Limitations of the Vickrey Auction in Computational Multiagent Systems

Limitations of the Vickrey Auction in Computational Multiagent Systems Tuomas Sandholm * sandholm~cs.wustl.edu Washington University Department of Computer Science One Brookings Drive, Campus Box 1045 St. Louis, MO 63130-4899

Abstract 1 Introduction Auctionsprovide an efficient distributed Auctions provide efficient, distributed and autonomy mechanism for solving problems such as task preserving ways of solving task and resource allocation and resource allocation in muitiagent sys- problems in computational multiagent systems [16; 21; tems. In the Vickrey auction--which has 5; 24; 2; 8]. Auctions can be used amongcooperative been widely advocated for automated auc- agents, but they also work in open systems consisting tions [22; 1; 3; 5; 24; 2; 8; 9; 20; 13J--the of self-interested agents. They can be analyzed no~na- best bid wins the auction, but at the second tiuel~ what strategies are self-interested agents best best price. In certain settings this promotes off using (and therefore will use), and will desirable truthful bidding and avoids counterspecula- social outcomes--e.g, efi~cient allocation--still follow. tion. This paper analyses the circumstances The goal is to design the protocoLs (mechanisms) whenthis protocol is appropriate, and expli- the interaction so that desirable soda] outcomes follow cates the desirable properties and lack thereof even though each agent acts based on self-interest. in varied settings. The first part of the pa- Auction theory analyzes protocols and agents’ per discusses knowndeficiencies of the Vick- strategies in auctions. An auction consists of an auc- rey auction: bidder collusion, a lying auction- tioneer and potential bidders I. Auctions are usually eer, promotion of lying in non-private-value discussed regarding situations where the auctioneer auctions, lower revenue than alternative pro- wants to sell an item and get the highest possible pay- tocols, and the necessity to reveal sensitive ment for it while each bidder wants to acquire the item information. The second part of the paper at the lowest possible price. However, settings where presents our results regarding new limitations the auctioneer wants to subcontract out a task at the of the protocol, which arise especially among lowest possible price and each bidder wants to handle computational agents. These include ineffi- the task at the highest possible payment, are totally cient allocation and lying in sequential suc- analogous. tions of interrelated items, untruthful bid- There are three qualitatively different auction set- ding whena risk averse agent has local uncer- tings depending on how an agent’s value of the item tainty, and the need for counterspeculation is formed. In private value auctions, the value of the to make deliberation control (or information good depends only on the agent’s own preferences. An gathering) decisions when an agent has local example is auctioning off a cake that the winning bid- uncertainty. der will eat. The key is that the winning bidder will not resell the item in which case the value would de- *This material is based upon work supported by the pend on other agents’ valuations. On the other hand, National Science Foundation under Grant No. ]RI- in commonvalue auctions, an agent’s value of an item 9523419. Additional support came from a University of Massachusetts at Amherst Graduate School Fellow- depends entirely on other agents’ values of it, which ship. Also supported by the Technical Research Cen- are identical to the agent’s by symmetryof this cri- tre of Finland, Finnish Culture Foundation, Finnish Sci- terion. For example, auctioning treasury bills fulfills ence Academy, Leo and Regina Wainstein Foundation, this criterion. Nobodyinherently prefers having the Jenny and Antti Wihuri Foundation, Honkanen Founda- bills, and the value of the bill comesentirely from re- tion, Ella and George Ehrnrooth Foundation, Finnish In- selling possibilities. In correlated value auctions, an formation Technology Research Foundation, Transporta- tion EconomicSociety, and Thanks to Scandinavia Foun- 1Thereare also auctions with multlple bid takers, i.e. dation. auctioneers.

agent’s value depends partly on its own preferences egy result of Vickrey auctions means that an agent is and partly on others’ values. For example, an auction best off bidding truthfully no matter what the other in a task contracting setting fulfills this criterion. An bidders are like: what are their capabilities, operating agent mayhandle a task itself in which case the agent’s environments, bidding plans, etc. This has two desir- local concerns define the cost of handling the task. On able sides. First, the agents reveal their preferences the other hand, the agent can recontract out the task truthfully which allows globally efficient decisions to in which case the cost depends solely on other agents’ be made. Second, the agents need not waste effort in valuations. Next, I discuss four different auction pro- counterspeculating other agents, because they do not tocols [12]. matter in making the bidding decision. In the English (first-price open-cry) auction, each Vickrey auctions have been widely advocated and bidder is free to raise his bid. Whenno bidder is will- adopted for use in computational multiagent sys- ing to raise anymore, the auction ends, and the highest tems [22; 1; 3; 5; 24; 2; 8; 9; 20; 13]. For example, bidder wins the item at the price of his bid. An agent’s versions of the Vickrey auction have been used to allo- strategy is a series of bids as a function of his private cate computation resources in operating systems [24; value, his prior estimates of other bidder’s valuations, 2], to allocate bandwidth in computer networks ’L22; and the past bids of others. In private value English 8; 9; 20], and to computationally control building en- auctions, an agent’s dominant strategy is to always bid vironments [5]. On the other hand, Vickrey auctions a small amount more than the current highest bid, and have not been widely adopted in auctions among hu- stop whenhis private value price is reached. In corre- mans [14; 15] even though the protocol was laid out lated value auctions the rules are often varied to make 25 years ago [23]. the auctioneer increase the price at a constant rate There are severe limitations to the applicability of or at a rate he thinks appropriate. Secondly, some- the Vickrey auction protocol. This paper explores times open-ezit is used where a bidder has to openly these limitations. It is important to understand thcsc declare exiting without a re-entering possibility. This limitations in order not to ascribe desirable character- provides the other bidders more information regarding istics to a protocol when the protocol really does not the agent’s valuation. guarantee them. On the other hand, the Vickrey auc- In the first-price sealed-bid auction, each bidder sub- tion protocol may well be a good choice in situations mits one bid without knowing the others’ bids. The that do not exceed the applicability limits. highest bidder wins the item and pays the amount of The first part of the paper details the knownprob- his bid. An a8ent’s strategy is his bid as a function of his private value and prior beliefs of others’ valuations. lems regarding the Vickrey auction. These problems In general there is no dominant strategy for acting in have been discovered by auction theorists and practi- this auction. With common knowledge assumptions tioners, and they have led to the lack of deployment regarding the probability distributions of the agents’ of Vickrey auctions among humans. The problems in- values, it is possible to determine Nash equilibrium clude bidder collusion (Section 2), an untruthful auc- strategies for the agents [12]. tioneer (Section 3), lying in non-private-value auctions In the Dutch (descending) auction, the seller contin- (Section 4), lower revenue than alternative protocols uously lowers the price until one of the bidders takes (Section 5), and the necessity to reveal sensitive in- the item at the current price. The Dutch auction formation (Section 6). The wide application plans is strategically equivalent to the first-price sealed-bid Vickrey auctions in computational multiagent systems auction, because in both games, an agent’s bid matters suggest that these limitations may have been forgot- only if it is the highest, and no relevant information is ten. The first part of the paper serves as a reminder. revealed during the auction process. The second part of the paper presents our results re- In the Vickrey (second-price sealed-bid) auction, garding new limitations of the Vickrey auction proto- each bidder submits one bid without knowing the oth- col. The settings that suffer from these new problems ers’ bids. The highest bidder wins, but at the price of arise especially among computational agents. The the second highest bid [23; 11; 12; 7; 4]. An agent’s newly discovered problems include inefficient alloca- strategy is his bid as a function of his private value tion and lying in auctions of interrelated items (Sec- and prior beliefs of others’ valuations. The dominant tion 7), untruthful bidding when an agent has local strategy in private value Vickrey auctions is to bid uncertainty (Section 8), and the need for counterspecu- one’s true valuation. If an agent bids more than that, lation whenan agent has local uncertainty (Section 9). and the increment made the difference between winning or not, he will end up with a loss if he wins. If duce the same allocation at the sameprices. Onthe other he bids less, there is a smaller chance of winning, but hand, in correlated value auctions, the bids of other agents the winning price is unaffected 2. The dominant strat- in the English auction provide information to the agent about its ownvaluation. Therefore English and Vickrey In private value auctions, the Vickreyauction is strate- auctions are not strategically equivalent in general, and gically equivalent to the English auction. They will pro- maylead to different results.

300 ICMAS-96 From: Proceedings of the Second International Conference on Multiagent Systems. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. 2 Vulnerability to bidder collusion the auctioneer because the highest bidder gets the item One problem with all four of the auction mechanisms at the price that it stated in the hid. (English auction, Dutch auction, first-price sealed-bid Cheating by the auctioneer has been suggested to be auction, and Vickrey auction) is that they are not col- one of the main reasons why the Vickrey auction pro- lusion proof. The bidders could coordinate their bid tocol has not been widely adopted in auctions among prices so that the bids stay artificially low. In this humans [15]. In another paper, two formal models manner, the bidders get the item at a lower price than of cheating by the auctioneer are discussed [14]. The they would without colluding. first model is gametheoretic. It analyses the situation The English auction and the Vickrey auction actu- where the auctioneer can choose to use a first-price ally self-enforce someof the most likely collusion agree- sealed-bid protocol or a Vickrey protocol. The bidders’ ments. Therefore, from the perspective of deterring equilibrium behavior creates positive incentives for all collusion, the first-prlce sealed-bid and the Dutch auc- auctioneers, except the type most prone to cheat, to tions are preferable. The following example from [12] choose standard first-price sealed-bid auctions. The shows this. second model assumes simple (not rational) bidders. Let bidder Smith have value 20, and every other bid- They bid honestly as long as the auctioneer has not der have value 18 for the auctioned item. Say that the been caught cheating, but after catching a cheating bidders collude by deciding that Smith will bid 6, and auctioneer, the bidders will bid as if the auctioneer everyone else will bid 5. In an English auction this always cheats. The result is that a seller with prob- is self-enforcing, because if one of the other agents ex- abilistic opportunities to cheat, and finite abilities to ceeds 5, Smith will observe this, and be willing to go all resist cheating, will cheat and be caught in finite time the way up to 20, and the cheater will not gain any- and thereafter have no reason to conduct Vickrey auc- thing from breaking the coalition agreement. In the tions. Vickrey auction, the collusion agreement can just as well be that Smith bids 20, because Smith will get the 4 Lying in non-private-value auctions item for 5 anyway. Bidding 20 removes the incentive Most auctions are not pure private value auctions: an from any bidder to break the coalition agreement by agent’s valuation of a good depends at least in part on bidding between 5 and 18, because no such bid would the other agents’ valuations of that good. For exam- win the auction. On the other hand, in a first-price ple in contracting settings , a bidder’s evaluation of a sealed-bid auction, if Smith bids anything below 18, task is affected by the prices at which the agent can the other agents have an incentive to bid higher than subcontract the task or parts of it out to other agents. Smith’s bid and to win the contract. The same holds This type of recontracting is commonlyallowed in au- for the Dutch auction. tomated versions of the contract net protocol also [16; However, for collusion to occur under the Vickrey 21]. auction, the first-price sealed-bid auction, or the Dutch Commonvalue (and correlated value) auctions suf- auction, the bidders need to identify each other be- fer from the winner’s curse. If an agent bids its valu- fore the submission of bids--otherwise a non-member ation and wins the auction, it will knowthat its valu- of the coalition could win the auction. On the other ation was too high because the other agents bid less. hand, in the English auction this is not necessary, be- Therefore winning the auction amounts to a monetary cause the bidders identify themselves by shouting bids. loss. Knowingthis in advance, agents should bid less An auctioneer can organize a computerized English than their valuations [11; 12]. This is the best strategy auction where the bidding process does not reveal the in this type of Vickrey auctions also. So, even though identities of the bidders. Vickrey auctions promote truthful bidding in private- value auctions where an agent’s valuation is totally 3 Vulnerability to a lying auctioneer determined locally, it fails to induce truthful bidding The insincerity of the auctioneer may be a problem in most auctions. in the Vickrey auction. The auctioneer may overstate the second highest bid to the highest bidder unless that 5 Lower revenue than with the bidder can verify it. An overstated second offer would give the bidder a higher bill than it would receive if English auction the contractor were truthful. In other words, the the- Whenconsidering one auction in isolation, each one of ory classically assumes a truthful auctioneer. Alterna- the four auction protocols (English, Dutch, first-price tively, cryptographic electronic signatures could per- sealed-bid, and Vickrey) allocates the auctioned item haps be used by the bidders so that the auctioneer Pareto efHciently to the bidder whovalues it the most. could actually present the second best bid to the win- One would imagine that the first-price auctions give ning bidder--and would not be able to alter it. higher expected revenue to the auctioneer because in The other three auction protocols (English, Dutch, second-price auctions the auctioneer only gets the sec- and first-price sealed-bid) do not suffer from lying by ond price. This is not the case however, because in

Sandholm 301 From: Proceedings of the Second International Conference on Multiagent Systems. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. ~st-price auctions the bidders are motivated to lle by items of a homogeneousgood [11]. However, the case biasing their bids downward. The revenue-equivalence of auctioning heterogeneous interrelated goods has re- theorem [23; 11; 12] states that all four auction proto- ceived little attention. On the other hand this is the cols produce the same expected revenue to the auction- setting of many real world problems where computa- eer in private value auctions where the values are inde- tional agents are used [18; 17; 19; 16; 13]. pendently distributed. Although all four are Pareto ef- This section discusses cases where heterogeneous ficient in the allocation, the ones with dominant strate- items are auctioned one at a time, and the agents’ gies (Vickrey auction and English auction) are more valuations of these items are not additive. This occurs efficient in the sense that no effort needs to be wasted for example in task allocation in transportation prob- in counterspeculating the other bidders. lems. Figure 1 presents a simple example of such a However, most auctions are not pure private value problem with just two delivery tasks: tl and t2. The auctions as discussed earlier. In correlated value auc- former task is auctioned before the latter. The auctions with at least three bidders, the open-exit English auction leads to higher revenue than the Vickrey auction. The reason is that other bidders willing to go high up in price causes a bidder to up its ownvaluation of the auctioned item. In this type of auctions, both English and Vickrey auction protocols produce greater revenue to the auctioneer than the first-price i .0 sealed-bid auction--or its equivalent, the Dutch auction. Put together, the English auction seems to be the right choice by the auctioneer because it creates the Agent 1 Agent 2 greatest revenue, allocates the item optimally, avoids ~ t2 ...... bidders wastefully counterspeculating each other, and has no question of a lying auctioneer. I- ...... -[ ...... 0.5 0.5 6 Undesirable private information Figure 1: Small ezample problem with two agents and revelation two delivery tasks. Because the Vickrey auction has truthful bidding as the dominant strategy in private value auctions, agents tioneerwants to get the taskshandled while paying often bid truthfully. This leads to the bidders reveal- agentsI and2 as littleas possiblefor handling them. ing their true valuations. Sometimesthis information Theinitial locations of thetwo agentsare presented is sensitive, and the bidders would prefer not to reveal in the figure.To handlea task,an agentneeds to it. For example, after winning a contract with a low moveto the beginningof the arrow,and take a parcel bid, a company’s subcontractors figure out that the fromthere to the end of the arrow.An agent’smove- company’s production cost is low, and therefore the mentincurs the samecost irrespective of whetherit companyis making larger profits than the subcontrac- is carryinga parcel.The agentsneed not returnto tors thought. It has been observed that when such theirinitial locations. The costsfor handlingtasks auction results are revealed, the subcontractors will (subscriptedby the nameof the agent)can be mea- want to renegotiate their deals to get higher payoff [15]. suredfrom the figure: c1({tl}): 2, cx({t2}): I, This has been suggested--along with the problem of a cx(~tl,ta}) : 2, ca({tl}) : 1.5, c2({ta}) : 1.5, lying auctioneer--as one of the main reasons why the c2(~tl,t2}) -- 2.5. These costs axe commonknowledge Vickrey auction protocol has not been widely adopted to the agents. Clearly the globally optimal allocation in auctions among humans [15]. is the one where agent 1 handles both tasks. This First-price auction protocols do not expose a bid- allocation is not reached if agents treat the auctions der’s valuation as clearly because the bid is subject to independently and bid truthfully: strategic lying. Therefore, these auction types maybe Theorem 7.1 Suboptimal allocation in interre. more desirable than the Vickrey auction when valua- lated auctions. If agents toith deterministic valuations are sensitive. tions treatVickrey auctions of interdependent goods The next sections present our results regardin8 pre- viously unvoiced limitations of the Vickrey auction. ~#ithout lookahead regarding later auctions, and bid truthfully, the resulting allocation maybe suboptimal. 7 Inefficient allocation and lying in Proof. Example of Figure 1. In the first auction, task tl is allocated. Agent 1 bids cx({tl}) = 2, and interrelated auctions agent 2 bids ca({tx}) = 1.5. The task is allocated In addition to single-item auctions, Vickrey auctions to agent 2. In the second auction, task t2 is allo- have been widely studied in the allocation of multiple cated. Agent 1 bids c1({t2}) = 1, and agent 2 bids

302 ICMAS-96 From: Proceedings of the Second International Conference on Multiagent Systems. Copyright © 1996, AAAI (www.aaai.org). All rights reserved. c2({t2}) = 1.5, so t2 is allocated to agent 1. The re- the original reasons suggested for adopting the Vickrey sulting allocation of the two tasks is suboptimal. auction, look.ahead requires counterspeculation in the If agent 2 takes the ownership of tx into account sense of trying to guess which items axe going to be when bidding for t~, then it will bid ca({tx, t2}) auctioned in the future. Other speculative issues in c2({tl}) = 2.5- 1.5 = 1. In this case t2 may be al- sequential Vickrey auctions have been discussed for located to either agent. In both cases the resulting example in [6]. al]ocation of the two tasks is still suboptimal, o 8 Untruthful bidding with local Alternatively, the agents can incorporate fun looks- uncertainty head into their auction strategies. This way the optimal allocation is reached, but agents do not bid their Agents often have uncertainty about the worth of the true costs: auction item to themselves. This valuation maybe in- herently uncertain. On the other hand, computational Theorem 7.2 Untruthful bidding in interrelated agents mayhave uncertainty regarding Local evaluation auctions. If agents with deterministic valuations because computing the valuation may be complex, and treat Vickrey auctions of interdependent goods with the computation may not have finished by the time of full lookahead regarding later auctions, their dominant the auction. Such computational complexities arise for strategy bids can differ from the truthful ones of the example in task aUocation auctions where evaluating corresponding isolated auctions. a task set requires solving NP-complete problems [18; Proof. Example of Figure 1. In the last auction, an 17;19; le; 13]. agent is best off bidding its owncosts that takes into Risk neutral agents--i.e, agents with linear utility account the tasks that the agent already has. Let us functions--are best off bidding the expected value of look at the auction of tz. If agent 1 has tl, it will bid their valuation in a single-shot private value Vicl~ey c1(f1, 2})- cl(ftl}) : 2 - 2 = o, andc1( t2}) auction. This is a dominant strategy. However, many otherwise. If agent 2 has tl, it will bid c2({tl,tz}) agents axe risk averse, i.e. their utility is a concave c2(~tl}) : 2.5 - 1.5 = 1, and c2({t2}) : 1.5 otherwise. function of payoff. For example, many people would So, if agent 1 has tl, it will win t2 at the price 1.5, prefer $100,000,000for certain over a fifty-fifty chance and get a payoff of 1.5- 0 : 1.5 in the second auction, of receiving $200,000,001. Computational agents take while agent 2 gets zero. On the other hand, if agent 2 on the preferences of the real world parties that they has tl, the bids for t2 axe equal, and both agents get a represent. Therefore many computational agents will zero payoff in the second auction--irrespective of who be risk averse. The following theorem states the result t2 gets allocated to. that risk averse agents are not best off bidding truth- Therefore it is knownthat getting tl in the first fully when the Vickrey auction protocol is used. Thus auction is worth an extra 1.5 to agent 1 while nothing it is nonobvious that the Vickrey auction protocol can extra to agent 2. So, in the auction for tl, agent l’s really be used in computational systems to avoid lying. dominant strategy is to bid cl(~tl}) - 1.5 = 2- 1.5 Theorem 8.1 Untruthful bidding. It is not the 0.5. This is lower than agent 2’s bid c2({tl}) - 0 case that in a single-shot private value Vickrey auction 1.5 - 0 : 1.5, so agent one gets tl. In the second with uncertainty about an agent’s own valuation, it is auction agent 1 gets t2 as discussed above. So the a risk averse agent’s best Cdominant or equilibrium) globally optimal allocation is reached. strategy to bid its ezpected value. However,agent 1 bids 0.5 for tl instead of 2, which would be the truthful bid if the auctions were treated Proof. Counterexample. We will anaiyse an auc- independently without lookahead. [] tion where the auctioneer wants to get a high price for a good, but task allocation auctions where the Put together, lookahead is a key feature in auctions auctioneer wants to allocate the task at a low price of multiple interrelated items. Up to date it has not are anaiogous. Let the agent’s utility function be been adequately addressed in computational multia- 2z if z _~ 0 U(z) z if z > 0. The concavity of this func- gent systems that use Vickrey auctions. In auctions t by humans, this issue is sometimes addressed by allow- tion represents risk aversion of the agent. Let the ing a bidder to pool all of the interrelated items under agent’s own valuation v be uniformly distributed be- one entirety bid [11]. Another method for enhancing tween 0 and 1. We now show that the agent can in- the efllciency of interrelated auctions is to allow agents crease its expected utility by bidding E[v] - e instead to backtrack from commitments by paying penalties. of E[v]. We analyze the situation based on what the This allows a winning agent to beneficially decom- highest bid b coming from other agents might be. mit from an auctioned item in case that agent does Case 1: b _~ E[v]- e. In this case, the agent wins not get synergic items from other related auctions [I0; the auction at price b when bidding E[v] or E[v] - e. 19]. Therefore the expected utility is unaffected by bidding While avoidance of counterspeculation was one of E[v] - ¯ instead of E[v].

Case 2: b ~ EIv]. In this case, the agent loses valuation ~2 be commonknowledge. Let us restrict the auction when bidding Etv] or E[~] - ~. Therefore ourselves to the situation where 0 _< v2 < ½, which the expected utility U(0) = 0 is unaffected by bidding implies E[v,] > v2. E[~]- ~ insteadof E[~] Let agent 1 have the choice of finding out its exact Case 8: E[~]-e < b < E[~]. In this case, the agent valuation Vl before the auction by paying a cost c. loses the contract when bidding E[v] - c, but wins it Now, should agent 1 take this informative but costly when bidding E[v]. Therefore, the utility from bidding action? E[v]-e is U(0) : 0. The expected utility from bidding No matter what agent 1 chooses here, agent 2 will bid v2 because bidding ones valuation is a dominant strategy in a single-shot private value Vickrey auction. = If agent 1 chooses to not pay c, agent 1 should bid fo 2(v - b)dv + v - bdv E[vl] : I, because bidding ones expected valuation is a risk neutral agent’s dominant strategy in a single- = - b2-b+~ shot private value Vickrey auction. Nowagent 1 gets the item at price v2. If agent l’s valuation vl turns which is less than zero when b > ~ ~, 0.41 (i.e. out to be less than v2, agent 1 will suffer a loss. Agent in the range of case 3). So, bidding E[v] has smaller 1%expected payoff is expected utility than bidding E[v] - e. E[II,~oi.jo] = vl - v~dvl = ~ - v2 9 Wasteful counterspeculatlon On the other hand, if agent 1 chooses to pay c for One of the main original motivations for using the the exact information, it should bid vl because bidding Vickrey auction was that an agent has a dominant ones valuation is a dominant strategy in a single-shot strategy (of telling the truth), i.e. an agent’s best ac- private value Vickrey auction. Agent 1 gets the item tion does not depend on other agents. Therefore the if and only if vl >_ vs. Note that now the agent has bidders will not waste effort in counterspeculating each no chance of suffering a loss, but on the other hand it other. This would lead to global savings. has invested c in the information. Agent l’s expected This section presents a new result which states that payoff is there are cases where the Vickrey auction fails to have this desirable property. Let us look at a situation where an agent has uncertainty regarding its own val- E[IIi,,lo] = -cdvl + ~I - r2 - cdvl uation of the auction item, but can pay to remove // /1 2 1 2 1 this uncertainty. This situation often occurs among = ~v2-v2+ ~-c computational agents, where the value of a good (or task contract [18; 17; 19; 16; 13]) can only be deter- Agent 1 should choose to buy the information iff mined via carrying out a costly computation--e.g, a solution of a combinatorial problem. Alternatively the E[II,~So]> ]E[I-I.~So paymentcan be viewed as the cost of solving a predic- I z 1 1 tion problem, or as the cost of performing an information gathering action, or as the cost paid to an expert oracle. The following theorem states that in such a setting, the Vickrey auction protocol does not avoid counterspeculation. ¢~ v2 > V~c (because v~ >__0) Theorem g.1 Incentive to eounierspeculate, In So, agent l’s best choice of action depends on agent 2’s a single-shot private value Vickrey auction ~oith uncer- valuation v2. Therefore, agent 1 benefits from coun- tainty about an agent’s own valuation, a risk neutral terspeculating agent 2. v agent’s best (deliberation or information gathering) action can depend on the other agents. Proof. Proof by example. We will analyse an auc- 10 Conclusions tion where the auctioneer wants to get a high price for Vickrey auctions have been widely suggested and a good, but task allocation auctions where the auc- adopted for use in computational mnltiagent sys- tioneer wants to allocate the task at a low price are tems [22; 1; 3; 5; 24; 2; 8; 9; 20; 13]. This auction pro- analogous. Let there be one auctioneer, and two bid- tocol has certain desirable properties--such as truth- der agents: 1 and 2. Let agent l’s ownvaluation vl for promotion and counterspeculation avoidance---in lim- the auctioned item be uniformly distributed between ited settings. It is important to clearly understand 0 and 1, i.e. agent 1 does not know its own valua- these limitations in order not to use the protocol when tion exactly. On the other hand, let agent 2% exact inappropriate, and in order not to trust the protocol

to have certain desirable properties whenit really does [11] P. R. Milgrom. The economics of competitive not have them in the particular setting. bidding: a selective survey. In L. Hurwics, The first part of the paper detailed knownproblems D. Schmeidler, and H. Sonnenschein, editors, regarding the Vickrey auction. These include bidder Social goals and social organization: Essays in collusion, a lying auctioneer, promotion of lying in memory of Elisha Pusher, chapter 9, pages 261- non-private-value auctions, lower revenue than alter- 292. Cambridge University Press, 1985. native protocols, and the necessity to reveal sensitive [12] E. Rasmusen. Games and Info~nation. Basil information. BlackweH, 1989. The second part of the paper presented our results regarding new limitations of the protocol. The settings [13] J. S. Rosenschein and G. Zlotkin. Rules o.f En- that suffer from these new problems arise especially counter. MITPress, 1994. among computational agents. The problems include [14] M. H. Rothkopf and R. M. Harstad. Two models inefficient allocation and lying in sequential auctions of bid-taker cheating in Vickrey auctions. Journal of interrelated items, untruthful bidding when a risk o/Business, 68(2):257-267, 1995. averse agent has local uncertainty, and the need for [15] M. H. Rothkopf, T. J. Teisberg, and E. P. Kahn. counterspeculation to makedeliberation control (or in- Whyare Vickrey auctions rare? Jou~al of Po- formation gathering) decisions when an agent has local litical Economy,98(1):94-109, 1990. uncertainty. [16] T. W. Sandholm. An implementation of the contract net protocol based on marginal cost calcula- References tions. In Proc. l lth National Conference on Arti- [1] Agorics, Inc. Going,going, gone! A surveyof ficial Intelligence (AAAI-93), pages 256-262, July auctiontypes, http://wlvw.agorics.com/agorics 1993. /auctions/bibliography.html, 1996. [17] T. W. Sandholm and V. R. Lesser. Coalition [2] K. E. Drexler and M. S. Miller. Incentive engi-" formation among bounded rational agents. In neering for computational resource management. Proceedings of the Fourteenth International Joint In B. A. Huberman, editor, The Ecology of Com- Conference on Artificial Intelligence, pages 662- putation. North-Holland, 1988. 669, Montreal, Canada, Aug. 1995. Extended ver- [3] J. Edelman. sion appeared as University of Massachusetts at Webpage mention of the use of Vickrey auction in Amherst, Computer Science Department techni- AgentTCL. http: //www.cs.dartmouth. edu/"jze/ cal report 95-71. resume.html, 1996. [18]T. W. Sandholmand V. R. Lesser.Issues in [4] D. Fudenberg and J. Tirole. Game Theory. MIT automatednegotiation and electroniccommerce: Press, 1991. Extendingthe contractnet framework.In Pro- ceedings of the First International Conference on [5] B. I-Iuberman and S. H. Clearwater. A multi- Multi-Agent. Systems (ICMAS-95), pages 328- agent system for controlling building environ- 335, San Francisco, CA, June 1995. ments. In Proceedings of the First International Conference on Multi-Agent Systems (ICMAS-g5), [19] T. W. Sandholm and V. R. Lesser. Advantages pages 171-176, San Francisco, CA, June 1995. of a leveled commitmentcontracting protocol. In Proceedings of the National Conference on Artifi- [6] M. Jackson and J. Peck. Speculation and price cial Intelligence, Portland, OR, Aug. 1996. Ex- fluctuations with private, extrinsic signals. Jour- tended version appeared as University of Mas- nal o/Economic Theory, 55:274-9.95, 1991. sachusetts at Amherst, Computer Science Depart- [7] D.M. Kreps. A Course in Microeeonomic Theory. ment technical report 95-72. Princeton University Press, 1990. [20] Smart Market. Web page mention of the [8] J.K. MacKie-Mason and H. R. Varian. Pricing use of Vickrey auction in Smart Market. the internet. In Proceedings o/the Public Access http://mortadeUo, wu-vTien, ac. at/’iuJork/pricing to the Internet Conference. JFK School of Gov- /smart_marhet.htnd, 1996. ernments May 1993. [21] R. G. Smith. The contract net protocol: High- [9] J. K. MacKie-Mason and H. R. Varian. level communication and control in a distributed Some FAQs about usage-based pricing, ftp: problem solver. IEEE Transactions on Comput- //alfred.sims.berkeley.edu /pub /Papers /use- ers, C-29(12):1104-1113, Dec. 1980. FA Qs.html, 1994. [22] Sun Microsystems. Web page mention of [10] R. P. McAfeeand J. McMillan. Analysing the air- the use of Vickrey auction in Webmart. waves auction. Journal of Economic Perspectives, http://www.sun.com:80 /960£01 /cover /web- 10(I):159-175,1996. mart.html, 1996.

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