Claremont McKenna College. Stochastically Equivalent Sequential Auctions with Multi-Unit Demands. Submitted to. Professor Yaron Raviv.

Transcription

1 Claremont McKenna College Stochastically Equivalent Sequential Auctions with Multi-Unit Demands Submitted to Professor Yaron Raviv and Dean Nicholas Warner by Tongjia Shi for Senior Thesis Spring 2015 April 23, 2015

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3 Acknowledgement I sincerely thank Dr. Yaron Raviv for extending valuable advice and expertise to me on the thesis. The thesis would not have been possible without his support. I am also grateful to Dr. Heather Antecol for her guidance and encouragement. I would also like to express sincere gratitude to the Robert Day School of Economics and Finance at the Claremont McKenna College for providing me excellent education and resources. 2

4 Abstract Past empirical analysis show that in contrast to the theory predictions; prices tend to decline in some sequential auctions, a puzzle known as the declining price anomaly. Several theoretical explanations were proposed demonstrating the possibility of a declining price pattern under certain assumptions. In this paper, we demonstrate that when bidders have private values and multi-unit demand, expected selling price can be increasing, constant, decreasing or even non-monotonic. In our model, price pattern depends on the distributions from which bidder valuations are drawn (including the size of the bidders demand reduction), and the number of bidders. 3

5 Contents 1 Introduction 6 2 The Model 8 3 Equilibrium Bidding Strategy 10 4 Steady Reduction Assumption 15 5 Demand Reduction and Expected Selling Price No Demand Reduction Full Demand Reduction Uniform Distribution Triangular distribution Steady Demand Reduction Various Level of Demand Reduction Additional Examples Exponential Distribution with Divisibility Assumption Normal Distribution with Divisibility Assumption Conclusion 28 4

6 List of Figures 1 A game tree of the three-stage sequential auction Bidders with two-unit demand under steady demand reduction Bidders with two-unit demand under steady demand reduction: Expected selling price Bidder valuations for various levels of demand reduction: Bid shading versus number of bidders at various levels of demand reduction Expected selling price versus number of bidders at various levels of demand reduction Expected selling price versus level of demand reduction Exponential Distribution: Bidder Valuations Exponential Distribution: Bid Shading Exponential Distribution: Expected Selling Price Normal Distribution: Bidder Valuations Normal Distribution: Bid Shading Normal Distribution: Expected Selling Price List of Tables 1 Expected selling price of a 3-stage sequential auction with unit-demand distributed uniformly Expected selling price of a 4-stage sequential auction with unit-demand distributed uniformly Expected selling price of a 4-stage sequential auction with unit-demand distributed triangularly Expected selling price pattern for different number of bidders and different level of demand reduction

7 1 Introduction It has been shown theoretically that when bidders have unit-demand and the goods are homogenous, the equilibrium selling prices in sequential auctions are either upwarddrifting when bidders have interdependent values [Milgrom and Weber, 1982][Milgrom and Weber, 2000] or form a martingale when bidders have independent private values. In contrast, starting with Ashenfelter [1989], empirical research found that prices tend to drift downward, a phenomenon known as the declining price anomaly (see Raviv [2006] for further discussion). Several modifications to the original model are subsequently made in attempt to explain the declining price pattern. For example, McAfee and Vincent [1993] have shown that this anomaly can be explained with the assumption of nondecreasing absolute risk aversion. Jeitschko [1999] found that declining price patterns may be explained if the supply is unknown ex ante However, empirical evidence suggests that declining price patterns may exist even if the supply if fully known beforehand. Thus this theory cannot completely explain this anomaly. Other modifications to the theory that may yield a declining price pattern include time impatience [von der Fehr, 1994] and auction of many stochastically equivalent objects [Engelbrecht-Wiggans, 1994]. Another family of solutions involves relaxing the assumption of the bidders unit demand. If bidders have multiple units of demand, and if bidders display diminishing marginal utility, it may be possible that the expected selling prices decline. The analysis on sequential auctions with multiunit demands is difficult because of the asymmetry: different types of bidders coexist in the auction due to winning in earlier stages, even though the bidders are symmetric ex ante. Different approaches have been taken to overcome this difficulty. Katzman [1999] restricts the analysis on a two-stage sequential auction. In this case, the asymmetry which only appears in the second stage does not matter as all bidders have a domi- 6

8 nant strategy to bid their valuation in the final round. Rodriguez [2009] was able to construct equilibrium in a sequential auction with more than two stages, but it assumes the complete information setting where all bidders know in advance everyone s valuation for each additional item (and the valuations are the same for each bidder). Therefore, there is a lack of research on sequential auctions under incomplete information with more than two stages. Relaxing this two-stage assumption may be important, because interesting price patterns may be neglected if there are only two periods. For example, it could be possible that prices display non-monotonic patterns, and such patterns may not be discovered if we restrict our models to twostage sequential auctions. This paper contributes to the literature by looking for a subgame perfect Nash equilibrium in a private value second-price three-stage sequential auction with incomplete information and demand reduction. 1 The items in different stages of the auction are assumed to be stochastically equivalent, meaning that bidders draw their valuations from certain distributions for each item in the auction, instead of holding the same valuation for all items. This assumption of stochastic equivalence has been made in Bernhardt and Scoones [1994]. In our model, the distribution from which valuations are drawn depend on the number of items a bidder already has. The demand reduction is modeled by the stochastic dominance between distributions. We show that the expected selling price at equilibrium can either be increasing, decreasing, constant or non-monotonic, depending on the set of valuation distributions. Intuitively, there are two forces driving the expected selling price pattern away from constant when there exists demand reduction. On one hand, when demand reduction exists, a bidder always shades his bid (compared to the scenario without demand reduction) in earlier rounds of auctions because he can expect greater profit if 1 Although the method we use can be applied to sequential auctions with more stages, we focus on three-stage auctions because more stages will provide negligible additional insight while complicating the calculations. 7

9 he does not win the current auction. When demand reduction is higher, bid shading in earlier rounds becomes more attractive, putting downward pressure on bids in earlier rounds, and thus contributing to a increasing price pattern. On the other hand, when demand reduction exists, the overall valuation of all bidders decline as auctions continue, because more and more bidders obtain items and demand less. This puts an downward pressure on bids in later rounds, which contributes to a declining price pattern. The magnitude of each force depends on the shape of the distribution function, the level of demand reduction, and the number of bidders. The interaction between these two forces gives us various expected selling price patterns. Given the variety of price patterns that could be observed when the unit demand assumption is relaxed, the notion of declining price anomaly seems overstated. Since we do not observe bidders demand in real-world auctions, different empirically observed price patterns could be explained different underlying valuation distributions of the bidders. The rest of the paper organizes as follows. Section 2 describes the model. Section 3 derives the Nash equilibrium. Section 4 discusses an assumption under which the equilibrium solution can be greatly simplified. In section 5, we use analytic derivations and Monte Carlo simulations to obtain expected selling price patterns under different set of valuation distributions. Section 6 concludes. 2 The Model Assume that there are n+1 bidders each with three units of demand facing sequential three second-price private value auctions (n 2). Valuations for the bidders are independent random draws from a distribution with cumulative distribution F i (x), where i is the number of items already won by a particular bidder. If F i (x) first-order dominates F j (x) when i < j, we have decreasing marginal utility. Bidders obtain 8

10 their valuation of an item at the display of that item, and the next item will not be displayed until the previous item has been sold. The partial game tree of the auction process is depicted below. Each node in the tree represents an auction. S k,i represent the auction stage k (equal to either 1, 2 or 3) for a particular bidder that has so far won i items. For example, S 1,0 represents the situation the bidders face at the beginning of the first auction when no one has won any item yet. S 2,1 represents the situation the winner of the first round faces at the beginning of the second auction. S 3,0 represents the situation where the winners of the first two rounds are the same person. S 3,0 (lose to same person) lose lose S 3,0 lose S 2,0 win S 1,0 win lose S 3,1 S 2,1 win S 3,2 Figure 1: A game tree of the three-stage sequential auction Notice the difference between this setup and the setup in Katzman [1999]. In Katzman [1999], a bidder s valuations are updated when she wins an item. Here, a bidder redraws her valuation at the display of each item. In Katzman [1999], a bidder s high valuation and low valuation (since each bidder only has two units of demand) are ordered statistics of the same distribution. Here, the high valuations and low valuations are drawn from two arbitrary independent distributions. This relaxation gives us freedom in choosing the degree of decreasing marginal utility. 9

11 3 Equilibrium Bidding Strategy In this section we derive a subgame perfect Nash equilibrium for the model described in the previous section. Let U k,i (z, x) denote the expected utility of being in stage S k,i for a bidder if she has true valuation x but bids at z for this round of auction (but plays equilibrium strategies in later stages), and all other bidders play equilibrium strategies. Let EU k,i denote the expected utility prior to the display of the item when all players play equilibrium strategy. This is the utility of entering the stage S k,i. Let b k,i (x) denote the equilibrium bid function of a bidder in the stage S k,i. Similar to Bernhardt and Scoones [1994], we assume that the minimum valuation of a bidder is always greater than the amount by which he shades his bid, and we look for bidding functions that are monotonically increasing over the range of possible valuations. Let r k,i = b 1 k,i be the inverse function of the bid function. We call this the revealing function." Given a bid, this function reveals the true valuation of the bidder of a particular type. The existence of bidders with multiple bidding functions at the same stage was the main difficulty to a full analysis of the model. We will use backward induction. In the last round, everyone bids truthfully because it is a dominant strategy for everyone. r 3,0 (z) = r 3,0 (z) = r 3,1 (z) = r 3,2 (z) = z. Consider the expected utility of a bidder being in stage S 3,0. The bidder has a non-zero payoff only if the highest valuation among all other bidders, t, is no more than the bidder s valuation, x. Since there are n 2 other bidders with zero items and 2 bidders with one item in stage S 3,0, the highest valuation among all other bidders follows a cumulative distribution function of F 0 (t) n 2 F 1 (t) 2. Therefore, we have the 10

12 following expression for the expected utility of being in stage S 3,0 : EU 3,0 = x 0 0 (x t) d[f 0 (t) n 2 F 1 (t) 2 ] df 0 (x) (1) Here, the outer integral integrates over the valuation of bidder of interest, and the inner integral integrates over the highest valuation of all other bidders. Similarly, we can calculate the expected utility of being in stages S 3,0, S 3,1 and S 3,2. EU 3,0 = EU 3,1 = EU 3,2 = x 0 0 x 0 0 x 0 0 (x t) df 0 (t) n 1 F 2 (t) df 0 (x) (2) (x t) df 0 (t) n 1 F 1 (t) df 1 (x) (3) (x t) df 0 (t) n df 2 (x) (4) In the second to last round, we have to deal with asymmetry. Let s analyze from the perspective of a bidder, Bob. First suppose Bob is in the state S 2,1. Then all other bidders have valuations drawn from the distribution F 0. We will now calculate Bob s utility U 2,1 (z, x) when his valuation is x but bids at z. Bob will enter stage S 3,1, unless he wins the current round of the auction. This happens if the highest bid submitted by all other bidders, which follows a cumulative distribution function of F0 n (r 2,0 (t)), is no more than z. If this happens, Bob will get an immediate payoff of (x t), plus the difference in expected utility between stage S 3,2 and S 3,1 as a result of winning. Thus we have the following expression for U 2,1 (z, x). U 2,1 (z, x) = EU 3,1 + z 0 (x t + EU 3,2 EU 3,1 ) df n 0 (r 2,0 (t)). (5) Here, the first term represents his expected utility for future rounds if he does not win the current round, and the second term represents how much more utility he will gain if he wins the current round. At equilibrium, Bob must have x = r 2,1 (x) by 11

13 definition of r 2,1, and he must have no incentive to deviate. So we have 0 = d dz U 2,1(z, x) x=r2,1 (z) = (r 2,1 (z) z + EU 3,2 EU 3,1 ) d dz F n 0 (r 2,0 (z)), which gives us r 2,1 (z) = z + EU 3,1 EU 3,2 (6) This shows that a bidder in stage S 2,1 shades his bid by a constant amount, EU 3,1 EU 3,2, equal to the expected utility drop in the next round of auction due to winning. Now suppose that Bob is in the state S 2,0 (i.e., he lost in the first round). Let p(t) denote the probability that the first-round winner submits the highest bid among all other bidders, given that the highest bid of all n other bidders (except Bob) is t. 2 The density function that the highest bid of all n other bidders (except Bob) is t is given by d n 1 [F dt 0 (r 2,0 (t))f 1 (r 2,1 (t))], and the density function that the first-round winner submits a bid of t and that all other bidders submit bids no more than t is given by F n 1 0 (r 2,0 (t)) d dt F 1(r 2,1 (t)). Therefore, we have p(t) = F 0 n 1 (r 2,0 (t)) d F dt 1(r 2,1 (t)) n 1 [F0 (r 2,0 (t))f 1 (r 2,1 (t))] d dt (7) Suppose Bob has valuation x but bids z. If he wins, his payoff for the current round will be (x t), where t is the highest bid of all other bidders, and his future payoff will be EU 3,1 because he will enter the stage S 3,1. If he loses, he will either enter stage S 3,0 or S 3,0, depending on whether the first-round winner wins again, which is 2 If we want to study a sequential auction with more than three stages, then it is possible that in some stages losing will lead to more than two different outcomes, and p(t) here will become a vector function. 12

14 governed by the function p(t). Hence the expected utility Bob gains is given by U 2,0 (z, x) = z + (x t + EU 3,1 ) d[f0 n 1 (r 2,0 (t))f 1 (r 2,1 (t))] 0 z [p(t)eu 3,0 + (1 p(t))eu 3,0 ] d[f n 1 0 (r 2,0 (t))f 1 (r 2,1 (t))] Here, the first integral represents the partial expectation of the utility if Bob wins the auction, and the second integral represents the partial expectation of utility if Bob loses the auction. By a similar argument, at equilibrium we must have 0 = d dz U 2,0(z, x) x=r2,0 (z) = (r 2,0 (z) z + EU 3,1 p(z)eu 3,0 + (1 p(z))eu 3,0 ) d n 1 [F0 (r 2,0 (z))f 1 (r 2,1 (z))] dz Since both F i and the revealing functions must be nondecreasing and non-constant, we have r 2,0 (z) = z + EU 3,0 + p(z)[eu 3,0 EU 3,0 ] EU 3,1 (8) Notice that the amount by which a bidder in stage S 2,0 shades his bid depends on his true valuation. Substituting with Equation 7, we have r 2,0 (z) = z + EU 3,0 + F 0 n 1 (r 2,0 (z)) d F dz 1(r 2,1 (z)) d dz [F n 1 0 (r 2,0 (z))f 1 (r 2,1 (z))] [EU 3,0 EU 3,0] EU 3,1 (9) Since r 2,1 is already known (Equation 6), the solution for r 2,0 (z), if exists, is then given by the implicit equation above. 3 The inverses of the two revealing functions are the equilibrium bidding functions, and we can calculate EU 2,0 and EU 2,1 by integrating 3 One can use the iterative method to look for a solution: Let p (j) (z) and r (j) 2,0 (z) be the function estimates of p(z) and r 2,0 (z) at the j th iteration, and set p (0) (z) = 0. One can then update the 13

15 over the bidder s valuation: EU 2,0 = EU 2,1 = 0 0 U 2,0 (b 2,0 (x), x) df 0 (x) (10) U 2,1 (b 2,1 (x), x) df 1 (x) (11) Now we apply the same reasoning to the first round: The total utility of a bidder who has valuation x but bids at z in the first round auction equals the expected utility of being in stage S 2,0, except if the bidder wins the auction. U 1,0 (z, x) = EU 2,0 + z 0 (x t + EU 2,1 EU 2,0 ) df n 0 (r 1,0 (t)). At equilibrium, we must have 0 = d dz U 1,0(z, x) = (r 1,0 (z) z EU 2,0 + EU 2,1 ) d x=r1,0 (z) dz F 0 n (r 1,0 (z)) Rearranging the terms, we obtain the revealing function for the first stage of the auction: a bidder in the first round shades his bid by an amount equal to the expected utility drop due to winning. r 1,0 (z) = z + EU 2,0 EU 2,1. So far, we have obtained the equilibrium bidding function for all bidders at each stage of the auction. This allows us to calculate the expected selling price, EP k, at each function estimates by r (j+1) 2,0 (z) = z + EU 3,0 + p (j) (z)[eu 3,0 EU 3,0 ] EU 3,1 p (j) (z) = F 0 n 1 (r (j) 2,0 (z)) d dz F 1(r 2,1 (z)) d n 1 dz [F0 (r (j) 2,0 (z))f 1(r 2,1 (z))], Then lim j p (j) (z) = p(z) and lim j r (j) 2,0 (z) = r 2,0(z) if this process converges. 14

16 round. 4 Steady Reduction Assumption Equation 9 is unsatisfactory because it is implicit and difficult to calculate. However, the formula for the equilibrium bidding strategy can be greatly simplified if F 0 (x) F 1 (x) = F 1(x) F 2 (x) (12) We call this the steady reduction assumption" of demand. Roughly speaking, this is intuitively saying that the effect of gaining an additional item does not depend on the number of items already owned. To give an example, suppose there exists F (x) such that F 0 (x) = F 3 (x), F 1 (x) = F 2 (x), F 2 (x) = F (x). (13) Then this set of demand distribution functions satisfies the steady reduction assumption. We call such set of demand distribution functions satisfying the divisibility assumption." It has an intuitive explanation. When a bidder already has i items, she has a demand of (3 i) items. We may think of these (3 i) units of demand being generated from (3 i) sub-bidders in her mind. When the item is displayed, each of the (3 i) sub-bidders draws a valuation from F (x) for this item, and the bidder bids the highest among these valuations, which has a cumulative distribution of F 3 i (x). The steady reduction assumption, including the more stringent divisibility assumption, gives us the bidding function explicitly. With Equation 12, we have EU 3,0 = EU 3,0 because F n 2 0 (t)f 2 1 (t) = F n 1 0 (t)f 2 (t). Thus, from the perspective of any bidder, the two states S 3,0 and S 3,0 in the reduced game tree (Figure 1) col- 15

17 lapse into one. Therefore, Equation 8 simplifies to r 2,0 (z) = z + EU 3,0 EU 3,1. (14) Together with Equation 6 and 8, this means that bidders in the same game state always shade their bids by the same amount, independent of their true valuation. 4 For convenience, let s denote the amount of bid shading by d k,i = EU k+1,i+1 EU k+1,i. The bid function of a bidder in state S k,i is thus b k,i (x) = x d k,i. As we see, the steady reduction assumption allows us to explicitly calculate the equilibrium of the game. The disadvantage is that we restrict the degree of demand reduction of the bidders, which may possibly prevent us from observing interesting expected selling price patterns. We will observe this effect in the next section. 5 Demand Reduction and Expected Selling Price In this section we explore the effect of various demand reduction levels on the expected selling price at equilibrium of a sequential auction. We will first start with two extreme cases: when bidders have no demand reduction and when bidders only have single unit of demand. Special cases of these two extreme cases are known in the literature, with which we will compare our results. We will then explore what happens if we vary the level of demand reduction in between. 5.1 No Demand Reduction If there is no demand reduction, a bidder s future utility does not depend on whether he wins the current round of the auction, so it is a dominant strategy to bid his true 4 In fact, this result generalizes to sequential auctions with more than 3 rounds, with r k,i (z) = z + (EU k+1,i EU k+1,i+1 ) at equilibrium. 16

18 valuation at each round of the auction. Thus the expected selling price at equilibrium will be the same for all rounds of the auction. 5.2 Full Demand Reduction In this section we will explore what happens when the demand reduction is taken to the maximum. In this case, bidders who already possess any number of items will for sure value any item at 0, i.e., F i (x) = 1 for x > 0 for any i = 1, 2,. Thus, df i (x) = 0 for any x > 0 for any i = 1, 2,. So the bidder s valuation is fully defined by a single distribution F 0 (x). In fact, for this section we are able to give closed form solution for sequential auction with any number of stages. In this section, we let K be the number of stages in the sequential auction. We will study two common choices of distributions for F 0 (x), the uniform distribution and the triangular distribution Uniform Distribution Following Bernhardt and Scoones [1994], we analyze an example where F 0 (x) is the cumulative density function of a uniform distribution on [1, 2]. The interval [1, 2] instead of [0, 1] is chosen to ensure that bidder never shades their bids to zero. The case K = 2 corresponds exactly to one of the results in Bernhardt and Scoones [1994]. Following the same procedure as in Section 3, it can be shown that the bid function of a bidder in the stage k is b k,0 (z) = z 1 n + 2 K + 1, k = 1, 2,, K. n + 2 k 17

19 It can also be shown that the expected selling price in stage k, EP k, is EP k = 2 1 n + 2 K 2 n + 3 k + 1, k = 1, 2,, K. n + 2 k The case when K = 2 agrees with the result presented in Bernhardt and Scoones [1994]. Notice that EP k EP k 1 = 1 n + 2 k 3 n + 3 k + 2 n + 4 k which is negative unless n = k (which results in EP k = EP k 1 ). This suggests that the expected selling price is strictly decreasing except from the second-to-last stage to the last stage when the total number of bidders is exactly one more than the number of stages. (Recall that there are a total of n + 1 bidders.) This agrees Bernhardt and Scoones [1994], which says that in a two-stage sequential auction we always have a declining price pattern if there are at least 4 bidders, and that if there are 3 bidders, both rounds have expected selling price of 4 3. These results for three-round and four-round sequential auctions are summarized in Table Triangular distribution In the results above, we have assumed that F 0 (x) is the cumulative density function of a uniform distribution on [1, 2], i.e., F 0 (x) = x 1 for x [1, 2]. Alternatively, we can assume that F 0 (x) = (x 1) 2 for x [1, 2], which gives us a triangular distribution. Bidders with this valuation distribution tend to value an item more than uniform distribution case we described above. 18

20 # bidders EP 1 EP 2 EP Table 1: Expected selling price of a 3-stage sequential auction. Bidders have unitdemand with valuation drawn uniformly on [1, 2] for each round. # bidders EP 1 EP 2 EP 3 EP Table 2: Expected selling price of a 4-stage sequential auction. Bidders have unitdemand with valuation drawn uniformly on [1, 2] for each round. With this valuation distribution, the equilibrium bidding function is b k,0 (z) = z 1 2n 2K n 2k + 3, k = 1,, K and the equilibrium selling price is EP k = 2 1 2n 2K (2n 2k + 5) + 1 2(2n 2k + 3), k = 1,, K 19

21 Notice that EP k EP k 1 = 1 ( ) 1 2 2n 2k n 2k n 2k + 7 which is always negative for k n. Therefore, the expected selling price is always declining. # bidders EP 1 EP 2 EP 3 EP Table 3: Expected selling price of a 4-stage sequential auction. Bidders have unitdemand with valuation drawn from F 0 (x) = (x 1) 2 on x [1, 2]. 5.3 Steady Demand Reduction In this section we study the scenario where the bidders have two-unit demand. For the first unit, bidders draw valuations from the triangular distribution described above; for the second unit, bidders draw valuations from the uniform distribution described above. It can be shown that this set of valuations satisfies Equation 12, the steady reduction assumption. The expected selling price is estimated numerically using Monte Carlo simulations. 5 As we can see, when there are 3 or 4 bidders, the expected selling price first increases and then decreases. When there are at least 5 bidders, the expected selling prices always increase. In summary, when there is no demand reduction, the expected selling price is 5 We apply the central limit theorem to ensure the accuracy of results at high confidence level 20

22 density Has 0 items Has 1 item valuation Figure 2: Bidders with two-unit demand under steady demand reduction Expected Selling Price nd round 3rd round # bidders Figure 3: Bidders with two-unit demand under steady demand reduction: Expected selling price constant; when there is maximum demand reduction, the expected selling price is declining; when there is steady demand reduction, the expected selling price is increasing, or even non-monotonic for small number of bidders. From these results we realize that we can theoretically observe increasing price or non-monotonic pattern under moderate level of demand reduction, despite that price pattern is always non-increasing for zero or maximum demand reduction. 5.4 Various Level of Demand Reduction In this section, we take a closer look at how increasing demand reduction gradually affect expected selling price pattern. Let F 0, F 1, F 2 be the cumulative distribution of 21

23 uniform distributions on [1, 2], [1, 1 + v], and [1, 1 + v 2 ], respectively, where v controls the level of demand reduction. When v 0, we are close to the maximum reduction case in Section When v = 1, we have the no reduction case in Section 5.1. When v varies continuously from 1 to 0, we are able to continuously vary the level demand reduction from weak to strong. The density functions of the valuations for bidders with different levels of demand reduction are shown in Figure 4. v=0.8 v= Has 0 items Has 1 item Has 2 items 0.5 Has 0 items Has 1 item Has 2 items v=0.4 Has 0 items Has 1 item Has 2 items v=0.2 Has 0 items Has 1 item Has 2 items Figure 4: Bidder valuations for various levels of demand reduction: The steady reduction assumption is no longer satisfied with this set of valuation functions, and the amount by which a bidder shades his bid in the second round can depend on his valuation (Equation 8). For convenience, below we plot the amount of bid shading for a bidder with a high valuation so that p(z) 0 in Equation 8, because we are interested mostly in the behavior of the winner of an auction who usually has a high valuation. We see that bidders shade their bids more in the first round than in the second 22

24 0.5 v=1 0.5 v= nd round ( loss) 2nd round ( win) nd round ( loss) 2nd round ( win) v= v= nd round ( loss) 2nd round ( win) nd round ( loss) 2nd round ( win) v= v= nd round ( loss) 2nd round ( win) nd round ( loss) 2nd round ( win) Figure 5: Bid shading versus number of bidders at various levels of demand reduction. Horizontal axis represents the number of bidders. round. With stronger demand reduction (smaller v), the bidders underbid a lot more in the second round if they won than if they lost in the first round. This is because the expected utility of being in the third stage with a high demand is much greater than the expected utility of being in the third stage with a low demand. Figure 6 shows how the expected selling price depends on the number of bidders for various levels of demand reduction. As we expect, the expected selling price is higher if there are more bidders. When demand reduction is strong (v = 0.2), the expected selling prices are declin- 23

25 1.8 v=1 1.8 v= nd round 1.2 3rd round nd round 1.2 3rd round v= v= nd round 1.2 3rd round nd round 1.2 3rd round v= v= nd round 1.2 3rd round nd round 1.2 3rd round Figure 6: Expected selling price versus number of bidders at various levels of demand reduction. Horizontal axis represents the number of bidders. ing even for small n; when demand reduction is weak (v = 0.8), the expected selling prices increase for small n and decline when n is sufficiently large, but the change in expected selling price also becomes smaller; when there is no demand reduction (v = 1), the expected selling prices become constant. Figure 7 shows how expected selling price patterns change with the level of demand reduction. As expected, the expected selling prices are higher when demand reduction is lower (corresponding to a higher v). However, the level of demand reduction affects the selling price in different rounds differently, leading to different expected selling 24

26 1.5 3 bidders bidders nd round 3rd round nd round 3rd round bidders bidders nd round 3rd round nd round 3rd round Figure 7: Expected selling price versus level of demand reduction with various number of bidders. Horizontal axis represents the discount factor of demand (v). Notice the different scales for vertical axes. v 3 bidders 4 bidders 5 bidders 6 bidders 7 bidders 8 bidders Table 4: (Qualitative) expected selling price pattern for different number of bidders and different level of demand reduction (v = 0 corresponds to complete reduction and v = 1 corresponds to no reduction) 25

27 price patterns. When there are at least 4 bidders, increasing the level of demand reduction will first lead to an increasing price pattern, and then a declining price pattern if the demand reduction is strong enough. This trend is also summarized in Table Additional Examples In this section, we look at the expected selling price at equilibrium for several other popular valuation distributions Exponential Distribution with Divisibility Assumption Let F 0 (x) = F 3 (x), F 1 (x) = F 2 (x), F 2 (x) = F (x), where 1 e x x 0 F (x) =. 0 x < 0 density Has 0 items Has 1 item Has 2 items valuation Figure 8: Exponential Distribution: Bidder Valuations The distributions of the valuation for three types of bidders are shown in Figure 8. This set of distributions satisfies the divisibility assumption. Thus, the amounts by which bidders shade their bids depend solely on the game states they are in. These 26

28 amounts are shown in Figure 9. Notice that the bidders underbid a lot more in the first round than in the second round. shade amount nd round ( loss) 2nd round ( win) # bidders Figure 9: Exponential Distribution: Bid Shading The expected selling price is non-decreasing for all n (Figure 10). Expected Selling Price nd round 3rd round # bidders Figure 10: Exponential Distribution: Expected Selling Price Normal Distribution with Divisibility Assumption Our second example still satisfies the divisibility assumption with F (x) being the cumulative normal distribution with mean 10 and variance 1. (Figure 11) Notice that the bidders still underbid a lot more in the first round than in the second round. This time, the expected selling price is non-decreasing for small n (Figure 13), and the change of expected selling price between rounds of auctions is small. 27

29 density Has 0 items Has 1 item Has 2 items valuation Figure 11: Normal Distribution: Bidder Valuations shade amount nd round ( loss) 2nd round ( win) # bidders Figure 12: Normal Distribution: Bid Shading Expected Selling Price nd round 3rd round # bidders 6 Conclusion Figure 13: Normal Distribution: Expected Selling Price Past empirical investigations of sequential auction reveal a variety of price patterns. Since the baseline auction theory predicted either an increasing or a constant price pattern, it had to be modified to include the possibility of a declining price pattern. In this paper we demonstrate that when one relaxes the assumption of unit demand and when bidders have a demand reduction, price patterns could be constant, 28

30 increasing, decreasing or even non-monotonic in sequential private value auctions. We have shown that the selling price pattern depends on different shapes of demand reduction, captured by the underlying distributions of bidders valuations. Even for the same distribution, the selling price pattern could be different depending on the number of bidders. Since in many sequential auctions bidders may have multiunit demand (in some auctions different bidders win multiple items), our results imply that different price patterns could be supported as a Nash Equilibrium. To comprehend the expected selling price pattern, we notice that there are two forces driving expected selling price over time: the direct effect of demand reduction and the option value of waiting as a result of demand reduction. The direct effect of demand reduction implies that bidders on average have lower valuations in later rounds of the sequential auction if they win. The option value of waiting implies that bidders shade bids a lot more in earlier rounds. The relative strength and interaction of the two forces generates various expected selling price patterns which many empirical studies have observed. 29

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