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On the Lowest Unique Bid Auction with Multiple Bids Nawapon Nakharutai, and Parkpoom Phetpradap*

TABLE I Abstract—This study focuses on the lowest unique sealed-bid EXAMPLEOF LUBA WITH 10 BIDDERS auctions in which the winning bidder is the one who places Bidder Bid 1 Bid 2 the unique bid that has the lowest value where exactly m bids 1 1.00 1.50 per bidder are allowed. The problem can be seen as injecting a 2 1.50 2.50 minimum into a random subset of a larger subset. By assuming 3 2.00 2.25 that bids are identical and independently placed according 4 1.75 3.00 to a given probability distribution, we obtain various exact 5 4.00 5.00 probabilities for the auctions, both as a bidder and an observer, 6 1.00 1.75 for m = 1, 2. The results are obtained via the inclusion- 7 1.50 2.50 exclusion principle. The computational results and algorithms 8 2.25 3.25 to calculate the probabilities are also given. 9 2.00 2.75 10 1.00 1.75 Index Terms—Unique minimum, inclusion-exclusion princi- ple, combinations, partitions, identical and independence, game theory, Internet auctions, reverse auctions, sealed unique-bid auctions 3.25, 4.00 and 5.00. Hence 2.75 is the lowest unique bid and Bidder 9 is the winner of the 2-bid LUBA. I.INTRODUCTION It is clear, for m ≥ 2, that each bidder must place non- UCTION is a process of buying or selling goods or identical bids for his own, otherwise he will reduce his A services. For a classical auction, the winning bidder chance of winning the auction. is determined by the bidder who makes the highest value We can see that, to win LUBAs depends on several factors bid. Nowadays, there are many types of auctions ([7], [8]), such as a number of bidders involved and strategies of such as open ascending price auctions (English auctions), participated bidders. The auction winners do not necessarily open descending price auctions (Dutch auctions), first-price have to place low bids. This somehow creates increased sealed-bid auctions, second-price sealed-bid auctions (Vick- attention from general participants, especially in the auctions rey auctions) and unique bid auctions. In this article, we over the internet. In general, bidders in LUBAs need to pay consider the lowest unique bid auctions(LUBAs), type of participation fees since the winner is justified in the opposite auctions which are recently introduced in the last few years. way as most of the other auctions where the winning bidders Moreover, we introduce the lowest unique bid auctions normally need to pay high fees to obtain the auctioned items. where multiple bids from each bidder are compulsory. We Additional rules may be added to attract further interests, for summarised the basic rules of m-bid LUBAs as follow: instant the winning bidder has to pay the exact amount of 1) Each bidder makes exactly m different sealed bids, money he bids. Such rules may effect strategies of bidders. assumed to be discrete and positive value. The LUBA problems can be modelled and studied in 2) The winning bidder is the bidder who makes the bid many aspects including Mathematics, Statistics, Engineering, that is unique, and has the lowest value amongst the Economics and Computer Sciences. However, all previous unique bids. results are for the classical LUBA problems. Bruss et al.[3] We call the auction classical LUBA when m = 1, which considered the classical LUBA models where the bidder’s means that each bidder makes exactly one bid. We demon- strategies and the number of participated bidders, which strate LUBAs by using an example here: In the LUBA with could be a random variable, were assumed to be known. ten participated bidders where each bidder places two bids The main aim of their article was to study the chance according to Table 1. of winning the auctions, and their approximated solutions • For the classical case where Bids 1 from each bidder were obtained via Poisson approximation theory. Pigolotti are considered, we can see that the unique bids are 1.75, et al. [11] and Ostling et al. [9], in different approaches, 2.25 and 4.00. Hence 1.75 is the lowest unique bid, and studied equilibrium strategies of the auctions and compare Bidder 4 is the winner of the classical LUBA. their developed results with the selected data set from the • For m = 2 case where both bids from each bidder are internet auctions. Note that, urn models can also be used for considered, we can see that unique bids are 2.75, 3.00, the problems where balls represent bids from participated bidders and urns represent values of bids( [2], [5]) and the Manuscript received January 13, 2015; revised April 8, 2015. This work lowest unique bid is represented by the left-most urn that is supported by Chiangmai University. Parkpoom Phetpradap, Corresponding Author, Department of Mathemat- contains exactly one ball. However, we do not use such ics, Faculty of Science, Chiangmai University, Chiangmai 50200, Thailand models here. (e-mail: [email protected]). In this article, when we participate in a LUBA, we refer Nawapon Nakharutai, Department of Mathematics, Faculty of Science, Chiangmai University, Chiangmai 50200, Thailand (e-mail: winning probability at bid k as the probability that we are [email protected]). the winning bidder given that we place bid k. We refer bid

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distribution or bid strategy of Bidder i as the probability III.CLASSICAL LUBAS distribution of the value(s) of bid(s) placed by Bidder i. In this section, we display exact winning probabilities When we do not participate in a LUBA, we refer winner in classical LUBAs, which was proved in [10]. Then, we probability at bid k as the probability that the lowest unique give an example and use the result to compute the winning bid occurs at the bid k, and we refer winner distribution or probabilities. We also give comments on the algorithm that equilibrium distribution as the probability distribution of the generates the probabilities. value of lowest unique bid. The main aim of this article is to derive exact winning Recall from Assumption 1 that the number of bidders probabilities for classical LUBAs and 2-bid LUBAs under excluding us is n and P1(k) is the probability that our bid k the assumptions that the number of participated bidders is is the lowest unique bid. Let Xi be the bid placed by Bidder fixed and the bid distribution of each bidder is independent i and define pk := P (Xi = k) to be the probability that and identically distributed. Our results are obtained via the Bidder i places the bid k inclusion-exclusion principle. The result and the proof of the classical LUBA had been shown in Phetpradap [10]. Theorem 2. (Winning probabilities in a classical LUBA) ∈ N However, we repeat those arguments again to ease readers With Assumption 1. For k , to understand other results in this article. For each result, ( k∑−1 [ ∑ ]) n i−1 we propose the algorithm that generate winning probabilities P1(k) = (1 − pk) 1 − (−1) A(I) , i=1 along with the comments. The discussion on the algorithms, I⊂{1,...,k−1}

as well as the validity of our LUBA models, will be discussed |I|=i in the conclusion. Furthermore, we extend our approaches to (1) obtain winner distributions when we do not participate in where, such auctions. An example of the use of the results of this ( )( ) ∏ p ∑ p n−i article is in dynamic task allocation in robot teams [13]. A(I) = P(n, i) j 1 − j , 1 − pk 1 − pk The rest of this article are presented as follow: In Section j∈I j∈I

2, we propose main notations and assumptions that are used k1! and P(k1, k2) = − with P(k1, k2) = 0 for k2 > k1. throughout the article. The results and proofs of the winning (k1 k2)! probabilities in classical LUBAs and 2-bid LUBAs are dis- Proof: We re-write winning probability in terms of played in Section 3 and 4 respectively. For both sections, we events of bids. For i = 1, 2,..., let

give examples and calculation of winning probabilities along • Ei be the event that no bidder places the bid i, with comments on algorithm. In Section 5, we act as an • Ui be the event that exactly one bidder places the bid observer in the LUBAs and obtain the winner distributions. i, Finally, we discuss pros and cons of our methods in Section C C with Ei and Ui are their complement events. Now, it can 6. The validity of the models and future study are also be seen that interpreted.

P ∩ C ∩ C ∩ · · · ∩ C II.LUBAMODELSAND MAIN NOTATIONS 1(k) = P (Ek U1 U2 Uk−1) In this section, we propose Mathematical models and k∩−1 C | assumptions for classical LUBAs and 2-bid LUBAs. For both = P (Ek)P ( Ui Ek). (2) models, we assume the following i=1 Obviously P (E ) = (1 − p )n, while Assumption 1. (1) The number of participated bidders, k k excluding us, is fixed and equals to n. k∩−1 k∪−1 C | − | (2) Bids from each bidder are independent and identically P ( Ui Ek) = 1 P ( Ui Ek). (3) distributed according to a given distribution. i=1 i=1 By the inclusion-exclusion principle, we get Also, without loss of generality, we may assume that bids k∪−1 k∑−1 ∑ are integers. i−1 P ( Ui|Ek) = (−1) P (UI |Ek), (4) For a classical LUBA, and for an integer k, let P1(k) i=1 i=1 be the probability that our bid k is the lowest unique bid I⊂{1,...,k−1} ˆ when we participate in the auction. Also, let P1(k)be the |I|=i probability that the bid k is the lowest unique bid when we ∩ where U = U . Note that P (U |E ) is the probability do not participate in the auction. I j∈I j I k that unique bids occur at bids j ∈ I given that no one bids Similarly for a 2-bid LUBA, and for integers k , k , let 1 2 on the bid k. Hence, we get, for I ⊂ {1, 2, . . . , k − 1}, P2(k1, k2) be the probability of winning the auction when ∩ the bids k1 and k2 are placed when we participate in the P (UI |Ek) = P ( Ui|Ek) ˆ auction. Also, let P2(k) be the probability that the bid k j∈I ( ) is the lowest unique bid when we do not participate in the ( ) n−|I| ∏ p ∑ p auction. = P(n, |I|) j 1 − j . 1 − pk 1 − pk Last but not least, in order to be able to calculate the j∈I j∈I probabilities by using computers, we need to restrict an upper (5) price limit of the auction. This can be seen as support of the Combining (2) - (5), Theorem 2 follows. set of bids.

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Example 3. Define the probability mass functions of 4) For an index set I ⊂ N, we define p˜j\I as the geometric(p) and Poisson(λ) random variables, Y and Z probability that exactly one bid is j and the other bid respectively, as must not be from the index set I. That is ∑ − i−1 ∀ ∈ Z+ P (Y = i) = (1 p) p, i , p˜j\I = (pjk + pkj). e−λλk−1 k∈N P (Z = k) = , ∀k ∈ Z+. k∈ /I (k − 1)! We can see that price limits have the role to make the prob- The plots of the winning probabilities of LUBA with 50 abilities computable. To give an example of these notations, bidders, where bid distributions are geometric(1/2), geomet- suppose the bids distribution of each bidder is, for i1, i2 ∈ N, { ric(1/3), Poisson(2) and the price limits are 200, are given in − (2 − q)q2(1 − q)i1+i2 3, i < i ; Figure 1. The optimum places to make the bid should be at 5, p = 1 2 (6) i1,i2 ≥ 7 and 7 respectively. For large k, the winning probabilities 0, i1 i2. seems to converge to a constant. This is because, by the We call the random variables with mass functions defined choice of bid distributions we choose, high bids are rarely in (6) as geometric(q, q). Note that (6) refers to the situation occur. where each bidder places the joint geometric distribution with the same parameter q and with the restriction that both bids cannot be the same. We can work out from the equation that 0.4 j−2 j−1 p˜j = (2 − q)q(1 − q) [1 − q(1 − q) ]. (7) geometric(1/2) ⊂ N 0.3 geometric(1/3) 5) For an index set I , let J(I) be a collection of Poisson(2) disjoint subsets where each subset contains exactly two ⌊ |I| ⌋ k elements from I. For k = 0, 1,..., 2 , define J := k 0.2 J (I) to be the set of the disjoint subsets of J(I) that have exactly k subsets. Clearly, 0 ∪ 1 ∪ ∪ ⌊|I|/2⌋ 0.1 J(I) = J J ... J .

Moreover, we define nk as the number of distinct elements in Jk. It can be proved that 0 0 5 10 15 20 |I|! n = . k (|I| − 2k)!k!2k Fig. 1. The winning probabilities of the classical LUBAs with n = 50. The price limits are 200. We may write the collection Jk as k = {J k,J k,...,J k }. J 1 2 nk Algorithm and comments To generate winning probabilities k from Theorem 2, the input required are the number of That is, for j = 1, . . . , nk, we have Jj as sets of k participated bidders, the bid distribution and the price limit. disjoint subsets where each subset contains exactly two The algorithm to generate (1) is straightforward. elements from I. In other words, k { k k k } Jj = Jj,1,Jj,2,...,Jj,k . IV. TWO-BID LUBAS Now, for l = 1, . . . , k, note that J k is the set which In this section, we derive exact winning probabilities j,l has two elements and J k ⊂ I. We set in 2-bid LUBAs where each bidder makes two (different) j,l bids. Recall from Assumption 1 that the number of bidders ∪k ˜k k excluding us is n, which means that the total number of Jj = Jj,l. bids is 2n. Then, we give an example and use the result to l=1 compute the winning probabilities, We also give comments For example, if I = {1, 2, 4, 5}, then on the algorithm that generates the probabilities. { } We start the section by defining the following notations: J0 = ∅ , { 1) Let X = (i , i ) ∈ N2 be the bids made by Bidder i. { } { } { } { } { } i 1 2 1 { } { } { } { } { } Without loss of generality, we assume that i < i . J = 1, 2 , 1, 4 , 1, 5 , 2, 4 , 2, 5 , 1 2 { }} 2) For i < i , we define 1 2 {4, 5 , {{ } { } { }} pi ,i := P (Xi = (i1, i2)) 1 2 J2 = {1, 2}, {4, 5} , {1, 4}, {2, 5} , {1, 5}, {2, 4} . to be the probability that Bidder i places bid i1 and i2. ≥ Now, we are ready for the main results of this section. For We set pi1,i2 = 0 for i1 i2. k < k , recall that P (k , k ) is the probability of winning 3) For j ∈ N, define p˜j to be the probability that exactly 1 2 2 1 2 a 2-bid LUBA when bids k and k are placed. To derive one bid of Xi is j. In other words, 1 2 ∑ the winning probability, we express it in terms of events of p˜j = (pjk + pkj ). bids. With the similar set up as in the proof of Theorem 2, ∈N k we remind that Ei is the event that no bidder places the bid

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i and Ui is the event that exactly one bidder places the bid Proof: To apply a similar technique as in Theorem 2, i. we write all the terms in Lemma 4 in the conditional probabilities form: Lemma 4. (Inclusion-exclusion principle of winning proba- bilities) ∈ N − With Assumption 1, for k1, k2 with k1 < k2, ( k∩1 1 ) ∗ C | P(k1, ) =P (Ek1 )P Ui Ek1 (13) P2(k1, k2) = P(k1, ∗) + P(∗, k2) − P(k1, k2), (8) i=1 − where, ( k∩2 1 ) C P(∗, k2) =P (Ek )P U |Ek (14) • P(k1, ∗) is the probability that our bid k1 is the winning 2 i 2 ̸ bid, i=1,i=k1 − • ∗ ( k∩2 1 ) P( , k2) is the probability that our bid k2 is the winning C P(k1, k2) =P (Ek ∩ Ek )P U |Ek ∩ Ek bid ignoring what happens at k1, 1 2 i 1 2 ̸ • P(k , k ) is the probability that both of our bids k and i=1,i=k1 1 2 1 (15) k2 can be the winning bid. Proof: Note that to win a 2-bid LUBA, we either win − at the bid k1 or at the bid k2. Hence, Note that (13) (15) can be determined in a similar way with P2(k1, k2) = − − k∩1 1 k∩2 1 ( ) ( ) P (E ) = (1 − p˜ )n, for i = k , k , (16) P E ∩ U C + P EC ∩ E ∩ U C . i i 1 2 k1 i k1 k2 i (9) i=1 i=1,i≠ k1 Note that the second term in (9) can be written as and, − ( k∩2 1 ) C C P E ∩ Ek ∩ U = k1 2 i ∩ − − n P (Ek1 Ek2 ) = (1 p˜k1 p˜k2 + pk1,k2 ) , (17) i=1,i≠ k1 − − ( k∩2 1 ) ( k∩2 1 ) P E ∩ U C − P E ∩ E ∩ U C , k2 i k1 k2 i therefore, we only show the calculation of the second term ̸ ̸ i=1,i=k1 i=1,i=k1 on the right hand side of (13). By the similar arguments as as required in the lemma. in Theorem 2, and (4), we get Theorem 5. (Winning probabilities in a 2-bid LUBA) ∈ N − With Assumption 1. For k1, k2 with k1 < k2, define k∩1 1 C | − − − P ( Ui Ek1 ) = qk1 = 1 p˜k1 , qk1,k2 = 1 p˜k1 p˜k2 + pk1,k2 , i=1 − and k∑1 1 ∑ 1 − (−1)i−1 P (U |E ). (18) I = {1, 2, . . . k − 1},I = {k + 1, k + 2, . . . k − 1}. I k1 1 1 2 1 1 2 i=1 I⊂{1,...,k1−1}

Then, |I|=i n (1) n (1) P2(k1, k2) = q P (k1) + q P ∪ (k2) k1 I1 k2 I1 I2 ∩ − qn P(2) (k , k ), (10) where U = U . Finally, we derive P (U |E ). Note k1,k2 I1∪I2 1 2 I j∈I j I k1 that we can not apply the result in(5) immediately since the where, for J ⊂ N and (c (1), . . . , c (d)) := c ∈ Nd, for d d d situations on unique bids are more complicated than on the d = 1, 2, classical LUBA. However, a similar approach can still be ∑|J| ∑ applied. (d) − − i−1 ∗ PJ (cd) = 1 ( 1) A (UI ), (11) For the index set I, since all bids in the set I must be unique, i=1 I⊂J we therefore categorise the bidders into three types:

|I|=i (1) Bidders who make exactly two unique bids amongst and the set I, ⌊ |I| ⌋ (2) Bidders who make exactly one unique bid amongst the ∑2 ∑nk [ ( ∏ ) ∗ pr1,r2 set I, A (UI ) = P(n, |I| − 2k) qc I k=0 j=1 { }∈ k d (3) Bidders who make no unique bid amongst the set . r1,r2 Jj ,r1

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elements in the set IC . With this idea, we get Algorithm 1: Creating combinations of index set P (U |E ) Input: Index set I and integer value k (must be less I k1 than ⌊I/2⌋) ⌊ |I| ⌋ ∑2 ∑nk [( ∏ ) Output: Jk, set of disjoint subsets pr1,r2 = P(n, k) Create C, set of all possible subsets of I that contain 1 − p˜k k=0 j=1 { }∈ k 1 r1,r2 Jj ,r1

0.3 that maximise the chance of winning by calculating winning probabilities where we make one bid, while 0.2 the others still make two bids. The probabilities are actually P(1)(k) for all k. We calculate winning prob- 0.1 I1 abilities using the same model as in Example 6. The 14 12 0 10 first two optimum bids are 7 and 8 with the chance of 14 8 12 10 6 8 6 4 winning 0.2527 and 0.2525 respectively, agrees with 4 2 2 Second bid First bid the calculation in Example 6. The plot of winning probabilities is given in Figure 3.

Fig. 2. The winning probabilities in the 2-bid LUBA with n = 40 and bids distribution is geometric(1/2, 1/2). The price limit is 100. V. WINNINGPROBABILITIESASANOBSERVER In this section, we derive winner distributions in classical LUBAs and 2-bid LUBAs when we do not participate in the Algorithm and comments auctions. We use similar ideas of the proofs of Theorem 2 and ˆ ˆ 1) The algorithm that generates results from Theorem 5 Theorem 5. Recall that P1(k) and P2(k) are the probability is far more complicated than in Theorem 2. This is that the winner bid occurs at the bid k given that n bidders are because every possible combinations in (12) is needed involved in a classical LUBA and a 2-bid LUBA respectively. to be considered, which has no easy way to deal with. Corollary 7. (Winner probabilities as an observer) We therefore need an algorithm that generates the combinations of subsets in the way of our set up. The ∈ N algorithm to generated required combinations works (a) For the classical LUBA with Assumption 1, and k as follow Then, the algorithm to generate winning k∑−1 ∑ Pˆ − i probabilities in (10) can be done as follow: 1(k) = ( 1) B(I), (19) i=1 a) Declare the number of participated bidders, the I⊂{1,...,k−1} bids joint distribution and the price limit. |I|=i

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0.3 deduce that geometric(1/2, 1/2) ∪k k∪−1 geometric(1/2) ( ) ( ) 0.25 P Ui − P Ui i=1 i=1 0.2 k∑−1 ∑ i = (−1) P (UI ∩ Uk) 0.15 i=0 I⊂{1,...,k−1}

|I|=i 0.1 k∑−1 ∑ [ ( ∏ ) i 0.05 = (−1) P(n, i + 1) pj i=0 j∈I∪{k} I⊂{1,...,k−1} 0 0 2 4 6 8 10 12 14 |I|=i ( ∑ )n−(i+1)] 1 − pj . j∈I∪{k} Fig. 3. The winning probabilities by placing one bid in 2-bid LUBA compared with the winning probabilities in the classical LUBAs with n = (b) With a similar idea as in the proof of Corollary 7(a), 40. The price limits are 100. we can see that k∑−1 ∑ i Pˆ2(k) = (−1) P (UI ∩ Uk) where, i=0 I⊂{1,...,k−1} B(I) = |I|=i − ( ∏ )( ∑ )n−(i+1) k∑1 ∑ P − i (n, i + 1) pj 1 pj . = (−1) P (UJ ), j∈I∪{k} j∈I∪{k} i=0 I⊂{1,...,k−1} (b) For the 2-bid LUBA with Assumption 1, and k ∈ N |I|=i where J = I ∪ {k}. Finally, by a similar approach as k∑−1 ∑ ˆ i in Theorem 5, we get P2(k) = (−1) P (UJ ), (20) i=0 ⌊ |J| ⌋ J=I∪{K} ∑2 ∑nk [ I⊂{1,...,K−1} P (UJ ) = P(n, |J| − 2k) |I|=i (k=0 j=1∏ )( ∏ ) p p˜ \ where r1,r2 s J { }∈ k ∈ \ ˜k r1,r2 Jj ,r1

pr1,r2 p˜s\J { }∈ k ∈ \ ˜k Example 8. We derive winner distributions for the classical r1,r2 Jj ,r1

r1,r2∈J,r1

Proof: VI.CONCLUSIONAND FUTUREWORK (a) We display the probability in terms of events as In this article, we introduce one of the ways to obtain in- ( k∩−1 ) teresting probabilities of LUBAs. We also propose computer Pˆ ∩ C 1(k) = P Uk Ui algorithms to obtain these probabilities, at which upper price i=1 limits of the auctions are required.The limits are acted as the − ( k∩1 ) ( ∩k ) supports of the random set of bids. C − C = P Ui P Ui From the results in previous sections, it is clear that the i=1 i=1 number of participated bidders as well as the bid strategies ∪k k∪−1 ( ) ( ) effect the winning probabilities. Therefore, we believe that = P U − P U . i i bid distribution should not be fixed as it can be changed due i=1 i=1 to human factors. As long as we gain further information of Now, by using the idea of the inclusion-exclusion bid distributions, we can update the data to get more accurate principle similar to the proof of Theorem 2, we can results. One choice of the distribution that can be used is

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0.3 Lemma 9. (Inclusion-exclusion principle for m-bid LUBA) geometric(1/2, 1/2) ∑m ∑ geometric(1/2) i+1 0.25 Pm(k1, k2, . . . , km) = (−1) P(I), ∏ ∈ m {∗ } i=1 I ( j=1 ,kj ) 0.2 |I|=i where 0.15 • I = (x1, x2, . . . , xm) is an m-dimensional vector, and ∗ 0.1 xi must be either or ki for i = 1, . . . , m, • |I| is the number of xi’s which are not ∗.

0.05 ACKNOWLEDGEMENT 0 0 2 4 6 8 10 12 14 This research is supported by Chiang Mai University. The authors would like to thank Prof. Sompong Dhom- pongsa, Supachai Awiphan and Wanchalerm Promduang for Fig. 4. Winner distributions with n = 40 and the price limit 100. their helps and suggestions.

REFERENCES an empirical distribution from real world data set over the [1] M. Archibalda, A. Knopfmachera and H. Prodingerb, The number of internet. distinct values in a geometrically distributed sample, European Journal of Combinatorics, 27 (2006), pp. 1059–1081. One important assumption from our model is that the [2] L. Bogachev, A. Gnedin and Y. Yakubovich, On the variance of the number of bidders and the bid strategy need to be fixed, number of occupied boxes, Advances in Applied Mathematics, 40 (2008), pp. 401–432. which is, in general, unlikely to happen in the real world. [3] F. Bruss, G. Louchard and M. Ward, Inverse Auctions: Injecting Unique However, it does not affect our models on the condition Minima into Random Sets, ACM Transactions on Algorithms, 6:1 that we have information on the number of bidders and bid (2009), Article 21. [4] J. Dossey, A. Otto, L. Spence and C. Vanden Eynden, Discrete Math- strategies. Indeed, suppose that the number of participated ematics (5th ed.), Pearson, 2006. bidders, N, follows a probability distribution with density g [5] M. Dutko, Central limit theorems for infinite urn models, Annals of P PN Probability 17:3 (1989), pp. 1255–1263. and define gn = (N = n). In a classical LUBA, let 1 (k) [6] G. Grimmett and D. Stirzaker, Probability and random processes (3rd be the probability that we win the auction when we place ed.), Oxford University press, 2001. PˆN the bid k and with N participated bidders, and define 1 (k) [7] V. Krishna, Auction theory, USA: Academic Press, San Diego, 2002. similarly for the winner probability. Then, [8] P. Milgrom, Putting Auction Theory to Work, Cambridge University Press, Cambridge, United Kingdom, 2004. ∑ ∑ [9] R. Ostling, J. Wang, E. Chou, C. Camerer, Testing Game Theory in PN (k) = g P (k), PˆN (k) = g Pˆ (k). the Field: Swedish LUPI Lottery Games, American Economic Journal: 1 n 1 1 n 1 Microeconomics, 3 (2011), pp. 1–33. n n [10] P. Phetpradap, On the lowest unique bid auction, Proceedings of AMM 2013 (2013), pp. 241–246. [11] S. Pigolotti, S. Bernhardsson, J. Juul, G. Galster and P. Vivo, Equi- Similarly for the bid strategies, let s1, s2, . . . , sd be pos- librium Strategy and Population-Size Effects in Lowest Unique Bid sible bid strategies with, respectively, h1, h2, . . . , hd proba- Auctions, Phys. Rev. Lett. 108, (2012) 088701. bilities of being chosen. Then, we can again apply the law [12] A. Rapoport, H. Otsubo, B. Kim and W. Stein, Unique bid auctions: Equilibrium solutions and experimental evidence, MPRA Paper 4185, of total probability to find a represented bid distribution. University Library of Munich, Germany, 2007. The main advantage of this work is to get exact cal- [13] W. Zhu and S.Choi, A closed-loop bid adjustment approach to dynamic task allocation of robots, Engineering letters, 19:4 (2011), pp. 279 - culations of winning probabilities, which are obtained via 288. inclusion-exclusion principle technique. However, the draw- back of our results is the use of all possible combinations of unique bids that come out during the calculation of the prob- abilities, which make the problem uncomputable by hands. The algorithm to calculate results, especially Algorithm 1, take a considerable time to process. Nevertheless, we believe that the idea of the proofs and the use of the inclusion- exclusion principle will benefit the readers who want to tackle similar kind of problems. Last but not least, we would like to introduce the idea to obtain the winning probabilities for the m-bid LUBAs when m ≥ 3. By defining similar notations as in the beginning of Section 3 properly, we expect similar results and proofs as in Theorem 5. Lemma 9 describe a result in the similar way as in Lemma 4. The lemma can be proved via Mathematical induction. Let Pm(k1, k2, . . . , km) be the probability of winning the bid when bids k1, k2, . . . , km are placed.

(Advance online publication: 10 July 2015)