Lowest Unique Bid Auctions with Resubmission Opportunities

Yida Xu and Hamidou Tembine Learning & Game Theory Laboratory, New York University Abu Dhabi, United Arab Emirates Tandon School of Engineering, New York University, U.S.A.

Keywords: LUBA, Auction, Game Theory, Imitative Learning, Reinforcement Learning.

Abstract: The recent online platforms propose multiple items for bidding. The state of the art, however, is limited to the analysis of one item auction. In this paper we study multi-item lowest unique bid auctions (LUBA) in discrete bid spaces under budget constraints. We show the existence of mixed Bayes-Nash equilibria for an arbitrary number of bidders and items. The equilibrium is explicitly computed in two bidder setup with resubmission possibilities. In the general setting we propose a distributed strategic learning algorithm to approximate equilibria. Computer simulations indicate that the error quickly decays in few number of steps by means of speedup techniques. When the number of bidders per item follows a Poisson distribution, it is shown that the seller can get a non-negligible revenue on several items, and hence making a partial revelation of the true value of the items.

1 INTRODUCTION (first, second, English or Deutch). Most of research articles (Bang-Qing, 2003; C. Yi, 2015 ; N. Wang, With the increasing impact of information technology, 2014; Rituraj and Jagannatham, 2013) on multi-item the traditional structure of economic and financial auctions works provide computer simulation or nu- markets has been revolutionized. Today, the market merical experiments results. However, no analysis includes millions of economic agents worldwide and of the outcome of the multi-item auction is available. reaches an annual transaction of billions U.S. dollars. There is no analysis of the equilibrium seeking algo- In this paper, we study a little-studied type auction rithm therein. which is called lowest unique bid auction (LUBA). LUBA: LUBA is different from the lowest cost auc- Because of a limited number of items provided by the tion called procurement auction which is widely used auctioneers and budget restrictions, bidders need to in e-commerce(J. Zhao, 2015), hybrid cloud comput- manage their behaviors in a strategic way. ing or in demand-supply matching in power grids (R. Zhou, 2015). As a special case of unmatched bid auc- Literature Review tions, the single-item LUBAs have been studied by other researchers (H. Houba, 2011; Stefan and Nor- Auctions: In 1961 in the work of Vickrey (Vick- man, 2006; Rapoport, 2007; Erik, 2015; J.Eichberger, rey, 1961), the theory of auctions as games of in- 2008). complete information is first proposed. Auctions The authors (Stefan and Norman, 2006) run lab- with homogeneous valuation distributions (symmet- oratory experiments with one bid min bid auctions. ric auctions) and self-interested non-spiteful bidders They consider the results of a Monte Carlo simula- are well-investigated in the literature. However, auction under the restriction of one bid per player. The tions with asymmetric bidders remain a challenging work in (Erik, 2015) considers a lowest unique posi- open problem (see (Maskin and Riley, 2000; Lebrun, tive integer experiment in a single bid per player set- 1999; Lebrun, 2006) and the references therein). In ting. The observed behaviors are compared with the asymmetric auction scenarios, the expected “revenue solution of a Poisson game. The authors in (Rapoport, equivalence theorem” ( Myerson 1981) does not hold, 2007) consider both high and low unique bid auctions, i.e., the revenue of the auctioneer depends on the and they also assume that bidders are restricted to a auction mechanism employed. In addition, there is single bid. A numerical approximation of the solu- no ranking revenue between the auction mechanisms tion for a game-theoretic model is provided. The so-

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Xu, Y. and Tembine, H. Lowest Unique Bid Auctions with Resubmission Opportunities. DOI: 10.5220/0006548203300337 In Proceedings of the 10th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2018) - Volume 2, pages 330-337 ISBN: 978-989-758-275-2 Copyright © 2018 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved Lowest Unique Bid Auctions with Resubmission Opportunities

lutions are compared with the results of a laboratory time, the winner can be calulated by j N . ∗ i,infBi∗ experiment. (J.Eichberger, 2008; J.Eichberger, 2015; ∈ J. Eichberger, 2016) conduct ﬁeld experiments on 2.1 The Payoff of Participants Lowest-Unmatched Price Auctions with high prizes involving large numbers of participants (tens and hun- The cost of bidder j on item i consists of the registra- dreds of thousands). The work in (Dong-Her, 2011) tion fee cr, the submission fee B ji ci, and the bid fee discusses security and privacy issues of multi-item re- which is conditional on moves| by|∗ the other bidders. If verse Vickrey auction. They provide more secure pro- he/she is the winner, the bid fee is infBi∗ calculated by tocol design. Most of the above works restrict the the proceeding subsection. Losing the LUBA yields number of submissions per bidder to one. The re- no bid fee. Thus, the payoff of bidder j on item i at cent focus within the auction ﬁeld has been multi-item a round would be r = v B c infB c , if j ji ji − | ji| i − i∗ − r auctions. The bidder can also place multiple bids for is a winner on item i, and r ji = B ji ci cr, if j is each item. It has been of practical importance in In- not a winner on item i. The payoff−| of bidder| − j on item ternet auction sites and has been widely executed by i is zero if B ji is reduced to 0 (or equivalently the them. empty set). Thus, it is easy to{ conclude} that

r ji(B) (1)

= [ cr ci B ji (v ji b ji)1l b ji=infB∗ ]1l B ji= 0 , 2 PROBLEM STATEMENT − − | | − − { i } { 6 { }} where the infininum of the empty set is zero. Over- all, the total payoff of bidder j is r j(B) = ∑ r ji(B). A LUBA operates under the following three main i I The instant payoff of the auctioneer of item∈i can be rules: Whoever bids the lowest unique positive bid calculated by wins; If no unique bid exits then no one wins; No participant can see bids placed by other participants. r = c 1l + infB + B c v , “Lowest unique bid” means the lowest amount that a,i ∑ r B ji=0/ i∗ ∑ ji i a,i j { 6 } j=1 | | ! − nobody else has bid. In the multi-item LUBA, there are n 2 bidders where va,i is the realized valuation of the auctioneer . exerting effort for m items (objects) proposed≥ by the for item i Obviously, the instant payoff of the auctioneer of a set of item I is ra,I = ∑i I ra,i. In terms of auctioneers. Each bidder has the information as fol- ∈ low. Valuation vector: Each bidder j has a vector of reward seeking, bidders are interested in optimizing their payoffs, and auctioneers are interested in their values v j = (v ji)i I , for assessing the worth of each object i offered for∈ the auction. The random variable revenue. v ji has support [v,v¯] where 0 < v < v¯. A bidder may have his/her own valuation vector but not the valuation vector of the others. Budget: Each bidder j has a 3 SOLUTION APPROACH initial total budget of b¯ j to be used for all items. Reg- istration fee: To participate in the LUBA, each bidder Since the game is of incomplete information, the needs to pay a one time registration fee cr. Submis- strategies of participating in the game must be spec- sion fee: To participate in the auction on item i, he/she ified as a function of the information structure. We needs to pay a submission fee ci for each submission. introduce the definition of pure strategy and mixed Note that each bidder can resubmit bids a certain strategy as follows. number of times subject to his or her available bu- Definition 1. A pure strategy of a bidder is a choice ject. If bidder j has (re)submitted n ji times on item i, of a subset of natural numbers contingent on the own- his/her total submmission/bidding cost would be n jici value and own-budget. Thus, given a bidder’s own in addition to the registration fee. The set that con- valuation vector v j = (v ji)i, the bidder j will choose tains all the bids of bidder j on item i is denoted by an action (B ji)i that satisfies the budget constraints, B ji N. Thus, we can obtain the set of bidders who which is are⊂ submitting b on item i by N = j J b B . m m i,b { ∈ | ∈ ji} ¯ cr1l B ji= 0 + [infBi∗]1lB ji [infB∗] + B ji ci b j. We use Ni,b to represent the cardinality of Ni,b. Ob- ∑ { 6 { }} ∑ ∩ i ∑ | | i i=1 i=1 | | ≤ viously, Ni,b = 1 yields that b was chosen by only one bidder.| Then,| the set of all positive natural num- The set of multi-item bid space for bidder j is bers that were unique on item i can be calculated by ¯ ¯ = > = . = B j(v j,b j) = (B ji)i B ji 0,1,...,b j cr , Bi∗ b 0 Ni,b 1 Bi∗ 0/ yields no winner { | m ⊂ { − } { | | | } c 1l + [infB∗]1l on item i at that round. If B∗ = 0/, there is a winner ∑i r B ji= 0 ∑i=1 i B ji [infBi∗] i 6 { 6 { }}m ¯ ∩ B +∑ = B ji ci b j . on item i, and the winning bid is inf i∗. At the same i 1 | | ≤ }

331 ICAART 2018 - 10th International Conference on Agents and Artiﬁcial Intelligence

A pure strategy is a mapping v j B j N. A con- Table 1: Payoff matrix of 2 bidders bidding 1 item. • 7→ ⊂ strained pure strategy is a mapping v j B j B j. 0 1 2 3 7→ ∈ 0 0, 0 (0,v c˜ 1) 0, v c˜ 2 0, v c˜ 3 − − ∗ − − − − A mixed strategy is a probability measure over the 1 (v c˜ 1,0) c˜, c˜ v c˜ 1, c˜ v c˜ 1, c˜ − − ∗ − − − − − − − − • 2 v c˜ 2,0 c˜,v c˜ 1 c˜, c˜ v c˜ 2, c˜ set of pure strategies. − − − − − − − − − − ¯ The action set B j(v j,b j) is finite because of bud- Proposition 4. When the budgets are identical and get limitations; hence, a bid b ji B ji is less than equal to k, and n = 2, There is a partially mixed equi- ¯ ∈ min(b j,v ji cr). We define a solution concept of librium which is symmetric and is explicitly given by the above game− with incomplete information: Bayes- c˜ c˜ Nash equilibrium. (X ,X ,X ,...,X ) = ( ,1 ,0,...,0). 0 1 2 k v 1 − v 1 Definition 2. A mixed Bayes-Nash strategy equilib- − − Three Participants. With three participants, the ac- rium is a profile (s (v )) such that for all bidders j j j j tion profile (1,1,1) is not an equilibrium anymore: if ( ( ), ) ( , ), Es j,s j r j B j v j B j v j Es0 ,s j r j B0j B j v j for − − | ≥ j − − | players 1 and 2 bid 1 cent then player’s 3 best re- any strategy s0j. sponse is to bid 2 and player 3 will be the winner The information structure of the auction is as fol- of the auction. The game is not a dominant solvable lows. Each bidder knows its own-value and own-bid game. but not the valuation of the other bidders. Each bidder has the valuation cumulative distribution of the others. Table 2: Payoff matrix of 3 bidders targetting 1 item. The structure of the game is common knowledge. We 0 1 0 0, 0, 0 (0,v c˜ 1,0) are interested in the equilibria, the equilibrium pay- − − ∗ 1 (v c˜ 1,0,0) c˜, c˜,0 offs of the bidders, and the revenue of the seller. − − ∗ − − 0 1 0 (0,0,v c˜ 1) 0, c˜, c˜ − − ∗ − − 1 c˜,0, c˜ c˜, c˜, c˜ Existence of Bayes-Nash Equilibrium − − − − − Proposition 5. When the budgets are identical and Proposition 1. The multi-item Bayesian LUBA game n 3, There is a partially mixed equilibrium which is (without resubmission) but with arbitrary number of symmetric≥ and is explicitly given by bidders has at least one Bayes-Nash equilibrium in mixed strategies under budget restrictions. n 1 c˜ n 1 c˜ (X ,X ,X ,...,X ) = ( − ,1 − ,0,...,0). 0 1 2 k v 1 − v 1 Proposition 1 provides the existence of at least one r − r − Bayes-Nash equilibrium. However, it does not tell us what are those equilibria or how can we target them. 3.1 Multi-item LUBA Game The next subsection computes some of the equilibria in specific setups. Now we investigate the property of multi-item LUBA game. First, we analyze a simple setup, then we ex- Computation of Bayes-Nash Equilibrium tend it to a general form. We assume n = 2, b1 = 4+c˜, and b2 = 3 + c˜. Thus, the action profiles of each bid- Proposition 2. Any strategy b j such that b ji > der can be expressed as A = b : b +b 4 and 1 { 1 11 12 ≤ } min v ji c˜ 1,b¯ j is strictly dominated by strat- A = b : b + b 3 . More specifically, − − 2 { 2 21 22 ≤ } egy“0”. If v ji < c, bidder j is better off of not participating on item i. A1 = (0,0),(1,0),(0,1),(2,0),(1,1),(0,2),(3,0), (0,3),{(2,1),(1,2),(4,0),(3,1),(0,4),(1,3),(2,2) Thus, the bid space of j can be reduced to } m ∏ 0,1,2,...,min(v ji c˜ 1,b¯ j) . Now, we ana- A = (0,0),(1,0),(0,1),(2,0),(0,2),(1,1),(3,0), i=1{ − − } 2 lyze some of the equilibria in specific setups. { (0,3),(1,2),(2,1) Two Participants. Let n = 2, and the action space is } restricted to 0,1,2,...,v c˜ , wherec ˜ = c + cr. From the action files, we can derive that { − } (0,0),(1,1) , (1,1),(0,0) , (0,1),(1,0) , Suppose v > c˜ + 1 and n = 2. With Proposition 3. {and (1,0),}(1,0){ are pure} Nash{ equilibria.} the non-participation option, i.e., when the bid can For the{ extended} version, the game can be de- be zero, (0,...,0,1,0,...,0) and its permutations are fined as G(J,(b j) j J,(c,cr),m,(Fj) j J). When equilibria. ∈ ∈ b j is given, we can derive that the action We can easily derive proposition 3 from the payoff profile A = D D D ... D , where j 0 1 2 b j matrix, which is shown in Table 1. ∪ m ∪ ∪ m∪ Dk = (a1,...,am) N ai 0, ∑i=1 ai = k is the set of{ all possible decomposition∈ | ≥ of the integer}k.

332 Lowest Unique Bid Auctions with Resubmission Opportunities

Proposition 6. If b j > 1 + c˜, the set of pure Nash equilibria of multi-item auction game G(J,(b j) j J,(c,cr),m,(Fj) j J), n 1,m 1 without resubmission∈ is the set∈ of matrices≥ ≥

푋2푖(1) b11 b12 b13 ... b1m b21 b22 b23 ... b2m b31 b32 b33 ... b3m ......  bn1 bn2 bn3 ... bnm   푋 (1)   1푖 n Figure 1: The vector field of ODEs where n = 2, v = 4, such that i, j,b ji 0,1 , i 1,...,m ,∑ j=1 b ji = ∀ ∈ { } m∀ ∈ { } c = 1. 1 and j 1,...,n , ∑ b ji b j c˜. The pure ∀ ∈ { } i=1 ≤ − Nash equilibria are also global optima of the game. are given by ˙ 3.2 Learning Algorithm X1i(1) = X1i(1)(1 X1i(1))(2 3X2i(1)) X˙ (1) = X (1)(1 − X (1))(2 − 3X (1)) 2i 2i − 2i − 1i Imitative learning plays a role in many applications Figure 1 shows the vector field of the ODEs. By an- (Tembine, 2012) such as security & reliability, cloud alyzing the vector field, we can observe that the algo- networking, power grid and evolution of protocols rithm always converges to one of pure Nash equilibria and technologies. It has been successfully utilized in except the symetric situation; however, the probabil- order to capture human learning and animal behav- ity of symmetric situation is zero. ior. We calculate the instant payoff of bidder j tar- t Proposition 8. In the symmetric case, the mixed equi- geting item i at tth round through R ji = cr c + librium is stable and the dynamical system converges t − − 1lj wins(v ji b ji). Bidder j wins item i at round t if to it. However, the probability of obtaining equal − t the placing bid b ji is the lowest unique bid on item i. numbers by stochastic processes is nearly zero. t R ji = 0 if bidder j does not participate to item i. This algorithm below describes how to update the reward ˆt and the strategy in the bid space. R ji(k) is the estima- 4 EXPERIMENT tion of the reward a bidder j can obtain by holding the t ˆt bid k. Denote the estimate of R ji(k) by R ji(k) . In this part, we do experiments to show the effective- ness of the imitative learning algorithm. First, we Algorithm 1: The Imitative learning algorithm. present a numerical investigation focusing on learn- ˆ0 ˆ0 ˆ0 ¯ Initialization: Generate R ji(0),R ji(1),...,R ji(b(0)) from the uniform distribution. ing symmetric equilibria for two bidders setting; then, For Round t + 1: ˆt ˆt+1 ˆt t t ˆt 1.Update the reward estimation:R ji(k) : R ji (k) = R ji(k)+1l bt =k α ji(R ji R ji(k)) we extend to the asymmetric situation. Table 3 shows { ji } − ˆt t+1 t t R ji(k) the experiment setting. We compare and contrast the 2.Update the mixed strategy: Xji (k) = Xji(k)(1 + λ ji) 3.Normalize the mixed strategy. results with the theory value presented in proposition Iterate until convergence. 4.

The next result provides a convergence of the al- Table 3: Summary of two bidders setting. gorithm to pure equilibria. Symbol Setting Proposition 7. If the algorithm starts from interior n 2 I 1 point, it converges to an ending point in a stable c 1 bmin 1 steady state of the replicator equation. This counts ˆ0 Initial of R ji [1 1 1] for an arbitrary number of participants. Therefore it Budget Static budget constraint is a Nash equilibrium. In the asymmetric case, the mixed equilibrium is unstable and the algorithm con- As shown in Figure 2, the proposed algorithm can verges to one of pure Nash equilibria. effectively learn the Nash equilibrium at a fast convergence rate. More speciﬁcally, for big v, the learning We give a simple example to show that the mixed algorithm converges in a few steps. For the asymmet- equilibrium is unstable, and the algorithm converges ric situation, the experiments in Figure 3 show that to one of pure Nash equilibria. We analyze the situa- the learning process can also converge to one of pure tion where n = 2, v = 4, c = 1. In this speciﬁc parame- equilibria in a few steps. ter setting, the ordinary differential equations (ODEs)

333 ICAART 2018 - 10th International Conference on Agents and Artiﬁcial Intelligence

0.04 0.02 0.1 0.04

Bidder1 Iteration 1 Iteration 1500 푣

0.02 0.01 0.05 0.02 =

Cumulativedistribution function 0 0 0 0 4 0 10 20 30 40 0 50 100 0 10 20 30 40 0 50 100 150 0.04 0.01 0.1 0.04 Bidder2

0.02 0.005 0.05 0.02 푣

0 0 0 0 = 0 20 40 60 0 50 100 150 0 20 40 60 0 50 100 150

10 0.04 0.02 0.1 0.05 Bidder3 Probability functionmass

0.02 0.01 0.05

0 0 0 0

0 10 20 30 40 0 20 40 60 80 0 10 20 30 40 0 50 100 푣

Bidder4 0.04 0.01 0.1 0.1 =

200 0.02 0.005 0.05 0.05

0 0 0 0 0 10 20 30 40 0 50 100 150 0 10 20 30 40 0 50 100 150 0.2 0.1 0.2 0.1 Bidder1

푣 Iteration 1800 Iteration 2000

0.1 0.05 0.1 0.05 =

0 0 0 0 1000 0 10 20 30 40 0 50 100 150 0 10 20 30 40 0 50 100 150 0.1 0.04 0.1 0.04 Bidder2 0.05 0.02 0.05 0.02

1푠푡 퐼푡푒푟푎푡𝑖표푛 10푡ℎ 퐼푡푒푟푎푡𝑖표푛 1000푡ℎ 퐼푡푒푟푎푡𝑖표푛 8000푡ℎ 퐼푡푒푟푎푡𝑖표푛 0 0 0 0 0 20 40 60 0 50 100 150 0 20 40 60 0 50 100 150 Bid (푛 = 2) 0.1 0.1 0.2 0.1 Bidder3 0.05 0.05 0.1 0.05

0 0 0 0 0 10 20 30 40 0 50 100 0 10 20 30 40 0 50 100 Figure 2: Nash equilibrium. The red squares present the 0.1 0.1 0.2 0.1 Bidder4 0.05 0.05 0.1 0.05

0 0 0 0 strategy learned by the proposed algorithm and the green 0 10 20 30 40 0 50 100 150 0 10 20 30 40 0 50 100 150

Item1 Item2 Item1 Item2 stars represent the theoretical equilibrium. Bid

Figure 4: Probability mass function of four players and two 푣

= items in the auction system with budget update.

4 Probability mass function

푣 Table 5: Summary of Experiment Setting.

= 10 Symbol Setting

푣 N 4 =

200 I 1 α 0.1

λ 0.1 푣

= c 1

1000 bmin 1 0 Initial of Xji Uniform distribution ˆ0 10푡ℎ 퐼푡푒푟푎푡𝑖표푛 80푡ℎ 퐼푡푒푟푎푡𝑖표푛 Initial of R ji Unidrnd(20,1,Budget) v for j 1,2,3,4 [147,165,170,177] Bid 푛 = 2 ji ∈ { } b¯ i for j 1,2,3,4 [100,100,100,100] j ∈ { } Figure 3: Learned strategies. The red squares present bidder Budget Static Budget Constraint 1’s learned strategy under the proposed algorithm and the green stars represent bidder 2’s learned strategy. are utilized to calculate the RMSE. RMSEt,t 1 = ¯ − N b j(t) t t 1 2 ∑ ∑ (X (k) X − (k)) . In order to analyze For multi-item under budget constrains, we also j=1 k=1 ji − ji do a simulation experiment to show the performance theq statistic property of the experiment results, we re- of the proposed learning algorithm. The experiment conduct the experiment 200 times using the experi- settings are shown in Table 4. We utilize a dynamic ment setting in the Table 5. The analysis of the statis- setting with changing budget in this experiment by tic properties are shown in Figure 5. setting different v ji and different initial budget. Con- sidering the resources of item is limited, we also as- sign resources of each item. Figure 4 shows the results.

Table 4: Summary of Multi-Items Experiment Setting.

Symbol Setting N 4 I 2 Figure 5: Statistic information on the RSME of the strate- c 1 gies. bmin 1 Resource for item 1 and 2 [2000 1500] Initial of Budget [100 120 80 90] ˆ0 Figure 5 presents the distribution of RMSE (per- Initial of R ji 0.0001 v for j 1,2,3,4 [40,102],[42,109],[38,100],[36,110] ∈ { } centiles P5 P25 P50 P75 P90) obtained from the pro- b¯ ji min(v ji,budget ji) Assumption Description posed learning algorithm. The results in this ﬁgure Budget Budget evolves with the iteration. show all the repeat experiment converge to zero and a narrow distribution of RMSE over the whole period. 4.1 Statistic Properties 4.2 Impact of Parameters In order to show the robustness of the proposed algorithm, we do an experiment analysis under the static We investigated the impact of the parameters in the budget constraint and analyze its statistic properties. proposed learning algorithm described in Sect.3.2. Table 5 presents the experiment setting. Table 6 shows experiment setting details. Figure 6 We evaluate the convergence rate by introduc- and 7 present the statistic properties of RMSE evo- ing the root-mean-square error (RMSE). The prob- lution obtained by the proposed learning algorithm. ability distributions of sequential round strategies Compared to α = 0.1 and λ = 0.1, the results show

334 Lowest Unique Bid Auctions with Resubmission Opportunities

a fast convergence rate. The Boxplot shows that the mission possibilities). If B = 0/ then there is a winner i∗ 6 large parameter setting results less outliers in the ex- on item i and the winning bid is infBi∗ and winner is periment results. According to the results in Figure j∗ N . The payoff of bidder j on item i at that ∈ i,infBi∗ 6 and 7, the parameter α and λ inﬂuence the conver- round would be r = v B c infB , if j is a win- ji ji −| ji| i − i∗ gence of the proposed algorithm equally and a large α ner on item i, and r ji = B ji ci, if j is not a winner can reduce disturbance and outliers more effectively. on item i. A pure strategy−| of a| bidder is a choice of a subset of natural numbers. That set is ﬁnite because Table 6: Summary of Experiment Setting in Parameters Im- of budget limitation. Thus, given its own valuation pact Investigation. vector (v ji)i bidder j will choose an action (B ji)i that respect the budget constraints Symbol Original Setting Compared Setting α 0.1 1 m m λ 0.1 1 j, c + [infB∗]1l + B c b¯ . Others Same as Table 5 r ∑ i B ji [infBi∗] ∑ ji i j ∀ i=1 ∩ i=1 | | ≤

A bid is hence less than min(b¯ j,v ji c). The instant payoff of the auctionner of item i is −

= + + . ra,i ∑cr1l B ji=0/ infBi∗ ∑ B ji ci va,i j { 6 } j=1 | | ! − The instant payoff of the auctionner of a set of item I is ra,I = ∑i I ra,i. The following result provides existence of equilibria∈ in behavioral mixed strategies. Proposition 9. The multi-item LUBA game with re- Figure 6: Statistic information on RSME on strategies. submission has at least one equilibrium in behavioral mixed strategies. Proposition 10. Let n = 2 and each bidder can resubmit a certain number of bids, each resubmission will cost c < v 1. The game has a partially mixed equilibrium which− is explicitly given by

k 1 c c c − c y∗ = ( , ,..., ,1 ,0,...,0) v 1 v 2 v k − ∑ v (l + 1) − − − l=0 − ¯ v min(b, c ) c˜ where k is the maximum number such that ∑l=0 v l < 1. − Figure 7: Statistic information on RSME. We now investigate how much money the online platform can make by running multi-item LUBA. Since the platform will be running for a certain time 5 LUBA WITH RESUBMISSION before the auction ends, each bidder is facing a a random number of other bidders, who may bid in In this section, we assume that each bidder can resub- a stochastic strategic way. Their valuation is not mit bids a certain number of times subject to her avail- known. We need to estimate the set of bids Bi and able budget, each resubmission for item i will cost ci. the bid values on item i. We denote by nib the random If bidder j has (re)submitted n ji times on item i, his number of bidders who bid on b. If ni,b follows a Pois- or her total submission cost would be n jici in addition son distribution with parameter λi,b, and that all vari- to the registration fee. Denote by the set that con- ables ni,b are independent. The value λi,b is assumed tains all the bids of bidder j on item i by B ji. Thus, to be non-decreasing with b. The expected payoff of n ji = B ji is the cardinality of the strictly positive bids the seller on item i is ∑ j cr +∑b Eni,bci +EinfB∗ vi. | | i − by j on item i. The set of bidders who are submitting vi 1 Proposition 11. Let λi,b = c (1+b)z , with z > 0. b on item i is denoted by Ni,b = j N b B ji . In i order to get the set of all unique{ bids,∈ we| ∈ introduce} The expected revenue of the seller on item i is + + [ 1 ], the following: The set of all positive natural num- ∑ j cr EinfBi∗ vi ∑b (1+b)z exceeds the value bers that were chosen by only one bidder on item i vi of the item i for small value of z whenever 1 ∑ j cr+EinfBi∗ is B∗ = b > 0 Ni,b = 1 . If B∗ = 0/ then there is vi z > 1 i i ∑b min(b¯, ) (1+b) vi no winner{ on item| | i at| that} round (after all the resub- ≤ ci −

335 ICAART 2018 - 10th International Conference on Agents and Artiﬁcial Intelligence

6 CONCLUSION Conference (WMNC), 2013 6th Joint IFIP, Dubai, 2013, pp. 1-6. We propose and analyze a multi-item LUBA game J. Zhao, X. Chu, H. Liu, Y. W. Leung and Z. Li, Online with budget constraint, registration fee and resubmis- procurement auctions for resource pooling in client- sion cost. We show that the analysis can be reduced assisted cloud storage systems, IEEE Conference on Computer Communications (INFOCOM), 2015, pp. into a finite game (with incomplete information) by 576-584. eliminating the bids that are higher than the value of R. Zhou, Z. Li and C. Wu, An online procurement auc- the item or by the bid that are higher the total available tion for power demand response in storage-assisted budget. Using classical fixed-point theorem, there is smart grids, IEEE Conference on Computer Commu- at least one Bayes-Nash equilibrium in mixed strate- nications, INFOCOM, 2015, pp. 2641-2649. gies. Next, we address the question of computation H. Houba, D. Laan, D. Veldhuizen, Endogenous entry in and stability of such an equilibrium. We provide ex- lowest-unique sealed-bid auctions,Theory and Deci- plicitly the equilibrium structure in simple cases. In sion, 71, 2, 2011, pp. 269-295 the general setting, we provide a learning algorithm Stefan De Wachter and T. Norman, The predictive power that is able to locate equilibria. We propose an im- of Nash equilibrium in difficult games: an empirical analysis of minbid games, Department of Economics itative combined fully distributed payoff and strat- at the University of Bergen, 2006 egy learning (imitative CODI- PAS learning) that is Rapoport, Amnon and Otsubo, Hironori and Kim, Bora adapted to LUBA. We examine how the bidders of the and Stein, William E.: Unique bid auctions: Equi- game are able to learn about the online system output librium solutions and experimental evidence, MPRA using their own-independent learning strategies and Paper 4185, University Library of Munich, Germany, own-independent valuation. The revenue of the auc- Jul 2007. tionner is explicitly derived in a situation where a ran- J.Eichberger and D. Vinogradov, Least Unmatched Price dom number of bids are placed. Auctions : A First Approach, Discussion Paper Series 471, 2008 J. Eichberger, Dmitri Vinogradov: Lowest-Unmatched Price Auctions, International Journal of Industrial Or- REFERENCES ganization, 2015, Vol. 43, pp. 1-17 J. Eichberger, Dmitri Vinogradov: Efficiency of Lowest- Vickrey, W.: Counterspeculation, auctions, and competitive Unmatched Price Auctions, Economics Letters, 2016, sealed tenders. J. Finance 16, 837.1961. Vol. 141, pp. 98?102 Maskin E. S., Riley J. G. Asymmetric auctions. Rev. Erik Mohlin, Robert Ostling, Joseph Tao-yi Wang, Lowest Econom. Stud. 67, 413-438, 2000. unique bid auctions with population uncertainty, Eco- Lebrun B: First-price auctions in the asymmetric n bid- nomics Letters, Vol. 134, Sept. 2015, Pages 53-57. der case. International Economic Review 40:125-142, Huang, G.Q., Xu, S.X., 2013. Truthful multi-unit (1999) transportation procurement auctions for logistics e- Lebrun B.: Uniqueness of the equilibrium in first-price marketplaces. Transport. Res. Part B: Methodol. 47, auctions. Games and Economic Behavior 55:131-151, 127-148. 2006. Dong-Her Shih, David C. Yen, Chih-Hung Cheng, Ming- Myerson R., Optimal Auction Design, Mathematics of Op- Hung Shih, A secure multi-item e-auction mechanism erations Research, 6 (1981),pp. 58-73. with bid privacy, Computers and Security 30 (2011) 273-287 Bang-Qing Li, Jian-Chao Zeng, Meng Wang and Gui-Mei Xia, A negotiation model through multi-item auction Tembine, H., Distributed strategic learning for wireless en- in multi-agent system, Machine Learning and Cyber- gineers, Master Course, CRC Press, Taylor & Francis, netics, 2003 International Conference on, 2003, pp. 2012. 1866-1870 Vol.3. Harsanyi, J. 1973. Games with randomly disturbed payoffs: A new rationale for mixed strategy equilibrium points. C. Yi and J. Cai, Multi-Item Spectrum Auction for Recall- Internat. J. Game Theory 2, 1-23. Based Cognitive Radio Networks With Multiple Het- erogeneous Secondary Users, in IEEE Transactions Weibull, J., Evolutionary game theory, MIT Press, 1995. on Vehicular Technology, vol. 64, no. 2, pp. 781-792, Feb. 2015. N. Wang and D. Wang, Model and algorithm of winner PROOFS determination problem in multi-item E-procurement with variable quantities,” The 26th Chinese Control Proof of Proposition 2. By budget constraint, j0s and Decision Conference (2014 CCDC), Changsha, ¯ 2014, pp. 5364-5367. bids must fulfill ∑i b ji b j. If bi j > v ji c˜, j gets v c˜ b which is negative≤ (loss), and j −could guar- Rituraj and A. K. Jagannatham, Optimal cluster head se- ji − − ji lection schemes for hierarchical OFDMA based video antee zero by not participating. Therefore the strategy sensor networks,” Wireless and Mobile Networking 0 dominates any b higher than v c˜. ji ji −

336 Lowest Unique Bid Auctions with Resubmission Opportunities

Proof of Proposition 1. By Proposition 2 the con- Proof of Proposition 8. In the symmetric situation, strained game has a finite number of actions. By stan- using the methodology used in the proof of Proposi- dard fixed-point theorem, the multi-item Bayesian tion 7, we can derive that the eigenvalues of the matrix LUBA game has at least one Bayes-Nash equilibrium corresponding steady point have negative real parts. in mixed strategies. Thus, the steady point is stable. Proof of Proposition 3. Suppose v > c˜ + 1. Then Proof of Proposition 9. A proof can be obtained the payoff when only one agent places a bid of 1 on following similar lines as in Proposition 1 with a no- item i is v c˜ 1 > 0 for that agent, and others get table difference that here the action is a choice of sub- 0. When a− single− devient change any other bidders’ set of the budget-constrained bid space. decision to bid 1 instead of non-participation, there Proof of Proposition 10. Let k be the largest in- ¯ is a collision and the bid 1 is not unique anymore. teger such that yk = P( 0,1,...,k ) > 0, k b. The expected payoff of bidder{ 1 when} playing ≤0,...,l is Both agents gets c˜ < 0 and the rest of the agents { } gets nothing. − therefore given by Proof of Proposition 4. In order to find a strictly Action 0 : r ( 0 ,y) = 0.Action 01 : { } 1i { } { } mixed equilibrium for each bidder, we can introduce r1i( 01 ,y) = (v c 1)y0 cy1 c(y2 + ... + yk),... the indifference condition. Now we can easily cal- Action{ }012...l −: − − − { } culate Xm by its general form and the initial condi- r1i( 012...l ,y) = (v lc 1)y0 + (v lc 2)y1 c˜ { } − − − − tion. X = max(0,y y ) = max(0, +... + (v lc l)yl 1 lcyl lc(yl+1 + ... + yk) m m 1 m 2 v (m 1) − − − − − − − ... − − − − c˜ ) = 0, and X = 1 c˜ , where c˜ < 1. There- v m 1 − v 1 v 1 − c˜ c˜ − − It turns out that fore X∗ = ( v 1 ,1 v 1 ,0,...,0) is a partially mixed equilibrium,− and the− − expected equilibrium payoff at (v 1)y = c. (v 1)y + (v 2)y = 2c − 0 − 0 − 1 (X∗,X∗) is zero. ...  Proof of Proposition 5. As the payoff for bidder  (v 1)y0 + (v 2)y1 + ... + (v l)yl 1 = lc  − − − − i = 1 is equal to 0 when he or she bids 0, we can de-  ... duce that the first equation in the indifference condi- (v kc 1)y0 + (v kc 2)y1 + ... + (v kc k)yk 1 = kc.  − − − − − − −  tion is equal to 0. When bidder i = 1 bids 1, his or her  For l between 1 and k 1 we make the difference payoff can be presented by (v c˜ 1) P11(x) c˜ − −n 1− ∗ − ∗ between line l + 1 and line l to get: (1 P11(x)), where P11(x) = X0 − is the probability that− bidder 1 wins under bid 1. c c c c y0 = v 1 y1 = v 2 ... yl = v (l+1) ...yk 1 = v k − − − − − Proof of Proposition 7. In the proposed algo- yk = 1 (y0 + y1 + ... + yk 1) > 0 yk+1+s = 0. t+1 − − rithm, we update the strategy according to Xji (k) = ˆt Thus, the partially mixed strategy t t R ji(k) Xji(k)(1+λ ji) t Rˆt (k) . Rewriting it by subtracting Xji(k), k 1 Xt (k)(1+ t ) ji c c c − c ∑k ji λ ji y∗ = ( , ,..., ,1 ,0,...,0) t v 1 v 2 v k − ∑ v (l + 1) then dividing the result by λ ji, we can conclude − − − l=0 − Rˆt (k) Xt+1(k) Xt (k) (1+λt ) ji is an equilibrium strategy. The equilibrium payoff is ji − ji t ji ∑k t = Xji(k)[ t t ], where ∑k λ ji λ ji ∑k − λ ji ∑k zero. ˆt n t t R ji(k) (1+λ) 1 Proof of Proposition 11. denotes ∑k Xji(k)(1+λ ji) . As limλ 0 λ − = n n n → The expected payoff of the seller on item i is 1+( )λ+( )λ2+ ( )λn 1 1 2 ··· n − n = n, and limλ 0(1 + λ) = 1 equal to ∑ j cr +∑b Eni,bci +EinfB∗ vi. We can cal- λ → i − ˙ ˆt ˆt vi 1 we can get the Xji = Xji(R ji(k) ∑k XjiR ji(k)). Ac- culate that Eni,b = λi,b = z . Rewrite the ex- − ci (1+b) cording to Proposition 5, we only consider k = 0 pected payoff of the seller, we can get it is equal to and k = 1 and derive that X˙ji = Xji(1 Xji)[(v − − ∑ j cr + ∑b Eni,bci + EinfBi∗ vi. Then we can easily 1)∏p= j(1 Xpi) c]. Obviously, an point Xji = 1 − 6 − − − induce the condition for the expected revenue of the n 1 c seller exceeds the value of v is − v 1 for j [1,...,n] is a steady point of ordinary i − ∈ differentialq equations (ODEs), and its corresponding 1 ∑ j cr + EinfBi∗ n > . matrix is A. Assuming det(A λI) = ( 1) (λ ∑ z 1 v (1 + b) − vi − − − b min(b¯, i ) n 1 n 1 c ≤ ci a) − (λ + a(n 1)) = 0, where a = c(1 − v 1 ), − − − we can deduce the eigenvalues vector of A is [-(n-1)a,q a ,a,. . . ,a]. There is at least one positive eigenvalue, which means the steady point is not stable. So, the mixed Nash equilibria is not stable.

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