Asymptotically Optimal Multi-Object Auctions Dov Monderer
Faculty of Industrial Engineering and Management Technion { Israel Institute of Technology Haifa 32000, Israel e-mail:
[email protected]
Moshe Tennenholtz Computer Science Department Stanford University Stanford, CA 94305, USA e-mail:
[email protected]
Abstract Auctions are a basic tool for resource allocation in non-cooperative environments. Much work in computer science, and in arti cial intelligence in particular, has been concerned with algorithms for winner determination in auctions in order to maximize a designer's revenue. In this paper we prove that in an environment with many participants, e.g., the Internet, the Clarke mechanism is an asymptotically optimal multi-object auction. Our results provide foundations and justi cation for the approach taken in the AI literature for dealing with multi-object auctions.
1 Introduction Auctions are widely applied economic mechanisms. Auctions are used for selling cars, ight tickets, paintings, oil leases, real estate, new shares, computer appliances, owers, as well as many other commodities. Auctions have been extensively studied in economic theory for many years (see e.g, 1
[16, 19, 23, 45] for surveys). The study of auctions has recently been facilitated by the famous FCC auction [3, 18, 21, 20], as well as by the huge popularity of them as an electronic-commerce tool.1 As a result, in the recent years, there is an explosion in papers by AI researchers dealing with auctions (see [37, 5, 26, 1, 44, 12, 46, 32, 31, 30] for some, not necessarily completely representative, pointers). A central issue in the related line of research is that of winner determination, the selection of agents' bids that will maximize the social surplus. Many of the algorithms that have been designed assume that agents' bids re ect their true valuations for the goods, and concentrate on selecting a set of bids that will maximize the social surplus. The related work refers to the famous Clarke [2] mechanisms (to be described later at the paper) as the approach to tackle the fact that agents may submit bids which are di erent from their actual valuations2 . This is due to the fact that the Clarke's mechanisms guarantee that truth revealing is a dominant strategy, and that social surplus is maximized. Social surplus is taken as a proxy for revenue, and since the Clarke mechanisms consist of polynomially many executions of a procedure for winner determination given a set of agents' bids, a lot of work in AI has concentrated on dealing with the related winner determination problem3. However, in order that the related research will become meaningful, properties of optimal, revenue-maximizing, auctions should be discussed and be compared to the revenue obtained by the Clarke mechanisms. The results presented in this paper will show that under natural conditions, and in particular under the assumption that there are many potential buyers, the Clarke mechanisms are almost (i.e., asymptotically) optimal. Our results provide foundations and justi cation for the approach taken in the AI literature for dealing with multi-object auctions as The number of auctions carried out in the Internet is huge. The reader may wish to browse the famous and well-established sites such as http://www.ebay.com, http://www.amazon.com, and http://www.ubid.com, in order to observe some of this activity. A huge list of online auctions can be found in http://www.auctionlist.com. 2 The clarke mechanisms are special type of the Vickrey-Clarke-Groves (VCG) mechanisms ([42, 2, 6]).Amongst the VCG mechanisms the Clarke mechanisms are characterized by two important properties; They satisfy the participation constraint ( it is rational for every agent to participate), and they yield a non negative revenue for the seller ( i.e., it is rational for the seller to use them). 3 Recent work deals also with communication complexity in combinatorial auctions (see e.g. [29]), as well as with the tradeo s between communication complexity and economic eciency in this context (see e.g. [7]). 1
2
described above. In the next section we present our assumptions and results. Sections 3{7 will provide the reader with full formal details and discussion of all concepts, de nitions, and theorems required to establishing the above-mentioned contribution. Proofs appear in an appendix.
2 The Main Results In a typical auction setting, an organizer designs a mechanism in order to sell k objects to a set of potential buyers. The outcome of each such mechanism is an allocation of the objects to the agents and to the organizer herself, and a transfer of money from the agents to the organizer (we do not exclude negative transfers). A potential buyer is an agent who considers participation in the market organized by the organizer. That is, he collects information about the value of the objects and about the other agents, and he learns about the rules of trade. However, he may still decide not to participate. It is assumed that every agent knows his monetary value for each possible outcome, and that the transfers of the other agents do not enter his own preferences. However, the goods may have externalities; An agent may have preferences that depend on the allocation of goods to the other agents too. Hence, each agent i has a valuation function wi de ned over the set of allocations. Each agent is also assumed to have a von-Newmann-Morgenstern utility function ui for money, which is normalized to be zero at zero. The utility of agent i if the allocation is chosen and he transfers t monetary units to the organizer is ui(wi() ? t). The valuation functions of the agents are their private information. These valuation functions are drawn from a compact set of possible valuation functions. If the (vector valued) random variables that determine the agents' valuation functions are independent, then the model is known as the independentprivate-values model. If, in addition, these random variables are identically distributed, we refer to the model as a valuation-symmetric independentprivate-values model4. The behavior of the agents depends on the auction 4
We use the notion "valuation-symmetric" because all agents have the same distribution
3
mechanism, their utility and valuation functions, and their information about the value of the objects and about the types of the other agents. We initially consider the following additional assumptions:
A1 No externalities: The value of an allocation to an agent depends
only on the goods allocated to this agent. Hence the valuation function of i is a function vi : 2G ! R, where G is the set of goods and 2G is the set of subsets of G. We always use the normalization vi(;) = 0. A2 Free disposal: The valuation function of every agent is non decreasing5. A3 valuation-symmetric independent-private-values model. A4 Agents are risk averse (recall that a risk-neutral agent is a simple, extreme case of a risk-averse agent). In addition, we make use of the following terminology. For a xed number of agents, for each allocation of the goods to the agents, we de ne the surplus variable at this allocation to be the sum of the valuation functions of the agents, where each agent's valuation function is computed at the set of goods allocated to him. The maximum, over all allocations, of the surplus variable is a random variable which depends on the valuation functions of the agents. We refer to it as the maximal surplus variable. The maximal value of this random variable is denoted by S and is referred to as the ultimate surplus6. Note that if A1 holds , and the number of agents exceeds the number of goods, the ultimate surplus does not depend on the number of agents.7 The organizer is assumed to be a risk-neutral economic agent. Hence, the objective of the organizer is to select an optimal auction mechanism, that is over valuation functions, but the agents may have di erent attitude to risk, i.e., di erent utility functions for money. 5 Note that A1-A2 imply that valuations are non negative. 6 The maximal value of a random variable is not necessarily well-de ned. Therefore we ~ where S~ is the maximal surplus de ne the ultimate surplus S as the essential sup of S, variable. That is, S is de ned by the following two conditions: Prob(S~ S = 1), and for every " > 0, Prob(S~ S ? ") < 1. 7 For example, when k = 1, if all random valuations (~ v ) =1 are supported at some interval [a; b], then the n-th random maximal surplus is maxfv~1 ; v~2; : : :; v~ g, and the ultimate surplus equals b. n i i
n
4
an auction mechanism that will maximize her expected revenue. However, the form of an optimal auction in the setup described by A1-A4 is not known. The optimality problem was solved only for very special environments. In all of these environments, A4 is not satis ed. That is, the agents are assumed to be risk-neutral8: In his pioneering work [27], Myerson characterized optimal auctions in a model for a single good (k = 1) that satis es A1-A2, independent-privatevalues and ARN, where
ARN Agents are risk-neutral. In particular, he proved that if in addition the model is valuation symmetric (i.e., A3 is satis ed), an optimal auction mechanism is a second-price auction with an appropriate reservation price9. It can be deduced from Krishna and Perry [11] that in a multi-object model, which satis es assumptions A1-A3 and ARN the Clarke mechanism is optimal among the ecient mechanisms10. Our work is concerned with asymptotic optimality. We prove:
R1 If A1-A2 and A4 hold, then for a xed number of participants, in
every auction mechanism the organizer's expected revenue in equilibrium is bounded above by the expected maximal surplus. Consequently, the expected revenue is bounded above by the ultimate surplus. R2 Suppose there are in nitely many agents , and A1-A4 hold. Then when the Clarke mechanism is applied to an in nite sequence of auctions with an increasing number of agents, the organizer's revenue, as well as the maximal surplus, converge almost surely to the ultimate There are only few papers that deal with auctions with risk-averse agents (see, e.g., [35, 13, 39]). The issue of optimality in such auctions is discussed in [14]. 9 This result can be easily extended to the case where we have multiple units of a good, in which every agent wishes to purchase only one unit (see e.g. [43]). 10The Clarke mechanism is sometime referred to as the Vickrey-Clarke-Groves (VCG) mechanism. The Vickrey mechanism [42] is the Clarke mechanism for one good, and the Clarke mechanism is the Groves mechanism [6] with a particular vector of transfer functions. 8
5
surplus, when the number of participants converges to in nity. In particular, the upper bound established in R1 is almost achieved by the Clarke Mechanism if the number of agents is suciently large. Our method of proof of R2 shows that:
R3 The rate of convergence in R2 depends on the number of goods k, and on the distribution of types, but not on the utility functions.
Our results generalize [26]. In [26], we proved R1 for a single good model (satisfying A1-A4), and we proved a weaker version of R2 for a single good model. In this weaker version (which was independently proved by Neeman [28]), we established that the ratio between the expected revenue and the expected maximal surplus converges to 1, while in the current paper we deal with almost surely convergence in a multi-object model. Our theorem implies that in an environment in which A1-A4 are satis ed, and in addition
A5 There are "many" players, a monopolist can (with high probability) extract almost all social surplus. As the Clarke mechanism is ecient, R2 implies that it is also an asymptotically optimal ecient mechanism11. Note again that all optimality results (including R2) assume the independentprivate-values model12. On the other hand, the inequality R1 does not assume it, and it holds in the most general information structure. It may not be reasonable to discuss auctions with many participants in which the number of potential buyers and their utility functions are not 11Asymptotic eciency properties of auctions are discussed e.g., by Rustichini, Satterthwaite and Williams [36], in the framework of a model of double auctions, and by Swinkels[41, 40] in the framework of multi-unit auctions (i.e., k identical objects). These works prove asymptotic eciency in independent-private-values models with risk neutral agents. The main novelty in Swinkels's works is the removal of the symmetry assumption. 12See e.g., [22, 8, 15, 4, 34, 33] for discussions of models in which types are correlated. Neeman [28] discusses the asymptotic ratio between the expected revenue and the expected maximal surplus in such models.
6
commonly known. Indeed, in Section 6, we show how our results R1 and R2 in Sections 3 and 5, respectively, can be generalized to a model with a random number of potential participants, in which the type of each agent i has two components: a valuation function vi and a utility function ui.13 In Section 7 we deal with the issue of externalities. There are two steps in the proof of the asymptotic optimality of the Clarke mechanism. In the rst step we establish the upper bound R1 for every auction mechanism. It turns out, that this proof can be extended to a very general case. In Section 7 we prove R1 without assuming A1-A3. That is, we do not require independence, symmetry, no-externalities or free disposal. All these assumptions are replaced by a single assumption:
E The expected utility of each agent in equilibrium is non-negative. Thus we prove that if the agents are risk-averse (i.e. A4 holds), then R1 holds in any equilibrium pro le in which E is satis ed. We provide two examples that show that these assumptions (A4 and E) are also necessary; that is, the theorem may not hold if one of them is not satis ed. It is also possible to prove R2-R3 in a model with externalities, if we assume anonymous externalities and nonnegative valuation functions. Since we believe that the real interest with externalities is with negative externalities, we do not discuss this issue in this paper14. A model with a random number of participants is discussed, e.g., in [17, 41]. We do not know on any model in auction theory that deals with random utility functions. 14An independent-private-values model with one object, in which each agent is riskneutral and has a negative externality on the other agents was discussed by Jehiel, Moldovanu and Stacchetti in [9]. Their paper deals with auction mechanisms in which every agent can give only a single numerical bid. Under these assumption it characterizes an optimal auction. Krishna and Perry [11] discuss a general independent-private-values model with risk neutral agents, in which there are many objects and externalities are possible. However, they incorporate into their model the assumption that every agent has an exogenously given and type-dependent outside option function. Under these conditions they proved that an appropriate Groves mechanism is optimal among the ecient mechanisms. 13
7
3 An Upper Bound on the Seller's Expected Revenue in Equilibrium In this section we assume A1-A2 and A4. Consider a seller denoted by 0, who wishes to sell a set G of k goods to a group of potential buyers, N = f1; 2; ; ng, where a potential buyer is an agent who considers participation in the market organized by the seller. That is, he learns about the rules of trade and collects information about other agents. However, he may still decide not to participate. Let N 0 = N [ f0g. An allocation of goods is a function : G ! N 0. That is, if the allocation is chosen, i 2 N receives the good a 2 G if and only if (a) = i. Thus, if (a) = 0 it means that no agent gets this item. The set of all allocations is denoted by , and for a subset N 0 of N we denote by N 0 the set of all allocation : G ! N 0 [ f0g. For each allocation 2 and for each i 2 N 0 we denote by i the set of goods that are allocated to i by . That is, i = fa 2 A : (a) = ig: Every agent i 2 N has a valuation function vi : 2G ! R+ normalized by vi(;) = 0. In addition, i has a strictly increasing concave von-Neumann Morgenstern utility function for money, ui : R ! R, which is normalized by ui(0) = 0. The utility of i if the center chooses the allocation and i pays t is ui(vi(i) ? t). The valuation functions of the agents are private information. Let R be the set of all possible valuation functions of an agent. We assume that for every agent i there exists a compact subset, Vi of R of feasible valuation functions. Each agent i's valuation function vi 2 Vi is determined by a random (vector valued) variable, v~i. The random valuation functions (~vi)i2N are distributed on V = V1 V2 : : : Vn according to the probability measure q. That is for each measurable15 subset B of V , Prob((~v1; v~2; ; v~n) 2 B ) = q(B ). Without loss of generality we assume that each v~i is actually de ned on V , that is v~i : V ! Vi is de ned by v~i(v1; v2; : : :; vn) = vi. 15
See the remark at the end of this section.
8
Let S~ be the random variable de ned on V by n ~S (v) = max X vi(i): 2 i=1
(3:1)
That, is for each vector of valuation functions v = (v1; v2; : : :; vn), S~ is the maximal-surplus variable. The expected value of S~ is denoted by E (S~), that is Z ~ E (S ) = V S~(v)dq(v); and the essential sup of S~ is denoted by S . That is S is de ned by the following conditions: and
Prob(S~ S ) = 1; Prob(S~ S ? ") < 1; " > 0:
We refer to S as the ultimate surplus. An auction mechanism is de ned by a set of messages (Mi)i2N , one set for each agent, and by two functions (g; t). The outcome function g determines a random allocation, that is g : M ! (), where M = M1 M2 Mn and () is the set of all probability distributions p on , that is p = (p())2, where P2 p() = 1 and p() 0 for every 2 . The transfer function t determines the transfers of the agents to the organizer as a function of the messages and of the random choice of allocations. That is, t = (t1; t2; : : : ; tn), where ti : M ! R. Thus, if the agents send the vector of messages m = (m1; m2 ; mn) then the seller conducts a lottery over allocations with the distribution function g(m). The probability that the allocation is chosen is g (m). If the allocation is chosen, then agent i transfers the amount ti(m; ) to the center. We assume that each Mi contains a distinguished message denoted by ', which means "no participation". Note that ' is not a real message; it is just a notation for non-participation. Naturally we require that a non participating 9
agent does not receive any item and pays nothing. That is, if mi = ', i = ; for all 2 for which g (m) > 0, and ti(m; ) = 0 for every 2 . Every auction mechanism (M; g; t) de nes a Bayesian game. In this game a strategy of agent i is a function bi : Vi ! Mi. Let us denote the set of possible strategies of agent i by i , and let = 1 2 n be the set of strategy pro les. If the agents use the vector of strategies b = (b1; b2; ; bn) 2 , then the expected utility of i is denoted by Li (b), that is Z X Li(b) = ui[vi(i) ? ti(b(v); )]g(b(v))dq(v); (3:2) V 2
where b(v) = (b1(v1); b2(v2); : : :; bn(vn)). A vector of strategies be 2 is in equilibrium if for every i 2 N , maxbi2i Li(bi; be?i) is attained at bei . Note that our setup is general. It includes sequential auctions and parallel auctions, in which the interaction between the seller and the buyers cannot be summarized by a description of one message which is sent by every agent to the seller. With each such auction we associate a ctitious auction mechanism, in which the message spaces of the agents are the sets of strategies in the Bayesian game associated with the auction. Though, formally the agents are not required to submit their full strategies, it is clear that their equilibrium behavior in the true auction can be analyzed within the framework of the associated ctitious auction mechanism.
Remark
In order to guarantee that the right hand side of (3.2) is well-de ned, one has to impose measurability properties on the utility functions, outcome function and transfer functions, as well as on the strategies. However, since we don't have existence results in this paper, but rather we prove that certain properties hold in equilibrium, whenever an equilibrium exists, then all these assumptions can be captured by the assumption, that an equilibrium exists. Alternatively, we can assume a discrete model in which the valuation functions are restricted to take values in some nite set, and therefore all integrals make sense. The main theorem of this section will make use of the following notation: For a given auction mechanism and a given equilibrium be (of the corresponding Bayesian game), we denote by R~ the random revenue variable of 10
the organizer. That is,
R~ (v) =
n X X i=1 2
ti(be(v); )g (be(v))
!
;
and we denote the expected value of R~ by E (R~ ). We can establish the following theorem whose proof is given in section 8.
Theorem 1 Assume A1, A2, and A4. Then the expected revenue of the
seller in equilibrium is bounded above by the expected maximal surplus. That is, E (R~ ) E (S~): Consequently, the expected revenue in equilibrium is bounded above by the ultimate surplus, that is E (R~) S :
4 The Clarke mechanisms Consider the environment described in Section 3, de ned by the set of goods, the set of agents, the utility functions of the agents and the distribution of valuation-functions. In particular we assume A1-A2. In a Clarke Mechanism the set of real messages of i is its set of valuationfunctions, that is Mi = Vi [ f'g. For every m 2 M we denote by N (m) the set of all active agents, that is the set of all i 2 N for which mi 6= '. The outcome function g is deterministic, that is g : M ! , and for m 2; : : :; mn ) with N (m) 6= ;, g (m) is an allocation that maximizes P = (m1v; m i2N (m) i (i) over 2 N (m). If N (m) = ;, then all goods go to the organizer, that is g(m)(a) = 0 for every a 2 G. We proceed to de ne the transfer functions in a Clarke mechanism. For every i 2 N we de ne ti(m; ) only for = g(m), and we denote ti(m) = ti(m; g(m)). The transfer of i to the seller is de ned as follows. If N (m) = fig, then ti(m) = 0. Otherwise, let N?i (m) = N (m) n fig and de ne: X X ti(m) = 2max vj (j ) ? vj (g(m)j ): N?i (m) j 2N?i (m)
11
j 2N?i (m)
That is, i pays the loss of surplus of the other agents that it causes. For every i 2 N let bei be the truth revealing strategy of i. That is, bei (vi) = vi for every vi 2 Vi. It is well-known (see e.g. [2]) that be = (be1; be2; ; ben) is an equilibrium vector of strategies independently of the probability of types q and of the utility functions (ui)i2N . This is so, because bei is a weakly dominating strategy for each i. When we refer in this paper to the expected revenue of the seller in a Clarke mechanism, we implicitly assume that all participants reveal their true valuation functions. Hence, n X R~ = ti(~v; g(~v)); i=1
where v~ = (~v1; ; v~n). We will need the following representations for the revenue: For v 2 V , 0 1 n X X X R~ (v) = @2max vj (j ) ? vj (g(v))A : Note that, Hence
i=1
0 n X X @ i=1 j 6=i
N nfig j 6=i
j 6=i
1 vj (g(v)j )A = (n ? 1)S~(v);
0 1 n X X vj (j )A : R~ (v) = S~(v) ? @S~(v) ? 2max N nfig j 6=i
i=1
(4:1)
Note that because of the possibility of ties, there may be many Clarke mechanisms, where each one is de ned by the choices it makes for vector of types that yield more than one allocation that maximizes the sum of valuations. The utility of a participating agent may depend on the particular chosen Clarke mechanism. However, by (4.1), the organizer's revenue depends only on v (and not on g(v)). Hence, the revenue of the seller does not depend on the particular choice of the tie breaking rule16. From (4.1) we deduce: Lemma 1 In a Clarke mechanism, R~(v) S~(v) S ; v 2 V: 16This holds for every announcements of the agents and not only for their behavior in equilibrium.
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5 Asymptotic Optimality of Clarke Mechanisms In this section we assume A1-A3. We deal with asymptotic properties of the seller's revenue in the Clarke mechanism applied to a sequence of auctions with an increasing number of agents. It is convenient to have an in nite set of players N1 = f1; 2; ; g. The nth auction is conducted for the set of agents Nn = f1 : : : ; ng. Because we assume A3, we can assume without loss of generality that Vi = Vj for all agents i; j: We denote by V the set of feasible valuation functions for an agent, that is V = Vi for every i. Let (~vi)1i=1 be the sequence of independent identically distributed random valuation functions of the agents in N1. We assume that each v~i is distributed on V according to the probability measure . As V is compact, and in particular it is bounded, G 2 there exists c 2 R+ such that for every v 2 V v(A) c(A) for every A 2 2G . Without loss of generality, assume that the constants c(A), A 2 2G are tight. That is, for every " > 0, and for every agent i, Prob(~vi(A) c(A) ? ") < 1. We further assume that at least for one A G, c(A) > 0, otherwise, with probability 1, for every agent i, vi(A) = 0 for every A G. Let V 1 = VN1 be the in nite Cartesian product of the agents' type sets, and let q denotes the product probability on V 1. Let S~n denote the maximal surplus random variable, when the set of agents is Nn. It is convenient to consider S~n as a random variable de ned on V 1, which depends only on the rst n coordinates. That is, for v = (v1; v2; : : :) 2 V 1, n X vi(i); S~n (v) = max 2n i=1
where n is the set of all allocations : G ! Nn [ f0g. Let Sn be the ultimate surplus in the nth auction, that is, Prob(S~n Sn ) = 1 and Prob(S~n Sn ? ") < 1 for every " > 0. Let l X = max c(Gi ); i=1
13
where the max ranges over all ordered partitions (G1; : : : ; Gl) of G (that is, Gs 6= ; for every s, Gs \ Gj = ; for every s 6= j , and [ls=1 Gs = G). Because we assume c(A) > 0 for some A G, > 0. Because of our assumption on the tightness of the constants c(A), A 2 2G , the following lemma holds:
Lemma 2 In the valuation-symmetric private-values-model, Sn for every n 1. In addition, for every n k, Sn = : We will denote the revenue obtained in the n-th auction by R~n . The proof of the following theorem is given in Section 8:
Theorem 2 Consider the Clarke mechanism applied to a sequence of auctions, where the nth auction is conducted for the set of agents Nn , n 2. Assume A1-A4. Then, Consequently, and
nlim !1 Rn
~ = a.s.
(5:1)
R~ n = 1 a.s., lim n!1 S~n
(5:2)
E (R~n ) = 1: lim n!1 E (S~n )
(5:3)
6 Extensions: random utility functions, and random number of potential participants In Section 6.1 we show that an appropriate versions of theorems 1 and 2 hold in an extended model of the one given in Section 3. In this new model the utility functions of the agents are their private information. In section 6.2 we extend theorems 1 and 2 to a model with a random number of potential participants. Combining these two subsections show that theorem 1 and 2 are generalized to an extended model in which we have both, random utility functions and a random number of potential participants. 14
6.1 Random utility functions
In all previous sections it was assumed that the utility functions of the agents (in contrast to their valuation functions) are commonly known. This assumption does not make sense when the number of participants is large. To deal with this we assume that every agent i is characterized by a pair of random variables (~vi; u~i), which is distributed on Ti = Vi Ui, where Ui is a set of utility functions17, that is every ui 2 Ui satis es ui(0) = 0 and ui is strictly increasing. This additional feature does not change the conclusions of Theorem 1 and 2. However, it is necessary to modify the notations: Let T = T1 Tn, and assume that ((~vi; u~i))ni=1 is distributed on T according to the probability measure q. Every auction mechanism de nes a Bayesian game, in which a strategy of i is a function bi : Ti ! Mi. For each pro les of strategy b and each ui 2 Ui we denote the expected utility of i given that its utility function is ui by Li(bjui). Thus, be is in equilibrium, if for every agent i, for every strategy bi of i, L(bejui) L((bi; be?i)jui) ui-a.s. (6:1) Note that the de nition (6.1) is neither the ex post de nition in (3.2), nor the equivalent interim de nition, which would require that (6.1) holds given the full type (vi; ui) of i. However, it is obviously equivalent to these two common de nitions. The proof of the following theorem is given in Section 8:
Theorem 3 Assume A1, A2, and that every agent i is risk averse with
probability 1. Then the expected revenue of the seller in equilibrium is bounded above by the expected maximal surplus. That is, E (R~ ) E (S~):
Consequently, the expected revenue in equilibrium is bounded above by the ultimate surplus, that is E (R~) S :
We proceed to generalize Theorem 2 to the case of random utility functions. We replace assumption A3 with 17
Of course, U should be associated with a - eld of measurable sets. i
15
A3' Symmetric independent-private-values model. That is, the types are identically distributed and independent. As the surplus variable does not depend on the utility functions of the agents, the proof of the following theorem is analogous to the proof of Theorem 2, and therefore it is omitted.
Theorem 4 Consider the Clarke mechanism applied to a sequence of auctions, where the nth auction is conducted for the set of agents Nn , n 2. Assume A1-A2, A3', and that every agent is risk averse with probability 1. Then, ~ nlim !1 Rn = a.s. Consequently, R~ n = 1 a.s., lim n!1 ~ and
Sn
E (R~n ) = 1: lim n!1 E (S~n )
6.2 Random number of potential buyers
Let A be the set of agents in the world. If the seller organizes an auction, only some of these agents consider participation. Many of them are not aware of the existence of this auction, and some of those who know about the auction do not go to the auction place (that may be an home page), without making any strategic considerations. In our previous results we take the group of agents as the group of agents who consider participation. The number of agents in this group was implicitly assumed to be commonly known. However, this common assumption in the theory of auctions is not reasonable in many setups, and in particular in Internet auctions. Neither the organizer, nor the potential buyers can possibly know the number of potential buyers. However, the number of agents (that is, the number of all Internet users) is commonly known (or, at least , commonly estimated). In order to capture this feature of many auctions, we assume that there exists a sequence (~zi)i2A of f0; 1g-random variables, such that agent i considers participation if and only if z~i = 1. We do not assume that z~i and v~i are 16
independent. Adding this feature to the model does not change our main results or their proofs, if we de ne the surplus as the surplus of the agents in A that consider participation. Thus, the de nition of the maximal-surplus variable in (3.1) is changed as follows: S~ : V f0; 1gA ! R, is de ned by X S~(v; z) = max vi(); 2A(v) i2A(v)
where
A(v) = fi 2 A : z~i = 1g: To see this we de ne a setup, like in Section 3, in which every agent i 2 A considers participation, but in the modi ed model the new random valuation function of agent i with the random valuation function v~i is v~iz~i, We then apply our theorems to this new setup. Note that in our optimality theorem we have to assume that the sequence of random variables (~vi; z~i)1i=1 is identically distributed and independent. However, for every agent i, the value of z~i may depend on its valuation function.
7 Externalities: An Upper Bound on the Seller's Expected Revenue in Equilibrium In this section we prove that the upper bound on the organizer's revenue established in Theorem 1, under assumption A1-A2 and A4 remains valid, when A1-A2 are replaced by assumption E. That is, we show that the expected revenue in equilibrium is bounded above by the expected maximal surplus, if the expected utility in this equilibrium is non negative for every participating agent. As in Section 3, we consider a seller denoted by 0, who wishes to sell a set G of k goods to a group of potential buyers, N = f1; 2; ; ng, and we set N 0 = N [ f0g. An allocation of goods is a function : G ! N 0, and the set of all allocations is denoted by . We deviate from the notations established in Section 3 when de ning valuation functions. A valuation function of i depends on the full distribution of goods amongst the agents, and not only on the set of goods allocated to him. In such a case, it is not reasonable to assume non negativity of valuations or the free disposal 17
assumption18. Thus, a valuation function of i is a function wi : ! R. In addition, i has a strictly increasing von-Neumann Morgenstern utility function for money, ui : R ! R, which is normalized by ui(0) = 0. The utility of i if the center chooses the allocation and i pays t is ui(vi() ? t). The valuation functions of the agents are private information. Let R be the set of all possible valuation functions of an agent. We assume that for every agent i there exists a compact subset, Wi of R of feasible valuation functions. Each agent i's valuation function wi 2 Wi is determined by a random (vector valued) variable, w~i. The random valuation functions (w~i)i2N are distributed on W = W1 W2 : : : Wn according to the probability measure q. That is for each measurable19 subset B of W , Prob((w~1; w~2; ; w~n) 2 B ) = q(B ). Without loss of generality we assume that each w~i is actually de ned on W , that is w~i : W ! Wi is de ned by w~i(w1; w2; : : :; wn ) = wi. Let S~ be the random variable de ned on W by n X wi(): S~(v) = max 2 i=1
That, is for each vector of valuation functions w = (w1; w2; : : :; wn ), S~ is the maximal-surplus variable. The expected value of S~ is denoted by E (S~), and the essential sup of S~ is denoted by S~. Every auction mechanism (M; g; t), as de ned in Section 3 de nes a Bayesian game. In this game a strategy of agent i is a function bi : Vi ! Mi. Let us denote the set of possible strategies of agent i by i , and let = 1 2 n be the set of strategy pro les. If the agents use the vector of strategies b = (b1; b2; ; bn) 2 , then the expected utility of i is denoted by Li(b), that is Z X Li(b) = V ui[wi() ? ti(b(w); )]g (b(w))dq(w); 2
where b(w) = (b1(w1); b2(w2); : : : ; bn(wn)). A vector of strategies be 2 is in equilibrium if for every i 2 N , maxbi2i Li(bi; be?i) is attained at bei . 18An agent may prefer less goods if these goods are allocated to particular agents, because this may increase competition amongst the other agents. 19See the remark at Section 3.
18
For a given auction mechanism and a given equilibrium be in this mechanism, we denote by R~ the random revenue variable of the organizer. That is, ! n ~R(w) = X X ti(be(w); )g (be(v) ; i=1 2
and we denote the expected value of R~ by E (R~ ). We can establish the following:
Theorem 5 Assume all agents are risk averse (A4). Consider an auction mechanism and an equilibrium pro le be in the Bayesian game determined by this mechanism, in which Ui (be) 0 for every i 2 N , where Ui (be) is the
expected utility of agent i in be. Then the expected revenue of the seller in this equilibrium is bounded above by the expected maximal surplus. That is, E (R~ ) E (S~):
Consequently, the expected revenue in equilibrium is bounded above by the ultimate surplus, that is E (R~) S :
The proof of Theorem 5 is given in Section 8. Note that in Theorem 1 we prove that Ui(be) 0, while in Theorem 5 we have to assume it. The proof of Theorem 5 uses the risk-aversion of the agents and the nonnegativity of the agents' expected utilities in equilibrium. We proceed to show that these conditions are necessary for the conclusion of the theorem. In Example 1 we present an example to an auction mechanism and to a distribution of types that yield an equilibrium, in which the expected utilities of the agents are negative. In this example the expected revenue of the seller exceeds the upper bound E (S~).
Example 1
There is only one good (k = 1) and two risk neutral agents (n = 2). Hence contains only 3 allocations denoted by 1; 2; 0. That is, the allocation in which i gets the object is denoted by i. The distribution over types is concentrated at a single vector of valuation functions (w1; w2) (that is 19
q(f(w1; w2)g) = 1), which are de ned as follows: For i = 1; 2, wi(0) = 0, wi(i) = 1, and wi(3 ? i) = ?3.20. Consider the following auction mechanism: The message spaces are Mi = fwi; 'g, i = 1; 2. If both players participate, that is they reveal their valuation functions, the center chooses = 0, and each player pays 1. If only one player participates, it gets the object and pays 0. if both agents do not participate, the center chooses = 0. The (degenerate) Bayesian game associated with this mechanism has a unique equilibrium, in which each agent declares its true valuation function. Actually, choosing wi strongly dominates ' for i = 1; 2. In equilibrium, the center chooses the allocation = 0 and each agent i pays 1. Hence the expected revenue of the seller in this equilibrium is t1 + t2 = 2. However the expected surplus is E (S~) = max3j=0(w1(j ) + w2(j )) = 0. In Example 2, which is taken from [26], there is a single good and the auction mechanism is a third-price auction. In such an auction the winning bid is the highest bid, and the price paid for the good is the third highest bid (see e.g., [45, 10, 25]). In this example the agents are risk seeking, their expected utility in equilibrium is non-negative, but the seller's expected revenue in equilibrium exceeds E (S~). It is well-known that a risk-neutral seller can sell lottery tickets with negative expected gain to a risk-seeking agent and obtain as a result very high gains. As pointed out in [24], this example shows that a third-price auction can serve as an implicit lottery mechanism for risk seeking agents.
Example 2
We will discuss the equilibrium set of a third-price auction for a single good with three participants, N = f1; 2; 3g. We use the independent-privatevalues model, and we assume that there are no externalities, and that the valuation of each agent is distributed in the interval [0,1] according to the uniform distribution, F (x) = x for every x 2 [0; 1]. Every agent i 2 N uses the convex utility function u : R ! R, > 1, where u (x) = x when x 0, and u (x) = x for x > 0. By a slight modi cation in the proof of Theorem AT in [25] 21 it can be shown that a continuous function s de ned on [0; 1] 20
The situation described in this example is referred to by [9] as the negative externalities
case.
This theorem is proved under the assumption that the utility functions are twice continuously di erentiable. 21
20
constitutes a symmetric equilibrium strategy in the auction game if and only if s (0) = 0, s is increasing, and Zv u (v ? s (t))dt = 0 t=0
for every v 2 [0; 1]. Solving this integral equation yields the unique solution: p s (v) = (1 + )v; v 2 [0; 1]: It is easily veri ed that the expected utility of each agent in this equilibrium is positive. The expected revenue of the seller in this equilibrium is p E (R~) = (1 + ) E (v[3]); where E (v[3]) is the expected value of the third-order statistics (i.e., the expected value of the type of the agent with the third-highest evaluation). As lim !1E (R~ ) = 1, for suciently large , E (R~) > E (max(^v1; v^2; v^3)): Actually, in our case the expected rst-order statistics is 43 , while the thirdorder statistics is 14 . Hence, if > 4 the expected revenue of the seller exceeds the expected maximal surplus.
8 Proofs Proof of Theorem 1
Consider a xed auction mechanism and a xed equilibrium pro le be. Let i 2 N . Because i is risk averse, ui is concave. Therefore, for every v 2 V ! X X e e e e ui(vi(i)?ti(b (v); ))g (b (v)) ui (vi(i) ? ti(b (v); ))g (b (v)) : 2
2
Because i can deviate to the strategy of always not participating, its expected utility Li (be) is non negative. Therefore (3.2) yields: ! Z X e e ui (vi(i) ? ti(b (v); )g(b (v)) dq(v) 0: (8:1) V
2
21
By applying Jensen inequality (i.e., ui(E ()) E (ui())) to (8.1), ! Z X ui ( (vi(i) ? ti(be(v); ))g (be(v)))dq(v) 0: V 2
As ui is increasing and ui(0) = 0, Z X ( (vi(i) ? ti(be(v); ))g (be(v)))dq(v) 0: V 2
(8:2)
Let R~i be the expected payment of i, and let E (R~i ) denotes the expected value of R~ i. That is Z X i ~ E (R ) = ti(be(v); )g(be (v))dq(v): V 2
By (8.2),
Z X E (R~i ) ( vi(i)g (be(v)))dq(v):
As E (R~ ) = Pni=1 E (R~i ),
V 2
! Z X X n vi(i) g (be(v)))dq(v): E (R~ ) V ( 2 i=1
Because a convex combination of a set of numbers is less or equals the maximal number in this set, Z n X ~ vi(i))dq(v): E (R) (max 2 V
Hence
i=1
E (R~ ) E (S~):
proof of Theorem 2
We will prove (5.1). Hence, Lemma 1 implies limn!1 S~n = a.s., which implies (5.2), because > 0. By the Lebesque convergence theorem, (5.1) yields limn!1 E (R~ n) = limn!1 E (S~n ) = , and therefore (5.3) holds. 22
We proceed to prove (5.1). By Lemma 2 there exists an ordered partition (G1; G2; : : : ; Gl) of the set of goods such that Xl = c(Gs ): s=1
For a nite set of agents B , let S~B be the random variable de ned on V 1, that describes the surplus of the agents in B . That is, X S~B (v) = max vi(i); v 2 V 1: 2b i2B
When B = N n fig we denote S~B by S~n;?i . By (4.1), n X R~n = S~n ? (S~n ? S~n;?i): i=1
We will show that S~n ! a.s., and that Pni=1 (S~n ? S~n;?i ) ! 0 a.s. It can be easily deduced from Theorem 1 (Page 251) in [38], that a sequence of random variables (Xn)1n=1 converges to a random variable X if for every > 0, 1 X Prob(jX ? Xn j ) < 1: Let then 0 < < .
n=1
Obviously, Prob(j ? S~n j ) = Prob(S~n ? ): Note that if S~n (v) ? , then forPevery ordered set of l agents in Nn, F = fi1; i2; : : :; ilg, i1 < < il n, lj=1 vij (Gj ) ? : Let n k, and let rn be a positive integer satisfying rn k n < (rn + 1)k. Then Xl ~ Prob(Sn ? ) Prob( v~jk+i (Gi ) ? for every 0 j rn ? 1): i=1
Because of our independence assumption, Xl Prob(S~n ? ) (Prob( v~i(Gi ) ? ))rn : i=1
23
Therefore,
X Prob(S~n ? ) (Prob( v~i(Gi) ? )) nk ?1:
(8:3)
Xl (") = Prob( v~i(Gi) ? ):
(8:4)
l
i=1
Let
i=1
We now show that
(") < 1: which combined with (8.4) proves that 1 1 X X Prob(S~n ? ") < (") nk ?1 < 1; and hence that
(8:5)
n=k
n=k
1 X n=1
Prob(S~n ? ") < 1:
To prove (8.5), recall that = c(G1 ) + + c(Gl). Hence Xl (") = Prob( v~i(Gi) ? ) Prob(91 i l; v~i(Gi) c(Gi) ? l ):
Hence,
i=1
Yl (") 1 ? (1 ? Prob(~vi(Gi ) c(Gi ) ? l ) < 1; i=1
by our assumption on the tightness of c(A), A 2 2G . We proceed to show that M~ n = Pni=1(S~n ? S~n;?i ) converges to 0 almost surely. It suces to show that for every > 0 1 X Prob(M~ n ) < 1: (8:6) n=1
For every n k and for every 1 j k, let ~jn be the random f0; 1gvariable de ned on V 1 by : ~jn(v) = 1 if and only if g(v)j 6= ; in the auction conducted for Nn. Let F~n = fj 2 Nn : ~jn = 1g. Obviously, jF~nj k, and X Prob(M~ n ; F~n = F ): Prob(M~ n ") = F 22Nn ;jF jk
24
We will show that for every n k, for every " > 0, and for every F 2 2Nn with jF j k, Prob(M~ n "; F~n = F ) ( k" ) nk ?2; (8:7) where (:) is de ned in (8.4). As the number of F Nn with jF j k is bounded above by nk , (8.7) implies Prob(M~ n ") nk ( k" ) nk ?2: As ( k" ) < 1, 1 X nk ( k" ) nk ?2 < 1; n=k
which proves (8.6). Indeed, let F = fi1; : : :; isg Nn , and let D = Nn n F . Note that n X Prob(M~ n "; F~n = F ) = Prob( S~n;?i nS~n ? ; F~n = F ): i=1
If i 2 D, S~n;?i = S~n on F , hence n s X X ~ ~ Prob( Sn;?i nSn ? ; Fn = F ) = Prob( S~n;?ij sS~n ? ; F~n = F ): i=1
j =1
As S~n ,
X Prob(M~ n ; F~n = F ) Prob( Sn;?ij s ? ; F~n = F ): s
j =1
As S~n;D S~n;?ij for every 1 j s,
Prob(M~ n ; F~n = F ) Prob(S~n;D ? s ; F~n = F );
implying that
Prob(M~ n ; F~n = F ) Prob(S~n;D ? k ; F~n = F ): 25
Therefore, by partitioning D to rn ? 1 subsets with cardinality of k each, Prob(M~ n ; F~n = F ) ( k" ) nk ?2; which proves (8.7).
Proof of Theorem 3
Consider a xed auction mechanism and a xed equilibrium pro le be. Let i 2 N . For every v 2 V , every u?i 2 U?i , and every concave function ui 2 Ui, ! X X e e e e ui (vi(i)?ti(b (v; u); ) )g (b (v; u)) ui (vi(i) ? ti(b (v; u); ))g(b (v; u)) ; 2
2
where u = (ui; u?i). Because i can deviate to the strategy of always not participating, its expected utility Li (bejui) is almost surely non negative. Therefore (6.1) yields that ui-almost-surely, ! Z X e e ui (vi(i) ? ti(b (v; u); )g(b (v; u)) dq((v; u)jui) 0: (8:8) V U 2
By applying Jensen inequality (i.e., ui(E (jui)) E (ui()jui) to (8.8), ! Z X e e ui ( (vi(i) ? ti(b (v; u); ))g(b (v; u)))dq((v; u)jui) 0; ui-a.s. V U 2
As ui is increasing and ui(0) = 0, Z X ( (vi(i)?ti(be(v; u); ))g(be(v; u)))dq((v; u)jui) 0; ui-a.s. (8:9) V U 2
Therefore, the expected value of the left-hand side of (8.9) is non negative. that is, Z X ( (vi(i) ? ti(be(v; u); ))g(be(v; u)))dq((v; u)) 0: (8:10) V U 2
Let R~i be the expected payment of i, and let E (R~i ) denotes the expected value of R~ i. That is Z X E (R~ i) = ti(be(v; u); )g(be(v; u))dq(v; u): V U 2
26
By (8.10),
E (R~i )
Z
(
X
V U 2
vi(i)g (be (v; u)))dq(v; u):
As E (R~ ) = Pni=1 E (R~i ), ! Z n X X E (R~ ) vi(i) g (be(v; u)))dq(v; u): ( V U 2 i=1
Because a convex combination of a set of numbers is less or equals the maximal number in this set, Z n X ~ vi(i))dq(v; u): E (R) (max 2 V U
Hence
i=1
E (R~ ) E (S~):
Proof of Theorem 5
Consider a xed auction mechanism and a xed equilibrium pro le be, for which Ui (be) 0 for every i 2 N . The proof can be produced word by word from the proof of Theorem 1, by replacing the terms v~i, vi(i), v, Vi and V in the proof of Theorem 1 with the terms w~i, wi(), w,Wi, and W respectively. Note again that in Theorem 1 we prove that Ui (be) 0, while in Theorem 5 we have to assume it.
References [1] C. Boutilier and H.H. Hoos. Bidding Languages for Combinatorial Auctions. In The 17th international joint conference arti cial inteligence, pages 1211{1216, 2001. [2] E. Clarke. Multipart pricing of public goods. Public Choice, 18:19{33, 1971. 27
[3] P.C. Cramton. Money out of thin air: The nationwide narrowband pcs auction. Journal of Economics and Management Strategy, 1995. [4] P. Dasgupta and E. Maskin. Ecient auctions. Discussion Paper Harvard University, to appear in Quartely Journal of Economics, 1999. [5] Y. Fujishima, K. Leyton-Brown, and Y. Shoham. Taming the computational complexity of combinatorial auctions: Optimal and approximate approaches. In IJCAI-99, 1999. [6] T. Groves. Incentives in teams. Econometrica, 41:617{631, 1973. [7] R. Holzman, N. K r-Dahav, D. Monderer, and M. Tennenholtz. Bundling equilibrium. Working paper Technion http://ie.technion.ac.il/ dov/rndm6.pdf, 2001. [8] P. Jehiel and B. Moldovanu. Ecient design with interdependent valuations. Working Paper University of Mannheim, 2000. [9] P. Jehiel, B. Moldovanu, and E. Stacchetti. Multidimensional mechanism design for auctions wite externalities. Journal of Economic Theory, 85(2):258{294, 1999. [10] J.H. Kagel and D. Levin. Independent private value auctions: bidder behavior in rst-, second-, and third-price auctions with varying numbers of bidders. Economic Journal, 103:868{879, 1993. [11] V. Krishna and M. Perry. Ecient mechanism design. Working Paper, 1998. [12] D. Lehmann, L.I. O'Callaghan, and Y. Shoham. Truth revalation in rapid, approximately ecient combinatorial auctions. In ACM Conference on Electronic Commerce, 1999. [13] E. Maskin and J. Riley. Optimal auctions with risk-averse buyers. Econometrica, 52(6):1473{1518, 1984. [14] S.A. Matthews. Selling to risk-averse buyers with unobservable tastes. Journal of Economic Theory, 30:370{400, 1983. 28
[15] P.R. McAfee and P. Reny. Correlated information and mechanism design. Econometrica, 60:395{421, 1992. [16] R.P. McAfee and J. McMillan. Auctions and bidding. Journal of Economic Literature, 25:699{738, 1987. [17] R.P. McAfee and J. McMillan. Auctions with a stochastic number of bidders. Journal of Economic Theory, 43:1{19, 1987. [18] J. McMillan. Selling spectrum rights. Journal of Economic Perspectives, 8:145{162, 1994. [19] P. Milgrom. The economics of competitive bidding: A selective survey. In L. Horwicz, D. Schmeidler, and H. Sonneschein, editors, Social Goals and Social Organization. Cambridge University Press, 1985. [20] P. Milgrom. Chapter i: Auctioning the radio spectrum. from Auction Theory for Privatization, forthcoming, 1995. [21] P. Milgrom. Putting auction theory to work: The simultaneous ascending auction. Technical Report 98-0002, Department of Economics, Stanford University, 1998. [22] P. Milgrom and R.J. Weber. A thoery of auctions and competitive bidding. Econometrica, 50:1089{1122, 1982. [23] P.R. Milgrom. Auction theory. In T. Bewly, editor, Advances in Economic Theory: Fifth World Congress. Cambridge University Press, 1987. [24] D. Monderer and M. Tennenholtz. Internet Auctions: are they gamblers' attraction? Working paper, submitted, 2000. [25] D. Monderer and M. Tennenholtz. K-Price Auctions. to appear in Games and Economic Behavior, 2000. [26] D. Monderer and M. Tennenholtz. Optimal Auctions Revisited. Arti cial Intelligence, 120(1):29{42, 2000. [27] R. Myerson. Optimal auction design. Mathematics of Operations Research, 6:58{73, 1981. 29
[28] Z. Neeman. The e ectiveness of simple auctions. Working Paper, 1999. [29] N. Nisan. The communication complexity of combinatorial auctions. Working paper, the Hebrew University, 2001. [30] N. Nisan and A. Ronen. Computationally feasible vcg mechanisms. To appear, 2000. [31] N. Nisan and A. Ronen. Algorithmic mechanism design. Games and Economic Behavior, 35:166{196, 2001. [32] D. C. Parkes. ibundle: An ecient ascending price bundle auction. In ACM Conference on Electronic Commerce, 1999. [33] M. Perry and P. Reny. An ex post ecientmulti-unit ascending auction for agents with interdependent valuations. Working paper, University of Chicago, 1999. [34] M. Perry and P. Reny. An ex post ecientmulti-unit auction for agents with interdependent valuations. Working paper, University of Pittsburgh, 1999. [35] J.G. Riley and W.F. Samuelson. Optimal auctions. American Economic Review, 71:381{392, 1981. [36] A. Rustichini, M.A. Satterthwaite, and S.R. Williams. Convergence to eciency in a simple market with incomplete information. Econometrica, 62:1041{1063, 1994. [37] T. Sandholm. An algorithm for optimal winner determination in combinatorial auctions. In IJCAI-99, 1999. [38] A.N. Shiryayev. Probability. Springer-Verlag, 1984. [39] J.L. Smith and D. Levin. Ranking auctions with risk averse bidders. Journal of Economic Theory, 68:549{561, 1996. [40] J.M. Swinkels. Asymptotic eciency for dicriminatory private value auctions. to apear in Review of Economic Studies, 1998. 30
[41] J.M. Swinkels. Eciency of large private value auctions. Working Paper, 1998. [42] W. Vickrey. Counterspeculations, auctions, and competitive sealed tenders. Journal of Finance, 16:15{27, 1961. [43] R.J. Webber. Multiple-object auctions. In R. Engelbreach, M. Shubik, and J. Stark, editors, Auctions, Bidding, and Contracting, pages 165{ 191. 1983. [44] M.P. Wellman, P.R. Wurman, W.E. Walsh, and J.K. MacKie-Mason. Auction protocols for decentralized scheduling. Games and Economic Behavior, 35:271{303, 2001. [45] E. Wolfstetter. Auctions: An introduction. Journal of Economic Surveys, 10(4):367{420, 1996. [46] M. Yokoo, Y. Sakurai, and S. Matsubara. Robsut combinatorial auction protocol against false-name bids. In Proceedings of the 17th national conference on Arti cial Intelligence, pages 110{115, 2000.
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