Optimal Bidding Strategies for Simultaneous Vickrey Auctions with Perfect Substitutes

Transcription

1 Optimal Bidding Strategies for Simultaneous Vickrey Auctions with Perfect Sustitutes Enrico H. Gerding, Rajdeep K. Dash, David C. K. Yuen and Nicholas R. Jennings University of Southampton, Southampton, SO7 BJ, UK. ABSTRACT In this paper, we derive optimal idding strategies for a gloal idder who participates in multiple, simultaneous second-price auctions with perfect sustitutes. We first consider a model where all other idders are local and participate in a single auction. For this case, we prove that, assuming free disposal, the gloal idder should always place non-zero ids in all availale auctions, irrespective of the local idders valuation distriution. Furthermore, for nondecreasing valuation distriutions, we prove that the prolem of finding the optimal ids reduces to two dimensions. These results hold oth in the case where the numer of local idders is known and when this numer is determined y a Poisson distriution. In addition, y comining analytical and simulation results, we demonstrate that similar results hold in the case of several gloal idders, provided that the market consists of oth gloal and local idders. Finally, we address the efficiency of the overall market, and show that information aout the numer of local idders is an important determinant for the way in which a gloal idder affects efficiency. Keywords Simultaneous Auctions, Perfect Sustitutes, Bidding Strategies, Vickrey Auction, Multiple Sellers, Market Efficiency. INTRODUCTION In recent years, there has een a surge in the application of auctions, oth online and within multi-agent systems [3, 8, 3, 4, 5]. As a result, there are an increasing numer of auctions offering very similar or even identical goods and services. In ebay alone, for example, there are often hundreds or sometimes even thousands of concurrent auctions running worldwide selling such sustitutale items. Against this ackground, it is important to develop idding strategies that agents can use to operate effectively across a wide numer of auctions. To this end, in this paper we devise and analyse optimal idding strategies for a idder that participates in multiple, simultaneous second-price auctions for goods that are perfect sustitutes. To date, much of the existing literature on simultaneous auctions focuses either on complementarities, where the value of items together is greater than the sum of the individual items, or on heuristic strategies for simultaneous auctions (see Section 6 for more details). In contrast, here To illustrate, at the time of writing, over one thousand ebay auctions were selling the ipod mini 4GB. we consider idding strategies analytically and for the case of perfect sustitutes. In particular, our focus is on simultaneous Vickrey or second-price sealed id auctions. We choose these ecause they are communication efficient and well known for their capacity to induce truthful idding [], which makes them suitale for many multi-agent system settings. Within this setting, we are ale to characterise, for the first time, a idder s utility-maximising strategy for idding in any numer of such auctions and for any type of idder valuation distriution. In more detail, we first consider a market where a single idder, called the gloal idder, can id in any numer of auctions, whereas the other idders, called the local idders, are assumed to id only in a single auction. For this case, we find the following results: Whereas in the case of a single second-price auction a idder s est strategy is to id its true value, this is generally not the case for a gloal idder. As we shall show, its est strategy is in fact to id elow the true value. We are ale to prove that, even if a gloal idder requires only one item and assuming free disposal, the expected utility is maximised y participating in all the auctions that are selling the desired item. Finding the optimal id for each auction can e an arduous task when considering all possile cominations. However, for most common idder valuation distriutions, we are ale to significantly reduce this search space. Empirically, we find that a idder s expected utility is maximised y idding relatively high in one of the auctions, and equal or lower in all other auctions. We then go on to consider markets with more than one gloal idder. Due to the complexity of the prolem, we comine analytical results with a discrete simulation in order to numerically derive the optimal idding strategy. By so doing, we find that, in a market with only gloal idders, the dynamics of the est response do not converge to a pure strategy. In fact it fluctuates etween two states. If the market consists of oth local and gloal idders, however, the gloal idders strategy quickly reaches a stale solution and we approximate a symmetric Nash equilirium outcome. Finally, we consider the issue of market efficiency when there are such simultaneous auctions. Efficiency is an important system-wide consideration within multi-agent systems since it characterises how well the allocations in the

2 system maximise the overall utility [5]. Now, efficiency is maximised when the goods are allocated to those who value them the most. However, a certain amount of inefficiency is inherent to a distriuted market where the auctions are held separately. In this paper, we measure the inefficiency of markets with local idders only and consider the impact of gloal idders on this inefficiency. In so doing, we find that the presence of a gloal idder has a slight, ut positive, impact on the efficiency when the numer of local idders is known, ut is, in general, negative when there exists uncertainty aout the exact numer of idders. Therefore, information aout the market, such as the numer of idders, plays an important role in the social welfare of the system. The remainder of the paper is structured as follows. In Section 2 we descrie the idders and the auctions in more detail. In Section 3 we investigate the case with a single gloal idder and characterise the optimal idding ehaviour for it. Section 4 considers the case with multiple gloal idders and in Section 5 we address the market efficiency and the impact of a gloal idder. Finally, Section 6 discusses related work and Section 7 concludes. 2. BIDDING IN MULTIPLE VICKREY AUC- TIONS The model consists of M sellers, each of whom acts as an auctioneer. Each seller auctions one item; these items are complete sustitutes (i.e., they are equal in terms of value and a idder otains no additional enefit from winning more than one item). The M auctions are executed simultaneously; that is, they end simultaneously and no information aout the outcome of any of the auctions ecomes availale until the ids are placed 2. We also assume that all the auctions are symmetric, i.e., a idder does not prefer one auction over another. Finally, we assume free disposal and risk neutral idders. 2. The Auctions The seller s auction is implemented as a second-price sealed id auction, where the highest idder wins ut pays the second-highest price. This format has several advantages for an agent-ased setting. Firstly, it is communication efficient. Secondly, for the single-auction case (i.e., where a idder places a id in at most one auction), the optimal strategy is to id the true value and thus requires no computation (once the valuation of the item is known). This strategy is also weakly dominant (i.e., it is independent of the other idders decisions), and therefore it requires no information aout the preferences of other agents (such as the distriution of their valuations). 2.2 Gloal and Local Bidders We distinguish etween gloal idders and local idders. The former can id in any numer of auctions, whereas the latter only id in a single auction. Local idders are assumed to id according to the weakly dominant strategy and id their true valuation 3. We consider two ways of modelling 2 Although this paper focuses on sealed-id auctions, where this is the case, the conditions are similar for last-minute idding in iterative auctions such as ebay [5]. 3 Note that, since idding the true value is optimal for local idders irrespective of what others are idding, their strategy is not affected y the presence of gloal idders. local idders: static and dynamic. In the first model, the numer of local idders is assumed to e known and equal to N for each auction. In the latter model, on the other hand, the average numer of idders is equal to N, ut the exact numer is unknown and may vary for each auction. This uncertainty is modelled using a Poisson distriution (more details are provided in Section 3.). As we will later show, a gloal idder that ids optimally has a higher expected utility compared to a local idder, even though the items are complete sustitutes and a idder only requires one of them. Nevertheless, we can identify a numer of compelling reasons why not all idders would choose to id gloally: Participation Costs. Although the idding itself may e automated y an autonomous agent, it still takes time and/or money, such as entry fees and time to setup an account, to participate in a new auction. Occasional users may not e willing to make such an investment, and they may restrict themselves to sellers or auctions that they are familiar with. Information. Bidders may simply not e aware of other auctions selling the same type of item. Even if this is known, however, a idder may not have sufficient information aout the distriution of the valuations of other idders and the numer of participating idders. Whereas this information is not required when idding in a single auction (ecause of the dominance property in a second-price auction), it is important when idding in multiple simultaneous auctions. Such information can e otained y an expert user or e learned over time, ut is often not availale to a novice. Risk Attitude. Although a gloal idder otains a higher utility on average, such a idder runs a risk of incurring a loss (i.e., a negative utility) when winning multiple auctions. A risk averse idder may not e willing to take that chance, and so may choose to participate only in a single auction to avoid such a potential loss. Budget Constraints. Related to the previous point, a udget constrained idder may not have sufficient funds to make a loss in case it wins more than one auction. In more detail, for a fixed udget, the sum of ids should not exceed, therey limiting the numer of auctions a idder can participate in and/or lowering the actual ids that are placed in those auctions. Bounded Rationality. As will ecome clear from this paper, an optimal strategy for a gloal idder is harder to compute than a local one. A idder will therefore only id gloally if the costs of computing the optimal strategy outweigh the enefits of the additional utility. From the aove, we elieve it is reasonale to expect a comination of gloal and local idders, and for only a few of them to e gloal idders. In this paper, we analyse the case of a single gloal idder theoretically, and then use a computational approach to address the case with at least two such idders.

3 3. A SINGLE GLOBAL BIDDER In this section, we provide a theoretical analysis of the optimal idding strategy for a gloal idder, given that all other idders are local and simply id their true valuation. After we descrie the gloal idder s expected utility in Section 3., we show in Section 3.2 that it is always optimal for a gloal idder to participate in the maximum numer of auctions availale. Susequently, in Section 3.3 we discuss how to significantly reduce the complexity of finding the optimal ids for the multi-auction prolem, and we then apply these methods to find optimal strategies for specific examples. 3. The Gloal Bidder s Expected Utility We use the following notation. The numer of sellers (or auctions) is M 2 and the numer of local idders is N. A idder s valuation v [, v max] is randomly drawn from a cumulative distriution F with proaility density f, where f is continuous, strictly positive and has support [, v max]. F is assumed to e equal and common knowledge for all idders. A gloal id B is a set containing a id i [, v max] for each auction i M (the ids may e different for different auctions). For ease of exposition, we introduce the cumulative distriution function for the first-order statistics G() = F() N [, ], denoting the proaility of winning a specific auction conditional on placing id in this auction, and its proaility density g() = dg()/d = NF() N f(). Now, the expected utility U for a gloal idder with gloal id B and valuation v is given y: U(B, v) = v i B ( G( i)) i B i yg(y)dy () Here, the left part of the equation is the valuation multiplied y the proaility that the gloal idder wins at least one of the M auctions and thus corresponds to the expected enefit. In more detail, note that G( i) is the proaility of not winning auction i when idding i, i B ( G(i)) is the proaility of not winning any auction, and thus i B ( G(i)) is the proaility of winning at least one auction. The right part of equation corresponds to the total expected costs or payments. To see the latter, note that the expected payment of a single second-price auction when idding equals yg(y)dy (see []) and is independent of the expected payments for other auctions. Clearly, equation applies to the model with static local idders, i.e., where the numer of idders is known and equal for each auction (see Section 2.2). However, we can use the same equation to model dynamic local idders in the following way: Lemma. By replacing the first-order statistic G(y) with Ĝ(y) = e N(F(y) ), (2) and the corresponding density function g(y) with ĝ(y) = dĝ(y)/dy = N f(y)en(f(y) ), equation ecomes the expected utility where the numer of local idders in each auction is descried y a Poisson distriution with average N, i.e., where the proaility that n local idders participate is given y P(n) = N n e N /n!. Proof. To prove this, we first show that G( ) and F( ) can e modified such that the numer of idders per auction is given y a inomial distriution (where a idder s decision to participate is given y a Bernoulli trial) as follows: G (y) = F (y) N = ( p + p F(y)) N, (3) where p is the proaility that a idder participates in the auction, and N is the total numer of idders. To see this, note that not participating is equivalent to idding zero. As a result, F () = p since there is a p proaility that a idder ids zero at a specific auction, and F (y) = F () + p F(y) since there is a proaility p that a idder ids according to the original distriution F(y). Now, the average numer of participating idders is given y N = p N. By replacing p with N/N, equation 3 ecomes G (y) = ( N/N +(N/N)F(y)) N. Note that a Poisson distriution is given y the limit of a inomial distriution. By keeping N constant and taking the limit N, we then otain G (y) = e N(F(y) ) = Ĝ(y). This concludes our proof. The results that follow apply to oth the static and dynamic model unless stated otherwise. 3.2 Participation in Multiple Auctions We now show that, for any valuation < v < v max, a utilitymaximising gloal idder should always place non-zero ids in all availale auctions. To prove this, we show that the expected utility increases when placing an aritrarily small id compared to not participating in an auction. More formally, Theorem. Consider a gloal idder with valuation < v < v max and gloal id B, where i v for all i B. Suppose j / B for j {, 2,..., M}, then there exists a j > such that U(B { j}, v) > U(B, v). Proof. Using equation, the marginal expected utility for participating in an additional auction can e written as: U(B { j}, v) U(B, v) = vg( j) ( G( i B i)) j yg(y)dy j Now, using integration y parts, we have j yg(y) = jg(j) G(y)dy and the aove equation can e rewritten as: U(B { j}, v) U(B, v) = G( j) v i B ( G( i)) j + j G(y)dy (4) Let j = ɛ, where ɛ is an aritrarily small strictly positive value. Clearly, G( j) and j G(y)dy are then oth strictly positive (since f(y) > ). Moreover, given that i v < v max for i B and that v >, it follows that v i B ( G( i)) >. Now, suppose j = 2 v i B ( G(i)), then j U(B { j}, v) U(B, v) = G( j) 2 v i B ( G(i)) + G(y)dy > and thus U(B {j}, v) > U(B, v). This completes our proof. 3.3 The Optimal Gloal Bid A general solution to the optimal gloal id requires the maximisation of equation in M dimensions, an arduous task, even when applying numerical methods. In this section, however, we show how to reduce the entire id space to two dimensions in most cases (one continuous, and one discrete), therey significantly simplifying the prolem at

4 hand. First, however, in order to find the optimal solutions to equation, we set the partial derivatives to zero: U ( G( j)) i = (5) j B\{ i } i = g( i) v j B\{ i } Now, equality 5 holds either when g( i) = or when ( G(j))v i =. In the dynamic model, g( i) is always greater than zero, and can therefore e ignored (since g() = Nf()e N and we assume f(y) > ). In case of the static model, g( i) = only when i =. However, theorem shows that the optimal id is non-zero for < v < v max. Therefore, we can ignore the first part, and the second part yields: i = v ( G( j)) (6) j B\{ i } In other words, the optimal id in auction i is equal to the idder s valuation multiplied y the proaility of not winning any of the other auctions. It is straightforward to show that the second partial derivative is negative, confirming that the solution is indeed a maximum when keeping all other ids constant. Thus, equation 6 provides a means to derive the optimal id for auction i, given the ids in all other auctions Reducing the Search Space In what follows, we show that, for non-decreasing proaility density functions, such as the uniform and logarithmic distriutions, the optimal gloal id consists of at most two different values for any M 2. That is, the search space for finding the optimal id can then e reduced to two continuous values. Let these values e high and low, where high low. More formally: Theorem 2. Suppose the proaility density function f is non-decreasing within the range [, v max], then the following proposition holds: given v >, for any i B, either i = high, i = low, or i = high = low. Proof. Using equation 6, we can produce M equations, one for each auction, with M unknowns. Now, y comining these equations, we otain the following relationship: ( G( )) = 2( G( 2)) =... = m( G( m)). By defining H() = ( G()) we can rewrite the equation to: H( ) = H( 2) =... = H( m) = v ( G( j)) (7) j B In order to prove that there exist at most two different ids, it is sufficient to show that = H (y) has at most two solutions that satisfy v max for any y. To see this, suppose H (y) has two solutions ut there exists a third id j low high. From equation 7 it then follows that there exists a y such that H( j) = H( low ) = H( high ) = y. Therefore, H (y) must have at least three solutions, which is a contradiction. Now, note that, in order to prove that H (y) has at most two solutions, it is sufficient to show that H() is strictly concave 4 for v max. The function H is strictly 4 More precisely, H() can e either strictly convex or strictly concave. However, it is easy to see that H is not convex since H() = H(v max) =, and H() for < < v max. concave if and only if the following holds: 2g() d 2 H d = d ( g() G()) = dg 2 d d + < By performing standard calculations, we otain the following condition for the static model: (N ) f()n F() + N f > 2 for v f() max, (8) and similarly for the dynamic model we have: N f() + f > 2 for v f() max, (9) where f () = df/d. Since oth f and F are positive, conditions 8 and 9 clearly hold for f (). In other words, conditions 8 and 9 show that H() is strictly concave when the proaility density function is non-decreasing for v max, completing our proof. Note from conditions 8 and 9 that the requirement of nondecreasing density functions is sufficient, ut far from necessary. Although we are as yet not ale to make a more precise formal characterisation, in practice even most density functions with decreasing parts satisfy these conditions. Moreover, the requirement for H() to e strictly concave is also stronger than necessary in order to guarantee only two solutions. As a result, for practical purposes, we expect the reduction of the search space to apply in most cases. Given there are at most 2 possile ids, low and high, we can further reduce the search space y expressing one id in terms of the other. Suppose the uyer places a id of low in M low auctions and high for the remaining M high = M M low auctions, equation 6 then ecomes: low = v( G( low )) M low ( G( high )) M high, and can e rearranged to give: high = G low v( G( low )) low M M high () Here, the inverse function G ( ) can usually e otained quite easily. Furthermore, note that, if M low = or M high =, equation 6 can e used directly to find the desired value. Using the aove, we are ale to reduce the id search space to a single continuous dimension, given M low or M high. However, we do not know the numer of auctions in which to id low and high, and thus we need to search M different cominations to find the optimal gloal id. Moreover, for each comination, the optimal low and high can vary. Therefore, in order to find the optimal id for a idder with valuation v, it is sufficient to search along one continuous variale low [, v], and a discrete variale M low = M M high {, 2,..., M} Empirical Evaluation In this section, we present results from an empirical study and characterise the optimal gloal id for specific cases. Furthermore, we measure the actual utility improvement that can e otained when using the gloal strategy. The results presented here are ased on a uniform distriution of the valuations with v max =, and the static local idder model, ut they generalise to the dynamic model and other distriutions (not shown due to space limitations). Figure

5 id fraction (x) valuation (v) expected utility (U).5 M= 2 M= 4 M= 6. local.5.5 valuation (v) Figure : The optimal id fractions x = /v and corresponding expected utility for a single gloal idder with N = 5 static local idders and varying numer of auctions (M). In addition, for comparison, the solid line in the right figure depicts the expected utility when idding locally in a randomly selected auction, given there are no gloal idders (note that, in case of local idders only, the expected utility is not affected y M). illustrates the optimal gloal ids and the corresponding expected utility for various M and N = 5, ut again the id curves for different values of M and N follow a very similar pattern. Here, the id is normalised y the valuation v to give the id fraction x = /v. Note that, when x =, a idder ids its true value. As shown in Figure, for idders with a relatively low valuation, the optimal strategy is to sumit M equal ids at, or very close to, the true value. The optimal id fraction then gradually decreases for higher valuations. Interestingly, in most cases, placing equal ids is no longer the optimal strategy after the valuation reaches a certain point. At this point, a so-called pitchfork ifurcation is oserved and the optimal ids split into two values: a single high id and M low ones. This transition is smooth for M = 2, ut exhiits an arupt jump for M 3. In all experiments, however, we consistently oserve that the optimal strategy is always to place a high id in one auction, and an equal or lower id in all others. In case of a ifurcation and when the valuation approaches v max, the optimal high id ecomes very close to the true value and the low ids go to almost zero 5. As illustrated in Figure, the utility of a gloal idder ecomes progressively higher with more auctions. In asolute terms, the improvement is especially high for idders that have an aove average valuation, ut not too close to v max. The idders in this range thus enefit most from idding gloally. This is ecause idders with very low valuations have a very small chance of winning any auction, whereas idders with a very high valuation have a high proaility of winning a single auction and enefit less from participating in more auctions. In contrast, if we consider the utility relative to idding in a single auction, this is much higher for idders with relatively low valuations (this effect cannot e seen clearly in Figure due to the scale). In particular, we notice that a gloal idder with a low valuation can improve its utility y up to M times the expected utility 5 Note in Figure that the low ids are significantly higher than zero at this point. This is ecause as v approaches v max, the low ids have very little impact on the utility and finding the optimum numerically at this point requires an extremely high precision. of idding locally. Intuitively, this is ecause the chance of winning one of the auctions increases y up to a factor M, whereas the increase in the expected cost is negligile. For high valuation uyers, however, the enefit is not that ovious ecause the chances of winning are relatively high even in case of a single auction. 4. MULTIPLE GLOBAL BIDDERS As argued in section 2.2, we expect a real-world market to exhiit a mix of gloal and local idders. Whereas so far we assumed a single gloal idder, in this section we consider a setting where multiple gloal idders interact with one another and with local idders as well. The analysis of this prolem is complex, however, as the optimal idding strategy of a gloal idder depends on the strategy of other gloal idders. A typical analytical approach is to find the symmetric Nash equilirium solution [6, 8, 4, 6], which occurs when all gloal idders use the same strategy to produce their ids, and no (gloal) idder has any incentive to unilaterally deviate from the chosen strategy. Due to the complexity of the prolem, however, here we comine a computational simulation approach with the analytical results from section 3. The simulation works y iteratively finding the est response to the optimal idding strategies in the previous iteration. If this should result in a stale outcome (i.e., when the current and previous optimal idding strategies are the same), the solution is y definition a (symmetric) Nash equilirium. In more detail, the simulation is ased on the oservation that the valuation distriution F of the local idders corresponds to the distriution of ids (since local idders id their true valuation). Therefore, y maximising equation we find the est response given the current distriution of ids. Now, we first discretize the space of possile valuations and ids. Then, y performing this maximisation for each idder type, where a idder type is defined y its (discrete) valuation v, we find a new distriution of ids. Note that this distriution can include ids from any numer of oth gloal and local idders, where the latter simply id their true valuation. This distriution of ids can then e used to find a new est response, resulting in a new distriution of ids, and so on, for a fixed numer of iterations or until

6 8 8 id () 6 4 id () valuation (v) (a) valuation (v) () Figure 2: Best response strategy for 2 auctions and 3 gloal idders without local idders (a), and with local idders (), averaged over iterations and 2 runs with different initial conditions. The measurements are taken after an initialisation period of iterations. The error-ars indicate the standard deviation. a stale solution has een found 6. In what follows, we first descrie the simulation settings, and then apply the simulation to settings with gloal idders only, followed y settings with oth gloal and local idders. 4. The Setting The simulation is ased on discrete valuations and ids. The valuations are natural numers ranging from to v max N, where v max is set to. Each valuation v {, 2,..., v max} occurs with equal proaility, equivalent to a uniform valuation distriution in the continuous case. Note, however, that even though the idder valuations are distriuted uniformly, the resulting distriution of ids is typically not uniform (since gloal idders typically id elow their valuation). The numer of different id levels that a idder is allowed is set to L N. Thus, a idder with valuation v can place the ids {v/l, 2v/L,..., v}. For the results reported here, we use L = 3. The initial state can play an important role in the experiments. Therefore, to ensure our results are roust, experiments are repeated with different random initial id distriutions. In the following, we assume the numer of local idders to e static and use N G and N L to denote the numer of gloal and local idders respectively. 4.2 The Results First, we descrie the results with no local idders (i.e., N L = ). For this case, we find that the simulation does not converge to a stale state. That is, when the numer of (gloal) idders is at least 2, the est response strategy keeps fluctuating, irrespective of the numer of iterations, and of the initial state. The fluctuations, however, show a distinct pattern and more or less alternate etween two states. Figure 2a depicts the average est response strategy for N G = 3 and M = 2. Here, the standard deviation is a gauge for the amount of fluctuation and thus the instaility of the strategy. In general, we find that the est response for low valuations remain stale, whereas the strategy for idders with high valuations fluctuates heavily, as is shown in Figure 2a. These results are roust for different initial 6 This approach is similar to an alternating-move estresponse process with pure strategies [7], although here we consider symmetric strategies within a setting where an opponent s est response depends on its type. conditions and simulation parameters. If we include local idders, on the other hand, we oserve that the strategies stailise. Figure 2 shows the simulation results for the same settings as efore except with oth local and gloal idders. As can e seen from this figure, the variation is very slight and only around the ifurcation point. We note that these outcomes are otained after only a few iterations of the simulation. The results show that the principal conclusions in case of a single gloal idder carry over to the case of multiple gloal idders. That is, the optimal strategy is to participate in all auctions and to id high in one auction, and equal or lower in the others. A similar ifurcation point is also oserved. These results are also otained for other values of M, N L, and N G. Moreover, the results are very roust to changes to the parameters of the simulation. To conclude, even though a theoretical analysis proves difficult in case of several gloal idders, we can approximate a (symmetric) Nash equilirium for specific settings using a discrete simulation in case the system consists of oth local and gloal idders. Our experiments show that, even in the case of multiple gloal idders, the est strategy is to id in multiple auctions. Thus, our simulation can e used as a tool to predict the market equilirium and to find the optimal idding strategy for practical settings where we expect a comination of local and gloal idders. 5. MARKET EFFICIENCY Efficiency is an important system-wide property since it characterises to what extent the market maximises social welfare (i.e. the sum of utilities of all agents in the market). To this end, in this section we study the efficiency of markets with either static or dynamic local idders, and the impact that a gloal idder has on the efficiency in these markets. Specifically, efficiency in this context is maximised when the idders with the M highest valuations in the entire market otain a single item each. More formally, we define the efficiency of an allocation as: Definition. Efficiency of Allocation. The efficiency η K of an allocation K is the otained social welfare proportional to the maximum social welfare that can e achieved in

7 efficiency (ηk) M Local Bidders Gloal Bidder 2 Static No 2 Static Yes 2 Dynamic No 2 Dynamic Yes 6 Static No 6 Static Yes 6 Dynamic No 6 Dynamic Yes (average) numer of local idders (N) Figure 3: Average efficiency for different market settings as shown in the legend. The error-ars indicate the standard deviation over the runs. the market and is given y: N η K = T i= vi(k) T i= vi(k ), () N where K = arg max K K N T i= vi(k) is an efficient allocation, K is the set of all possile allocations, v i(k) is idder i s utility for the allocation K K, and N T is the total numer of idders participating across all auctions (including any gloal idders). Now, in order to measure the efficiency of the market and the impact of a gloal idder, we run simulations for the markets with the different types of local idders. The experiments are carried out as follows. Each idder s valuation is drawn from a uniform distriution with support [, ]. The local idders id their true valuations, whereas the gloal idder ids optimally in each auction as descried in Section 3.3. The experiments are repeated 5 times for each run to otain an accurate mean value, and the final average results and standard deviations are taken over runs in order to get statistically significant results. The results of these experiments are shown in Figure 3. Note that a degree of inefficiency is inherent to a multiauction market with only local idders [4]. 7 For example, if there are two auctions selling one item each, and the two idders with the highest valuations oth id locally in the same auction, then the idder with the second-highest value does not otain the good. Thus, the allocation of items to idders is inefficient. As can e oserved from Figure 3, however, the efficiency increases when N ecomes larger. This is ecause the differences etween the idders with the highest valuations ecome smaller, therey decreasing the loss of efficiency. Furthermore, Figure 3 shows that the presence of a gloal idder has a slightly positive effect on the efficiency in case the local idders are static. In the case of dynamic idders, however, the effect of a gloal idder depends on the numer of sellers. If M is low (i.e., for M = 2), a gloal idder 7 An exception is when N = and idders are static, since the market is then completely efficient without a gloal idder. However, since this is a very special case and does not apply to other settings, we do not discuss it further here. significantly increases the efficiency, especially for low values of N. For M = 6, on the other hand, the presence of a gloal idder has a negative effect on the efficiency (this effect ecomes even more pronounced for higher values of M). This result is explained as follows. The introduction of a gloal idder potentially leads to a decrease of efficiency since this idder can unwittingly win more than one item. However, as the numer of local idders increase, this is less likely to happen. Rather, since the gloal idder increases the numer of idders, its presence makes an overall positive (aleit small) contriution in case of static idders. In a market with dynamic idders, however, the market efficiency depends on two other factors. On the one hand, the efficiency increases since items no longer remain unsold (this situation can occur in the dynamic model when no idder turns up at an auction). On the other hand, as a result of the uncertainty concerning the actual numer of idders, a gloal idder is more likely to win multiple items (we confirmed this analytically). As M increases, the first effect ecomes negligile whereas the second one ecomes more prominent, reducing the efficiency on average. To conclude, the impact of a gloal idder on the efficiency clearly depends on the information that is availale. In case of static local idders, the numer of idders is known and the gloal idder can id more accurately. In case of uncertainty, however, the gloal idder is more likely to win more than one item, decreasing the overall efficiency. 6. RELATED WORK Research in the area of simultaneous auctions can e segmented along two road lines. On the one hand, there is the game-theoretic analysis of simultaneous auctions which concentrates on studying the equilirium strategy of rational agents [6,, 2, 4, 6]. Such analyses are typically used when the auction format employed in the simultaneous auctions is the same (e.g. there are N second-price auctions or N first-price auctions). On the other hand, heuristic strategies have een developed for more complex settings when the sellers offer different types of auctions or the uyers need to uy undles of goods over distriuted auctions [, 2, 9]. This paper adopts the former approach in studying a market of N second-price simultaneous auctions since this approach yields provaly optimal idding strategies. In this case, the seminal paper y Engelrecht-Wiggans and Weer [6] provides one of the starting points for the game-theoretic analysis of distriuted markets where uyers have sustitutale goods. Their work analyses a market consisting of couples having equal valuations that want to id for a dresser. Thus, the couple s id space can at most contain two ids since the husand and wife can e at most at two geographically distriuted auctions simultaneously. They derive a mixed strategy Nash equilirium for the special case where the numer of uyers is large and also study the efficiency of such a market and show that for local idders, the market efficiency is /e. Our analysis differs from theirs in that we study simultaneous auctions in which idders have different valuations and the gloal idder can id in all the auctions simultaneously (which is entirely possile for online auctions). Following this, Krishna and Rosenthal [] then studied the case of simultaneous auctions with complementary goods. They analyse the case of oth local and gloal idders and characterise the idding of the uyers and resultant

8 market efficiency. The setting provided in [] is further extended to the case of common values y Rosenthal and Wang [4]. However, neither of these works extend easily to the case of sustitutale goods which we consider. This case is studied in [6], ut the scenario considered is restricted to three sellers and two gloal idders and with each idder having the same value (and therey knowing the value of other idders). The space of symmetric mixed equilirium strategies is derived for this special case, ut again our result is more general. 7. CONCLUSIONS In this paper, we derive utility-maximising strategies for idding in multiple, simultaneous second-price auctions. We first analyse the case where a single gloal idder ids in all auctions, whereas all other idders are local and id in a single auction. For this setting, we find the counter-intuitive result that it is optimal to place non-zero ids in all auctions that sell the desired item, even when a idder only requires a single item and derives no additional enefit from having more. Thus, a potential uyer can consideraly enefit y participating in multiple auctions and employing an optimal idding strategy. For most common valuation distriutions, we show analytically that the prolem of finding optimal ids reduces to two dimensions. This consideraly simplifies the original optimisation prolem and can thus e used in practice to compute the optimal ids for any numer of auctions. Furthermore, we investigate a setting with multiple gloal idders y comining analytical solutions with a simulation approach. We find that a gloal idder s strategy does not stailise when only gloal idders are present in the market, ut only converges when there are local idders as well. We argue, however, that real-world markets are likely to contain oth local and gloal idders. The converged results are then very similar to the setting with a single gloal idder, and we find that a idder enefits y idding optimally in multiple auctions. For the more complex setting with multiple gloal idders, the simulation can thus e used to find these ids for specific cases. Finally, we compare the efficiency of a market with multiple simultaneous auctions with and without a gloal idder. We show that, if the idder can accurately predict the numer of local idders in each auction, the efficiency slightly increases. In contrast, if there is much uncertainty, the efficiency significantly decreases as the numer of auctions increases due to the increased proaility that a gloal idder wins more than two items. These results show that the way in which the efficiency, and thus social welfare, is affected y a gloal idder depends on the information that is availale to that gloal idder. In future work, we intend to expand our analysis for markets with more than one gloal idder, and extend the model to investigate optimal strategies for purchasing multiple units of an item, and when the auctions are no longer symmetric. The latter arises, for example, when the numer of (average) local idders differs per auction or the auctions have different settings for parameters such as the reserve price. Acknowledgements This research was undertaken as part of the EPSRC funded project on Market-Based Control (GR/T664/). This is a collaorative project involving the Universities of Birmingham, Liverpool and Southampton and BAE Systems, BT and HP. Rajdeep K. Dash is funded y the ALADDIN project. In addition, we would like to thank Alex Rogers, Adam Prügel-Bennett, and the anonymous referees for their insightful comments and suggestions. 8. REFERENCES [] S. Airiau and S. Sen. Strategic idding for multiple units in simultaneous and sequential auctions. 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