Discussion Paper No. 743

AN EXPERIMENTAL STUDY OF
JAPANESE PROCUREMENT AUCTIONSWITH ENDOGENOUS MINIMUM PRICES
Kenju Akai
Tatsuyoshi Saijo
Shigehiro Serizawa

June 2009

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

An Experimental Study of Japanese Procurement
Auctions with Endogenous Minimum Prices
Kenju Akaia,b, *, Tatsuyoshi Saijoc,d,† , Shigehiro Serizawae,†aJapan Society for the Promotion of Science, 1-6 Chiyoda-Ku, Tokyo 1028471, Japanb
Department of Value and Decision Science, Graduate School of Decision Science andTechnology, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo
1528552, Japanc
Research Institute for Sustainability Science, Osaka University, 6-1 Mihogaoka, Ibaragi,
Osaka 5670047, Japand
CASSEL, University of California at Los Angeles, Los Angeles, CA90095, UnitedStateseInstitute of Social and Economic Research, Osaka University, 6-1 Mihogaoka, Ibaragi,
Osaka 5670047, Japan
November 17, 2007
Abstract
Several Japanese local governments started to add endogenous minimum prices to firstprice auctions in their public procurements. Any bid less than the endogenous minimum
price is referred to as abnormally low and is excluded from the procurement procedure.
The endogenous minimum price is generally calculated as 80% to 90% of the average ofsome of the lowest bids or all bids. Therefore, producers who join this new institution
have incentives to raise their bids and pull the endogenous minimum price to exclude
others. We experimentally evaluate the performance of this new institution relative tothe standard first-price auction which do not have any minimum price. We find thatwinning prices of this new institution (i) coincide with the ones of the standard first-priceauction and are close to the production cost under our identical cost condition, and (ii)are higher than the ones of the standard first-price auction and diverge from the lowest
production cost under our diﬀerent cost condition when subjects’ identifications and alltheir bids are revealed.
JEL classification: C92; D44; L15
Key words: Public procurement; First-price auction; Experiment; Abnormally low
tender; Endogenous minimum price*Corresponding author: k-akai@iser.osaka-u.ac.jp, Institute of Social and Economic
Research, Osaka University, 6-1 Mihogaoka, Ibaragi, Osaka 5670047, Japan. Tel.: +81-66879-8552; Fax: +81-6-6879-8584.
†
E-mail: saijo@iser.osaka-u.ac.jp (T. Saijo), serizawa@iser.osaka-u.ac.jp (S. Serizawa)1. Introduction
First-price auctions are widely employed in public procurements for constructing roads,
bridges, buildings, etc. But they have a risk that winning prices become much lower
than standard production costs and that the quality of public works becomes low forgovernments and taxpayers. The European Commission (1999) refers to such bids as"abnormally low tenders". Ganuza (2007) lists examples of procurement projects whose
costs overran winning prices and huge delay occurred in Europe and the US. Cox et al.
(1996) found that cost overruns resulted from too-low bids of the first-price auction in
procurement auction experiments with post-auction cost uncertainty.
According to the European Commission (1999), Belgium, Italy, Portugal, Spain and
Greece "recognize (and/or suspect) a tender as being abnormally low, the price oﬀered isless by a certain percentage than the average of the tenders submitted or discounts granted,
with various diﬀerences in the percentage and/or calculation of the average" 1 .
Japan also has employed the first-price auction in public procurements and has suﬀeredfrom the abnormally low tenders, particularly local governments have2 . Since many local
governments do not have specialists who investigate the quality of public works, the law of
the local government procurements allows local governments to set minimum prices which
automatically exclude any bids less than the minimum price in procurement procedures
to exclude abnormally low bids3 .
Since Japanese local governments have traditionally thought that the minimum priceshould be a benchmark to engage for the quality of public work, they have traditionally
set the minimum price as 80% to 90% of the reservation price. The reservation price isestimated as standard costs by the governments4 .
Many Japanese local governments have announced the reservation price and/or theminimum price in advance to avoid a corruption that civil servants leak these prices tosome construction companies by bribery5 . An announcement of the minimum price inadvance, however, leads to a problem that all producers who join the auction bid theminimum price, and a winner is selected by a lottery regardless of their real costs andqualities6 . Even if they do not announce the minimum price but only announce thereservation price in advance, same problem occurs because the minimum price is set
traditionally 80% to 90% of the reservation price, which makes producers estimate theminimum price easily.
To avoid deciding the winner by a lottery and exclude abnormally low bids, several
1

Calveras, Ganuza, and Hauk (2004) point out the possibility of the failure of this type of minimum
prices briefly.
2
In Japan, this problem is generally called “dumping”.
3
Since the central government has the specialist who investigates the quality of products, it does not
employ the minimum price.
4
In Japan, the law of local government procurement requires reservation prices to avoid a situationthat winning prices are over their budgets.
5
According to the Ministry of Land, Infrastructure, Transport and Tourism’s survey (2007) covered
1,874 local governments, 64% of all local governments announced the reservation price in advance, and
65% of all local governments employed the minimum price. 20% of local governments which employed
the minimum price announced the minimum price in advance.
6
According to the Ministry of Land, Infrastructure, Transport and Tourism’s survey (2006) covered
1,826 local governments which employed the minimum price on April in 2003 to July in 2004 (thisfrequency is about 56% of 3,228 local governments existing at that time), 145 local governments decided
the winner by a lottery during that period. This frequency is 61% of 236 local governments whichannounced the minimum price in advance.

1Japanese local governments started to determine minimum prices endogenously like European countries. For example, Nagano Prefecture set the minimum price as 80% of averageof the lowest five bids, and Yokosuka city set the minimum price as 90% of the lowest ten
bids7 . We refer to these new minimum prices as endogenous minimum prices.
Similar types of endogenous minimum prices have become popular in the past several
years, and now the Ministry of Internal Aﬀairs and Communications in Japan (2006)
recommends endogenous minimum prices. Total contract prices of the public works from2005 to 2007 for all local governments are about 24 trillion yen (229 billion US dollars)8 ;the ones for the central government are about 7 trillion yen (67 billion US dollars)9 . In
the future, the first-price auction with endogenous minimum price is likely to become
one of major institutions to transfer large amounts of money from local governments toconstruction companies.
This new institution, however, has a feature that producers who join this procurement
institution raise their bids and pull the minimum price to exclude others, so that winning
prices are likely to be higher than their real cost. It implies that taxpayers waste theirmoney. To confirm this feature, first we theoretically analyze Nash equilibria in this new
institution.
Then we employ an experimental method to evaluate the theoretical result. Experimental studies had been used to evaluate the performance of markets with price controls
such as price ceilings and floors. Smith and Williams (1981), Isaac and Plott (1981),
Coursey and Smith (1983), and Gode and Sunder (2004) find that even theoretically nonbinding price controls aﬀect trading prices in the laboratory. Not only for evaluating pricecontrols but for evaluating new auction institutions, the experimental method had beenused widely. Cason (1995) and Cason and Plott (1996) use it to evaluate emission permit
auctions in the US. Olson et al. (2003), Plott and Salmon (2004), Banks et al. (2005)
and Salmon and Iachini (2007) use it for spectrum auctions in the UK and US. Abbink
et al. (2006) use it for government security auctions in Spain.
Our experiment features three treatment variables — minimum prices (the endogenous
minimum price vs. no-minimum price), production costs (identical vs. diﬀerent), and theinformation of subjects’ identifications and all their bids (disclosure vs. nondisclosure). In
the previous procurement auction experiments, Saijo et al. (1996) find that winning prices
are close to the cost regardless winning prices are revealed or not, but Dufwenberg andGneezy (2002) find that winning prices diverge from the cost when all bids are revealed.
But they do not disclose subjects’ identifications. We disclose of the seating chart ofsubjects and all their bids for all subjects to see who bids what price and examine thisinformation eﬀect.
In the theoretical analysis, we find that the most eﬃcient type of minimum price forgovernments and taxpayers to minimize their expenditures is the certain percents of theaverage of the some of the lowest bids. The winning price of the first-price auction withthis type of minimum price coincides with the cost (lowest cost) in the identical (diﬀerent)
cost condition.
In the laboratory, we find that winning prices of the first-price auction with this7
They are first local governments which employ the endogenous minimum price. They employed theseminimum prices in 2005.
8
1 US dollar = 105 yen.
9
Original data are from websites of East Japan Construction Surety Co.,Ltd, Hokkaido Construction
Surety Co.,Ltd, and West Japan Construction Surety Co.,Ltd.

2

minimum price (i) coincide with the ones of the standard first-price auction which does
not have any minimum prices and close to the cost in the identical cost condition, and (ii)are higher than the ones of the standard first-price auction and diverge from the lowest
cost in the diﬀerent cost condition when subjects’ identifications and their all bids are
revealed.
The paper is organized as follows. Section 2 develops the theoretical model and hypotheses. Section 3 details the experimental design and procedures. Section 4 analyzes
the results. Section 5 summarizes the concluding remarks.
2. Theoretical model and hypotheses
In Section 2.1, we classify the endogenous minimum prices into four types and analyze
Nash equilibria in the first-price auction (FPA) with each type of the minimum price.
We also consider the most eﬃcient type of the minimum price for the governments andtaxpayers to minimize their expenditures. In section 2.2, we provide testable hypotheses
in the parameter values used in the experiment.
2.1 Theoretical model
A rule of the FPA with endogenous minimum price is simply described as follows.
Consider n producers who join this procurement auction. Let p be the reservation pricedetermined by a government in advance. Denote producer i’s bid by bi ∈ [0, p] and thisbid function is continuous. A bid profile is a n-tuple b = (b1 , . . . , bn ) ∈ [0, p]n . Given a
bid profile b = (b1 , . . . , bn ), denote the first lowest bid by b1 , the second lowest bid by
b2 , and so on. Given a bid profile b, 2 ≤ m ≤ n, and k ∈ (0, 1], denote the average bid
am (b) = [b1 + · · · + bm ]/m of the m lowest bids and the minimum price d(b) = k · am (b)10 .
When a bid profile is b, a winner is a producer whose bid is the lowest one between d(b)and p inclusively, and a winning price is her bid. In case of tie, such producers win withequal probability.
We classify endogenous minimum prices d(b) = k · am (b) into four types below.
Type 1 : average of the lowest m bids (k = 1 and m < n)Type 2 : average of the all bids (k = 1 and m = n)Type 3 : k × average of the lowest m bids (k < 1 and m < n)Type 4 : k × average of the all bids (k < 1 and m = n)
In Japan, Type 3 minimum price is the most popular and Type 4 minimum price isthe second popular. Nobody employs Type 1 and Type 2 minimum prices. Procurement
auctions for Japanese local governments are held with the same set of producers who
work in the same cities or prefectures. Since wages and rental costs are not so varied inlocal areas, we assume that every producer has identical cost, c, to produce a homogenous
good, and the cost is common knowledge among producers11 .
In the following, we divide Nash equilibria of the FPA with endogenous minimum
prices into two cases to compare with Nash equilibria of the standard FPA which does
not have any minimum prices (simply we call it standard FPA). In Case I, we show Nashequilibria such that all bids are not less than the cost, that is, bi ≥ c, for every i. Thiscondition is satisfying if the bid profile is a Nash equilibrium in the standard FPA because
10

In 2005, m was determined in advance. Since n is uncertaintiy, several local goverments have started
to set m as rougly 60% of n. Genearally, they do not use the endogneous minimum price if n < 5.
11
We have two other projects. One is that we assume costs are common value among producers, andthe other is that producer can decide the quality of the good. We may leave the detail to the other paper.

3

any producer bidding less than the cost obtains negative payoﬀs. In Case II, we show
Nash equilibria such that some bids are less than the cost, that is, bi < c, for some i.
Although there is no Nash equilibrium satisfying this condition in the standard FPA, there
are many Nash equilibria satisfying this condition in the FPA with endogenous minimum
prices as described in Case II.
Case I: Nash equilibria such that bi ≥ c, for every i.
First we analyze Nash equilibria in the FPA with Type 1 minimum price and the onesin the FPA with Type 2 minimum price in Propositions 1 and 2 below, respectively, withtechnical details and proofs contained in Appendix A.
Proposition 1. In the FPA with Type 1 minimum price (k = 1 and m < n), a bid
profile b is a Nash equilibrium if and only if (i) there is b∗ ∈ [c, p] such that b1 = · · · = bn =b∗ , or (ii) there is N 0 ⊆ N such that #N 0 ≥ m + 1 and bi = c for all i ∈ N 0 .
In the FPA with Type 1 minimum price, since the winning price is more than or equal
to the cost, governments and taxpayers have a risk to pay more than the cost and waste
their money.
Proposition 2. In the FPA with Type 2 minimum price (k = 1 and m = n), only
the bid profile b such that b1 = · · · = bn = p is a Nash equilibrium.
In the FPA with Type 2 minimum price, since the winning price coincides with thereservation price, governments and taxpayers need to pay more than the cost and waste
their money.
Next we analyze Nash equilibria in the FPA with Type 3 minimum price (k < 1 andm < n) and FPA with Type 4 minimum price (k < 1 and m = n). Since there are many
Nash equilibria, it is complicated to describe all of them in each institution. First we
analyze the winning price in the Nash equilibrium in each institution in Proposition 3
below, with this proof contained in Appendix A. Then we present an example of the Nashequilibrium in each institution. For general case, see Appendix B.
Proposition 3. Both in the FPA with Type 3 minimum price (k < 1 and m < n)and FPA with Type 4 minimum price (k < 1 and m = n), if a bid profile b is a Nashequilibrium, the winning price is the cost.
Example 1 (Type 3 ). Let n = 5, k = 0.8, m = 3, p = 243, and c = 97. Consider a bid
profile b = (97, 97, 130, 150, 160). The minimum price is d(b) = 0.8·(97+97+130)/3 = 86.4,
and the winner is one of producers bidding 97. If one of the producers bidding 97 raises her
bid to 130, then since the minimum price becomes d(b) = 0.8·(130+97+130)/3 = 95.2 andstill remains less than 97, she cannot exclude the other producer bidding 97. If she raises
her bid to 150, then since the minimum price becomes d(b) = 0.8·(150+97+130)/3 = 100.5
and still remains less than 130, she cannot exclude producers bidding prices less than 130.
She cannot aﬀect the minimum price by raising her bid to a price more than 150. If
the producer bidding 130 raises her bid to 150, then since the minimum price becomesd(b) = 0.8 · (97 + 97 + 150)/3 = 91.7 and still remains less than 97, she cannot exclude
producers bidding 97. In this bid profile, nobody has an incentive to raise her bid and4exclude others. Hence, this bid profile is a Nash equilibrium, and the winning pricecoincides with the cost.
Next consider a bid profile b = (97, 97, 130, 220, 230). Since b4 = 220 is much higher
than b3 = 130, the producer bidding b3 = 130 can pull the minimum price above 97 andexclude producers bidding 97 by raising her bid to b4 = 220. Thus, this bid profile is not aNash equilibrium. In this institution, since b4 is the same as a price ceiling for producers
bidding prices less than b4 , b4 needs to be not so high that producers bidding prices lessthan b4 cannot exclude others if a bid profile is a Nash equilibrium.
In the FPA with Type 3 minimum price, the winning price coincides with the cost.
This minimum price is the most eﬃcient for the government and taxpayers to minimize
their expenditures.
Example 2 (Type 4 ). Let n = 5, k = 0.8, p = 243, and c = 97. Consider a bid profileb = (97, 97, 100, 130, 150). The minimum price is d(b) = 0.8·(97+97+100+130+150)/5 =91.8, and the winner is one of the producers bidding 97. If one of the producers bidding 97
raises her bid to 100, then since the minimum price becomes d(b) = 0.8 · (100 + 97 + 100 +130+150)/5 = 92.3, she cannot exclude the other producer bidding 97. If she raises her bid
to 130, then since the minimum price becomes d(b) = 0.8·(130+97+100+130+150)/5 =97.1, she cannot exclude the producer bidding 100. If she raises her bid to 150, then since
the minimum price becomes d(b) = 0.8· (150 +97+100+130 +150)/5 = 100.3, she cannot
exclude the producer bidding 130. If she raises her bid to 243, then since the minimum
price becomes d(b) = 0.8 · (243 + 97 + 100 + 130 + 150)/5 = 115.2, she cannot exclude
producers bidding prices less than 243.
If the producer bidding 100 raises her bid to 130, then since the minimum pricebecomes d(b) = 0.8 · (97 + 97 + 130 + 130 + 150)/5 = 96.6, she cannot exclude the producer
bidding 97. If the producer bidding 130 raises her bid to 130, then since the minimum
price becomes d(b) = 0.8 · (97 + 97 + 100 + 150 + 150)/5 = 99.8, she cannot exclude theproducer bidding 100. If the producer bidding 150 raises her bid to 243, then since theminimum price becomes d(b) = 0.8 · (97 + 97 + 100 + 130 + 243)/5 = 106.7, she cannot
exclude the producer bidding 130. In this bid profile, nobody has an incentive to raise her
bid and exclude others12 . Hence, this bid profile is a Nash equilibrium, and the winning
price coincides with the cost.
Next consider a bid profile b = (97, 97, 99, 100, 150). If the producer bidding 150 raises
her bid to p = 243, then since the minimum price becomes d(b) = 0.8 · (97 + 97 + 99 +100 + 243)/5 = 101.7, she can exclude others and be the winner. Thus, this bid profile is
not a Nash equilibrium. In this institution, since all producers aﬀect the minimum price,p needs to be not so high that all producers cannot excludes others if a bid profile is aNash equilibrium.
Remark. Let p = 600 in Example 2. Since everybody has an incentive to raise her
bid to p = 600, there is no pure Nash equilibrium (see Claim and Proof in Appendix
Let d(b) ≤ 97 = b1 = b2 . A bid profile b is a NE if and only if (1) k5 (b3 + b2 + b3 + b4 + b5 ) ≤ b2 ,(2) k5 (b4 + b2 + b3 + b4 + b5 ) ≤ b3 , (3) k5 (b5 + b2 + b3 + b4 + b5 ) ≤ b4 , (4) k5 (p + b2 + b3 + b4 + b5 ) ≤ b5 ,(5) k5 (b1 + b2 + b4 + b4 + b5 ) ≤ b2 , (6) k5 (b1 + b2 + b5 + b4 + b5 ) ≤ b4 , (7) k5 (b1 + b2 + p + b4 + b5 ) ≤ b5 ,(8) k5 (b1 + b2 + b3 + b5 + b5 ) ≤ b2 , (9) k5 (b1 + b2 + b5 + p + b5 ) ≤ b5 , (10) k5 (b1 + b2 + b3 + b4 + p) ≤ b4 . Noticethat (4) implies (7) and (9) and (3) implies (6).
125B). In this institution, if the reservation price is much higher than the cost, then since
everybody has an incentive to bid the reservation price and exclude others, there is nopure Nash equilibrium.
By the remark above, since it is not ensured that the winning price coincides with thecost, governments and taxpayers have a risk to waste their money.
Case II: Nash equilibria such that bi < c, for some i.
In this case, in each type of minimum price, there are many Nash equilibria such thatsome bids are less than the cost. We show such a Nash equilibrium in each type of theminimum price in the example below.
Example 3. Let n = 5, p = 243, c = 97, k = 0.8 if k < 1, and m = 3 if m < n.
Type 1 (k = 1 and m = 3). Consider a bid profile b = (1, 97, 97, 97, 150). The
minimum price is d(b) = (1 + 97 + 97)/3 = 52, and the winner is one of the producers
bidding 97. If the producer bidding 1 raises her bid to 97, then since her payoﬀs still
remain 0, she does not have an incentive to deviate from this bid profile. She can notaﬀect the minimum price by raising her bid more than 97. The producer bidding 97 can
not aﬀect the minimum price by raising her bid more than 97, either. In this bid profile,nobody has an incentive to raise her bid. Hence, this bid profile is a Nash equilibrium,
and the winning price coincides with the cost. Similarly, there are many Nash equilibria
such that some bids are less than the cost.
Type 2 (k = 1 and m = 5). Consider a bid profile b = (1, 1, 97, 97, 140). The minimum
price is d(b) = (1 + 1 + 97 + 97 + 140)/5 = 67.2, and the winner is one of the producers
bidding 97. If one of the producers bidding 1 raises her bid to 97, then since her payoﬀs still
remain 0, she does not have an incentive to deviate from this bid profile. If she raises her
bid to 140, then since the minimum price becomes d(b) = (140 + 1 + 97 + 97 + 140)/5 = 95
and still remains less than 97, she cannot exclude producers bidding 97. If she raises her
bid to 243, then since the minimum price becomes d(b) = (243+1+97+97+140)/5 = 115.6
and still remains less than 140, she cannot exclude the producer bidding 140. Since one ofthe producers bidding 1 does not exclude producers bidding 97, nobody exclude producers
bidding 97. In this bid profile, nobody has an incentive to raise her bid. Hence, this bid
profile is a Nash equilibrium. Similarly, there are many Nash equilibria such that somebids are less than the cost.
Type 3 (k = 0.8 and m = 3). Consider a bid profile b = (1, 97, 97, 150, 200). The
minimum price is d(b) = 0.8 · (1 + 97 + 97)/3 = 52, and the winner is one of the producers
bidding 97. If the producer bidding 1 raises her bid to 97, then since her payoﬀs still
remain 0, she does not have an incentive to deviate from this bid profile. If she raises her
bid to 150, then since the minimum price becomes d(b) = 0.8 · (150 + 97 + 97)/3 = 91.7
and still remains less than 97, she cannot exclude producers bidding 97. If the producer
bidding 97 raises her bid to 150, then since the minimum price becomes d(b) = 0.8 · (1 +97 + 150)/3 = 66.1 and still remains less than 97, she cannot exclude the other producer
bidding 97. Nobody can aﬀect the minimum price if she raises her bid more than 150. In
this bid profile, nobody has an incentive to raise her bid. Hence, this bid profile is a Nashequilibrium. Similarly, there are many Nash equilibria such that some bids are less thanthe cost.
Type 4 (k = 0.8 and m = 5). Consider a bid profile b = (1, 97, 97, 130, 150). The
minimum price is d(b) = 0.8 · (1 + 97 + 97 + 130 + 150)/5 = 76.0, and the winner is6one of the producers bidding 97. If the producer bidding 1 raises her bid to 97, thensince her payoﬀs still remain 0, she does not have an incentive to deviate from this bid
profile. If she raises her bid to 130, then since the minimum price becomes d(b) =0.8 · (130 + 97 + 97 + 130 + 150)/5 = 96.6 and still remains less than 97, she cannot exclude
producers bidding 97. If she raises her bid to 150, then since the minimum price becomesd(b) = 0.8 · (150 +97 + 97 + 130 +150)/5 = 99.8 and still remains less than 130, she cannot
exclude producer bidding 130. If she raises her bid to 243, then since the minimum pricebecomes d(b) = 0.8 · (243 + 97 + 97 + 130 + 150)/5 = 114.7 and still remains less than 130,she cannot exclude producer bidding 130. If the producer bidding 130 raises her bid to150, then since the minimum price becomes d(b) = 0.8 · (1 + 97 + 97 + 150 + 150)/5 = 79.2
and still remains less than 97, she cannot exclude producers bidding 97. If the producer
bidding 150 raises her bid to 243, then since the minimum price becomes d(b) = 0.8 ·(1 + 97 + 97 + 130 + 243)/5 = 90.8 and still remains less than 97, she cannot exclude
producers bidding prices less than 243. In this bid profile, nobody has an incentive toraise her bid13 . Hence, this bid profile is a Nash equilibrium. Similarly, there are many
similar Nash equilibria such that some bids are less than the cost.
In both cases of Nash equilibria, since winning prices of the FPA with Type 3 minimum
price coincide with the cost regardless of the value of the reservation price, this institution
is best for the governments and taxpayers to minimize their expenditures. Therefore, we
focus on this institution and evaluate this performance in the laboratory.
2.2 Hypotheses
In the experiment, we employ the FPA with Type 3 minimum price. We set theminimum price as 0.8 × average of the lowest 5 bids (k = 0.8 and m = 5) out of
10 producers (n = 10)14 . In the experiment, we allow subjects to bid only nonnegative
integers. To evaluate the performance of the FPA with Type 3 minimum price, we compare
winning prices of this institution with the ones of the standard FPA in the identical costcondition and diﬀerent cost condition.
First we propose hypotheses in the identical cost condition. In the identical costcondition, we set every producer’s cost as 97 (c = 97). By Proposition 3, the winning
price coincides with the cost of 97. Notice that since bids are only integers, a bid profile
b1 = · · · = b10 = 98 is also a Nash equilibrium. Therefore, we obtain the hypothesis below.
Hypothesis 1. In the identical cost condition, the winning price is 97 or 98.
In the standard FPA, the winning price also coincides with the cost of 97. Additionally,
since bids are only integers, a bid profile b1 = · · · = b10 = 98 is also a Nash equilibrium.
Therefore, we obtain the hypothesis below.
Hypothesis 2. There is no diﬀerence in the winning prices between the FPA withType 3 minimum price and standard FPA.
Let b1 < d(b) ≤ 97 = b2 = b3 . A bid profile b is a NE if and only if (1) k5 (b4 + b2 + b3 + b4 + b5 ) ≤ b3 ,(2) k5 (b5 + b2 + b3 + b4 + b5 ) ≤ b4 , (3) k5 (p + b2 + b3 + b4 + b5 ) ≤ b5 , (4) k5 (b1 + b2 + b4 + b4 + b5 ) ≤ b2 = b3 ,(5) k5 (b1 + b2 + b5 + b4 + b5 ) ≤ b4 , (6) k5 (b1 + b2 + p + b4 + b5 ) ≤ b5 , (7) k5 (b1 + b2 + b3 + b5 + b5 ) ≤ b2 = b3 ,(8) k5 (b1 + b2 + b5 + p + b5 ) ≤ b5 , (9) k5 (b1 + b2 + b3 + b4 + p) ≤ b4 . Notice that (3) implies (6) and (8). (1)implies (4). (2) implies (5).
14This minimum price is the same as the first one of Nagano prefecture.
13

7

Next we propose hypotheses in the diﬀerent cost condition. Although wages andrental costs are not so varied in the limited area, some huge nationwide companies whojoin procurement auctions for local governments have a power to cut wages and rental
costs much more than small localized companies do. We capture this situation into thelaboratory and set producer i’s cost as 97 (ci = 97, i = 2, 5, 8) and producer j’s cost as
150 (cj = 150, j = 1, 3, 4, 6, 7, 9, 10)15 .
Since there are many Nash equilibria in the diﬀerent cost condition, and it is complicated to describe all of them. First we analyze the winning price in the Nash equilibriumin Proposition 4 below, with proofs in Appendix A. Then we present an example of the
Nash equilibrium.
Proposition 4. Let c1 = c2 ≤ · · · ≤ cn . In the FPA with Type 3 minimum price(k < 1 and m < n), if a bid profile b is a Nash equilibrium, the winning price is the lowest
cost, c1 .
Fact. In the diﬀerent cost condition, there is no Case I Nash equilibrium such thatbi ≥ ci , for every i.
To explain the fact above intuitively (see Lemmas 1 to 4 and Fact in Appendix A),
consider a bid profile b such that every producer whose cost is 97 bids 97 and every
producer whose cost is 150 bids 150. If the producer bidding 97 raises her bid to 149, she
can pull the minimum price above 97 and be the winner. So that producers whose costs
are 97 have incentives to raise their bids more than 97 not to be excluded. But they also
keep incentives to reduce their bids to win. Hence, they trade oﬀ raising and reducing
their bids. Therefore, there is no Case I Nash equilibrium such that every producer bidsa price more than or equal to the cost. There are only Case II Nash equilibria such thatbi < ci , for some i in the example below.
Example 4. Consider a bid profile b such that producers 2, 5, and 8 whose costs are
97 bid 97, producer 1 whose cost is 150 bids 1, and remaining producers whose costs are
150 bid 150, that is, b = (1, 97, 150, 150, 97, 150, 150, 97, 150, 150). In this bid profile, theminimum price is 0.8 · (97 + 97 + 97 + 1 + 150)/5 = 70.7, and the winner is one of the
producers bidding 97. Since nobody can pull the minimum price above 97, this bid profile
is a Nash equilibrium. Similarly, there are many Case II Nash equilibria such that somebids are extremely low.
Notice that since bids are only integers in the experiment, there are Nash equilibria
such that wining price is 98. For example, a bid profile b such that producers 2, 5, and 8
whose costs are 97 bid 98, producer 1 whose cost is 150 bids 1, and remaining producers
whose costs are 150 bid 150, that is, b = (1, 98, 150, 150, 98, 150, 150, 98, 150, 150) is aNash equilibrium16 . Therefore, we obtain the hypothesis below.
15

Dufwenberg and Gneezy (2000) find that price competition does work well when the number ofcompetitors is three. We follow their idea and allocate the lower cost to three producers.
16
Since bids are only integers in the experiment, there are several nonessential Nash equilibria.
For example, consider a bid profile b such that producer 2 bids 120, producer 5 bids 130, producer
8 bids 145, producer 1 bids 97, producer 3 bids 121, and remaining producers bid 145, that is,b = (97, 120, 121, 145, 130, 145, 145, 145, 145, 145). The minimum price is d(b) = 0.8 · (97 + 120 + 121 +
130 + 145)/5 = 98. 0, and the winner is the producer bidding 121. If the producer bidding 97 raises her

8

Hypothesis 3. In the diﬀerent cost condition, the winning price is 97 or 98.
In the standard FPA, the winning price also coincides with the lowest cost of 97.
Additionally, since bids are only integers, a bid profile such that producers 2, 5 and 8 bid
98 (b2 = b5 = b8 = 98) and others bid prices more than 98 is also a Nash equilibrium.
Therefore, we obtain the hypothesis below.
Hypothesis 4. There is no diﬀerence in the winning prices between the FPA withType 3 minimum price and standard FPA.
From above hypotheses, there is no diﬀerence in winning prices between the FPA withType 3 minimum price and standard FPA in either cost condition.
3. Experimental design and procedures
3.1 Design
Each session proceeded through a sequence of 10 procurement periods. Each session
consisted of 10 producers who had the same homogenous good. Each subject bid only one
selling price per period, and the experimenter bought only one unit of good per periodwith the reservation price of 243 points of experimental cash. The winner produced agood and earned profits equal to her selling price less her cost; other producers did not
produce a good and earned zero profits for that period17 .
Our experiment features three treatment variables — minimum prices, production costs,
and the information of subjects’ identifications and all their bids.
The minimum prices of our most interest have two conditions — the no-minimum pricecondition and Type 3 minimum price condition. In the no-minimum price condition, we
employed the standard FPA which does not have any minimum prices. In this condition,
the winner was the producer whose bid was the lowest one among bids less than or equal
to the reservation price, and the winning price was her bid. In case of tie, such producers
won with equal probability. This was the benchmark control condition.
In the Type 3 minimum price condition, we employed the FPA with Type 3 minimum
price. The minimum price was 80% of the average of the lowest five bids among 10
producers. The winner was the producer whose bid was the lowest one between thereservation price and Type 3 minimum price inclusively, and the winning price was herbid. In case of tie, such producers won with equal probability.
The production costs have two conditions — the identical cost condition and diﬀerent
cost condition. In the identical cost condition, we announced that production costs for
all producers were 97 points. In the diﬀerent cost condition, we announced that production costs for three producers with identification numbers 2, 5 and 8 were 97 points,
and the ones of the remaining producers were 150 points. In each condition, subjects
were randomly assigned one of the identification numbers and remained in the same role
throughout the session.
bid to 120 and win, since her cost is 150, she obtains negative payoﬀs. If the producer bidding 121 reduces
her bid to 120 and win, since her cost is 150, she obtains negative payoﬀs. If the producer bidding 130
reduces her bid to 120, since the minimum price d(b) = (97 + 120 + 121 + 120 + 145)/5 = 120. 6, shecannot be the winner. Thus, nobody has an incentive to change her bid, so that this bid profile is a Nashequilibrium. But, essentially, the winning price in the FPA with Type 3 minimum price coincides withthe lowest cost of 97 in the Nash equilibrium.
17
We avoid terms “winners” and “winning prices” in the instructions (see Appendix C for more detail).

9

The information of subjects’ identifications and all their bids has two conditions —the disclosure condition and nondisclosure condition. Procurement auctions for Japanese
local governments are held with the same set of producers. They work in the same cities
or prefectures and know each other very well. We designed our experiments to capture
this environment in the disclosure condition. To this end, in the disclosure condition, atthe beginning of the session, we wrote the seating chart of all producers on a whiteboard
for all subjects to see which producer had which seat18 .
After the winner was decided, we called each bid along with writing it in each seatof the seating chart19 . After writing the all bids, in the no-minimum price condition, thewinning price, winner’s identification number, and her own profits were displayed on eachsubject’s screen. In the Type 3 minimum price condition, the winning price, winner’s
identification number, the average of the lowest five bids, the minimum price, and her
own profits were displayed on each subject’s screen.
Since we examined the eﬀect of information of subjects’ identifications and all theirbids, we did not disclose the seating chart and all bids in the nondisclosure condition.
After the winner was decided, the same kind of information provided in the disclosure
condition were displayed in each subject’s screen.
We used a full factorial design with 2 minimum price conditions × 2 production costconditions × 2 information conditions of subjects’ identifications and all their bids. Table
1 summarizes our 8 cells. Each cell included 2 sessions. The No- and T3- prefix inthe session name indicate the no-minimum price condition and Type 3 minimum pricecondition, respectively. The -I- and -D- at the middle of the session name indicate theidentical cost condition and diﬀerent cost condition, respectively. The -D- and -N- suﬃx
indicate the disclosure condition and nondisclosure condition, respectively.
––––––––––
Table 1 is around here––––––––––
3.2 Procedures
The experiments were conducted in the PC laboratory in Institute of Social and Economic Research at Osaka University. A computerized interface was programmed with thesoftware z-Tree. (Fischbacher 2007). Subjects were undergraduate and graduate students
at Osaka University20 . They were invited to sign up at designated websites by flyers
posted around campus and email solicitations sent to students who had signed up forother experiments before. No one participated in more than one session.
Upon arrival, the subjects were seated at separate computer terminals and no communication was permitted throughout the session. Appendix C contains the instructions andPC operation manuals. Subjects listened to prerecorded instructions, while they followed
along on their own copies. Then, they privately read questions and answers about instruc18

In our laboratory, subjects seating rearward can see backs of subjects seating fore.
Yokosuka city discloses all bids along with companies’ names. We capture this information disclosure
rule into the laboratory.
20
Subjects were a mixture of economics majors and noneconomics majors.
19

10

tions in 10 minutes to confirm the rules of the experiment21 . After that, they listened toprerecorded PC operation manuals, while they followed along on their own copies.
After above procedures, the producers’ identification numbers 1 to 10 were determinedby a lottery, and each subject received a record sheet. Then, we provided 10 minutes for
subjects to consider the strategy of subsequent auctions.
After that, period 1 started, and all subjects entered their selling price22 on the computer and their record sheets within 3 minutes in each period. As soon as all subjects
bid, the auction was automatically closed and the winner was decided even if 3 minutes
did not elapse.
In the disclosure condition, we called each bid with the identification number and wrote
it in each seat of the seating chart on the whiteboard. In the nondisclosure condition, we
did not announce that information. In the no-minimum price condition, subjects entered
the winning price, winner’s identification number, and their own profits on their record
sheets. In the Type 3 minimum price condition, all subjects entered the winning price,
winner’s identification number, the average of the lowest five bids, the minimum price,and their own profits on their record sheets.
After period 10 was completed, record sheets were collected, and their total profits
were calculated. Subjects were paid their experimental points in cash under the conversion
rate of 1 point equal to 30 yen (29 cents)23 . All sessions lasted roughly 1.5 hour, andearnings of the subjects ranged from 900 yen ($8.57) to 5,070 yen ($48.29), with a meanof approximately 1,710 yen ($16.29).
4. Results
We divide the results into two subsections. Section 4.1 presents results in the identicalcost condition, and Section 4.2 presents the ones in the diﬀerent cost condition. The raw
data from the entire experiment are available on the supplementary materials.
4.1 The identical cost condition
First we test Hypothesis 1. Figure 1 presents all bids in sessions 1 and 2 in cell No-I-N.
The horizontal axis represents the number of periods, and vertical axis represents the priceof bids. Diamonds with numbers 1 to 10 represent bids of producers 1 to 10, respectively.
Winning prices are connected by a straight line, and these values are presented right next
to the winning prices. Figure 2 presents all bids in sessions 1 and 2 in cell T3-I-N. Squares
connected by a straight line represent the minimum prices. Figure 3 presents all bids insessions 1 and 2 in cell No-I-D. Figure 4 presents all bids in sessions 1 and 2 in cell T3-I-D.
––––––––––––––––––––––––––
Figures 1, 2, 3 and 4 are around here
21

Subjects sometimes ask the detailed strategy of experiments, which has a possibility to give a bias totheir decision makings in the subsequent auction periods. Since we prohibited any questions and excluded
any biases, we prepared questions and answers about instructions and distributed them to each subject.
22
We used the term “selling price” instead of “bid”.
23
In cells No-I-N and T3-I-N, although conversion rate of 1 point was 30 yen (29 cents), we did notinform of it. We announced that their initial experimental points were zero and that the higher theirexperimental points were, the higher their earnings of the experiments were in the instructions. We
ensured minimum payments even if final experimental points were less than or equal to zero in thequestions and answers. The minimum payments were 1500 yen ($14.28), which was equal to 50 points ×30 yen (29 cents).

11

––––––––––––––––––––––––––
Table 2 reports the summary statistics across the sessions in each cell. The medianof pooled winning prices across sessions is 97 in both cells No-I-N and T3-I-N. Wilcoxon
signed-rank tests do not reject the null hypothesis that the median of pooled winning
prices across sessions coincides with the theoretical prediction of 97 in both cells No-I-N
and T3-I-N (two-tailed p-values are 1.00 and 0.146, respectively). On the other hand, themedian of pooled winning prices across sessions is 98 in both cells No-I-D and T3-I-D.
Wilcoxon signed-rank tests do not reject the null hypothesis that the median of pooled
winning prices across sessions coincide with the theoretical prediction of 98 in both cells
No-I-D and T3-I-D (two-tailed p-values are 1.00 and 0.453, respectively). These results
support Hypothesis 1. Hence, we obtain the result below.
Result 1. In the identical cost condition, wining prices of each institution coincidewith the theoretical prediction of 97 or 98, regardless of the information condition.
––––––––––––––––––––––––––
Table 2 is around here––––––––––––––––––––––––––
Next we test Hypothesis 2. Wilcoxon rank sum tests do not reject the null hypothesis of
no diﬀerence in medians of pooled winning prices across sessions between cells No-I-N and
T3-I-N (z-value=-1.25, two-tailed p-value=0.209) and do not reject the null hypothesisof no diﬀerence in medians of pooled winning prices across sessions between cells No-I-D
and T3-I-D (z-value=1.12, two-tailed p-value=0.264). These results support Hypothesis
2. Hence, we obtain the result below.
Result 2. In the identical cost condition, winning prices of the FPA with Type 3
minimum price coincide with the ones of the standard FPA, regardless of the information
condition.
Next we focus on the information eﬀect of subjects’ identifications and all their bids
within the same institution. Wilcoxon rank sum tests do not reject the null hypothesisof no diﬀerence in medians of pooled winning prices across sessions between cells T3-I-N
and T3-I-D (z-value=1.52, two-tailed p-value=0.128) but reject the null hypothesis of nodiﬀerence in medians of pooled winning prices across sessions between cells No-I-N and
No-I-D (z-value=3.67, two-tailed p-value=0.002). Hence, we obtain the result below.
Result 3. In the identical cost condition, winning prices of the FPA with Type 3
minimum price in the disclosure condition coincide with the ones in the nondisclosure
condition, but winning prices of the standard FPA in the disclosure condition are slightly
higher than the ones in the nondisclosure condition.
Dufwenberg and Gneeze (2002) find, in their FPA experiment, that although winning prices converge to the theoretical prediction when they disclose only winning prices,winning prices diverge from the theoretical prediction when they disclose all bids. In
12contrast, winning prices of our standard FPA coincides with the theoretical prediction,and the diﬀerences of winning prices between two information conditions are very small.
These slightly diﬀerences seem to due to the diﬀerent information of payments scheme.
In cells No-I-N and T3-I-N, we did not provide a calculation formula from experimental
points to cash rewards. It was likelihood that some subjects misunderstood their rewards
when they sold the good than when they did not sell it even if their profits were negative.
This misunderstanding motivates some subjects to bid prices less than or equal to thecost of 97 to win.
In cells No-I-D and T3-I-D, on the other hand, we provide the calculation formulaof their rewards. It was clearer for subjects to understand that negative payoﬀs would
reduce their rewards, so that they tried to earn profits to bid prices more than the costof 97.
Since the number of the bids less than or equal to 97 in cell No-I-N is more thanthe ones in cell No-I-D, the diﬀerent information on the payment scheme seem to induce
the diﬀerence of winning prices between these cells. But we do not have such diﬀerences
between cells T3-I-N and T3-I-D. This is an issue in the future.
4.2 The different cost condition
First we test Hypothesis 3. Figure 5 presents all bids in sessions 1 and 2 in cell No-DN. Diamonds with 2 (97), 5 (97), and 8 (97) represent bids of producers 2, 5 and 8 whose
costs are 97, respectively. Triangles with identification numbers 1, 3, 4, 6, 7, 9, and 10
represent bids of each producer whose cost is 150. Figure 6 presents all bids in sessions 1
and 2 in cell T3-D-N. Figure 7 presents all bids in sessions 1 and 2 in cell No-D-D. Figure
8 presents all bids in sessions 1 and 2 in cell T3-D-D.
––––––––––––––––––––––––––
Figures 5, 6, 7 and 8 are around here––––––––––––––––––––––––––Medians of pooled winning prices across sessions in cells No-D-N and T3-D-N are
106 and 102, respectively. On the other hand, medians of pooled winning prices acrosssessions in cells No-D-D and T3-D-D are 100 and 119.5, respectively. By Table 2, Wilcoxon
rank-sum tests reject the null hypothesis that the median of pooled winning prices acrosssessions coincides with 97 or 98 at 1% significance level in each cell..
But winning prices in the nondisclosure condition appear to converge toward the theoretical prediction of 98; the ones in the disclosure condition do not. We test whether
winning prices converge to the theoretical prediction of 98. We define that winning prices
converge to the theoretical prediction if the slopes of the OLS regression of winning prices
on the number of periods are negative and winning prices in last five periods coincidewith the theoretical prediction.
Table 3 presents results of the OLS regression of winning prices on the number of
periods in each session in cells No-D-N, T3-D-N, No-D-D, and T3-D-D:
W inning prices = a + b · P eriod + ε
13In cells No-D-N and T3-D-N, slopes are negative and constants are positive in eachsession. They are significant at 1% level, and joint F-statistic tests of slopes and constants
are also significant at 1% level.
––––––––––
Table 3 is around here––––––––––Medians of pooled winning prices in the last five periods across the sessions are 100.5
and 98 in cells No-D-N and T3-D-N, respectively. Wilcoxon signed-rank tests do not
reject the null hypothesis that the median of pooled winning prices in the last five periods
across the sessions coincide with 98 in both cells No-D-N and T3-D-N (two-tailed p-values
are 0.062 and 1.000, respectively). These results support that winning prices converge to98.
In cells No-D-D and T3-D-D, however, slopes are not significant at 1% level, and joint
F-statistic tests of slopes and constants are significant at 1% level. Medians of pooled
winning prices in the last five periods across the sessions are 100 and 116 in cells No-D-D
and T3-D-D, respectively. Wilcoxon signed-rank tests reject the null hypothesis that themedian of pooled winning prices in the last five periods across sessions coincide with 98
in both cells No-D-D and T3-D-D (two-tailed p-values are 0.015 and 0.002, respectively).
These results does not support that winning prices converge to 98. Therefore, Hypothesis 3
is supported in the nondisclosure condition; it is not supported in the disclosure condition.
Hence, we obtain the result below.
Result 4. In the diﬀerent cost condition, winning prices in each institution converge
to the theoretical prediction of 98 in the nondisclosure condition but do not converge tothe theoretical prediction in the disclosure condition.
Next we test Hypothesis 4. Wilcoxon signed-rank tests do not reject the null hypothesis of no diﬀerence in medians of winning prices between cells No-D-N and T3-D-N(z-value=0.733, two-tailed p-value=0.463) but reject the null hypothesis of no diﬀerence
in medians of winning prices between cells No-D-D and T3-D-D (z-value=-4.67, two-tailed
p-value=0.000). Therefore, Hypothesis 4 is supported in the nondisclosure condition; it
is not supported in the disclosure condition. Hence, we obtain the result below.
Result 5. In the diﬀerent cost condition, winning prices of the FPA with Type 3
minimum price coincide with the ones of the standard FPA in the nondisclosure condition,
but winning prices of the FPA with Type 3 minimum price are higher than the ones ofthe standard FPA in the disclosure condition.
The result above implies that the information of subjects’ identifications and all theirbids aﬀects the winning prices. Next we focus on this information eﬀect within the same
institution. Wilcoxon rank sum tests do not reject the null hypothesis of no diﬀerence
in medians of pooled winning prices between cells No-D-N and No-D-D (z-value=-1.28,
two-tailed p-value=0.199) but reject the null hypothesis of no diﬀerence in medians ofpooled winning prices between cells T3-D-N and T3-D-D (z-value=3.88, two-tailed pvalue=0.000). Thus, we obtain the result below.
14

Result 6. In the diﬀerent cost condition, winning prices of the FPA with Type 3
minimum price in the disclosure condition are higher than the ones in the nondisclosure
condition, but winning prices of the standard FPA in the nondisclosure condition coincidewith the ones in the disclosure condition.
Although winning prices in cell No-D-D do not converge to nor coincide with thetheoretical prediction, since winning prices ranged from 98 to 100 and from 99 to 119
in sessions 1 and 2, respectively, apart from period 9 in session 1, they are not so far
from the theoretical prediction. Winning prices in cell T3-D-D, however, are higher thanthe ones in cells T3-D-N and No-D-D and diverge from the theoretical prediction. Next,
comparing T3-D-N and T3-D-D, we consider why winning prices in cell T3-D-N diverge
from the theoretical prediction.
As described in Fact in Section 2.1, in the FPA with Type 3 minimum price, there is
no Case I Nash equilibrium such that bi ≥ ci , for any i. If all producers bid prices morethan their costs, producers whose costs are 97 have incentives to bid prices more thantheir cost not to be excluded. But, as described in Example 4 in Section 2.2, there aremany Case II Nash equilibria such that bi < ci , for some i. Specifically, if a producer bids
extremely low price such as 1, then since the minimum price becomes low, there is very
little possibility that producers whose costs are 97 are excluded when they bid prices close
to their cost.
Comparing cells T3-D-N and T3-D-D, although there are three 1 point bids in cell T3D-D, there are many 1 point bids in cell T3-D-N. Specifically, in cell T3-D-N, producers
7, 9 and 10 in session 1 and producer 4 in session 2 repeated bidding 1, and the minimum
prices became less than 80 from period 5, apart from period 10 in session 1. In cell T3D-D, however, nobody repeated bidding 1 from period 5, and the minimum prices werehigher than 105, apart from periods 6, 9 and 10 in session 1.
According to the questionnaire sheets we distributed after the experiment, producers
4 and 10 who repeated bidding 1 in cell T3-D-N changed their strategies, along the way ofthe experiment, to reduce payoﬀs of the producers whose cost are 97 by bidding extremely
low prices. Their motivations seem to be similar to spiteful strategies defined by Cason
et al. (2004) in the non-excludable public good experiments. They define the spitefulstrategy "if she selects a strategy reducing both her own payoﬀ and the other subject’s
payoﬀ in comparison to the payoﬀs when she takes an own payoﬀ-maximizing strategy,
given an expected strategy of the other subject"24 .
We expand this definition in the meaning of the expected payoﬀs and apply to our
result of repeated bidding 1. Since bidding a price less than the cost has a risk to obtain
negative payoﬀs, expected payoﬀs of the producer whose cost is 150 are smaller when shebids 1 than when she bids 151, given any bid profiles. Her strategy of bidding 1, however,reduces not only her own expected payoﬀs but the ones of the producer whose cost is 97.
Given bidding prices more than 150 of producers whose costs are 150, if producers
whose costs are 97 bid prices more than 97 and less than 150, they can win and the winner’s
payoﬀs ranged from 1/3 to 5225 . On the other hand, given bidding 1 of producers whose
costs are 150, the expected payoﬀ-maximizing strategies of the producers whose costs are24

Cason et al. (2004), p.89, l.4-6.
The minimum expected payoﬀs are 1/3 if three producers whose costs are 97 bid 98 and the maximum
ones are 52 if only one producer whose cost is 97 bid 149 and win.
25

15

97 are bidding 98, and their expected payoﬀs are 1/3 at maximize. Thus, bidding 1 ofproducers whose costs are 150 is the spiteful strategy in the meaning of expected payoﬀs.
Since the disclosure of subjects’ identifications and all their bids erases repeated 1
bids as the spiteful strategy, it seems to be easier for subjects to play the spiteful strategy
when their identifications and actions are hidden than when this information is revealed.
Thus, the nondisclosure condition leads the spiteful strategy which motivates producers
whose costs are 97 to bid prices close to their costs, so that winning prices converge to thetheoretical prediction of 98. The disclosure condition, however, does not lead the spitefulstrategy, which motivates producers whose costs are 97 to raise their bids more than 97
not to be excluded, so that wining prices diverge from the lowest cost.
Finally, we consider the market eﬃciency defined by total surplus of the local government and producers. In the view of the market eﬃciency, the most eﬃcient case is thatthe producer whose cost is 97 produces the good and sells it to the government. In cellT3-D-D, producer 10 whose cost is 150 won at her bid of 151 at period 10 in session 1,
and producer 7 whose cost is 150 won at her bid of 155 at period 8 in session 2. Theseobservations imply that the FPA with Type 3 minimum price has a risk to lose the market eﬃciency when the production costs are diﬀerent if governments disclose producers’
identifications and all their bids. We compare the frequency of achieving the most eﬃcient case between cells No-D-D and T3-D-D. The proportion test does not reject the null
hypothesis of no diﬀerence in frequencies of achieving the most eﬃcient case between cells
No-D-D and T3-D-D (two-tailed p-value=0.146). Hence, we obtain the result below.
Result 7. In the diﬀerent cost condition, there is no diﬀerence in market eﬃciencies
between the FPA with Type 3 minimum price and standard FPA.
5. Concluding remarks
In this paper, we evaluate the performance of the FPA with endogenous minimum
prices relative to the standard FPA. Theoretically, the minimum price calculated as thecertain percents of the average of some of the lowest bids is the most eﬃcient for governments and taxpayers to minimize their expenditures. Winning prices in the FPA withthis type of minimum price coincide with the cost.
In the laboratory, winning prices of the FPA with the most eﬃcient type of minimum
price (i) coincide with the ones of the standard FPA and close to the cost in the identicalcost condition, but (ii) are higher than the ones of the standard FPA and diverge fromthe lowest cost in the diﬀerent cost condition when subjects’ identifications and their allbids are revealed.
Higher winning prices of the FPA with endogenous minimum price due to the information condition. The nondisclosure of subjects’ identifications and their all bids leads
bidding 1 point as the spiteful strategy which motivates producers with the lowest costs
to bid prices close to their costs, so that winning prices converge toward the lowest cost.
The disclosure condition, however, does not lead bidding 1 point as the spiteful strategy,
which motivates producers with the lowest costs to raise their bids not to be excluded, sothat wining prices diverge from the lowest cost. These results suggests for local governments to pay attention to the information control of producers’ identifications and theirbids when they employ the endogenous minimum price.
Acknowledgements
16This research was partially support by the Japanese Ministry of Education, Culture,
Sports, Science and Technology (the COE program of Behavioral Macrodynamics basedon Surveys and Experiments; the Grant-in-Aid for Scientific Research on Priority Areas).
We wish to acknowledge the helpful comments of Akihiko Matsui, Charles Noussaire,
Vesteg Robert, Junyi Shen, and Thakehiko Yamato and participants at meetings of the
Asia Pacific Economic Science Association and conferences of the Experimental Social
Sciences in Japan. We also thank Keiko Aoki, Atsushi Hirata, and Tsuyoshi Nihonsugi
for assistance conducting the experiment.
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18Appendix A.
When a bid profile is b, producer i’s probability Xi (b) of selling the good is : Xi (b) = 0
if bi > min{bj : bj ≥ d(b)}, and Xi (b) = 1/n0 if bi = min{bj : bj ≥ d(b)}, where n0 is thenumber of bids of arg min{bj : bj ≥ d(b)}. When a bid profile is b, producer i’s (expected)
payoﬀ is ui (b) = (bi − ci ) · Xi (b), where ci is producer i’s cost for the good considered.
Proof of proposition 1. Part I: “if”. Let b be a bid profile such that there isb ∈ [c, p] and b1 = · · · = bn = b∗ . We show that b is a Nash equilibrium. Let i ∈ N,
and bbi ∈ [c, p]. We need to show ui (b) ≥ ui (bbi , b−i ). Note that ui (b) = (b∗ − c)/n ≥ 0.
If bbi < bi , then since 2 ≤ m, together with k = 1, implies bbi < d(bbi , b−i ), we haveui (bbi , b−i ) = 0 ≤ ui (b). If bbi > bi , then since m < n implies bbi > d(bbi , b−i ), and there isj ∈ N\{i} such that bj = d(bbi , b−i ), we have ui (bbi , b−i ) = 0 ≤ ui (b). Thus, b is a Nashequilibrium.
Let b be a bid profile such that there is N 0 ⊆ N such that #N 0 ≥ m + 1 and bi = c
for all i ∈ N 0 . Note that ui (b) = 0 for all i ∈ N. Also note that for all i ∈ N and allbbi ∈ [c, p], since #N 0 ≥ m + 1 and bj = c for all j ∈ N 0 , whether i ∈ N 0 or i ∈/ N 0,
d(bbi , b−i ) = c and ui (bbi , b−i ) = 0. Thus, b is a Nash equilibrium.
Part II: “ only if”. Let b be a Nash equilibrium. We show that (1) there is b∗ ∈ [c, p]such that b1 = · · · = bn = b∗ , or (2) there is N 0 ⊆ N such that #N 0 ≥ m + 1 and bi = c
for all i ∈ N 0 . Suppose that neither (1) nor (2) holds. Let i ∈ N be such that bi = b1 (b).
Since (2) does not hold, #{j ∈ N : bj = c} < m or #{j ∈ N : bj = c} = m. We derive acontradiction in each of the following cases.
Case 1: producer i is not a winner. Note that ui (b) = 0, and by bi = b1 (b),bi < d(b) ≤ bm (b) ≤ bm+1 (b). Since p ≥ d(p, b−i ) and d(·, b−i ) is continuous in bi , there isbbi ∈ (bi , p] such that bbi = d(bbi , b−i ). Since bi < bm+1 (b), d(·, b−i ) is increasing in bi up tobi = bm+1 (b). Thus, d(bbi , b−i ) > d(b) ≥ c. Therefore, ui (bbi , b−i ) = (bbi − c)/n0 > 0 = ui (b),where n0 is the number of producers whose bids are equal to d(bbi , b−i ). Since b is a Nashequilibrium, this is a contradiction.
Case 2: producer i is a winner and #{j ∈ N : bj = c} < m. Since producer i isa winner, d(b) ≤ bi . Since (1) does not hold, there is j ∈ N\{i} such that bj > bi . Noteuj (b) = 0.
If #{j 0 ∈ N : bj 0 = c} < m − 1, since c ≤ d(c, b−j ) and d(·, b−j ) is continuous in bj ,there is bbj ∈ (c, bj ) such that bbj = d(bbj , b−j ). Then, uj (bbj , b−j ) = (bbj − c)/n0 > 0 = uj (b),where n0 is the number of producers whose bids are equal to d(bbj , b−j ). Since b is a Nashequilibrium, this is a contradiction.
If #{j 0 ∈ N : bj 0 = c} = m − 1, let bbj ∈ (c, bm (b)). Then, c < d(bbj , b−j ) < bbj , and so
producer j can be a single winner by bidding bbj . Thus, uj (bbj , b−j ) = (bbj − c) > 0 = uj (b).
Since b is a Nash equilibrium, this is also a contradiction.
Case 3: producer i is a winner and #{j ∈ N : bj = c} = m. Note thatd(b) = bi = c, and ui (b) = 0. Also note #{j ∈ N : bj = c} = m implies bm+1 (b) > c. Letbbi ∈ (c, bm+1 (b)). Then, c < d(bbi , b−i ) < bbi , and so producer i can be a single winner bybidding bbi . Thus, ui (bbi , b−i ) = (bbi − c) > 0 = ui (b). Since b is a Nash equilibrium, this isa contradiction.
QED∗

Proof of proposition 2. Let b be a Nash equilibrium. Similarly to Part II of theproof of Proposition 1, we can show that there is b∗ ∈ [c, p] such that b1 = · · · = bn = b∗
19

because (ii) cannot happen by m = n. Suppose b∗ < p. Note that ui (b) = (b∗ − c)/n.
Let bbi = p. Since by m = n and k = 1, b∗ < d(bbi , b−i ). Thus, the bids of all producers
other than i are excluded, and so ui (bbi , b−i ) = (p − c) > (b∗ − c)/n = ui (b). This is acontradiction.
Finally, we show that the bid profile b such that b1 = · · · = bn = p is a Nash equilibrium. Let i ∈ N, and bbi < bi = p. Note that ui (b) = (p − c)/n > 0. By m = n and k = 1,
bbi < d(bbi , b−i ) < p. Since bbi is excluded, ui (bbi , b−i ) = 0 < (p − c)/n = ui (b). Thus, b is aNash equilibrium.
QED
Proof of proposition 3.
Step 1: More than one producers bid b∗ . Suppose that only one producer bids b∗ . Thiswinner can raise her bid to the price slightly less than the second lowest bid, b∗ + ε < b2 ,and increase her payoﬀs. This is a contradiction.
Step 2: b∗ = c. Suppose b∗ > c.
Case 1: d(b) < b∗ . The producer bidding b∗ has an incentive to reduce her bid to∗b − ε and increase her payoﬀs. This is a contradiction.
Case2: d(b) = b∗ . If the producer bidding b∗ raises her bid to the price slightly lessthan the second highest bid, b2 −ε, then since she can pull the minimum price above b∗ andexclude other producers bidding b∗ , she can increase her payoﬀs. This is a contradiction.

Proof of proposition 4.
Step 1: More than one producers bid b∗ . Suppose that only one producer bids b∗ . Thiswinner can raise her bid to the price slightly less than the second lowest bid, b∗ + ε < b2 ,and increase her payoﬀs. This is a contradiction.
Step 2: The winner is a producer whose cost is c1 . Suppose that the winner is a
producer whose cost is more than or equal to c3 . Since b∗ > c3 , the producer whose costis c1 can raise her bid to b∗ and increase her payoﬀs. This is a contradiction.
Step 3: c1 ≤ b∗ < c3 . Suppose b∗ < c1 . Since the winner is the producer whose costis c1 and her payoﬀs are less than zero, she has an incentive to raise her bid to c1 andincrease her payoﬀs. This is a contradiction. Suppose b∗ ≥ c3 . Case 1: d(b) < b∗ . The
producer whose cost is c1 has an incentive to reduce her bid to b∗ − ε, and increase her
payoﬀs. This is a contradiction. Case2: d(b) = b∗ . If the producer whose cost is c1 raises
her bid to the price slightly less than the second highest bid, b2 − ε, then since she canpull the minimum price above b∗ and exclude other producers bidding b∗ , she can increase
her payoﬀs. This is a contradiction.
Step 4: b∗ = c1 . Suppose b∗ > c1 . Case 1: d(b) < b∗ . The producer whose cost is c1 has
an incentive to reduce her bid to b∗ − ε and increase her payoﬀs. This is a contradiction.
Case2: d(b) = b∗ . If the producer whose cost is c1 raises her bid to the price slightly lessthan the second highest bid, b2 −ε, then since she can pull the minimum price above b∗ andexclude other producers bidding b∗ , she can increase her payoﬀs. This is a contradiction.
In the following, we assume that bids are nonnegative integers and that producer 1 has
the lowest cost, producer 2 has the second lowest cost, and so on, that is, c1 ≤ c2 ≤ · · · ≤cn . Given a bid profile b, let b∗ (b) be the winning bid of b, that is, b∗ (b) = min{bi : i ∈ N
and bi ≥ d(b)}.
Lemma 1: In a Nash equilibrium b such that bi ≥ ci for every i, no bid is excludedby d(b), that is, b1 (b) ≥ d(b).
20

Proof. Suppose that there is a bid bi excluded by d(b), that is, bi < d(b). Notethat ui (b) = 0. Let bbi = b∗ (b). If producer i can be a winner by bidding bbi , that is,bbi = min{bj : j ∈ N and bj ≥ d(bbi , b−i )}, then since bbi = b∗ (b) ≥ d(b) > bi > ci , andXi (bbi , b−i ) > 0, it follows that ui (bbi , b−i ) = (bbi − ci ) · Xi (bbi , b−i ) > 0 = ui (b). This isa contradiction to Nash equilibrium. Let ebi = bm (b), that is, ebi is equal to the m−thlowest bid among {b1 , . . . , bn }. Note that ebi > d(ebi , b−i ) ≥ d(bbi , b−i ) ≥ d(b). Thus, if
producer i cannot be a winner by bidding bbi , then producer i can be a winner by bidding
some bi ∈ [b∗ (b), bm (b)], and similarly he can obtain a positive ui (bi , b−i ). This is also acontradiction to Nash equilibrium.
QED∗
Remark: In the Nash equilibrium b such that bi ≥ ci for all i, b (b) = min{bi : i ∈ N}.
Lemma 2: Let bi ≥ ci for all i. In the Nash equilibrium b, for all i ∈ N, if b∗ (b) > ci ,bi = b∗ (b).
Proof. Suppose that b∗ (b) > ci , and bi 6= b∗ (b) for some i ∈ N. By Lemma 1, for alli ∈ N, bi ≥ b∗ (b) ≥ d(b). Thus, bi > b∗ (b) > ci . By bi > b∗ (b), ui (b) = 0. Let bbi = b∗ (b).
Note that d(bbi , b−i ) ≤ d(b) ≤ b∗ (b) = bbi . By Lemma 1, for all j ∈ N, bj ≥ b∗ (b) = bbi . Thus,
producer i can be a winner by bidding bbi , that is, bbi = min{bj : j ∈ N and bj ≥ d(bbj , b−1 )}.
Since bbi = b∗ (b) > ci , and Xi (bbi , b−i ) > 0, it follows that ui (bbi , b−i ) = (bbi − ci ) · Xi (bbi , b−i ) >0 = ui (b). This is a contradiction to Nash equilibrium. Thus, for all i ∈ N, if b∗ (b) > ci ,bi = b∗ (b).
QED∗
Lemma 3: In the Nash equilibrium b such that bi ≥ ci for all i, if b (b) > max{c1 +2, c2 }, then d(b) ∈ (b∗ (b) − 1, b∗ (b)].
Proof. Suppose b∗ (b) > max{c1 + 2, c2 } and d(b) ≤ b∗ (b) − 1. Since Lemma 2 implies
b1 = b2 = b∗ (b), X1 (b) ≤ 1/2. Thus, u1 (b) = (b1 − c1 ) · X1 (b) ≤ (b1 − c1 )/2. Noteb∗ (b) > c1 + 2 ⇒ 2 · {b∗ (b) − c1 } − 2 > b∗ (b) − c1
⇒ {b∗ (b) − 1 − c1 } > (b1 − c1 )/2.
By bidding b∗ (b) − 1, producer 1 can be a single winner and can obtain the payoﬀ (b∗ (b) −1 − c1 ) > (b1 − c1 )/2 = u1 (b). This is a contradiction to Nash equilibrium. Thus, d(b) ∈(b∗ (b) − 1, b∗ (b)].
QED
1
∗
2
Lemma 4: In the Nash equilibrium b such that bi ≥ ci for all i, b (b) = b (b) ≤ b (b) ≤∗b (b) + 1.
Proof. By Lemma 1, b1 (b) = b∗ (b) ≥ d(b). By definition, b1 (b) ≤ b2 (b). Thus, weshow b2 (b) ≤ b∗ (b) + 1. Suppose b2 (b) > b∗ (b) + 1. Let i ∈ N be such that bi = b1 (b). Let
bbi = b∗ (b) +1. Then, ui (bbi , u−i ) = b∗ (b) +1 − ci > b∗ (b) − ci = ui (b). This is a contradiction
to Nash equilibrium.
QED
Claim: Let n = 10, m = 5, k = 4/5, c1 = c2 = c3 = 97, c4 = · · · = c10 = 150. Thereis no Nash equilibrium such that every producer bids a price more than or equal to her
cost.
Proof. Suppose there is a Nash equilibrium b such that bi ≥ ci for all i. Since bi ≥ ci
for all i, it follows that for all i ∈ N, ci < bi and ci < bi (b). Note that b∗ (b) ≥ b1 (b) ≥c1 = 98. We derive a contradiction in each of the following cases.
Case 1. b∗ (b) ≥ 151. (b∗ (b) ≥ c5 + 1.)
21

By Lemma 2 and b∗ (b) > c5 , b1 = · · · = b5 = b∗ (b). Thus, by Lemma 1, b1 (b) = · · · =b5 (b) = b∗ (b), and so d(b) = (4/5) · b∗ (b). By Lemma 3, (4/5) · b∗ (b) > b∗ (b) − 1. Thisinequality implies b∗ (b) < 5. This is a contradiction to b∗ (b) ≥ 151.
Case 2. 98 ≤ b∗ (b) ≤ 150. (c3 + 1 ≤ b∗ (b) ≤ c4 .)By Lemma 2 and b∗ (b) > c3 , b1 = b2 = b3 = b∗ (b). Thus, by Lemma 1, b1 (b) = b2 (b) =b3 (b) = b∗ (b), and so d(b) = (4/5) · [3 · b∗ (b) + b4 (b) + b5 (b)]/5. Since bi ≥ ci , b4 (b) ≥ 151
and b5 (b) ≥ 151. Note that u3 (b) = [b∗ (b) − 97]/3 ≤ [150 − 97]/3.
By Lemma 3, b∗ (b) − 1 < (4/5) · [3 · b∗ (b) + b4 (b) + b5 (b)]/5. Thus, [13 · b∗ (b) − 25]/4 <b4 (b) + b5 (b). Let bb3 = b4 (b).
We show d(bb3 , b−3 ) > b∗ (b) = b1 = b2 . Suppose d(bb3 , b−3 ) ≤ b∗ (b). Since [13 · b∗ (b) −25]/4 < b4 (b) + b5 (b), [13 · b∗ (b) − 25]/4 − b5 (b) < b4 (b). Thus,b∗ (b)≥ d(bb3 , b−3 )4 2 · b∗ (b) + b4 (b) + b4 (b) + b5 (b)
=
·
5
5
¤
4 £
· 2 · b∗ (b) + 2 · b4 (b) + b5 (b)=25 ∙¸
13 · b∗ (b) − 25
4∗55
· 2 · b (b) + 2 ·− 2 · b (b) + b (b)
>
25
4
¤
1 £
· 8 · b∗ (b) + 2 · {13 · b∗ (b) − 25} − 4 · b5 (b)
=
25
¤
1 £
=
· 34 · b∗ (b) − 50 − 4 · b5 (b) .
25

by b4 (b) >13 · b∗ (b) − 25
− b5 (b)4Therefore,¤
1 £· 34 · b∗ (b) − 50 − 4 · b5 (b)25∗25 · b (b) > 34 · b∗ (b) − 50 − 4 · b5 (b)4 · b5 (b) > 9 · b∗ (b) − 50
b5 (b) > [9 · b∗ (b) − 50]/4.
b∗ (b) >Let bb4 = b5. Then,

4 3 · b∗ (b) + 2 · b5 (b)d(bb4 , b−4 ) =
·
5 ∙
5
¸
9 · b∗ (b) − 50
4∗
· 3 · b (b) + 2 ·>25
41· [30 · b∗ (b) − 100] .
=25

by b5 (b) >9 · b∗ (b) − 50
41
Thus, if d(bb4 , b−4 ) ≤ b∗ (b), then 25· [30 · b∗ (b) − 100] < b∗ (b). This inequality impliesb∗ (b) < 20. This is a contradiction to b∗ (b) ≥ 98. Thus, d(bb4 , b−4 ) > b∗ (b) = b1 = b2 = b3 .
Since d(bb4 , b−4 ) ≤ b5 (b) = bb4 , X4 (bb4 , b−4 ) ≥ 1/2, and sou4 (bb4 , b−4 ) ≥ [b5 (b) − c4 ]/2 ≥ [151 − 150]/2 > 0 = u3 (b).
22

Since b is a Nash equilibrium, this is a contradiction. Therefore, d(bb3 , b−3 ) > b∗ (b) = b1 =
b2 .
Note
4 2 · b∗ (b) + b4 (b) + b4 (b) + b5 (b)d(bb3 , b−3 ) =
·
5
5
¤
4 £
=
· 3 · b∗ (b) + {b4 (b) − b∗ (b)} + b4 (b) + b5 (b)
25
¤
4
4 £
· 3 · b∗ (b) + b4 (b) + b5 (b) +· {b4 (b) − b∗ (b)}
=
25
25
4
= d(b) +· {b4 (b) − b∗ (b)}25
4≤ b∗ (b) +· {b4 (b) − b∗ (b)}25
< b∗ (b) + {b4 (b) − b∗ (b)} = b4 (b).
Therefore, d(bb3 , b−3 ) ≤ b4 (b) = bb3 . Thus, X3 (bb3 , b−3 ) ≥ 1/2, and sou3 (bb3 , b−3 ) ≥ [b4 (b) − c3 ]/2 ≥ [151 − 97]/2 > [150 − 97]/3 ≥ u3 (b).

Since b is a Nash equilibrium, this is a contradiction..

23

QED

Appendix B.
Here, we show the Nash equilibrium in the FPA with Type 3 minimum price (k < 1and m < n) and the FPA with Type 4 minimum price (k < 1 and m = n) in Propositions
1 and 2 below, respectively.
Proposition 1. In the FPA with Type 3 minimum price (k < 1 and m < n), a bid
0
profile b is a Nash equilibrium if and only if there is a set of positive integers {nl }ml=1 suchthat(1) b1 = · · · = bn1 < bn1 +1 = · · · = bn1 +n2 < · · · < bn1 +···+nm0 −1 +1 = · · · = bn1 +···+nm0 ,(2) n1 + · · · + nm0 −1 ≤ m, n1 + · · · + nm0 ≥ m + 1,(3) b1 = c,(4) d(b) ≤ c,(5) n1 ≥ 2,(6) for all l ∈ {1, 2, . . . , m0 },6-i) if nl = 1, thenk · (b1 + · · · + bn1 +···+nl−1 + bn1 +···+nl +1 + bn1 +···+nl−1 +2 + · · · + bm )/m ≤ bn1 +···+nl−1 ,6-ii) if nl ≥ 2, thenk · (b1 + · · · + bn1 +···+nl−1 + bn1 +···+nl +1 + bn1 +···+nl−1 +2 + · · · + bm )/m ≤ bn1 +···+nl ,and 6-iii) for all l0 ∈ {l + 1, . . . , m0 },k · (b1 + · · · + bn1 +···+nl−1 + bn1 +···+nl0 +1 + bn1 +···+nl−1 +2 + · · · + bm )/m ≤ bn1 +···+nl0 .
Proof. Part I: “if”. Let b be a bid profile such that there is a set of positive integers
0{nl }ml=1 such that Conditions (1)-(6) hold. We show that b is a Nash equilibrium. Leti ∈ N, and bbi ∈ [c, p]. We need to show ui (b) ≥ ui (bbi , b−i ). Since (3) and (4) imply
b1 = c is a winning bid, ui (b) = 0. If bbi < bi , then since d(bbi , b−i ) ≤ d(b) ≤ c, b1 = c isstill a winning bid, and so we have ui (bbi , b−i ) = 0 = ui (b). If bbi > bi , then since (5) and(6) imply that when bbi raises the minimum price, bbi cannot be a winning bid, we haveui (bbi , b−i ) = 0 = ui (b). Thus, b is a Nash equilibrium.
Part II: “only if”. Let b be a Nash equilibrium. It is trivial to construct a set of0m0
positive integers {nl }ml=1 satisfying (1) and (2). We show that {nl }l=1 satisfies (3)-(6).
Let i ∈ N be such that bi = b1 (b). We show i is a winner. Suppose not. Then,ui (b) = 0, and by bi = b1 (b), c ≤ bi < d(b) ≤ bm (b) ≤ bm+1 (b). Since p ≥ d(p, b−i )and d(·, b−i ) is continuous in bi , there is bbi ∈ (bi , p] such that bbi = d(bbi , b−i ). Sinced(bbi , b−i ) ≥ d(b) > c, ui (bbi , b−i ) = (bbi − c)/n0 > 0 = ui (b), where n0 is the number of
producers whose bids are equal to d(bbi , b−i ). Since b is a Nash equilibrium, this is acontradiction.
We show (3). Suppose b1 > c. First, consider the case that b1 = · · · = bn . Note thatu1 (b) = (b1 −c)/n > 0. Since k ∈ (0, 1), d(b) < b1 . Note that for all bb1 ∈ (max{d(b), c}, b1 ),d(bb1 , b−1 ) < d(b) < bb1 , and u1 (bb1 , b−1 ) = (bbi − c). Thus since (b1 − c)/n > 0, for bb1 close tob1 , u1 (bb1 , b−1 ) = (bb1 − c) > (b1 − c)/n = u1 (b). This is a contradiction. Next, consider thecase that bj > b1 for some j ∈ N. Since j is not a winner, uj (b) = 0. Since b1 is a winningbid, by bidding bbj = b1 , j can be a winner, and obtain uj (bbj , b−j ) = (bbj − c)/(n1 + 1) >0 = uj (b). Since b is a Nash equilibrium, this is a contradiction. Therefore, (3) holds.
Since b1 is a winning bid, (3) implies (4).
We show (5) n1 ≥ 2. Suppose n1 = 1. Then, b2 > b1 . Note that (3) implies ui (b) = 0.
Let bbi ∈ (c, b2 ). Then, since (4) implies d(bbi , b−i ) ≤ bbi , by bidding bbi , i is still a single
24winner and obtains ui (bbi , b−i ) = (bbi − c) > 0 = ui (b). Since b is a Nash equilibrium, thisis a contradiction. Therefore, (5) holds.
Let l ∈ {1, . . . , m0 }. We show 6-i). Suppose that nl = 1, andk · (b1 + · · · + bn1 +···+nl−1 + bn1 +···+nl +1 + bn1 +···+nl−1 +2 + · · · + bm )/m > bn1 +···+nl−1 .
Let j = n1 + · · · + nl−1 + 1. Note that by (3) and (4), uj (b) = 0. If j bids bbj = bn1 +···+nl +1 ,then since k · (b1 +· · · +bn1 +···+nl−1 + bn1 +···+nl +1 + bn1 +···+nl−1 +2 +· · · +bm )/m > bn1 +···+nl−1 ,all the producers whose bids are lower than or equal to bn1 +···+nl−1 are excluded, and j
can be a winner and obtain uj (bbj , b−i ) = (bbj − c)/(nl+1 + 1) > 0 = uj (b). Since b is aNash equilibrium, this is a contradiction. Therefore, 6-i) holds. Similarly, we can show6-ii) and 6-iii).
QED
Proposition 2. In the FPA with Type 4 minimum price (k < 1 and m = n), a0
bid profile b is a Nash equilibrium if and only if there is a set of positive integers {nl }ml=1satisfying (1), (3), (4), (5) of Proposition 2, and (20), (60) below:(20) n1 + · · · + nm0 = n.
(60) for all l ∈ {1, 2, . . . , m0 − 1},
6 − i0) if nl = 1, thenk · (b1 + · · · + bn1 +···+nl−1 + bn1 +···+nl +1 + bn1 +···+nl−1 +2 + · · · + bn )/n ≤ bn1 +···+nl−1 ,6 − ii0) if nl ≥ 2, thenk · (b1 + · · · + bn1 +···+nl−1 + bn1 +···+nl +1 + bn1 +···+nl−1 +2 + · · · + bn )/n ≤ bn1 +···+nl ,6 − iii0) for all l0 ∈ {l + 1, . . . , m0 − 1},
k · (b1 + · · · + bn1 +···+nl−1 + bn1 +···+nl0 +1 + bn1 +···+nl−1 +2 + · · · + bn )/n ≤ bn1 +···+nl0 ,6 − iv0) k · (b1 + · · · + bn1 +···+nl−1 + p + bn1 +···+nl−1 +2 + · · · + bn )/n ≤ bn ,6 − v0) if nm0 = 1, thenk · (b1 + · · · + bn1 +···+nm0 −1 + p)/n ≤ bn−1 ,and 6 − vi0) if nm0 ≥ 2, thenk · (b1 + · · · + bn1 +···+nm0 −1 + p + bn1 +···+nm0 −1 +2 + · · · + bn )/n ≤ bn .
Proof is similar to Proposition 2.
Claim. Let n = 10, m = 5, k = 4/5, c = 97, and p = 600. There is no pure Nashequilibrium satisfied with Proposition 2.
Proof. Let n = 5, k = 8/10, c = 97, and p = 600. Suppose that for a bid profile
0b, a set of positive integers {nl }ml=1 satisfying (1), (20), (3), (4), (5), (60) exists. Without
loss of generality, let b1 ≤ b2 ≤ b3 ≤ b4 ≤ b5 . By (3) and (5), b1 = b2 = 97. Thus, by(4), d(b) = (97 + 97 + b3 + b4 + b5 ) · k/m ≤ c, that is, b3 + b4 + b5 ≤ c · m/k − 194 =
485/0.8 − 194 ' 412. By (6 − v0), d(b) = (97 + 97 + b3 + b4 + p) · k/m ≤ b4 , thatis, b4 ≥ (194 + b3 + p) · k/(m − k) = (194 + b3 + 600) · 0.8/(5 − 0.8). Since 97 ≤ b3
and b4 ≤ b5 , b5 ≥ b4 ≥ (194 + 97 + 600) · 0.8/(5 − 0.8) ' 169. Thus, b3 + b4 + b5 ≥97 + 2 · (194 + 97 + 600) · 0.8/(5 − 0.8) ' 436. This is a contradiction. Hence, for no bid
0
profile, a set of positive integers {nl }ml=1 satisfying (1), (20), (3), (4), (5), (60) exists, andthere is no Nash equilibrium.
QED

25

Appendix C. Instructions and PC operation manuals
Instructions (of cell T3-I-D)
In this experiment, every subject makes a decision on selling the goods to theexperimenter. Please understand the rules of the experiment well, make an appropriate
decision, and earn as much rewards as possible.
Outline of the experiment
In this experiment, you will be assigned a role of a producer, and sell a “good”
to the experimenter. The number of subjects participating in this experiment is 10
people in total. Each subject acts the role of the producer. The computer acts as theexperimenter. The experiment will be repeated 10 periods. In each period, theexperimenter purchases a good from a producer according to pre-determined certain
rules. Following explains the rules of purchasing a good for the experimenter acted by
the computer and the role of a producer every subject acts.
The purchasing rules of the experimenterFirst, we explain the purchasing rules of the experimenter acted by thecomputer. The experimenter purchases one unit of “good” from one of 10 producers
according to the rules below. The purchasing rules of the experimenter are as follows.
First of all, the experimenter asks all 10 producers to submit the selling prices
for the good through computer display. Money used in this experiment is measured by“point” of a fictitious monetary unit. The experimenter purchases a good at the selling
price less than or equal to 243 points. Therefore, the maximum selling price of you as
the producer can submit to the experimenter is 243 pt. Note that selling prices you can
submit are only integers. You have three minutes to input your selling price on yourcomputer screen. Note that you cannot cancel the selling price once you havesubmitted it to the experimenter.
Next, the experimenter selects five producers in the order of the lowest selling
prices. The experimenter calculates the average selling price of these five producers.
Then, this average is multiplied by 0.8. The experimenter selects one producer whose
selling price is the lowest among all the producers who have submitted the selling
prices higher than or equal to “0.8 × (average selling price of lowest five producers)”.
The experimenter purchases the good from that producer at the selling price he/she
has submitted.

26

If there are two or more producers who have submitted that price, then theexperimenter randomly selects one of them and purchases the goods from that
producer.
Roles of producers
We explain the role of the producer you act. Before the beginning of theexperiment, every subject will be assigned to a producer’s identification number from
1 to 10 by a lottery. Once you receive a producer’s number, that number will notchange throughout the experiment.
Each producer can produce one unit of good at “production costs”. The
production cost of a good are common among all producers and it is 97 pt. The qualityof the good any producers produce is same.
When a producer is selected by the experimenter, he/she is to produce a goodand sell it to the experimenter to get “sales revenues”. Since the unit of goods you cansell is one unit, the sales revenue is equal to the selling price to the experimenter, i.e.
equal to the purchase price of the experimenter. The difference between the sales
revenue and the production cost of the good is a “profit” the producer earns from theproduction and sales. That is, “profit = purchase price of the experimenter – production
cost of the good.”
However, producers who do not sell the good to the experimenter do not
produce a good. In this case, these producers do not earn sales revenues but have no
production cost incurred either, so his/her profit is 0 pt.
Purchasing rules of the experimenter and an example of the way to calculate
producers’ profits
This section explains the purchasing rules of the experimenter and the exampleof way to calculate producers’ profits by using actual numbers. Although theexperimenter do not purchase any goods with the selling price higher than 243 points
in the experiment, here we assume that the experimenter can purchase a good at theprice higher than 243 points, and explain the example.
For example, producers number 1 to 10 submit the following selling prices
respectively as their selling prices: 3000 pt, 4000 pt, 5000 pt, 6000 pt, 7000 pt, 8000 pt,
5000 pt, 3000 pt, 4000 pt and 9000 pt. Five lowest prices are 3000 pt = 3000 pt < 4000
pt = 4000 pt < 5000 pt. The average of these prices is (3000 + 3000 + 4000 + 4000 +5000) / 5 = 3800 pt. Multiplying this with 0.8, we get 3800 × 0.8 = 3040 pt. All theselling prices equal to or higher than this number are 4000 pt = 4000 pt < 5000 pt =27

5000 pt < 6000 pt < 7000 pt < 8000 pt < 9000 pt. The lowest price is 4000 pt among
them. However, there are two producers who have submitted 4000 pt of selling price.
Therefore, one of them will be selected with the probability of 1/2, and that producer
sells the good to the experimenter.
At this point, the producer selected by the experimenter is to produce a goodat the production cost of 97 pt so that this producer’s profit is 4000 – 97 = 3903 pt. Allother producers’ profits are 0 pt.
Let us repeat that, in the actual experiment, the experimenter purchases thegood at the price lower than or equal to 243 pt. Every subject decide what selling priceyou are to submit to the experimenter, while taking account of what selling prices otherproducers submit, and try to earn as much rewards as possible.
Completion of the experiment and the calculation of rewards
Once all the producers submit their selling prices, the experimenter notify all
producers from whom the experimenter purchases a good according to the rules as we
explained earlier.
The experimenter discloses the following 4 types of information to all
producers: “producer’s identification number from whom the experimenter purchase agood”, “the purchasing price of the experimenter”, “average of five lowest selling
prices”, and “0.8 × (average of lowest five selling prices)”. At this point, the first period
finishes. Once the first period finishes, the second period will start with sameprocedures. At the time 10th period finishes this experiment is completed.
After the end of the experiment, the rewards each subject receives is calculated
from total profits of the producer each subject acts. Specifically, the rewards to eachsubject after the end of the experiment are calculated as the following formula.
Rewards after the end of the experiment = (50 pt + Gross profit) × 30 Yen
We explain about this formula in detail. At the beginning of the experiment, eachsubject is given 50 pt only once. Total profits as the sum of the profits of producer you
act for 10 periods are added to the 50 pts. However, profits can be minus in theexperiment. If you submit the selling price below the production cost and theexperimenter purchases the goods at that selling price, your profit will be minus. If thetotal profits as the sum of profits for 10 periods are minus, it will be subtracted from 50
pts given to you at the beginning of the experiment. Each subject receives the rewards incash under the conversion rate of 1 pt = 30 yen. Please try to earn as much rewards as28

possible.
Above is the content of today’s experiment. During the experiment, do nottalk to others, and follow orders of the experimenter. You make a decision in theexperiment by the software that operates on the personal computer assigned to eachsubject. The attached “PC Operation Manual” describes how to operate the PC. We
will explain about this manual after the next reading period of “questions and answers
on instructions”.

Different production cost condition
Roles of producers
We explain roles of producers you act. Before the beginning of the experiment,
every subject will be assigned to a producer identification number from 1 to 10 by alottery. Once you receive a producer number, that number will not change throughout
the experiment.
Each producer can produce one unit of good at “production cost”. Production
costs of a good are in the table below.
Producer No.

1

2

3

4

5

6

7

8

9

10

Cost

150 pt

97 pt

150 pt

150 pt

97 pt

150 pt

150 pt

97 pt

150 pt

150 ptProducer number 1 has the production cost of 150 pt, No. 2 has 97 pt, No. 3
150 pt, No. 4 150 pt, No. 5 97 pt, No. 6 150 pt, No. 7 150 pt, No. 8 97 pt, No. 9 150 pt,
and No. 10 150 pt. The quality of goods any producers produce is the same.
When a producer is selected by the experimenter, he/she is to produce a goodand sell it to the experimenter to get “sales revenue”. Since the unit of goods sold is one
unit, the sales revenue is equal to the selling price to the experimenter, i.e. equal to thepurchase price of the experimenter. The difference between the sales revenue and the
production cost of the good is the “profit” the producer earns from the production andsales. In other words, “profit = purchase price of the experimenter – production cost of
the good.”
However, producers who do not sell the good to the experimenter do not
produce a good. In this case, these producers do not earn sales revenue but have no
production cost incurred either, so his/her profit is 0 pt.

29

Purchasing rules of the experimenter and an example of the way to calculate
producers’ profits
This section explains the purchasing rules of the experimenter and the exampleof way to calculate producers’ profits by using actual numbers. Although theexperimenter do not purchase any goods with the selling price higher than 243 points
in the experiment, here we allow the experimenter to purchase a good at the price higher
than 243 points and explain the example.
For example, producers number 1 to 10 submit the following selling prices
respectively as their selling prices: 3000 pt, 4000 pt, 5000 pt, 6000 pt, 7000 pt, 8000 pt,
5000 pt, 3000 pt, 4000 pt and 9000 pt. Five lowest prices are 3000 pt = 3000 pt < 4000
pt = 4000 pt < 5000 pt. The average of these prices is (3000 + 3000 + 4000 + 4000 +5000) / 5 = 3800 pt. Multiplying this with 0.8, we get: 3800 × 0.8 = 3040 pt. All theselling prices equal to or higher than this number are 4000 pt = 4000 pt < 5000 pt =5000 pt < 6000 pt < 7000 pt < 8000 pt < 9000 pt. The lowest price is 4000 pt among
them. However, there are two producers who have submitted 4000 pt of selling price.
Therefore, one of them will be selected with the probability of 1/2, and that producer
sells the good to the experimenter.
At this point, if the producer with production cost of 97 pt is selected, thenhe/she produces a good at 97pt, so his/her profit is 4000 – 97 = 3903 pt. If the selected
producer has the production cost of 150 pt, then he/she manufactures goods at the costof 150 pt, and his/her profit is 4000 – 150 = 3850 pt. All other producers’ profits are 0pt.
Let us repeat that, in the actual experiment, the experimenter purchases thegood at the price lower than or equal to 243 pt. Every subject decide what selling priceyou are to submit to the experimenter, while taking account of what selling prices otherproducers submit, and try to earn as much rewards as possible.
Nondisclosure condition
Completion of the experiment and the calculation of rewards
Once all the producers submit their selling prices, the experimenter notify all
producers from whom the experimenter purchases a good according to the rules as we
explained earlier.
The experimenter discloses the following 4 types of information to all
producers: “producer’s identification number from whom the experimenter purchase agood”, “the purchasing price of the experimenter”, “average of five lowest selling
prices”, and “0.8 × (average of lowest five selling prices)”. At this point, the first period30finishes. Once the first period finishes, the second period will start with sameprocedures. At the time 10th period finishes this experiment is completed.
After the end of the experiment, the rewards each subject receives arecalculated from total profits of the producer each subject acts. Specifically, the rewards
to each subject after the end of the experiment are calculated as the following formula.
Rewards after the end of the experiment = (50 pt + Gross profit) × 30 Yen
We explain about this formula in detail. At the beginning of the experiment, eachsubject is given 50 pt only once. Total profits as the sum of the profits of producer you
act for 10 periods are added to the 50 pt. However, profits can be minus in theexperiment. If you submit the selling price below the production cost and theexperimenter purchases the goods at that selling price, your profit will be minus. If thetotal profits as the sum of profits for 10 periods are minus, it will be subtracted from 50
pts given to you at the beginning of the experiment. Each subject receives the rewards incash under the conversion rate of 1 pt = 30 yen.
The experimenter will not disclose to other subjects whom is assigned to which
producer number during and after the experiment. Therefore, try to earn as muchrewards as possible without concern for others. Please try to earn as much rewards as
possible.

Cells T3-I-N and No-I-N
Completion of the experiment and the calculation of rewards
After the end of the experiment, the rewards for each subject are calculated
from total profits of the producer each subject acts. Although every subject cannot
know the formula to calculate the rewards, that formula is common among all subjects
and have a feature that greater the total profit of the producer you act the greater the
rewards.
At the beginning of the experiment, the profit of each producer is 0 pt. If you
submit the selling price below the production cost and the experimenter purchases the
goods at that selling price, your profit will be minus. Even if the total profits are 0 pt orminus, each subject is guaranteed to receive the minimum rewards. We cannot tell you
how much the minimum rewards are. The rewards paid to the subject with the negative
total profits are the same regardless of how big the negative profits are. Moreover, thesubject with the negative total profits and the subject with 0 pt will receive the same31

minimum rewards.
The experimenter will not disclose to other subjects whom is assigned to which
producer number during and after the experiment. Therefore, try to earn as muchrewards as possible without concern for others.
PC operation manuals
In the experiment, each subject sells a good to the experimenter by using the PC
assigned to each of them. We explain screens of the software we use in the experiment and how
to manipulate it as follows.

PeriodRemaining time (sec.)

your producer’s number
your production costthe selling priceScreen 1
As soon as period 1 starts, the experimenter asks each producer to submit aselling price. Screen 1 is shown in the screen of the PC in front of every subject.
Please see screen 1. In the center of this screen, “your producer’s number”, “yourproduction cost”, and a cell you submit a “selling price” are displayed. In this example, the cellof “your producer’s number” displays producer 1. Producer 1’s production cost is 150 pt so that32

the cell of “your production cost” displays 150 pt. Any subjects submit integers between 1 pt
and 243 pt (limits included). Every subject inputs the selling price and then writes thatselling price in the cell in the record sheet. After you complete writing, click OK button in
the upper-right corner of the screen. Note you can neither cancel nor correct once you
click OK button. Please pay attention about it carefully.
Additionally, “period” in the upper-left corner of the screen displays what period is in
the experiment. In screen 1, it displays “1/10” as shown period 1. “Remaining period” in theupper-right corner of the screen displays how long you have left for submitting a selling price tothe experimenter. Any subjects input the selling prices within 3 minutes (180 seconds). You
necessarily input the selling price. The experimenter encourages subjects who do not input yet
after the remaining time elapse 0 to input the selling price. Notice once all subjects click OK
button, the experimenter start purchasing procedures even though 3 minutes do not elapse.

periodRemaining time (sec.)

your producer’s number
the experimenter buys a good from you
purchasing price of the experimenter
your production costyour profits

Screen 2
If the experimenter purchases the good from you, screen 2 is displayed. From the
top of the display, “your producer’s number”, “the experimenter buys a good from you”,“purchasing price of the experimenter”, “production cost”, and “your profits” are displayed. In

33this example, the cell of “your producer’s number” displays producer 1. Producer 1’s production
cost is 150 pt so that the cell of “your production cost” displays 150 pt. Numerical numbers are
displayed in the white box cells of the right side of the “purchasing price of the experimenter”
and “your profit”.
Every subject, in the actual experiment, transcribes the numerical number
displayed at the right side of “your profit” on the record sheet. After the transcription,please click OK button in the lower-left corner of the screen.

periodyour producer’s number
the experimenter buys a good from you
your profits

Screen 3
If the experimenter does not purchase the good from you, screen 3 is displayed. From
the top of the display, “your producer’s number”, “the experimenter does not buy a good fromyou”, “purchasing price of the experimenter”, “production cost”, and “your profits” aredisplayed. “Your profit” will be 0 pt.
Every subject, in the actual experiment, transcribes the numerical number
displayed at the right side of “your profit” on the record sheet. After the transcription,please click OK button in the lower-left corner of the screen.

34

period

your producer’s number
the experimenter buys a good from producer
the purchasing price of the experimenter
The average of five lowest selling price
0.8 × (average of lowest five selling prices)

Screen 4
After the experimenter decides from whom he buy the good, he disclose “producer’s
identification number from whom the experimenter purchase a good”, how much “thepurchasing price of the experimenter” is, how much “average of five lowest selling prices” is,and how much “0.8 × (average of lowest five selling prices)” is. These information aredisplayed from the top of screen 4. Here we show the case where we purchase the good fromproducer 1 as an example. Numerical numbers are displayed in the white box cells of the rightside of the “purchasing price of the experimenter”, “average of five lowest selling prices”, and“0.8 × (average of lowest five selling prices)”.
Every subject transcribes these information from the top on the cells of“producer’s number from whom the experimenter purchased the good”, “the purchasing
price of the experimenter”, “average of five lowest selling prices”, and “0.8 × (average oflowest five selling prices)” in the record sheet. After the transcription, please click OK
button in the lower-left corner of the screen. Once everybody click OK button, period 1 iscompleted and period 2 will start. The operation after period 2 is the same as one in period
1. The experiment is completed at the time when period 10 is completed.

35

This is the end of explain the PC operation manuals.

36

Table 1. The environment of each cell
NumberCellofsessions

Minimum priceconditions

Production

Information conditions ofcost

subjects’ identifications

conditions

and all their bids

No-I-N

2

No-minimum priceIdentical

Nondisclosure

T3-I-N

2

Type 3 minimum priceIdentical

Nondisclosure

No-I-D2

No-minimum priceIdentical

Disclosure

T3-I-D

2

Type 3 minimum priceIdentical

Disclosure

No-D-N

2

No-minimum priceDifferent

Nondisclosure

T3-D-N

2

Type 3 minimum priceDifferent

Nondisclosure

No-D-D2

No-minimum priceDifferent

Disclosure

T3-D-D

2

Type 3 minimum priceDifferent

DisclosureTable 2. Summary of statistics in winning prices(1)
Treatment(2)
Periods

(3)
Mini.

(4)
Median

(5)
Max.

(6)
Wilcoxon signed-rank tests(two-tailed p-values)
H0: (4) = 97

H0: (4) = 98

No-I-N

1-5
6-10
all

96
1
1

97
97
97

98
98
98

0.625
1.000
1.000

0.016
0.004
0.000

T3-I-N

1-5

90

97

98

1.000

0.031

6-10
all

97
90

97.5
97

98
98

0.063
0.146

0.063
0.001

1-5

97

98

106

0.004

0.375

6-10
all

76
76

98
98

99
106

0.070
0.000

0.625
1.000

T3-I-D

1-5
6-10
all

78
97
78

98
98
98

100
98
100

0.754
0.004
0.019

0.688
1.000
0.453

No-D-N

1-5

99

121

136

0.002

0.002

6-10
all

98
98

100.5
106

110
136

0.002
0.000

0.063
0.000

1-5

104

114

148

0.002

0.002

6-10
all

98
98

98
102

100
148

0.002
0.000

1.000
0.001

No-D-D

1-5
6-10
all

98
98
98

100
100
100

119
147
147

0.002
0.002
0.000

0.008
0.016
0.000

T3-D-D

1-5

119

123.5

151

0.002

0.002

6-10
all

110
110

116
119.5

155
155

0.002
0.000

0.002
0.000

No-I-DT3-D-N

Mini. is the minimum value of winning prices.
Max. is the maximum value of winning prices.

Table 3. Regression analysiscell
No-D-N
T3-D-N
No-D-DT3-D-DsessioncoefficientabProbability adjusted>F-statistic
R2

1

136*** -3.61***

0.00

0.90

2

120*** -2.86***

0.00

0.61

1
2
1
2
1

147***
116***
113***
93***
134***

0.00
0.00
0.06
0.26
0.07

0.83
0.85
0.29
0.05
0.28

2

129*** -0.99

0.55

-0.07

-5.98***-2.13***
-1.19
1.99
-2.08

* denote that the parameters are different from zero at the 1% significance level

1
2

3
4

5
6

7
8

9
10

1
2

5
6

7
8

9
10

98

98

97

97

97

96

97

97

97

Price
97

Price

243

Winning Price

243

Winning Price

3
4

97

2

3

4

5

6

7

8

9

97

97

97

97

97

96

98

97

97

0

0

1
1

98

10

1

2

3

4

5

Period

6

7

8

9

10

Period(a) Session 1(b) Session 2

Figure 1. No-I-N

1
2

3
4

5
6

7
8

1
2

3
4

5
6

7
8

Winning Price

9
10

Minimum Price

98

97

95

98

98

98

98

98

98

98

97

97

96

97

97

97

97

97

0

90

0

98

97

Price
97

Price

243

Minimum Price

243

Winning Price

9
10

1

2

3

4

5

6

7

8

9

10

1

Period

2

3

4

5

6
Period(a) Session 1(b) Session 2

Figure 2. T3-I-N

7

8

9

10

1
2

3
4

5
6

7
8

9
10

1
2

5
6

7
8

9
10

Price
97

106 101
99

98

99

98

98

98

100 98

97

Price

243

Winning Price

243

Winning Price

3
4

97

98

98

97

98

98

98

97

98

0

0

76

1

2

3

4

5

6

7

8

9

10

1

2

3

4

5

Period

6

7

8

9

10

Period(a) Session 1(b) Session 2

Figure 3. No-I-D

1
2

3
4

5
6

7
8

1
2

3
4

5
6

7
8

Winning Price

9
10

Minimum Price

97

100 99

98

98

98

98

97

98

98

98

97

Price

Price

243

Minimum Price

243

Winning Price

9
10

98

95

90

98

90

98

98

98

98

0

0

78

1

2

3

4

5

6

7

8

9

10

1

Period

2

3

4

5

6
Period(a) Session 1(b) Session 2

Figure 4. T3-I-D

7

8

9

10

1
2 (97)

3
4

5 (97)
6

7
8 (97)910

1
2 (97)

5 (97)
6

7
8 (97)

910

150
114 110106 107 105 103

129
118
106 102
99

98

98

98

98

98

0

0

97

125 127 124

97

Price

136

Price

150

243

Winning Price

243

Winning Price

3
4

1

2

3

4

5

6

7

8

9

10

1

2

3

4

5

Period

6

7

8

9

10

7
8 (97)

9
10

Period(a) Session 1(b) Session 2

Figure 5. No-D-N

1
2 (97)

3
4

5 (97)
6910

12 (97)

3
4

5 (97)
6

Winning Price

Minimum Price

Price

150

141
130

120
98

98

98

98

118

109 110 108 104
100 98

98

98

98

0

98

0

97

108

97

148

Price

150

243

Minimum Price

243

Winning Price

7
8 (97)

1

2

3

4

5

6

7

8

9

10

Period

1

2

3

4

5

6
Period(a) Session 1(b) Session 2

Figure 6. T3-D-N

7

8

9

10

1
2 (97)

3
4

5 (97)
6

7
8 (97)910

1
2 (97)

5 (97)
6

7
8 (97)

910150
106 111 106 107

100 105 99

99

98

100 100 100 98

98

98

98

0

100

0

97

107

147

97

119

Price

Price

150

243

Winning Price

243

Winning Price

3
4

1

2

3

4

5

6

7

8

9

10

1

2

3

4

5

Period

6

7

8

9

10

7
8 (97)

9
10

Period(a) Session 1(b) Session 2

Figure 7. No-D-D

12 (97)

3
4

5 (97)
6910

12 (97)

3
4

5 (97)
6

Winning Price

Minimum Price

150

150

243

Minimum Price

243

Winning Price

7
8 (97)

128

126

120

119 115111

115 1100097

120 115 121 117 117

97

121 119 119

155
135 132

Price

Price

151

1

2

3

4

5

6

7

8

9

10

Period

1

2

3

4

5

6
Period(a) Session 1(b) Session 2

Figure 8. T3-D-D

7

8

9

10

Supplementary materials
producer's ID (cost)
1
2
T3-I-N: session 1
1 (97)
98* 982 (97)
110 98
3 (97)
159 125
4 (97)
1
1
5 (97)
120 105
6 (97)
140 987 (97)
150 100
8 (97)
163 100
9 (97)
169 1
10 (97)
104 98*endogenous minimu
69.3 47.4
T3-I-N: session 2
1 (97)
120 110
2 (97)
116 100
3 (97)
168 984 (97)
150 985 (97)
104 98*6 (97)
125 115
7 (97)
140 988 (97)
133 989 (97)
98* 9810 (97)136 109
endogenous minimu
90.1 78.4
T3-I-D: session 1
1 (97)
160 99*2 (97)
196 150
3 (97)
130 115
4 (97)
126 126
5 (97)
100* 99
6 (97)
200 127
7 (97)
105 105
8 (97)
158 104
9 (97)
160 160
10 (97)117 115
endogenous minimu
92.5 83.5
T3-I-D: sesion 2
1 (97)
150 105
2 (97)
194 143
3 (97)
49
130
4 (97)
153 78*5 (97)
98* 105
6 (97)
153 120
7 (97)
147 122
8 (97)
160 138
9 (97)
130 9810 (97)
127 101
endogenous minimu
88.2 77.9* shows the winning price.

3

4

Round
5
6

7

8

9

10

98
98
98
98
100
97*
99
70
98
98
73.8

98
98
98
98
99
99
99
95*
100
98
77.9

98
98
98
98
90* 98*
98
1
98
98
98
243
97
98
98
98
100 100
98
98
77
62.9

98
98
101
98
98*
1
98
98
1
98
47.4

98
98
98
98*
98
98
98
98
98
98
78.4

98
98
98*
98
100
243
98
98
100
98
78.4

98
98
98
98
98
98
99
98*
100
98
78.4

100
98
98
98
97*
98
98
98
98
98
78.2

99
98
98
98
97
98
98
98
98
96*
77.9

98
98
98
98
97*
98
98
98
98
97
78.1

97
98
97*
98
97
98
98
98
98
97
77.8

97
98
98
98
98
98
98
98
98
97*
78.1

97*
98
98
98
98
98
98
98
98
2
62.9

97*
98
98
98
98
98
97
98
98
97
77.9

97
98
98
98
98
97*
97
98
98
98
77.9

98
243
130
243
98
110
105
102
243
98*
80.2

98*
243
130
243
98
147
105
99
243
107
81.1

98*
243
200
243
98
137
105
98
243
127
84.2

243
243
243
243
98
98
105
98
150
97*
79.4

98
147
130
243
98
137
105
98
150
98*
79.5

98* 194147 127
243 243
222 11198 98
147 140
105 10598
98
1
222
118 98*64 81.6

100
127
104
7
98
101
105
98*
109
102
64.6

100
106
90*
100
110
100
107
104
102
104
78.7

101
125
95*
98
99
99
98
99
98
120
78.1

160
90*
95
98
98
98
98
98
98
115
76.6

115
127
99
110
98
98
130
98*
98
115
78.6

101 145120 120100 9898
98*
98* 9898
98
100 243
103 100
150 140
114 98
79
78.4

98
145
98*
98
98
98
243
150
176
98
78.4

98
175
98
98
98
98
98*
98
111
98
78.4

Round
producer's ID (cost)
1
2
3
4
5
6
7
8
9
10
T3-D-D: session 1
1 (150)157 153 200 152 165 151 170 151 240 160
2 (97)
148* 141* 150 128 108* 104 102 98* 98 98*
3 (150)193 243 243 243 243 243 243 243 243 243
4 (150)152 151 151 243 243 243 151 243 243 160
5 (97)
112 107 130* 129 119 98* 98 102 98* 103
6 (150)156 153 155 155 155 155 155 155 155 155
7 (150)159 151 155 150 1
1
153 155 200 151
8 (97)
150 150 138 120* 111 103 98* 98 98
98
9 (150)155 160 155 175 243 31
155 240 1
10 (150)151 151 243 243 243 1
11
1
1
endogenous minimu price 114 112 116 109 79 33 48 72 72
48.2
T3-D-N: session 2
1 (150)180 151 151 151 151 151 151 151 151 151
2 (97)
120 120 110* 108 104* 102 99 98 98* 98*
3 (150)157 170 184 198 190 185 188 180 178 181
4 (150)158 158 243 243 1
1
1
1
1
158
5 (97)
118* 109* 103 108* 104 100 98* 98 243 98
6 (150)151 151 151 200 200 160 160 160 160 160
7 (150)243 243 151 243 151 151 151 151 243 243
8 (97)
180 130 130 125 106 100* 102 98* 98 98
9 (150)181 155 170 151 155 161 240 155 153 241
10 (150)
175 155 190 151 149 140 135 130 125 120
endogenous minimu price 113 106 103 103 74.2 70.9 69.6 68 75.7 90.4
T3-D-D: session 1
1 (150)159 151 243 243 243 152 243 243 151 151
2 (97)
180 147 119* 119* 128* 135 119 121* 125 117*
3 (150)153 155 160 220 243 243 243 243 243 180
4 (150)160 151 151 151 151 151 151 151 151 151
5 (97)
165 121* 124 149 110 131 118 115 150 120
6 (150)165 155 180 200 220 225 190 243 240 230
7 (150)152 151 151 151 243 243 243 230 220 219
8 (97)
110 150 120 108 112 120* 115* 113 117* 118
9 (150)155 155 154 210 210 230 222 222 222 222
10 (150)
151* 152 151 151 242 243 243 243 225 200
endogenous minimu price 115 115 106 108 114 110 111 116 111 105
T3-D-D: session 2
1 (150)180 155 160 170 160 200 200 200 198 195
2 (97)
175 138 142 126* 119* 125 113 105 122 146
3 (150)158 155 243 243 243 1
243 243 243 155
4 (150)159 167 230 243 180 180 243 243 1
1
5 (97)
120* 200 200 140 130 115* 130 115 121 110
6 (150)163 195 160 243 243 243 160 190 243 154
7 (150)163 160 161 155 156 156 156 155* 159 163
8 (97)
135 135* 132* 127 137 180 111* 199 115* 110*
9 (150)170 200 200 225 243 243 243 243 243 180
10 (150)160 160 243 243 243 243 243 243 243 200
endogenous minimu price 117 119 121 115 112 92.3 107 122 82.9 83.4
* shows the winning price.

producer's ID (cost)
1
2
3
No-I-N: session 1
1 (97)
98* 9897
2 (97)
115 98* 98
3 (97)
130 9898
4 (97)
120 120 120
5 (97)
124 9898
6 (97)99
98
98
7 (97)
127 9898
8 (97)
117 117 98
9 (97)
177 243 97*
10 (97)143 105 98
No-I-N: session 2
1 (97)
199 97* 97*
2 (97)
222 190 150
3 (97)
199 169 98
4 (97)
147 9898
5 (97)
119 100 100
6 (97)
98* 9998
7 (97)
190 120 100
8 (97)
199 199 243
9 (97)
120 9898
10 (97)
110 9898
No-I-D: session 1
1 (97)
109 103 99*
2 (97)
143 101* 104
3 (97)
125 115 100
4 (97)
125 125 125
5 (97)
178 135 240
6 (97)
140 106 115
7 (97)
117 108 106
8 (97)
110 104 107
9 (97)
106* 112 114
10 (97)118 102 101
No-I-D: session 2
1 (97)
145 119 121
2 (97)
147 99147
3 (97)
106 104 99
4 (97)
117 117 117
5 (97)
104 98* 98
6 (97)
194 121 120
7 (97)
200 110 98
8 (97)
146 145 145
9 (97)
104 100 100
10 (97)100* 100 98*
price.
* shows the winning

4

Round
5
6

7

8

9

10

97*
98
98
120
98
98
98
98
100
98

97
98
98
120
98
98
96*
98
200
98

97*
98
98
120
98
98
97
98
98
98

98
98
98
120
98
98
97*
98
98
98

97*
98
98
120
98
98
97
98
98
98

98
98
98
120
98
98
97*
98
120
98

98
98
98
120
98
1*
96
98
97
98

98
97*
100
97
100
98
100
200
98
98

98
120
98
97
98
98
98
200
97*
98

200
120
98
96*
100
98
98
200
97
200

98
100
98
98
98
98
98
200
97*
98

97
100
98
97*
98
98
98
197
97
98

200
100
98
98
98
98
98*
197
98
98

97*
98
98
98
98
98
98
197
98
98

100
98*
100
123
100
107
105
100
121
100

98*
103
100
105
243
112
105
99
122
98

99*
99
100
126
200
102
105
100
121
100

99
99
100
156
98
99
98*
100
101
100

98
98
123
99
98
99
98
100
100
98*

98
98
123
98
98
97*
98
100
98
98

97
98
123
100
98
112
76*
100
98
98

163
147
97*
117
98
120
98
145
100
98

109
117
99
117
98
197
98
145
100
98*

243
107
98
117
98*
197
98
117
100
98

98
107
98*
117
98
98
98
117
200
98

99
107
97*
117
98
98
98
200
210
98

121
107
98
117
98
98
98
200
220
98*

106
107
98
117
98
98
98
105
242
98*

Round
producer's ID (cost)
1
2
3
4
5
6
7
8
9
10
No-D-N: session 1
1 (150)151 150 152 151 151 150 151 143 151 141
2 (97)
136* 149 127* 150 114* 140 119 109 109 103*
3 (150)155 155 151 151 151 151 151 200 151 151
4 (150)170 155 155 151 155 155 160 160 152 151
5 (97)
137 125* 135 124* 130 120 106* 117 105* 106
6 (150)180 151 151 151 151 151 151 151 151 151
7 (150)197 151 151 151 151 151 151 151 151 151
8 (97)
160 150 130 125 117 110* 125 107* 117 104
9 (150)200 170 151 150 230 240 199 240 240 240
10 (150)
185 160 170 155 151 243 170 243 155 160
No-D-N: session 2
1 (150)243 243 151 151 151 151 151 151 151 151
2 (97)
135 119 106* 108 100 98* 98* 99 98* 98
3 (150)151 150 151 151 151 151 151 151 151 151
4 (150)158 158 158 158 158 158 158 158 158 158
5 (97)
138 118* 107 102* 102 101 100 99 98 98*
6 (150)187 152 150 180 160 160 160 160 160 160
7 (150)200 180 230 170 190 210 220 160 215 200
8 (97)
129* 122 112 104 99* 106 99 98* 107 102
9 (150)151 151 151 151 151 151 151 151 151 151
10 (150)200 200 200 200 200 200 200 200 200 200
No-D-D: session 1
1 (150)152 155 151 153 151 152 152 151 151 243
2 (97)
127 107* 100* 106* 117 106* 107 100* 105* 99*
3 (150)200 160 160 160 160 160 160 160 160 160
4 (150)160 155 151 151 243 151 151 151 151 151
5 (97)
198 168 168 168 168 155 107* 128 106 243
6 (150)152 152 152 152 152 152 152 152 152 152
7 (150)160 151 158 155 160 151 243 161 151 155
8 (97)
119* 107 126 135 111* 111 128 108 111 109
9 (150)243 243 243 243 243 243 243 243 243 151
10 (150)151 151 200 155 155 155 155 155 200 200
No-D-D: session 2
1 (150)163 243 243 237 238 226 199 239 237 155
2 (97)
142 120 141 150 109 109 151 151 149 125
3 (150)150 150 150 150 150 150 150 150 150 149
4 (150)160 155 200 240 180 230 243 240 155 184
5 (97)
99* 99 98* 100* 109 105 99 98* 151 120
6 (150)200 160 160 243 243 243 242 240 239 160
7 (150)169 168 242 243 243 243 243 243 243 243
8 (97)
106 98* 114 111 100* 100* 98* 105 147* 98*
9 (150)160 243 243 243 243 243 243 243 243 243
10 (150)155 155 154 160 154 153 153 151 153 160* shows the winning price.