882 ECAI 2012 Luc De Raedt et al. (Eds.) © 2012 The Author(s). This article is published online with Open Access by IOS Press and distributed under the terms of the Creative Commons Attribution Non-Commercial License. doi:10.3233/978-1-61499-098-7-882
Multi-unit Double Auction under Group Buying1
Dengji Zhao2 and Dongmo Zhang3 and Laurent Perrussel4
Abstract. Group buying is a business model in which a number of All participants benefit from successful group buying deals: con- buyers join together to make an order of a product in a certain quan- sumers enjoy good services with lower prices, merchants promote tity in order to gain a desirable discounted price. Such a business their services and most likely more consumers will buy their services model has recently received significant attention from researchers in with normal prices in the future (i.e. group buying also plays a role economics and computer science, mostly due to its successful appli- of advertising), and the company providing the platform benefit from cation in online businesses, such as Groupon5. This paper deals with merchants’ revenue. the market situation when multiple sellers sell a product to a number Besides its simple concept and its successful business applications, of buyers with discount for group buying. We model this problem as a group buying is not well studied in academia [1, 3, 2, 5]. It is not be- multi-unit double auction. We first examine two deterministic mech- cause the idea is new, but the combination of collective buying power anisms that are budget balanced, individually rational and only one- and advertising challenges theoretical analysis. In this work, we ex- sided truthful, i.e. it is truthful for either buyers or sellers. Then we tend the simple concept, used by Groupon and most other similar find that, although there exists a “trivial” (non-deterministic) mech- platforms, to allow merchants (or sellers) and consumers (or buy- anism that is (weakly) budget balanced, individually rational and ers) to express more of their private information (aka type). More truthful for both buyers and sellers, such a mechanism is not achiev- specifically, instead of one single discounted price for selling a cer- able if we further require that both the trading size and the payment tain number of units of a product, sellers will be able to express dif- are neither seller-independent nor buyer-independent. In addition, we ferent prices for selling different amounts of the product. Buyers will show that there is no budget balanced, individually rational and truth- be able to directly reveal the amount they are willing to pay for a ful mechanism that can also guarantee a reasonable trading size. product, other than just show interest in buying a product coming with a fixed price. To that end, we do not just enhance the expres- sion of traders’ private information, but also reduce the number of 1 Introduction no-deal failures that happen when the number of buyers willing to Group buying (or collective buying power) is when a group of con- purchase a product does not reach the predetermined minimum on sumers come together and use the old rule of thumb, there is power the Groupon platform. Moreover, we will allow multiple sellers to in numbers, to leverage group size in exchange for discounts. Led by build competition for selling identical products. Groupon, the landscape for group buying platforms has been grow- Given the above extension, what we get is a multi-unit double auc- ing tremendously during last few years. Due to the advent of social tion, where there are multiple sellers and multiple buyers exchang- networks, e.g. facebook, this simple business concept has been lever- ing one commodity and each trader (seller or buyer) supplies or de- aged successfully by many internet companies. Taking the most suc- mands multiple units of a commodity. Different from the multi-unit cessful group buying platform Groupon for example, a group buying double auctions studied previously [7, 4], the focus of this model is deal is carried out in the following steps: group buying and we assume that sellers have unlimited supply and a seller’s average unit price is decreasing (non-increasing) when the 1. the company searches good services and products (locally) that number of units sold is increasing. The unlimited supply assumption normally are not well-known to (local) consumers, simplifies the utility definition of sellers, and it is not clear to us how 2. the company negotiates with a target merchant for a discounted to properly define sellers’ utility when their supply is limited. price for their services and the minimum number of consumers Due to revelation principle, we only consider mechanisms where required to buy their services in order to get this discount, traders are required to directly report their types. We will pro- 3. the company promotes the merchant’s services with the dis- pose/examine some mechanisms in terms of, especially, budget bal- counted price within a period, say two days, ance, individual rationality,andtruthfulness, which are three impor- 4. if the number of consumers willing to buy the services reaches the tant criteria we usually try to achieve in designing a double auction. minimum during that period, then all the consumers will receive Budget balance guarantees that the market owner running the auc- the services with the discounted price, and the company and the tion does not lose money. Individual rationality incentivises traders merchant will share the revenue. Otherwise, no deal and no loss to participant in the auction, as they will never get negative util- for any party, especially the merchant and consumers. ity/benefit for participating in the auction. Truthfulness makes the game much easier for traders to play, because the best strategy can 1 This research was supported by the Australian Research Council through Discovery Project DP0988750. be easily computed for each trader, which is just his true type. Truth- 2 University of Western Sydney, Australia & University of Toulouse, France, fulness also plays an important role for achieving other properties email: [email protected] based on traders’ truthful types, e.g. efficiency (i.e. social welfare 3 University of Western Sydney, Australia, email: [email protected] 4 maximisation). We will not measure social welfare in this model, University of Toulouse, France, email: [email protected] due to unlimited supply. However, we will consider the number of 5 www.groupon.com D. Zhao et al. / Multi-Unit Double Auction Under Group Buying 883 units exchanged, called trading size, which is part of market liquid- let R(v) be the set of all possible type profile reports of traders with B ity, indicating the success of an exchange market. type profile v.Wewillusev =(vi)i∈B to denote the type profile S We find that, even without considering other criteria, budget bal- of buyers, and v =(vi)i∈S for sellers. ance, individual rationality and truthfulness are hard to be satisfied Definition 1. An multi-unit double auction (MDA) M =(π,x) together in this model. It is shown that there is no budget balanced, consists of an allocation policy π =(πi)i∈T and a payment policy individually rational and truthful auction, given that both the trad- + x =(x ) ∈ , where, given traders’ type profile report v, π (v) ∈ Z ing size and the payment are neither seller-independent nor buyer- i i T i indicates the number of units that seller (buyer) i sells (receives), and independent, although we do get mechanisms that are budget bal- + x (v) ∈ R determines the payment paid to or received by trader i. anced, individually rational and one-sided truthful, i.e. truthful for i either buyers or sellers. We say a parameter of a mechanism is seller- Note that the above definition of MDA contains only determin- independent (buyer-independent) if its value does not depend on sell- istic MDAs, i.e. given a type profile report, the allocation and ers’ (buyers’) type reports. However, if we allow either the trading payment outcomes are deterministic. We will also consider non- size or the payment to be seller-independent or buyer-independent, deterministic/random MDAs where the outcomes are random vari- we will be able to design auctions satisfying budget balance, in- ables. A non-deterministic MDA can be described as a probability dividual rationality and truthfulness at the same time. In addition, distribution over deterministic MDAs. we prove that there is no budget balanced, individually rational and Given MDA M =(π,x) and type profile v, we say trader i π v > π truthful mechanism that can also guarantee trading size. wins if i( ) 0, loses otherwise. An allocation is feasible π v π v S B v This paper is organised as follows. After a brief introduction of the if i∈B i( )= i∈S i( ) and for all , and .AnMDA model in Section 2, we propose two budget balanced, individually ra- M =(π,x) is feasible if π is feasible. A non-deterministic MDA is tional and partially truthful (deterministic) mechanisms in Section 3 feasible if it can be described as a probability distribution over fea- and 4. Following that, we further check the existence of (weakly) sible deterministic MDAs. Feasibility guarantees that the auctioneer budget balanced, individually rational and truthful mechanisms in never takes a short or long position in the commodity exchanged in Section 5. Finally, we conclude in Section 6 with related and future the market. For the rest, only feasible MDAs are discussed. work. Given traders’ type profile v, their type profile report vˆ ∈ R(v) and deterministic MDA M =(π,x),theutility of trader i with type vi is defined as 2 The Model vi(πi(ˆv)) − xi(ˆv), if i ∈ B. u(vi, v,ˆ (π,x)) = We study a multi-unit double auction where multiple sellers and mul- xi(ˆv) − vi(πi(ˆv)), if i ∈ S. tiple buyers exchange one commodity. Each seller supplies an un- M limited number of units of a commodity and each buyer requires Considering might be non-deterministic, we use E u v , v, π,x i a certain number of units of the commodity. Each trader (seller [ ( i ˆ ( ))] to denote the expected utility of trader . or buyer) i has a privately observed valuation function (aka type) Definition 2. An MDA M =(π,x) is truthful (or incentive- v Z+ → R+ i : where the input of the function is the number of compatible)ifE[u(vi, (vi, vˆ−i), (π,x))] ≥ E[u(vi, v,ˆ (π,x))] for units of the commodity and the output is the valuation for those units all i ∈ T ,allvˆ ∈ R(v),allv. together. In other words, a mechanism is truthful if reporting type truth- We assume that sellers’ valuation is monotonic: vi(k) ≤ vi(k + ( ) ( +1) fully maximises each trader’s utility. We say an MDA M is buyer- 1), and satisfies group buying discount: vi k ≥ vi k .Thatis, k k+1 truthful (seller-truthful)ifM is truthful for at least buyers (sellers). a seller’s valuation is non-decreasing as the number of units to sell An MDA is budget balanced (BB) if the payment received from increases, while the mean unit valuation is non-increasing (so buyers buyers is equal to the payment paid to sellers, and it is weakly budget can get a discount if the mean valuation is decreasing). One intuition balanced (WBB) if the payment received from buyers is greater than for group buying discount constraint is that the average unit produc- the payment paid to sellers. An MDA is individually rational (IR) tion cost may decrease when many units can be produced at the same if it gives its participants non-negative utility. Because of unlimited time. For a buyer i of type v requiring c > 0 units, v satisfies i i i supply, we will not be able to measure social welfare in this model, v (k)=0for all k
M c-competitive S ( ) Definition 3. An MDA is if the (expected) trading vi k p k kopt(v) mini=w(k) k = ( ). In other words, the mean unit price is kM(v) M v size of is at least c , for all type profile report .We non-increasing as the number of units sold together increases. say M is competitive if M is c-competitive for a constant c. We refer Because of vi(k+1) ≥ vi(k) for each seller i, we conclude p(k+ to c as competitive ratio. S S 1) · (k +1)= mini=w(k+1) vi (k +1) ≥ mini=w(k) vi (k)= p(k) · k. Moreover, other than following Definition 2, we will use Proposi- tion 1 to analyse the truthfulness of an MDA. Proposition 1 is based Theorem 1. M2nd is buyer-truthful. on Proposition 9.27 of [9], and its proof directly follows the proof there. Proof. The auction result of M2nd for buyer i is either receiving one unit with certain payment or receiving nothing with no payment. Proposition 1 (Proposition 9.27 of [9]). An MDA M =(π,x) is B ∗ If i received one unit, then vi (1) ≥ p(k ) and the payment of i truthful if and only if it satisfies the following conditions for every ∗ B is p(k ) which is independent of vi (1). Otherwise, we know that ∗ trader i with type vi and every v−i B vi (1)
u v − p k∗ i M M+ = i(1) ( +), while the utility will get in 2nd is 2nd ∗ B ∗ ∗ ∗ ∗ 2. Let k =max{k|vk (1) ≥ p(k)}, and i = w(k ). u v − p k u − u p k − M2nd = i(1) ( ).Soweget M2nd M+ = ( +) K {k|vB ≥ p k } K∗ 2nd 3. Let = k (1) ( ) ,and is the least set such ∗− ∗ ∗ ∗ ∗ ∗ ∗ k k+ ∗ i ∈ K K ⊇{k|k K \ K ∧ w k p(k ) ≤ ∗ p(k ). that and =max( ) ( )= k+ S ∗ ∗ B v3rd(min K ) S i ∧v ∗ < } v k min K (1) min K∗ ,where 3rd( ) is the third lowest valuation of sellers for selling k units and it is ∞ if 5 Existence of (W)BB, IR and Truthful MDAs there are less than three sellers. ∗ S 4. Let k+ = max argmaxk∈K∗ (p(k) · k − vi∗ (k)). Following the results in previous sections, we demonstrate in this ∗ 5. The first k+ buyers, i.e. buyers of valuation section that there are multi-unit double auctions that are (weakly) B B B v1 ,v2 , ··· ,v ∗ , receive one unit of the commodity budget balanced, individually rational and truthful. However, we also k+ ∗ prove that there does not exist a (weakly) budget balanced, individ- each and each of them pays p(k+). ∗ ∗ ually rational and truthful MDA, in which both the trading size and 6. Seller i sells k+ units of the commodity and receives pay- ∗ ∗ the payment are neither seller-independent nor buyer-independent. ment p(k+) · k+. 7. The rest of the traders lose without payment. Proposition 3. There exists (weakly) budget balanced, individually rational, and truthful multi-unit double auctions.
∗ ∗ + Proof. The fixed pricing MDA described in Auction 1 is BB, IR and k and the winning seller i of M is the same as that in M2 . 2nd nd truthful. Given a predetermined transaction price p, M first cal- Set K contains all possible numbers of units that can be exchanged fixed ∗ culates the total number k1 of buyers whose valuations are at least p, without sacrificing budget balance. Set K contains all k points that ∗ ∗ then calculates the maximal number k of units that a seller can sell, seller i can manipulate and force the auctioneer to choose some ∗ ∗ ∗ ∗ with non-negative utility, under unit price p, given that k ≤ k1.Af- k ∈ K if M2nd is used. The reason is that, for all k ∈ K ex- ∗ ∗ ter it calculates all the winning candidates of both sides, candidates cept the minimum (min K ), seller i is the only winner, i.e. without ∗ from the same side win with the same probability. It is evident that seller i , there is no other seller who can win at those points. There- M+ k∗ ∈ K∗ this auction is budget balanced and individually rational. fore, 2nd chooses + , as the final trading size, such that ∗ ∗ Regarding truthfulness, firstly, payment p does not depend on any seller i ’s utility is maximised among all k ∈ K . It is evident that + trader. Secondly, all buyers whose valuation for one unit is at least p M is also budget balanced and individually rational. 2nd will win with the same probability with payment p, so their utilities ∗ M+ are maximised if their winning probability k is maximised. Buyer Theorem 3. 2nd is seller-truthful but not buyer-truthful. k1 B B i of vi (1) ≥ p will not report vˆi (1) Auction 2 (One-sided Pricing MDA Msingle). Given type profile Proof of Theorem 4. We first assume that there is such MDA M,and report v =(vB ,vS), then we end up with a contradiction. Let p and p be the payment (unit price) for winning sellers and 1. let p be the m -th highest of vB (1)s, where m is the total number s b 2 i winning buyers respectively. According to Lemma 3, without loss of buyers, B of generality, we assume that M selects at most one winning seller. 2. let k1 = |{i|vi (1) >p}|, B S Assume the trading size is k.Letvmin be the minimum valuation ∗ vi (k) k {k|k ≤ k1 ∧ ≤ p i} k2 B 3. let =max k for some , and = (for one unit) of all winning buyers, and v be the maximum val- S ∗ max |{i| vi (k ) ≤ p}| uation of all losing buyers (vB =0if there is no losing buyer). k∗ , max ∗ B S 4. randomly select k winning buyers from {i|v (1) >p},i.e.each Let vwin be the valuation of the winning seller for selling k units, i ∗ B k S i ∈{i|v (1) >p} and vmin be the minimum valuation of all losing sellers for selling k buyer i wins with probability k1 , S ∗ vS ∞ vi (k ) units ( min = if there is no losing seller). Because of individual 5. randomly choose one winning seller from {i| ∗ ≤ p},i.e. S k vwin B S ∗ rationality, we have ≤ ps ≤ p ≤ v .SinceM is truthful, vi (k ) 1 k b min each seller i ∈{i| ∗ ≤ p} wins with probability , S k k2 p ≤ vmin p ≥ vB p p 6. each winning buyer receives one unit of the commodity and pays we further get s k and b max and s and b should ∗ ∗ M k p, the winning seller sells k units and receives payment p ∗ k , not depend on any winning trader. Therefore, if chooses any satisfying any of the following four conditions, there will be proper and all the rest of the traders lose with no payment. B S payments ps ≤ pb only depending on vmax and vmin. Theorem 4. There is no (weakly) budget balanced, individually ra- S tional and truthful multi-unit double auction, where both the trad- vmin ≤ vB 1. k max, ing size and the payment are neither seller-independent nor buyer- S S S vmin >vB vB ≥ vmin vB ≥ vwin independent. 2. k max, min k ,and max k , S S S vmin >vB vB ≥ vmin vB < vwin 3. k max, min k ,and max k , Before we give the proof of Theorem 4, we first prove some lem- S S S vmin >vB vB < vmin vB ≥ vwin mas that are going to be used for the proof. Lemma 2 says that an 4. k max, min k ,and max k . IR and truthful MDA cannot have price discrimination. An MDA has S price discrimination if buyers (sellers) pay (receive) different pay- p ,p ∈ vmin ,vB p ≤ p For condition (1), b s [ k max] s.t. s b. For condition ments for identical goods or services. For instance, when two buyers S p ,p ∈ vB , vmin p ≤ p p (2), b s [ max k ] s.t. s b. For condition (3), b = pay different prices for receiving one unit of the same commodity in S p vmin p p vB a deterministic MDA, this is considered as price discrimination. s = k ,and b = s = max for condition (4). In other words, M chooses any k satisfying any of the above four Lemma 2. An individually rational multi-unit double auction with conditions can also get payments independent of winning traders and price discrimination is not truthful. satisfying (weakly) budget balance. Besides these four conditions, Proof. Because of individual rationality, the expected payments for we cannot choose any k under other conditions where we can still get all winning buyers (sellers) must not over (under) their valuations.6 (weakly) budget balanced and winning trader independent payments, If the expected payments are not the same between winning buy- given that both k and ps,pb are neither seller-independent nor buyer- ers/sellers, then a winning buyer (seller) with high (low) expected independent. payment will have a chance to manipulate the auction in order to get Therefore, in order to satisfy truthfulness, M has to choose a k a low (high) expected payment by, for example, reporting the same such that all traders’ utilities are maximised. For winning buyers, valuation as that of a winning buyer (seller) receiving relatively a they would prefer a bigger k as their payment will be lower compared lower (higher) expected payment. to the payment with a lower k, i.e. their utilities are maximised when 6 k is maximised. However, the winning seller might prefer a lower Note that we consider expected payment to check price discrimination, k k because if an MDA is non-deterministic and it can assign different pay- as her utility is not necessarily maximised with maximum (see ments to winning buyers/sellers. However, if a non-deterministic MDA is the proof of Theorem 2 for example). Thus, we may not always be individually rational and truthful, then the expected payment will be the able to choose a k maximising both buyers’ and sellers’ utilities. This same for all winning buyers/sellers and the prices should be randomly cho- contradicts the truthfulness of M, i.e. buyers may be incentivised to sen from some range independent of winning traders’ valuations. A non- k deterministic MDA is not considered price discrimination if the expected disable the above four conditions for lower s, while sellers may be payment is the same for all winning/losing buyers/sellers. motivated to disable that for higher ks. D. Zhao et al. / Multi-Unit Double Auction Under Group Buying 887 ∗ 5.1 Competitive MDAs i M k ·ci c of buyer in step 3 of fixed will be k1 .As is are not part of buyers’ private information, this extension will not affect any of the Corollary 1. There is no (weakly) budget balanced, individually ra- properties that hold in the single-unit demand case. tional, truthful multi-unit double auction that is also competitive. As closely related work, Huang et al. [7] proposed weakly budget Proof. From Theorem 4, we know that there is no (W)BB, IR, balanced, individually rational and truthful multi-unit double auc- truthful, and competitive multi-unit double auction, if both the trad- tions, under the model where each seller (buyer) supplies (demands) ing size and the payment are neither seller-independent nor buyer- a publicly known number of units, their valuation for each unit is not independent. In the following, we will prove that if the trading size changing and their requirements can be partially satisfied. Chu [4] or the payment of an MDA is either seller-independent or buyer- studied a multi-unit double auction model where there are multiple independent, the MDA will not be competitive. commodities, each seller supplies multi-units of one commodity and If the trading size of MDA M is seller-independent, say the ex- each buyer requires a bundle of different commodities. They pro- posed a method that intentionally creates additional competition in pected trading size is ke,thenke must be also buyer-independent, otherwise we can always find a example that violates budget balance, order to get budget balanced, individually rational and truthful mech- individual rationality and truthfulness. For instance, each seller’s unit anisms. Wurman et al. [10] also considered one-sided truthful dou- valuation for selling any number of units is larger than the highest ble auctions for optimising social welfare. Goldberg et al. [6] studied valuation of sellers, in which the trading size should be zero if BB, one-sided auctions where the seller has an unlimited supply without giving any valuation or reserve price for the commodity, and their IR and truthfulness are satisfied. Therefore, given ke > 0 is trader- independent, for any type profile report v with optimal trading size gaol is to design truthful mechanisms that guarantee the seller’s rev- kopt(v) enue. For group buying, Edelman et al. [5] considered the advertis- k (v), the competitive ratio c = . It is clear that c is not opt ke ing effect of discount offers by modelling the procedure with two bounded as k (v) can be any value approaching to infinite. opt periods, so traders can come back in the future after getting dis- If the payment of MDA M is seller-independent, then for any counted offers. Arabshahi [2] provided a very detailed analysis of payment determined without considering sellers, there exists a case the Groupon business model and Byers et al. [3] showed some pri- where all sellers’ unit valuation for selling any number of units are mary post-analysis of Groupon. A very earlier study of online group higher than the payment, which means that the trading size will be buying is provided by Anand and Aron [1]. zero if M is (weakly) budget balanced, individually rational, truth- There are many questions for considering group buying in multi- ful. Therefore, M cannot be competitive under this condition. This unit double auction worth further investigation. Especially, if sell- result also holds when the payment is buyer-independent. ers have limited supply, how do we calculate their utilities, as they should have valuation for the unsold units and the valuation for the 6Conclusion unsold units is not the same before and after the auction, raising the further question of how to optimise social welfare and guaran- In this paper, we studied a multi-unit double auction, where each tee other properties in this case. For instance, a seller supplies two seller has an unlimited supply, for exchanging one kind of commod- units with unit prices p1 >p2 for selling one and two units respec- ity. Different from the previous studies of multi-unit double auction, tively. If we end up with one unit left for the seller, we might consider we introduced group buying in the model. 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