact that the agent always loses utility C, since its opponents always gain a value C fro the top 1 ites. The total expected utility for bidder i when considering all possibilities is therefore EU i = C +ΔU 1 +ΔU 2 +ΔU 3. This iplies: EU i = ω g 1 ω Φ g 1 ω g 1 ω dω v i + 1 u i+ 1 v i Φg 1 v i Φ 1g 1 v i + vi 1 ui + 1 ω Φ 1g 1 ω g 1 dω ω 1 ω Φ jω dω 1 To find the value of v i that axiizes equation 1, we set deu i dv i =. If strategy u gives the equilibriu strategy, then it ust be the case that the value of v i that axiizes the total utility is given by u i.e. that v i = u i. By substituting this into the equation derived fro setting deu i dv i =, we finally get the differential equation: ui g u i Φ u i u =1 Φ u i Φ 1u i i 11 To siplify this equation we use lea 2 to get: u i= u i g N u i 1 F u i F u i 12 The solution of this equation is: u u =u F u N 1 F z N 1 dz 13 where c depends on the boundary condition. To select the appropriate boundary condition one should note that ters u i u i and 1 havethesae sign, since all other ters in equation 12 are positive. Therefore: i if <1, then u i u i > and the boundary condition is =, and the resulting strategy is given by equation 2, ii if =1,thenu i u i= u i=u i equation 3, and iii if >1, then u i u i < and the boundary condition is u h=u h, and the resulting strategy is given by equation 4. A.2 Proof of Theore 2 We assue that all bidders except i follow the strategy u. In this case, bidder i, is allowed to bid for any nuber of ites. In order to show a contradiction we assue that he bids for 2 ites, and the bids are equal to v i and ṽ i. Without loss of generality, we assue that v i ṽ i. By considering several possible cases that can occur depending on the relative values of v i and ṽ i to the bids of the opponents, we conclude that the total expected utility in this case is: ẼU i = EU i +ΔEU i where EU i depends only on v i not ṽ i and is given by equation 1 and ΔEU i depends only on ṽ i not v i. Therefore we need to axiize both ters in order to axiize the total utility. The value that axiizes the first ter is v i = u i. The second ter is equal to: c 78 The Sixth Intl. Joint Conf. on Autonoous Agents and Multi-Agent Systes AAMAS 7

ΔEU i = ṽi 1 ui+ 1 ω Φ 1g 1 ω g 1 dω ω + 1 u i+ 2 ṽ i Φ 1g 1 ṽ i Φ 2g 1 ṽ i + ṽi 1 ui + 2 ω Φ 2g 1 ω g 1 dω ω + ṽi g 1 ω Φ 1g 1 ω g 1 dω ω To siplify this equation, let us set τ = g 1 ṽ i. Then τ u i. Hence ΔEU i can be expressed as a function of τ and : τ ΔEU iτ, = ω gω Φ 1ω dω 14 τ + 2 ω Φ 1ω Φ 2ω dω If 2 it is easy to see that both integrals are negative, when τ>, so the axiizing value is τ =, which eans that one should not bid for the second ite. 11 If >2, then using lea 2 and equation 12, equation 14 can be rewritten as: τ ΔEU iτ, = N +1 ω ω 15 +N 2 F ω gω ω Φ 1 Φ 2 ω dω 1 F ω So even in this case > 2,τ : τ ΔEU iτ, N +1 ω ω +N 2 F ω gω ω Φ 1 Φ 2 ω dω 1 F ω ΔEU iτ, This shows that it is not beneficial to bid for a second ite, 12 when the opponents use the equilibriu strategy u of theore 1. Thus the best response to strategy u isbidding according to u forexactlyoneite. A.3 Proof of Theore 3 We assue that all bidders except i follow the strategy u. Bidder i bids v i, the bid that axiizes his objective function on expectation. Let C be the su on expectation of the top highest opponent valuations. In ost cases that we will exaine when bidder i does not win, the opponents with the top valuations will win an ite, and they will gain this aount C fro doing so. In the case that bidder i outbids the copetition case iii below, then we will subtract the th valuation fro the opponents gain to copensate. Now: 11 To be ore precise, τ = eans to bid as if it had the iniu valuation for the ite, which in practice is equivalent to not bidding. 12 And since bidding for a second is not beneficial, bidding for a second and third and/or ore, is not beneficial either. C = E U j = EU j = ω Φ jω dω is a constant and does not depend on the bid v i. We need to consider the following three cases: i When B +1 >v i,bidderidoes not win any ite and the closing price is B +1. Therefore bidder i s gain is and the opponents ust ake total payents of B +1. The total additional expected utility for bidder i in this case is hence: d ΔU 1 = ω Φ+1g 1 ω dω v i dω ii When B >v i B +1, bidder i does not win and the closing price is v i. Therefore bidder i s gain is and the opponents ust pay v i for the ites that they purchase. The total additional expected utility for bidder i in this case is: ΔU 2 = v i Φ +1g 1 v i Φ g 1 v i iii When v i B,bidderiwins an ite and the closing price is B. Therefore bidder i s gain is u i B and the opponents ust pay 1 B for the ites that they purchase. We also need to subtract the th highest valuation U = g 1 B frothetotalgainofthe opponents, since they only won 1 ites. The total additional vi expected utility for bidder i in this case is: ΔU 3 = 1 ui ω+ 1 ω + g 1 ω d Φg 1 ω dω dω The total expected utility for bidder i when considering all possibilities is therefore EU i = C +ΔU 1 +ΔU 2 +ΔU 3. This iplies: EU i = ω Φ +1 1 ω v i g 1 dω ω + v i Φ +1g 1 v i Φ g 1 v i + vi 1 ui+ 1 ω+ g 1 ω Φ g 1 ω g 1 ω dω ω Φ jω dω 16 To find the value of v i that axiizes equation 16, we set deu i dv i =. If strategy u gives the equilibriu strategy, then it ust be the case that the value v i that axiizes the total utility is given by u i.e. that v i = u i. By substituting this into the equation derived fro setting deu i dv i =, we finally get the differential equation: ui g u i Φ u i u = Φ +1u i Φ u i 17 i To siplify this equation we use lea 2 to get: g u i= 1 u i u i F u i 1 F u i 18 The solution of this equation is: u u =u 1 F u 1 1 F z 1 dz 19 where c depends on the boundary condition. To select the appropriate boundary condition one should note that ter u i u i is negative since is negative and all the other ters in equation 12 are positive. Therefore the boundary condition is u h=u h,andthe resulting strategy is given by equation 6. c The Sixth Intl. Joint Conf. on Autonoous Agents and Multi-Agent Systes AAMAS 7 79