GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.121

Efficiency versus Economy of Time inMulti-Unit Descending Auction:
The Role of “Mari” at Flower Markets in Japan

Minoru Kitahara and Ryo Ogawa

February 2010

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Eﬃciency versus Economy of Time inMulti-Unit Descending Auction:
The Role of “Mari” at Flower Markets inJapan∗
Minoru Kitahara†andRyo Ogawa‡First Draft: February 2006
This Version: February 2010

Abstract∗

The authors are grateful to Michihiro Kandori, Toshihiro Matsumura, Kuniyoshi
Saito, Kiri Sakahara, Daisuke Shimizu, and Takahiro Watanabe for comments and discussions. The authors are also grateful to seminar/conference participants at the 2006
Autumn Meeting of the Japanese Economic Association, the 2006 South and South-east
Asia Meeting of the Econometric Society in Chennai, and the 2008 Far Eastern Meetingof the Econometric Society in Singapore. Financial supports from Japan Society for the
Promotion of Science are gratefully acknowledged.
†
JSPS Research Fellow, Graduate School of Social Sciences, Tokyo Metropolitan University, 1-1 Minami-Osawa, Hachioji-shi, Tokyo 192-0397, Japan, and Visiting Research
Fellow, Advanced Research Institute for the Sciences and Humanities, Nihon University,
12-5 Goban-cho, Chiyoda-ku, Tokyo 102-8251, Japan (mkitahar@tmu.ac.jp).
‡
Corresponding author. Institute of Social and Economic Research, Osaka University,
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan (r-ogawa@iser.osaka-u.ac.jp).

1

The auctioning rule in Japanese flower markets is a slightly modified version of that of the original Dutch flower auction. At Japanese
flower markets, there is an additional stage, called “mari”, wherebuyers who lost in the previous auction can apply for purchasing theremainder of flowers at the same price as in the previous auction. We
investigate the role of “mari” in multi-unit descending auction, andshow that “mari” extensively speeds up the market procedure at thecost of suﬃciently small loss of eﬃciency, compared to the originalDutch sequential auction.

1 Introduction
Descending price auction (or, Dutch auction) is widely used at wholesale
flower markets in countries such as the Netherlands and Japan among
others. The Dutch market is leading in the worldwide market share, andOta Floriculture Auction in Tokyo is the second largest1 . The Dutch styleof the flower auction originates in the Netherlands (as is clear from thename) and was imported into Japan, but in the process of adoption of thedescending style, many of flower markets in Japan, including Ota, made aslight modification to the original rule. In this paper we call the modified
rule in Japan as mari, which comes from marketmen’s jargon referring to themodification. This paper investigates both advantages and disadvantagesof mari, and shows that mari extensively speeds up the market procedure
at the cost of suﬃciently small loss of eﬃciency, compared to the original1

Two organizations of flower auctions, Flora Holland and Aalsmeer, were competingin the Netherlands until 2007. In 2008, Flora Holland and Aalsmeer merged, and thenOta Floriculture in Tokyo became the second largest.

2

Dutch sequential auction.2
The key instrument at those markets is the “auction clock”, beginningwith a high price which is then gradually lowered. If a buyer desires tobuy the good at the price on the clock, he pushes the button then the clock
stops. The good is won by the buyer who was the first to stop the clock,and the winner pays the price displayed on the clock to the auctioneer.
When several identical goods are for sale, some options are open to theseller, even when the market sticks to the descending price style as the basic
procedure. For instance, the seller must decide how to sell the remainderof the goods after some buyer has won one good. The remainder is brought
up to the auction again at the Ducth flower auctions,3 whereas at many
wholesale flower markets in Japan, the remainder is sold by the peculiar
method called mari.
Roughly speaking, mari allows other buyers who did not win the goodat the previous auction to apply for purchasing the remainder (if any) at thesame price as the winner in the previous auction paid. If the number of applicants is less than the number of the remainder then all the applications are
accepted, while the remainder is randomly allocated between applicants
if the number of applicants exceeds the number of the remainder.
This paper investigates the eﬀect of mari to the equilibrium outcomeof the auction. One interesting nature would be that when the number ofapplicants exceeds the number of the remainder, the random allocation inmari may cause ineﬃciency: that is, some portion of the remaining goods
2

In an interview at the Ota Floriculture Auction, a director who organizes the information system of the auctioning room pointed out that they acutally have a “feeling” thatmari contributes to the economy of time at the market.
3
Such situations in which the goods are sold sequentially are called “sequential auctions” in the literature. The analysis of sequential auctions originates with Milgrom andWeber [3] and Weber [4].

3

may be allocated to the buyer whose value is lower among the applicants.
On the other hand, mari may contribute to the economy of time in the sensethat it decreases the number of auctions that should be held until all of theremainder be sold out. The purpose of the paper is to show that addition ofmari to the sequential descending auction extensively reduces the numberof auctions to be held, while the loss of eﬃciency is suﬃciently small.
In investigating the relationship between eﬃciency and economy oftime, there can be various ways of evaluating the trade-oﬀ. In this paper,
we define two measures, “eﬃciency loss rate” (LR) and “round reductionrate” (RR), and examines the asymtotic property of LR and RR when thenumbers of buyers and goods increase to infinity. LR is defined as the ratioof the expected welfare loss in mari to the expected welfare if the goods
were to be allocated eﬃciently. If such LR is closer to zero, we see that theloss of eﬃciency is smaller. RR, on the other hand, is defined as the ratio of
the expected number of goods sold in mari to the number of the remainder.
We regard that the mari contributes more to the economy of time when RR
is closer to one. Our result actually shows that LR converges to 0 while RR
converges to 1 when the number of buyers and that of goods increase toinfinity (Theorem in page 19).
The rest of the paper is organised as follows. We formulate the sequential descending auction with mari in Section 2. In Section 3, we investigate
the symmetric equilibrium of the game and provide some characteristics.
Main result on the trade-oﬀ between eﬃciency and economy of time ispresented in Section 4, in which we argue that mari extensively reduces thenumber of auctions while the loss of eﬃciency is suﬃciently small. Section
5 provides concluding remarks.

4

2

The Model

In this section, we introduce the sequential descending auction model withmari.
To formulate the problem, suppose that there are k identical goods forsale and N buyers. We assume that each buyer can buy only one item; thatis, we limit our attention to the unit-demand case. The goods are to be soldin the following procedure:
1. The “auction clock” starts from a suﬃciently high price and goes
down until one of the buyers stops it.
2. The buyer who stopped the clock wins the good at the price on theclock.
3. Other buyers can apply to purchasing the remaining goods at thesame price being paid by the winner in step 2 (mari).
(a) If the number of buyers who applied to the purchase is greaterthan the number of remaining goods (that is, k − 1), then the
goods are allocated randomly between the applying buyers. Putanother way, if the number of applying buyers is j ≥ k − 1, theneach buyer wins the good with (ex ante) probability j/(k − 1).
(b) Otherwise, all the applying buyers can win the good.
4. The remaining goods will be sold in the usual sequential descending
auction without mari.4
4

In the real Japanese flower auctions, mari may happen for several times. That is, after
mari (step 3), the auctioning procedure simply goes back to step 1. The existence of mariin and after the second round may aﬀect how mari in the first round works. In order toavoid such eﬀects, in this paper we focus on a simplified game in which mari can comeup only in the first round.

5

We let ti denote ith buyer’s type (value), and assume that the types are
independently drawn from the uniform distribution on [0, 1]. Each buyer
attains the payoﬀ of ti − p if the buyer wins the good and the winning priceis p, and the payoﬀ of 0 if the buyer fails to win the good. We assume thatthe buyers are risk-neutral.
The strategy of the buyers comprises following components. One is thedecision about on what price the buyer stops the “auction clock” in the firstround (depending on her type t), another is the decision about whether toapply in mari (depending on the price p in the first round and on her typet), and the other is how to act in the sequential auction for the remaining
goods if she either does not apply or loses in mari (depending on the pricep, her type t, and the numbers of remaining buyers and remaining goods).
In this paper we focus on the symmetric equilibrium as in the following
form: there exist βN,k and µN,k such that (i) type-t buyer stops the clock at
βN,k t in the first round, (ii) if the clock stopped at the price p = βN,k t̄ inthe first round, then other buyers participate in mari if and only if the typet is higher than µN,k t̄, and (iii) the buyers follow symmetric equilibrium
strategies in the sequential descending auction in 4.

3

Equilibrium Strategy

In this section, we investigate the equilibrium strategy of buyers in theauction with mari. We also provide some equations characterizing µN,k and
βN,k in the equilibrium, which will be extensively used in exlopring analysisof the trade-oﬀ between eﬃciency and economy of time in the next section.
In the beginning, we should obtain the symmetric equilibrium strategy
in the auctions after mari when there is still remainder of the goods. It

6

is easily shown that the bidding strategy in the equilibrium after mari is
identical to that in the equilibrium of the usual sequential auction withoutmari, which is provided as
bM,ℓ (t) =

M−ℓ
t,
M

(1)

where M is the number of remaining buyers and ℓ is that of remaining
goods.5
Given the equilibrium strategy in the auctions after mari, bM,ℓ (t), we
investigate the buyers’ decision about whether to apply in mari, for arbitrarily fixed bidding strategy in the first round, β. Suppose that type-t̄
buyer won the good in the first round, that is, the winner bid βt̄ and won
the good with payment βt̄. Then, in mari, each of remaining N − 1 buyers, whose types are independently distributed uniformly on [0, t̄), has achance to apply for purchasing the good at the price βt̄.
As was noted in the previous section, we are now interested in the linear
symmetric equilibrium in which each buyer apply in mari if and only if thebuyer’s type is no less than µt̄. To investigate the symmetric equilibrium,in the following, we study one typical buyer’s decision whether to apply
in mari, given other N − 2 buyers’ strategy µ.
If the buyer decides to apply in mari, the probability of winning the gooddepends on the number of other applicants, j. If j < k − 1, the buyer will
be able to win the good with probability one, whereas the buyer wins withprobability (k − 1)/( j + 1) if j ≥ k − 1. As each of other buyers apply in mariwith probability of 1 − µ ex ante, the probability that the number of other( )applicants equals to j is N−2(1 − µ) j µN−2− j , and therefore the probabilityj5

See Milgrom and Weber [3] and Weber [4].

7that the buyer wins the good can be written as
)
)
k−2 (
N−2 (
∑
∑
N−2
N
−
2
k−1
(1 − µ)i µN−2−i +.
(1 − µ)i µN−2−i
i
i+1
i
i=0
i=k−1

In the economy of notation, we write the expression as
[
1−µ
EN−2

k−1
1 j<k−1 +1 j≥k−1
j+1

]

1−µ

in the following, where the function Em [ f ( j)] is defined as6
1−µEm [ f ( j)] ≡

m ( )
∑
m
(1 − µ)i µm−i f (i),
i
i=0

(2)

and 1 is the indicator function.
To summarize, if the other buyers are following the mari participation
strategy µ, then the expected payoﬀ of the buyer with type t who decides
to apply in mari, denoted by πm (t) would be
1−µπm (t) = (t − βt̄)EN−2[]
k−1
1 j<k−1 +1 j≥k−1 ,j+1(3)

where t − βt̄ is the ex post payoﬀ when the buyer successfully wins thegood in mari and the E thing represents the probability that the buyer wins
the good in mari.

If the buyer does not apply in mari, on the other hand, the expected

1−µEm [ f (j)] can be understood as the expectation of function f ( j) when j follows the
binomial distribution with parameters 1 − µ and m.
6

8

payoﬀ, denoted by πw (t), can be written as

[(
)
]

1−µ


t
−
b
(µ
t̄)
1
if t ≥ µt̄,EN−1−
j,k−1−
j
j<k−1
 N−2
πw (t) = 
[(
(
)
)
]
∫t


s

E1−µ
x
ds 1 j<k−1 if t < µt̄.
N−2
0 N−1− j,k−1− j µt̄

(4)

We shall explain the two expressions in due order. For case of t ≥ µt̄,the buyer’s optimal action in the auctions after mari (if any) is to act as if
type-µt̄ buyer in the equilibrium. This is because there are no other buyers
with types more than µt̄ in the auctions after mari as long as other buyers
are following µ. Moreover, this buyer will win with probability one in theauction just after mari (if any). Hence, the expected payoﬀ when this buyer
does not participate in mari can be written as in (4), where bM,ℓ (t) is theequilibrium strategy in the auctions after mari as given in (1).
For case of t < µt̄, we define xM,ℓ (s) is asxM,ℓ (s) ≡

)
ℓ−1 (
∑
M−1
i=0

i

(1 − s)i sM−1−i .

(5)That is, xM,ℓ (s) denotes the probability that type-s buyer is at least ℓ-highest
among M buyers whose types are independently drawn from the uniform
distribution over [0, 1].7
Here we have the following characterization of µ in the equilibrium inmari stage.

7

Conceptually, the expected payoﬀ from deferring participation in mari is written in
more complex form than the one given in (4). In calculating the expected payoﬀ, we musttake into consideration all of the possibilities that the buyer wins the first after mari, thebuyer loses the first but wins the second after mari, and so forth. Given the symmetricequilibrium strategy in the usual sequential descending auction following mari, whichis already given in (1), we can rewrite the (complex) expected payoﬀ as the simple form
given in (4).

9Proposition 1. Suppose that the buyers are taking first-round bidding strategy
β and type-t̄ buyer won the good in the first round. Then there exists uniquelinear symmetric equilibrium in mari stage in which each of the remaining buyers
participates in mari if and only if the buyer’s type is no less than µt̄, where µ is
given as follows.
1. If β ≥ (N − k)/(N − 1), then µ = 1.
2. If 0 < β < (N − k)/(N − 1), then µ is the unique solution of1−µ(1 − β)EN−2

[
]
k−1
k−1
1 j<k−1 +1 j≥k−1 =
.
j+1
N−1

(6)

Proof. (Necessity). If µ constitutes an equilibrium when 0 < β < (N −k)/(N −1), it must be that the expected payoﬀ of applying in mari and that of not
applying ((3) and (4), respectively) are indiﬀerent when the buyer’s typeis µt̄. That is,[
1−µ(µt̄ − βt̄)EN−2

[
]
]
k−1
1−µ k − 1 − j
1 j<k−1 +1 j≥k−1 = µt̄EN−2
1 j<k−1 .
j+1
N−1− jApplying Lemma 2 in the Appendix, we have
[
1−µ
µEN−2

]
[
]
k−1− j1
1−µ
1 j<k−1 =EN−1 (k − 1 − j)1 j<k−1 ,N−1− j
N−1

and
[

]
k−1
1 j<k−1 +1 j≥k−1
j+1
]
[
[
]
1
k−1
1−µ
1−µ
1 j≥k−1 −EN−1 j1 j≤k−1 + (k − 1)1 j>k−1 .
= EN−2 1 j<k−1 +
j+1
N−1

1−µ
µEN−2

10

(7)

Substituting these two equations into (7) and rearranging yields[]
k−1
1 j<k−1 +1 j≥k−1
j+1[]
1−µEN−1 (k − 1)1 j<k−1 + j1 j=k−1 + (k − 1)1 j>k−1

1−µ(1 − β)EN−2

1
N−1
k−1
=
.
N−1
=

Thus we have equation (6). As the left-hand side is increasing in µ by
applying Lemma 3 in the Appendix, and
[

]
k
−
1
k−1
E1N−2 1 j<k−1 +1 j≥k−1 =
,
j+1
N−1

[
]
k
−
1
E0N−2 1 j<k−1 +1 j≥k−1 = 1,
j+1

equation (6) has a unique solution for µ in (0, 1) as long as 0 < β < (N −k)/(N − 1). For β ≥ (N − k)/(N − 1), µ = 1 follows from that the uniquesolution for (6) is µ = 1 when β = (N − k)/(N − 1).

(Suﬃciency). Next we show that the buyer whose type is more thanµt̄ actually participates in mari as long as other buyers are following theequilibrium strategy given in (6). If this buyer does not participate in mari,the expected payoﬀ is (from (4))[]
1−µπw (t) = t × EN−2 1 j<k−1 + B1 ,where B1 is the term irrelevant of t. If this buyer decides to participate inmari, on the other hand, the expected payoﬀ is (from (3))
1−µπm (t) = t × EN−2[]
k−1
1 j<k−1 +1 j≥k−1 + B2 ,j+1

11

where B2 is again the term irrelevant of t. Thus, we have that∂∂πm (t) > πw (t)∂t∂t
for all t > µt̄. As the two payoﬀs are indiﬀerent for type-µt̄ buyer (that is,πm (µt̄) = πw (µt̄) as shown in the necessity part), we have πm (t) > πw (t) for
all t > µt̄, and thus we can conclude that the buyer whose type is morethan µt̄ cannot be better oﬀ by absenting mari as long as other buyers arefollowing µ.
Finally we show that the buyer whose type t is less than µt̄ does notparticipate in mari as long as other buyers are following the symmetricequilibrium strategy µ. For such t < µt̄, we have
[
( )
]
∂
t
1−µ
× 1 j<k−1 ,πw (t) = EN−2 xN−1−j,k−1− j∂tµt̄
from (4). We should note here that xN−1− j,k−1− j (·) is a sort of probability (see(5)), and therefore we have
[
]
∂
1−µ
πw (t) ≤ EN−2 1 j<k−1∂t(8)

for all t < µt̄. As the two payoﬀs are indiﬀerent for type-µt̄ buyer (asshown in the necessity part), we can apply similar discussion as in the caseof t > µt̄, and conclude that the buyer whose type is less than µt̄ cannot
be better oﬀ by participating in mari as long as other buyers are followingµ.


An informal interpretation of the equilibrium is as follows. Withoutmari, a buyer who failed to win the good in the first round bids higher
in the next round than in the first, due to the deterioration of available
12

supply relative to current demand.8 Hence, the buyer whose value is close
to t̄ would prefer to win the good in mari with price βt̄ (although this ishigher than the buyer’s potential bid in the first round), rather to get into
the next round where the bid higher than βt̄ might be required due to theraise of competitiveness. This is what happens in 2. of the proposition. On
the other hand, if β is suﬃciently high as in 1. of the proposition so thatthe bidding higher than βt̄ would not be needed in the following roundafter mari even with the raise of competitiveness, then in the equilibrium
no buyers participate in mari (µ = 1).
In what follows, we investigate the bidding strategy β in the firstround in the equilibrium, given the mari participation strategy provided
in Proposition 1. As will be shown in Proposition 2, β is actually less than(N − k)/(N − 1), and therefore participations in mari occur with positive
probability in the equilibrium.
In the first-period auction the decision problem facing a buyer is slightly
more complex. Again let us take the perspective of one buyer with value t
and suppose that all other buyers are following the first-period strategy β
and mari participation strategy µ.
The equilibrium requires the buyer with type t to bid βt in the firststage, but consider what happens if the buyer bids βs instead. Then thebuyer’s payoﬀ is

∫z
∫1


N−1
N−1

s
[t
−
βs]
+
π
(t
|
u)du
+
πw (t | u)duN−1

m

s
z
Π(s, t) = ∫1sN−1 [t − βs] + πw (t | u)duN−1s
8

if s < z
if s ≥ z, (9)This eﬀect is one aspect of martingale price path result in the usual sequential descending auction without mari. See Milgrom and Weber [3].

13

where z = min{t/µ, 1}, and πm (· | u) and πw (· | u) are those payoﬀs givenin (3) and (4), respectively, emphasizing that the type of the winner in thefirst-round, u, is included in the function as t̄ = u. We shall explain the two
expressions in due order.
The first term results from the event that the buyer wins the good in
the first round. The second term of the first line results from the event thatthe buyer loses in the first round but participates in mari, and u stands forthe highest type among other N − 1 buyers (that is, the type of the buyer
who wins the good in the first round). The third term is the term resulting
from the event that the buyer loses the first round, does not participate inmari and then finally wins the good in the auctions after mari.9
It is possible, however, that the buyer might bid suﬃciently large pricein the first round so that when the buyer happens to lose in the first roundeven with this high bid, it never becomes optimal for him to participate inmari, as the type of the winner in the first round is supposed to be extremely
high. The second line in the expression (9) corresponds to such biddings.
The first term results from the event that the buyer wins the good in thefirst round, and the second term is the event that the buyer loses the firstround, does not (definitely) participate in mari and then finally wins thegood in the auctions after mari.
In the symmetric equilibrium, the buyer with type t bids βt in the firstround and therefore the payoﬀ is calculated according to the first line in (9).
An increase in the bid of this buyer will increase the probability of winning
the good, but at the same time this will reduce the gain from the winningin the first round (first term) as well as the gain from the event that theThe buyer with suﬃciently large type (t > µ) always finds it optimal to participate inmari whatsoever the first round price βt̄ could be. For such buyers, the expected payoﬀ
Π(s, t) is calculated with z = 1, and the third term vanishes.
914

buyer loses the first round but wins the good in mari (second term). To getsome idea about how these eﬀects balance oﬀ, we begin with a heuristic
derivation of symmetric equilibrium strategies.
Diﬀerentiating Π(s, t) with respect to s provides us with()∂Π(s, t) = (N − 1)sN−2 t − βs − βsN−1 − πm (t | s)(N − 1)sN−2
∂s
{
[
]
}
= sN−2 (N − 1) (t − βs) − πm (t | s) − βsAt a symmetric equilibrium, the first-order condition should be satisfied
in substituting s = t. That is, we have{}(N − 1) (t − βt) − πm (t | t) = βtfor all t.

(10)

The left-hand side corresponds to the marginal benefit by an incrementof s, whereas the right-hand side corresponds to the marginal cost (themarginal increment in payment), and the two eﬀects are balanced oﬀ at the
point s = t. Substituting πm (t | t) defined in (3), we have[{(N − 1)1−µ(1 − β) − (1 − β)EN−2

k−1
1 j<k−1 +1 j≥k−1
j+1

]}
= β,

(11)

and by substituting (6), we have β = (N − k)/N.
The derivation of β is only heuristic because (11) is merely a necessary
condition that will be satisfied in the equilibrium. The next proposition
verifies that this constitutes the equilibrium and provides the full characterization of µ.
Proposition 2. Symmetric equilibrium strategies in the sequential descending
auction with mari are given by the bidding strategy in the first round β = (N−k)/N

15

and mari participation strategy µ given as the unique solution of
[
1−µ
EN−2

]
k−1
k−1 N
1 j<k−1 +1 j≥k−1 =
.
j+1
N−1 k

(12)

Proof. Suppose that the other N−1 buyers follow the strategy β = (N−k)/N.
Consider a buyer with type t who is going to bid an amount βs instead ofβt. We are to show that
Π(t, t) > Π(s, t) for all s , t
by showing∂
Π(s, t) > 0 for all s < t,∂s∂Π(s, t) < 0 for all s > t.
∂s
For s < t, we have from (9) that∂
Π(s, t) = (N − 1)sN−2 (t − βs) − βsN−1 − πm (t | s)(N − 1)sN−2
∂s
{
[(
)
]
}
= sN−2 (N − 1) t − βs − πm (t | s) − βsAs long as the other buyers are following the strategy β, (10) is satisfied for{}
all t, and in particular, βs = (N − 1) (s − βs) − πm (s | s) . Therefore,{}∂Π(s, t) = (N − 1)sN−2 fs (t) − fs (s) ,∂swhere we defined fs (t) as fs (t) = t − βs − πm (t | s). It is easy to see thatfs′ (t) = 1 −∂πm (t | s)
∂t
16

(13)

[
1−µ
= 1 − EN−2]
k−1
1 j<k−1 +1 j≥k−1 > 0.
j+1

(14)

Thus we can conclude that ∂Π(s, t)/∂s > 0 for all s < t.
For t < s ≤ z, the partial derivative is the same form as in (13), and wehave ∂Π(s, t)/∂s < 0 for all t < s ≤ z by (14).
For s > z, we can prove the result in a similar manner. We have thederivative{}∂Π(s, t) = (N − 1)sN−2 gs (t) − gs (s) ,∂swhere gs (t) = t − βs − πw (t | s). As t/µ < s in this case, πw (t | s) corresponds
to the second line of the definition in (4), and therefore
[
]
∂
1−µ
πw (t | s) ≤ EN−2 1 j<k−1 < 1∂tfrom (8). Thus we have g′s (t) > 0 for all t, and ∂Π(s, t)/∂s < 0 for all s > z.
We have established that
Π(t, t) > Π(s, t)

for all s , t,that is, the type-t buyer finds it optimal to bid βt as long as other buyers
are following the strategy β.
Substituting β = (N − k)/N into (6) yields the equation (12).

4Eﬃciency vs. Economy of TimeIn this section, we argue that mari extensively speeds up the market procedure at the cost of suﬃciently small loss of eﬃciency.
As shown in Proposition 2, in the symmetric equilibrium, participation17

in mari occurs with strictly positive probability. The possibility of random
allocation in mari causes ex ante ineﬃciency, whereas selling several goods
at one time in mari may contribute to the economy of time at the market. To
formally evaluate the degree of ineﬃciency as well as the level of economyof time, we define two measures, the “eﬃciency loss rate” (LR, hereafter)
and the “round reduction rate” (RR, hereafter). Formal (mathematical)
definitions of those two concepts will be given in the proof of Theorem
with some relevant notions to smoothen the outset of the proof. In thefollowing, we present casual definitions of the concepts and argue how thetrade-oﬀ of our interest can be investigated mathematically.
We refer to LR as the ratio of the expected eﬃciency loss in mari tothe expected surplus when the goods are allocated eﬃciently between theparticipants. Thus, LR = 0 says that the goods will be allocated perfectly
eﬃciently between buyers and hence there will be no loss of surplus inmari (i.e., first best allocation), whereas LR = 1 would describe the situation where the latent surplus from the auction is totally vanished and theresulting surplus is going to be zero.
On the other hand, RR is defined as the ratio of the expected numberof the goods sold in mari to k − 1 (the number of remaining goods in the
beginning of mari). Thus, RR = 1 says that all of the remaining k − 1
goods will be “cleaned up” at one time in mari, whereas RR = 0 refers to
the case where none of the remaining goods will be sold and mari has nocontribution to the economy of time in the auction process.
Figure 1 plots the values of LR and RR for various sets of parameters
2 ≤ k ≤ 10 and k + 1 ≤ n ≤ 5k. The point on the upper-left corner is forthe case of k = 2 and n = 3, while the one on the lower-right corner isfor the case in which k = 10 and n = 11. The figure shows that for most

18

LR
0.03
0.025
0.02
0.015
0.01
0.005
0.76

0.78

0.82

0.84

RR

Figure 1: LR and RRof the parameters, the LR tends to be small with relatively large values of
RR. Put in another way, the figure indicates that mari extensively speeds
up the market procedure (high RR) at the cost of suﬃciently small loss ofeﬃciency (low LR). For instance, van den Berg et al. [2] reports that k is
5.98 on average at the Aalsmeer Flower Auction in the Netherlands. The
figure shows for k = 6 that we have more than 80% of RR in exchange forless than 1.5% of LR.
To formally investigate the tradeoﬀ between LR and RR, we have thefollowing convergence theorem.
Theorem. In the linear symmetric equilibrium of the sequential descending auction with mari, we have
LR → 0 and RR → 1 as N, k → ∞ with

k
= α.
N

∗
denote the expected surplus when the k − 1
Proof for LR → 0. Let WN,kmitems are allocated eﬃciently between N − 1 buyers10 , and WN,kdenote the10

In the beginning of the mari period, one unit out of k items is already sold to one
buyer out of N buyers, and therefore we are interested in how the remaining k − 1 items
are to be allocated between the remaining N − 1 buyers as far as we are concerned with19

expected surplus attained under the mari auction discussed in the paper.
∗m∗
Then the rate LR can be written as (WN,k− WN,k)/WN,k. In what follows, wem∗
are to prove LR → 0 by showing 1 − LR = WN,k/WN,k→ 1 in N, k → ∞.
∗∗In accordance with the definition of WN,k, we let WN,k( j) denote theexpected surplus when j buyers participate in mari and k − 1 goods aremallocated eﬃciently between them. In a similar way, we let WN,k( j) denotethe expected surplus when j buyers participate in mari and k − 1 goods areallocated under the mari rule. Note that we have
[
]
1−µ
∗
∗
WN,k
= EN−1N,k WN,k
(j)

[
]
1−µ
m
m
WN,k
= EN−1N,k WN,k
(j) ,

and

where µN,k is the mari-participation threshold strategy in the equilibriumwith N buyers and k items (see Propositions 1 and 2; in what follows weomit the subscripts N and k).
Our main interest is on j ≥ k − 1, in which case the resulting allocation
would be diﬀerent between eﬃcient allocation and mari. Suppose that, inthe first round, some buyer with value t̄ has pushed the button and won
the good. Now the values of remaining buyers are distributed on U[0, t̄],and those buyers with values in [µt̄, t̄] would participate in mari in theequilibrium. Then for j ≥ k − 1, we have
)
}
k
∗
+µ ,
WN,k
( j) = t̄(k − 1) (1 − µ) 1 −2(j + 1)
1+µ
m
.
WN,k
( j) = t̄(k − 1)
2{(

The first equation is the expected surplus when k − 1 goods are allocatedthe relative eﬃciency of mari auction.

20

eﬃciently to the top k − 1 buyers out of j participants11 , and the second
equation is the expected surplus when k − 1 goods are allocated randomly
between j participants, each of whose type is distributed on U[µt̄, t̄] withthe expected value of t̄(µ + 1)/2.
Firstly, we have

m
WN,k
∗
WN,k

≥

wm
N,k

(15)

∗
WN,k

as a lower bound for 1 − LR, where wm
denotes the expected surplus
N,kgenerated by the event j ≤ (1 + ϵ)(k − 1) in the mari auction for a given small
number ϵ > 0. In what follows, we decompose this lower bound as
wm
N,k
∗
WN,k

=

w∗N,k
∗
WN,k

×

wm
N,k
w∗N,k

,

(16)

where w∗N,k is similarly denoted as the expected surplus generated by theevent j ≤ (1 + ϵ)(k − 1) when the goods are allocated eﬃciently between
j participants, and investigate each term’s further lower bound and its
asymptotic behavior.
∗
For the relationship between w∗N,k and WN,k, we have

[
]
1−µ
∗
∗
WN,k
= w∗N,k + EN−1 WN,k( j)1 j>(1+ϵ)(k−1)

11Note that the expectation of ith order statistic in j draws from the uniform distribution
U[µt̄, t̄] is µt̄ + (t̄ − µt̄)(j − i + 1)/( j + 1), and therefore we have∗WN,k( j) =

k−1 {
∑
i=0

}
{
(
)
}
j−i+1
k
µt̄ + (t̄ − µt̄)
= t̄(k − 1) (1 − µ) 1 −+µ .
j+1
2(j + 1)

Note also that the (unconditional) expected surplus if allocated eﬃciently is calculated as()
k∗
WN,k = t̄(k − 1) 1 −
.
2N

21

∗
by the definition of w∗N,k . For (1 + ϵ)(k − 1) < j ≤ N − 1, it is clear that WN,k(j)

is at most)(}kt̄(k − 1) (1 − µ) 1 −
+µ ,
2N
{

∗
is given by
and therefore a lower bound of w∗N,k /WN,k

w∗N,k
∗
WN,k

≥1−

(
)
k
(1 − µ) 1 − 2N+µk
1 − 2N[]1−µEN−1 1 j>(1+ϵ)(k−1) ,

(17)

∗
=
where the denominator in the second term comes from the fact that WN,kt̄(k − 1)(1 − k/(2N)) (see footnote 11).
For wm/w∗N,k part in (16), we focus our attention to the event thatN,kj ≤ (1 + ϵ)(k − 1). It is clear that{∗WN,k( j) ≤ t̄(k − 1)

)
}
k
+µ
(1 − µ) 1 −2((1 + ϵ)(k − 1) + 1)(

for each j ≤ (1 + ϵ)(k − 1), and therefore a lower bound for wm/w∗N,k is given
N,k
by
1+µ

wm
N,k

t̄(k − 1) 2
}
{
)
(
≥
k
w∗N,k t̄(k − 1) (1 − µ) 1 −+µ2((1+ϵ)(k−1)+1))/(
1−µϵ(k − 1)=1 1+
1 + µ (1 + ϵ)(k − 1) + 1/()ϵ(k − 1)
k/N
≥1 1+
,
2 − k/N (1 + ϵ)(k − 1) + 1(18)

where the last inequality comes from the condition that µ ≥ β = (N − k)/Nin the equilibrium.

22

From (15), (16), (17), and (18), we have
1 − LR ≡

m
WN,k
∗
WN,k

1−

)
(
k
+µ
(1 − µ) 1 − 2N≥k1 − 2N[]1−µEN−1 1 j>(1+ϵ)(k−1)ϵ(k − 1)k/N
1+
2 − k/N (1 + ϵ)(k − 1) + 1,

and by applying Lemma 4 in the Appendix,
lim inf(1 − LR) ≥
N→∞

1
ϵ .
α
1+
2−α1+ϵAs ϵ > 0 can be arbitrarily small, this completes the proof.

Proof for RR → 1. Our proof proceeds as follows: 1. we establish that
lim infN→∞ (1 − µ)(N − 1)/(k − 1) ≥ 1, and 2. we show that RR → 1 under thecondition derived in step 1. by applying Arratia-Gordon [1] method as in
the proof for LR → 0. (that is, the expected number of buyers participatingin mari in the equilibrium, (1 − µ)(N − 1), converges to something slightly
greater than the number of remaining items, k − 1).
1. Firstly we should note that the left-hand side of the equilibrium
condition (12) represents the probability that “a buyer wins the item inmari if the buyer participates in it” in the equilibrium. Rearranging theequation in terms of the probability that “a buyer loses in mari if the buyer
participates in it”, we have[1−µ
1 − EN−2]
k−1
k−1 N 1−α 1
1 j<k−1 +1 j≥k−1 = 1 −
=
,
j+1
N−1 k
α N−1

where α = k/N. As the event that “a buyer loses in mari if the buyer
23participates in it” is included in the event that “the number of other buyers
participating in mari is larger than k − 1”,
1−µ
EN−2

[

1 j≥k−1

Thus we have

]

[
1−µ
≥ 1 − EN−2]
k−1
1 j<k−1 +1 j≥k−1 .
j+1[] 1−α 11−µEN−2 1 j≥k−1 ≥
.
α N−1

(19)

Given the inequality (19), we can show limN→∞ (1 − µ)(N − 2)/(k − 1) ≥ 1
by contradiction. Specifically, we are to show that “if we suppose that
limN→∞ (1 − µ)(N − 2)/(k − 1) < 1, then it contradicts the inequality (19)”.
Suppose that we have limN→∞ (1 − µ)(N − 2)/(k − 1) < 1. Then we canapply Lemma 1 (Arratia-Gordon Theorem) by seeing as p := 1 − µ andα := (k − 1)/(N − 2), and would have[]1−µEN−2 1 j≥k−1 ≤ e−(N−2)·H .
[]1−µThis inequality would say that EN−2 1 j≥k−1 is the order of 1/eN , which
contradicts the inequality (19) stating that the left-hand side has no lessorder than 1/N. Thus we have
(1 − µ)(N − 2)
≥ 1.
N→∞
k−1
lim

2. RR can be written as follows:
RR =

[
]
[
]
1−µ
1−µ
EN−1 j1 j<k−1 + EN−1 (k − 1)1 j≥k−1
k−1.

The first term in the numerator is regarding the event that j < k − 1 inwhich case the number of items sold in mari is only j, whereas the second24term in the numerator is regarding the event that j ≥ k − 1 in which casethe number of items sold in mari is k − 1, and the sum of these two terms isdivided by the number of remaining goods, k − 1. We are to show RR → 1
by firstly giving a lower bound for RR and then showing that the lower
bound converges to 1.
For any arbitrarily given number ϵ > 0, we have

RR ≥

[
]
1−µ
EN−1 0 · 1 j<(1−ϵ)(k−1) + (1 − ϵ)(k − 1) · 1 j≥(1−ϵ)(k−1)
k−1
]

,

[
1−µ
= (1 − ϵ)EN−1 1 j≥(1−ϵ)(k−1)[]
1−µ≥ (1 − ϵ)EN−2 1 j≥(1−ϵ)(k−1)
{
[
]}
1−µ
= (1 − ϵ) 1 − EN−2 1 j<(1−ϵ)(k−1) .
As we have limN→∞ (1 − µ)(N − 2)/(k − 1) ≥ 1 established in step 1., we canapply Lemma 1 (Arratia-Gordon Theorem) and have12[]1−µEN−2 1 j<(1−ϵ)(k−1) ≤ e−(N−2)·H → 0

as N → ∞.

As ϵ > 0 can be arbitrarily small, this completes the proof.

5Summary

In this paper we introduced the model that formulates sequential descending auction with mari, and showed that it speeds up the market procedure
at the cost of suﬃciently small loss of eﬃciency.
Such mari rules are used in reality at various wholesale flower mar12

From the symmetricity of binomial distributions, we can construct[ the ]“mirror” ofpArratia-Gordon Theorem as follows: If 0 < α < p < 1, then we have EN 1 j≤αN ≤ e−N·H(α,p) .

25

kets in Japan. We believe that the result in the paper provides a partial
justification for the introduction of the rule in other markets.

Appendix
Lemma 1 (Arratia-Gordon). If 0 < p < α < 1, then we have[]p
EN 1 j≥αN ≤ exp{−N · H(α, p)},where H(α, p) is the Kullback-Liebler distance satisfying
H(α, p) ≡ α ln

1−αα
+ (1 − α) ln.
p1−pProof. See Theorem 1 in Arratia and Gordon [1].
1−µLemma 2. For Em [ f ( j)] ≡∑m (m)i m−i
f (i), we havei=0 i (1 − µ) µ11−µE [(m − j + 1) f (j)],m + 1 m+111−µ1−µ(1 − µ)Em [ f ( j)] =Em+1 [ j f ( j − 1)].
m+1
1−µµEm [ f ( j)] =Proof.
1−µµEm [ f ( j)] =

m ( )
∑
m

i

i=0

=

)
m (
∑
m+1 m−i+1
i=0

=

m+1

i

(1 − µ)i µm−i+1 f (i))
m+1 m−i+1(1 − µ)i µm−i+1 f (i)
i
m+1

m+1 (
∑
i=0

=

(1 − µ)i µm−i+1 f (i)11−µE [(m − j + 1) f ( j)].
m + 1 m+1
26

Similarly,1−µ(1 − µ)Em [ f (j)] =

m ( )
∑
m
i=0

i

(1 − µ)i+1 µm−i f (i))
m (∑m+1 i+1(1 − µ)i+1 µm−i f (i)
i
+
1
m
+
1
i=0
)
m+1 (
∑
m+1
i
=
(1 − µ)i µm+1−i f (i − 1)
i
m+1
i=1
=

=

1
1−µ
Em+1 [ j f ( j)].
m+1

1−µ

Lemma 3. If f ( j) ≥ f ( j+1) for all j and f (j) > f (j+1) for some j, then Em [ f (j)]is (strictly) increasing in µ.
Proof.
m−1 ( ) {
∑
}
∂ 1−µ
m
Em [ f (j)] =−(1 − µ)i−1 µm−i + (m − i)(1 − µ)i µm−i−1 f (i)
i
∂µ
i=1

+ mµm−1 f (0) − (1 − µ)m−1 f (m)()}
m−1 {( )∑mm=(m − i) f (i) −(i + 1) f (i + 1) (1 − µ)i µm−i−1
i
i
+
1
i=0
=

m−1
∑

[
]
m!
(1 − µ)i µm−i−1 f (i) − f (i + 1)
i!(m − i − 1)!
i=0

> 0.

Lemma 4. If N is large enough to satisfy k/N < (1 + ϵ)(k − 1)/(N − 1) for a given
ϵ > 0, then][1−µEN−1 1 j>(1+ϵ)(k−1) → 0
27

as N → ∞.

Proof. In the equilibrium it should be that 1−µ ≤ 1−β = k/N, and therefore
we have

[
]
[
]
1−µ
1
.
EN−1 1 j>(1+ϵ)(k−1) ≤ Ek/N
j>(1+ϵ)(k−1)
N−1

If k/N < (1+ϵ)(k −1)/(N −1), then Lemma 1 can be applied, which provides
us with( (1+ϵ)(k−1) )
[
]
k
k/N
EN−1
1 j>(1+ϵ)(k−1) ≤ e−(N−1)·H N−1 , N ,where H is the Kullback-Liebler distance converging to a finite value as
N → ∞.



References[1] Arratia, R., and L. Gordon (1989), “Tutorial on Large Deviations for the
Binomial Distribution”, Bulletin of Mathematical Biology 51, 125–131.
[2] van den Berg, G. J., J. C. van Ours, and M. P. Pradhan (2001), “TheDeclining Price Anomaly in Dutch Rose Auctions”, American Economic
Review 91, 1055–62.
[3] Milgrom, P., and R. Weber (2000), “A Theory of Auctions and Competitive Bidding, II”, in P. Klemperer ed., The Economic Theory of Auctions,
Edward Elgar.
[4] Weber, R. J. (1983), “Multiple-Object Auctions”, in R. EngelbrechtWiggans ed., Auctions, Bidding, and Contracting: Uses and Theory, NewYork University Press.

28