Is Japanese Dutch Auction Unreasonable?: A Note on Dutch Auction with Mari Minoru Kitahara and Ryo Ogawa February 7, 2006 Dutch auction is a widely used auction system in flower markets, and Japanese flower markets are no exception. However, there is one unique difference in Japanese Dutch auction: it has Mari -stages. In a usual multi-unit Dutch auction, for each round, (i) clock drops continuously from sufficiently high price until a buyer stops the clock, (ii) the buyer gets the goods at the price pointed at by the clock, and then (iii) the auction goes to the next round if there still remain some units. In Japanese Dutch auction, Mari-chū (in the process of Mari) signal appears for a few seconds between (ii) and (iii), while other buyers are allowed to purchase the goods at the same price. In this short note, we investigate the effects of Mari. We find that Japanese Dutch auction seems reasonable in the sense that it speeds up the auction sufficiently at the cost of negligible efficiency loss. 1 Model and Results 1.1 Overall settings There are k units of homogeneous goods and n buyers with their types t independently drawn from U0, 1, which are their private information. Each buyer demands one-unit and is risk-neutral: his payoff is (v(t) the price) 1 {He gets the goods.}. We assume the uniform distribution for the valuation: v(t) = t. The auction system is almost the same as a usual multi-unit Dutch auction but has a Maristage in the first round. (After the Mari-stage, a usual Dutch auction follows.) We focus on simple symmetric equilibrium: there exist βn,k and µ n,k such that (i) type-t buyers stop the clock at βn,k t in the first round, (ii) if the clock stops at βn,k t in the first round, then other buyers participate in Mari if and only if their type t is higher than µ n,k t, and (iii) they play as in a usual Dutch auction from the second round on: if there remain l units and m buyers, then type-t buyers bid m l m t as well known. 1.2 Mari-Stage 1.2.1 Modeling Mari-Stage A type- t buyer has stopped the clock at β t. Other n 1 buyers, with their types independently drawn from U0, t), simultaneously decide whether to participate We are grateful to Kuniyoshi Saito, Kiri Sakahara, and Daisuke Shimizu. 1
in Mari. If the number of Mari-participants is no more than k 1, then all the participants become winners. Otherwise, k 1 winners are randomly selected among the participants. All winners pay β t for the goods. If there still remain some units after Mari, then the auction goes to the next round. 1.2.2 Equilibrium Mari-Participation Note that each buyer participates in Mari with probability 1 µ ex ante in the equilibrium. Let E 1 µ,m f(j) m i=0 f(i) mc i (1 µ) i µ m i. Then, if a type-µ t buyer participated in Mari, then his expected payoff would be (µ t β t)e 1 µ,n 2 1 j k 2 + k 1, and otherwise, the payoff would be ( E 1 µ,n 2 µ t n k ) n 1 j µ t 1 j<k 1 Thus, µ is characterized as follows. Proposition 1. If β n k n 1, then µ = 1. Otherwise, µ (uniquely) solves (µ β)e 1 µ,n 2 1 j k 2 + k 1 Proof. See that (1) implies (1 β)e 1 µ,n 2 1 j k 2 + k 1 k 1 j = µe 1 µ,n 2 n 1 j 1 j<k 1. = k 1 n 1. (1) (2) Note that it implies that Mari-participation occurs only if Mari-price is less than n k t n 1 < t. Thus, whether Mari-participation in reality is justified as an equilibrium behavior may not be clear until we investigate the first winner s equilibrium clock-stopping strategy. 1.3 Inefficient Allocation and Speeding Up The expected efficiency loss within this stage is ( 1 t(1 µ)(k 1)E 1 µ,n 1 1 k ) 1 j>k 1. (3) 2 j + 1 If goods are allocated efficiently, then the expected surplus is ( t(k 1) 1 k ). (4) 2n Denote by LR n,k (β) the loss rate (3)/(4) where µ is determined as in Proposition 1. For n = 3 and k = 2, the graph becomes as follows. 2
LR 0.25 0.2 0.15 0.1 0.05 0.1 0.2 0.3 0.4 0.5 Β The expected number of the goods sold within this stage is E 1 µ,n 1 (k 1)1 j>k 1 + j1 j k 1. (5) Similar to LR, denote by RR n,k (β) the reduction rate (5)/(k 1). For n = 3 and k = 2, the graph becomes as follows. 0.8 0.6 0.4 0.2 RR 1 0.1 0.2 0.3 0.4 0.5 Β Note that to suppress the loss rate below 5%, β should be more than 0.26. At the same time, to attain the reduction rate of more than 50%, β should be less than 0.42. These results may make one pessimistic about the good performance in the equilibrium. 1.4 Overall Equilibrium Characterization Suppose that a type-t buyer lowers his bid by small > 0. Then, his payment decreases by in the event he is the first to win the good, the probability of which is t n 1. At the same time, since βt is bid by type-(t /β) buyers, he additionally fails the possibility to be the first winner in the event the highest type among other buyers belongs to (t /β, t, the probability of which is ( /β)(n 1)t n 2, where his probability to win the good decreases from 1 to. In the equilibrium, these potential gains and E 1 µ,n 2 1 j k 2 + k 1 j+1 1 j>k 2 losses from deviation must offset one another. Consequently, the equilibrium β must satisfy: t n 1 = 1 β (n 1)tn 2 {(t βt) (1 E 1 µ,n 2 1 j k 2 + k 1 )}. (6) 3
In summary, the equilibrium clock-stopping β n,k and Mari-participation µ n,k are the solution of (1) and (6), which are characterized as follows. Proposition 2. βn,k = n k n, and µ n,k is the (unique) solution of Proof. See that (1) implies (2). E 1 µ,n 2 1 j k 2 + k 1 < n k n 1. = k 1 n n 1 k. (7) Note that βn,k Thus, the equilibrium Mari-price is consistent with Mari-participation. Moreover, the equilibrium performance looks good: β3,2 = 1/3 (0.26, 0.42). It seems very good: LR3,2 = 3.125% and RR3,2 = 75%. This result is not specific for these parameter values. We plot (RR, LR ) below for more broad parameter values, for 2 k 10 and k + 1 n 5k. 0.03 0.025 0.02 0.015 0.01 0.005 LR 0.76 0.78 0.82 0.84 van den Berg et al. (2001) reports that k is 5.98 on average in the Aalsmeer Flower Auction. Thus, we may expect more than 80% reduction in exchange for less than 1.5% efficiency loss. 2 Discussions It may be easy to understand the low efficiency-loss result. The expected value of the k 1th highest value among n 1 buyers is n k t. n Thus, β is a marketclearing price on average. As expected by this intuition (and law of large numbers), LR15,10 0.990%. We are unable to give clear intuition for the high round-reduction result. However, by (7), E 1 µ,n 1 1 j>k 1 1 µ 1 α n where α = k/n, (k 1) (n 1)(1 µ) = n 1 (n 1)µ(1 µ) k 1 n 1 RR (1 µ), (8) µ(1 µ) and for any ɛ > 0, 1 Φ(ɛ n 1) 1/n 0 as n. 4
where Φ is the cumulative distribution of N(0, 1). Thus, it seems that µ /α converges to 1 as the market size grows. If so, then since β /α 1 and RR = 1 µ 1 β, RR converges to 1. RR 15,10 83.798%, as expected by this intuition. Note that in more competitive (i.e., larger α) markets, the pure random allocation, which corresponds to β = 0, leads to larger efficiency loss: for example, LR 3,2 (0) = 25% but LR 11,2 (0) 45%. As indicated by the intuitions above, the equilibrium performances of Mari-stages seem not damaged by the competitiveness: LR 11,2 1.376% and RR 11,2 78.613%. 3 Summary Japanese Dutch auction seems reasonable: it speeds up the auction sufficiently in exchange for negligible efficiency loss. It seems more effective for larger or more competitive markets. References van den Berg, Gerald J., van Ours, Jan C. and Pradhan, Menno P. (2001): The Declining Price Anomaly in Dutch Dutch Rose Auctions. American Economic Review 91: 1055 1062. Milgrom, Paul and Weber, Robert J. (2000): A Theory of Auctions and Competitive Bidding, II. The Economic Theory of Auctions. P.Klemperer. Cheltenham: Edward Elgar Publishing, Ltd. 2: 179 94 5