Bidders’ Strategy for Multi-Attribute Sequential English Auction with a Deadline1 “Student paper” Esther David Rina Azoulay-Schwartz Sarit Kraus
Bar-Ilan University Department of Computer Science Ramat-Gan 52900 Israel {saadone,schwart,sarit}@macs.biu.ac.il
ABSTRACT General Terms In this paper we consider a procurement multi-attribute English auction with a deadline. This protocol can be used for agents who Economics. try to reach an agreement on an item or issue, which is characterized by several quality attributes in addition to the price. The protocol allows the specification of a deadline since in many Keywords real world situations it is essential to conclude a negotiation Bidding and bargaining agents. Electronic commerce market. among agents and to reach an agreement under a strict deadline. Currently, the deadline rules, which are mainly used for auction 1. INTRODUCTION mechanisms, result in an non-recommended and unstable bidding strategy, , i.e. the last minute bidding strategy, causing system Negotiating agents can efficiently use auction mechanisms for overhead and inefficient auction outcomes. Therefore, we define reaching agreements among agents [8,13,14], and they can also be another deadline rule which diminishes the phenomenon of last- used when the issue under consideration is associated with multi- minute bidding-strategy and thus prevents bottlenecks in the attributes items [5]. A multi attribute item is characterized by agents' network. We analyzed simultaneous and sequential multi- several attributes in which one of them is the price and the others attribute English auctions with a fixed deadline that can be used are non-price/quality attributes. For example, a service can be for negotiating agents. For each of these protocols combined with characterized by its quality, supply time, and the risk involved, in the deadline rule, we provide the bidders automated agent with the case the service is not supplied eventually. Telephone service optimal and stable bidding strategy. providers and Internet portals, as well as video-on-demand suppliers, would like to rent extra storage capacity from suppliers over the Internet. The attributes of the required item in this Categories & Subject Descriptors domain are the storage capacity, the access rates to the data, the availability period and time limit, the level of security, etc. In task Intelligent agents, Multiagent systems. allocation, the attributes of a deal include the size of a task, its starting time, its ending deadline, the accuracy level, etc. In the International Logistics Supply Chain (ILSC) domain, the required Permission to make digital or hard copies of all or part of this work for service moves a given cargo from one location to another. The personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that details of this service include arrival time, dispatching time, path copies bear this notice and the full citation on the first page. To copy length, weight, volume, etc. otherwise, or republish, to post on servers or to redistribute to lists, In many real world situations it is essential to conclude a requires prior specific permission and/or a fee. negotiation among agents and to reach an agreement within a AAMAS ’03, July 1-2, 2003, Melbourne, Australia. strict deadline. For example, if an auctioned service is to be Copyright 2000 ACM 1-58113-000-0/00/0000…$5.00 provided very soon then the negotiation among the agents must be concluded by a defined deadline. Another example can be the case . where video-on-demand suppliers need to rent an extra capacity
1This work is supported in part by NSF Grant # 9820657. The third author is also affiliated with UMIACS. from a supplier over the Internet. In this case, it is obvious that the is different from the bidders’ strategies in the traditional auction extra storage capacity is meaningless if it is not provided within protocol, without a deadline. Accordingly, any interested bidder is the time it is needed. Therefore, a deadline for a negotiation motivated to join the auction the minute discovers it since he is process or an auction mechanism is an essential issue. provided with an automated agent that will act optimally on his behalf and the human bidder does not need to be available even if Recently, the fundamental value of a deadline principle in an the deadline is close. auction mechanism, has been realized by auction house designers and has been applied to many e-commerce applications [10]. In section 2 we describe the model including the bidder agents There are primarily two types of deadline rules used in auction– and the auctioneer agent. We proceed to section 3 in which we houses: (1) the fixed deadline in which an exact time is defined provide the automated agents participating in a sequential multi- for a strict auction’s ending and in which no extensions are attribute English auction with stable and optimal strategies. allowed; and (2) a deadline, which is based on a predefined Similarly, in section 4 we consider the simultaneous protocol and interval with no activities from the last proposed bid. In this type provide the optimal bidding strategy. In section 5 we describe of rule, an ending time is defined but the auction is not closed related work on the topic of deadline rules and the multi-attribute until an interval of time with no activities has passed. issue and in section 6 we present our conclusions. eBay is an example of an auction house that uses a fixed deadline 2. THE MODEL rule. The advantage of this protocol is that the exact time the auction closes is known in advance. On the other hand its The auction model consists of one buyer agent, the auctioneer, disadvantage is that it encourages the last minute bidding strategy, and a fixed number of n seller agents, the bidders. The buyer which is not recommended in such environments since it causes agent that needs a particular item (service or product) begins the system overhead [1,11]. Moreover Roth [11] proved that “There auction process. At the beginning of the auction, the buyer does not exist a dominant strategy at which each bidder bids his announces its item request, which consists of the item’s desired true value at some time t< 1”. In practical terms, since many bids characteristics, and a scoring rule that describes its preferences are submitted at the last moment, the best bid may not reach the concerning the item properties. A seller agent, who decides to deadline and therefore, will not be considered. Sometimes the send a bid, has to specify the full configuration it offers. reason that the best bidder waits until the last minute to bid his Each buyer agent and each seller agent is characterized by a utility true value is to avoid a “bid war” which unnecessarily increases function that describes its preferences. The multi-attribute utility- his and the other’s following bids [10]. functions we refer to are based on the Simple Additive Weighting Amazon is an example of an on-line auction on the Internet that (SAW) method [16]. A utility or a score in the SAW method is uses the second deadline rule. The advantage of this rule is that it obtained by adding the contributions of each attribute. There are all interested participants gather at the same time to follow the other methods for obtaining multi-attribute utility functions (e.g. auction. Its disadvantage is that strategically; the deadline multiplying the contributions of the various attributes). However, principle does not play any role since the auction continues while the SAW method suits the examples (e.g. ILSC domain or renting there are bidding activities. However, in this auction the bidders storage capacity) we refer to. The utility function of the buyer are likely to use the last minute bidding strategy [10]. In associates a value with each bid, which is the sum of the buyer’s conclusion, the deadline principle is recommended and needed, level of satisfaction from the various attributes’ values. The utility though the phenomenon of the last-minute bidding-strategy function of each seller associates a utility value reflecting the should be prevented whenever possible. seller’s cost and profit from each bid. Therefore, we suggest designing the deadline rules in a manner The auctioneer announces a scoring rule at the beginning of the that provides the bidders with stable strategies and provides the auction. This scoring rule associates a score with each proposed auctioneer with more efficient auction results. The first step of bid and the auction protocol dictates the winner (best scored bid) this design is to define the last step/interval/round of the auction based on this scoring rule. We assume that each participant knows in which new bidders will not be permitted to join the auction and its utility function, and bidding is not costly. yet long enough to ensure that all interested bidders will be able to In our model, each seller agent has private information about the successfully submit their bids. Each bidder can bid only once in costs of improving the quality of the product it sells, or its this step. There can be two types of bidding in the last phase: performance. Each seller agent i (bidder) is assumed to be simultaneous or sequential. S characterized by a cost parameterθi , which is its private This work is an extension of our previous work on multi-attribute information. As θi increases, the cost of the seller to achieve an English auctions [5,6]. Our proposed protocol for multi-attribute item of a higher quality also increases, i.e., the seller is “weaker”. auctions with a deadline is based on a predefined number of rounds. In the sequential version of our protocol the bidders bid Similar to the model described by Che [4], we assume that θi is sequentially in each round according to a predefined order such independently and identically distributed over [θ ,θ ] that each bidder observes the bids of the bidders before him in the round. In the simultaneous protocol, all the bidders bid in each ( 0 <θ <θ < ∞ ) according to a distribution function F for which round without knowing the bids of the other bidder agents. a positive, consciously differentiable density f exists. Because of According to the deadline rule we use, the bidder knows exactly complete symmetry among agents, the subscript i is omitted when it is his last chance to act. In addition, we propose an throughout the rest of the paper. automated intelligent agent that uses a sophisticated and stable bidding strategy thus avoiding application of the non- We analyze a general case of multi-attribute auctions in which recommended late-bidding strategy. The bidders’ optimal strategy there is an arbitrary number of attributes (m+1), which is predefined and known to all the participants. One of the attributes Given the buyer’s private utility function, the buyer will announce is the price (p) and the others are quality attributes ( qi where a scoring rule, which is used for choosing among bids. The scoring rule of the buyer can be different than its real utility i ∈ [1,.., m]) for which the preferences of the buyer and the function in the sense that the announced weights wi may be sellers conflict. We assume that as qi increases, the quality of the different than the actual weights i . In particular, the scoring item increases. That is, as qi increases the cost of the seller to W m provide it increases since it is harder to provide higher quality items. In addition, the buyer’s utility from higher quality items rule is of the form: S( p, q1,..,qm) = − p + ∑ wi ⋅ qi , increases. i=1 where wi , are the weights that the buyer assigns to qi . From the For example, a multi-attribute service of providing a machine can scoring rule we infer that the announced bid’s value for the buyer be characterized by three attributes: the price p of the item, q1 can m denote the speed of the machine and q2 can denote its accuracy, is: V (q1,.., qm) = wi ⋅ qi . The announced values of the or the warranty period for this machine. And the seller’s cost ∑ i=1 increases if it provides a more accurate machine, and the buyer’s weights wi can be equal to or different from the real values of utility is higher if it obtains such a machine. Another example of a three-attribute service is video-on-demand supply, where p is the the weights Wi . For example, if w1
Sketch of proof: Since the proof is independent of the auction Assume that seller Si reaches the last round and he is not the last protocol, this Lemma holds also for the case of sequential multi- to bid. And assume that there are k sellers that may bid after him. attribute English auctions with a deadline assuming the model Since the qualities are determined according to Lemma 1 and described in section 2, which also was assumed in [6]. Lemma 2, the question is what should the price of his bid be. Consequently, the deadline does not affect the choice of the Assume that seller S i bids {q * 1,q * 2,...,q * m, p}which yields optimal qualities to be offered throughout the auction.■ him a score of B. In order to maximize the expected utility from Now we proceed by searching for the optimal price and the this bid, seller Si has to calculate the probability that all the other specific attributes values to be offered in each of the R-1 rounds sellers cannot afford to bid an equivalent bid (w.r.t the scoring of the auction. The optimal bid in each of the R-1 rounds is rule) plus D. That is, for each of the k sellers that bid after him, determined according to the feature of the English auction in the score of their best possible bid must be lower than B. The which the score of the proposed bid should be higher than the condition that ensures this is: S(q *1,q * 2,...,q * m, p,θj) < B + D score of the last proposed bid by the minimal increment allowed, for each . denoted D. We term the last propose bid selected. j ∈[1..k] Lemma 2 [6] The next step is to find the best price that seller S j can afford. This price actually yields utility 0 to this seller according to its Given the scoring rule, the seller’s utility functions, and the last utility function. proposed bid in a multi-attribute English auction, the seller’s best strategy in the first R-1 rounds is to bid the following bid if he m can afford it; Otherwise the seller drops out: Usj = p * −θj ⋅ ∑at ⋅ q * t = 0 t =1 where j ∈[1..k]. 2 m wi where i ∈[1,..,m]. ⇒ p* = θj ⋅ ∑at ⋅ q * t qi *(θ ) = , t =1 2⋅aiθ By assigning p* and the optimal qualities values as in Lemma 2 in 2 the scoring rule, to the above inequality and by isolating the cost m parameter θ j of seller S j we obtain the following condition wi p * (θ , selected) = ∑ − S(selected) − D. which specifies the case in which seller S i may overcome 2 ⋅ ai ⋅θ i=1 seller S j for each j ∈[1..k].
m 2 Sketch of proof: In [6] we proved it holds for the case of a 1 wt ⋅ ∑ < θj sequential multi-attribute English auction without a deadline. 4(B + D) t =1 at However, it holds also in case of an auction with a deadline, since in all the R-1 rounds the bidder only has to signal his participation In other words, this constraint means that if seller S j is weak and to follow the protocol which means to increment the last enough, then seller S i can beat him. The probability that this proposed bid by the minimal increment required D as in an situation will happen is: auction without a deadline. The deadline plays no role in each of m 2 the R-1 rounds since all the bidders can see all the proposed bids 1 wt and consequently can update their bid in the following sessions. In 1− F ⋅ ∑ 4 B + D at order to understand the intuition lets consider the other two ( ) t=1 possible strategies. If the seller proposes a price that eventually Now the main goal of seller S i is to ensure that this situation yields him a score less than the score of the last proposed bid than holds for all the k sellers, which may bid after him in order to win it will be rejected. On the other hand, if the seller proposes a price by proposing a bid that yields score B. So the expected utility of that yields him more than the score of the last bid +D than in the seller S i from bidding {q * 1,q * 2,...,q * m, p}is its utility from next turn he will have to propose more and he actually enforces all participants including himself to loose therefore the seller will bidding {q * 1,q * 2,...,q * m, p}multiplied by the probability that propose exactly as claimed.■ with this bid he will overcome all the k sellers, which may bid after him. That is, the probability to beat one of the k sellers in the The sellers that survive the first R-1 rounds and reach the last power of k. So the Expected utility of a bid {q * 1,q * 2,...,q * m, p} round have to change their strategy in the last round in order to ensure their winning, since they will not have another chance to is: improve their bid. Therefore, each of these sellers should take into consideration the last proposed bid and the number of sellers that k m m 2 Empirically, when we checked more than 1000 different 1 wt parameter values, sol1 was always a positive bid while sol2 was ExpUsi = p * −θ ⋅ ∑ at ⋅ q * t ⋅ 1− F ⋅ ∑ t=1 4(B + D) t=1 at negative. Moreover, sol1 was the unique maximum point (e.g. Figure 1). In Figure 1 we see that the maximum point of sol1 is about 14 (14.09). On the negative side, a minimum point sol2 In this phase seller S i has to determine a price to bid in the last exists (note that we assume that the score of the bid should be round, which will maximize his expected utility. By solving a higher than a minimal value, since an undefined interval of the maximization problem of the seller’s expected utility function ExUs function for very low values of sol1exists. with regard to price p, the seller can calculate the optimal price p*. In the following theorem we provide the optimal bid (qualities In the following Lemma we state that as θ decreases, the value of and the price) to be offered by seller S i , which he bids in the last sol1 increases. This is intuitively correct since as the seller is round and in which there are k sellers that may bid after him. stronger he can afford better bids. Theorem 1 Lemma 3 Given the sequential auction protocol with a fixed deadline, the Given k, if θi <θj then sol1(θi ,k)>sol1(θj ,k). optimal strategy of the bidder is to bid the minimal required bid in each of the R-1 rounds, and in the last round the bidder should Sketch of proof bid This can be derived from sol1’s expression, given the permitted 2 parameter values.■ wi where i ∈[1,..,m]. qi *(θ ) = , 2⋅aiθ and
m 2 1 wt p* = −s + ∑ 4θ t =1 atj where s is equal to the following scoring value of the buyer s = max{sol, S(selected) + D} where sol is the score of the optimal bid (including the qualities and the optimal price) and is one of the following values (sol1, or sol2) for which the second differentiation of the ExUsj is negative, Figure 1. The Seller’s expected utility given the scoring value where : is grater than 2(f=10,D=0.01.k=6. θ =0.1, θ =1, θ =0). 2 16θ θDfk In Figure 2 we show an example of the optimal bid of a seller in + k 2 f 2θ 2 the last round’s behavior as a function of the bidder's type θ . 1 2 2 That is as the seller efficiency decreases (θ increases) the score sol1 = ⋅ − 32θθD − 4kfθ + 4θf + 4 − 2kf θ of the optimal bid decreases because he can only offer worse bids. 32θθ 2 2 + f θ + 2 4θθkf
2 16θ θDfk + k 2 f 2θ 2 1 sol2 = ⋅ − 32θθD − 4kfθ + 4θf − 4 − 2kf 2θ 2 32θθ 2 2 + f θ + 2 4θθkf m wi 2 where f is: f = ∑ . i=1 ai Figure 2: The scoring of the optimal bid, given the bidder’s type. Sketch of proof Similarly, as k increases, the value of the optimal bid also The qualities are determined following Lemm 2 And the price is increases, as proved in the following Lemma and as demonstrated found by solving a maximization problem of the ExpUs with in Figure 3. regard to price p.■ Lemma 4 If k=0, i.e., the strongest bidder is the last one, then he will just bid the previous bid+D. In this case, if the type of the second best Given θ , if k1 Lemma 6 Suppose that the best bidder is of type θ 1 , and the second Figure 3: The scoring of the optimal bid of a bidder with type strongest is of type θ 2 . If the difference between them is at least θ (given the number of bidders after him is k). D, then ERbuyer(θ 1 ,θ 2 ) is between: To summarize, the optimal strategy of the bidder in the last round sol(θ 2 ,1)+D < ERbuyer(θ 1 ,θ 2 ) < sol(θ 1 ,n-1)+(n-1)*D. depends on its type, θ , and on k, which is the number of bidders Sketch of proof after him in the last round. The other parameter values are known to all participants and to the auctioneer. Given that the strongest seller is of type θ 1 , no seller after θ 1 will have an optimal bid higher than sol(θ 1 ,k). However, there 3.2 Buyer’s Strategy in a Sequential Multi- may be sellers weaker than the seller with θ 1 that will be able to attribute English Auction with a Deadline suggest bids better than sol(θ 1 ,k). The strongest seller can be the In this section we would like to reveal the expected revenue of the first bidder in the last round (bidder 1). In this case, the revenue of auctioneer, given the above protocol, and given the optimal bid of the buyer will be between sol(θ 1 ,n-1) and sol(θ 1 ,k)+(k-1)*D. On each bidder. the other hand, he may also be the last bidder (bidder n). In this case he will bid the score of the last previous bid+D. The score of Denote by sol(θ ,k) the best bid of a seller with a private cost the best bid at time n-1 depends on the best bid in the n-1 rounds. parameterθ , when there are k bidders after him in the last round. This bid depends on the second best type, θ 2 . In particular, it is Assuming that the real weights of the various attributes are between sol(θ 2 ,1) (if the turn of θ 2 is at time n-1) and specified in the scoring rule, then the real utility of the auctioneer sol(tθ 2 ,(n-1))+(n-2)*D (if the turn of θ 2 is at time 1, and all the from a bid that yields score sol(θ ,k) is equal to sol(θ ,k ). In other bidders are strong enough). Since sol(θ 1 ,n-1)>sol(θ 1 ,1), other words, Ubuyer(θ ,k)=sol(θ ,k) if k>0. and sol(θ 2 ,1)