DP2020-17

Implementation in Iterative Eliminationof Obviously Dominated Strategies:
An Experiment on King Solomon's Dilemma*
Makoto HAGIWARA
Fumihiro YONEKURA

April 20, 2020

* The Discussion Papers are a series of research papers in their draft form, circulated to encourage
discussion and comment. Citation and use of such a paper should take account of its provisional character.
In some cases, a written consent of the author may be required.

Implementation in Iterative Elimination of
Obviously Dominated Strategies: An Experiment on
King Solomon’s Dilemma∗
Makoto Hagiwara†

Fumihiro Yonekura‡

April 20, 2020

Abstract“King Solomon’s Dilemma” is based on a biblical story and this can be considered as an allocation problem for an indivisible object among two players. A social
planner wants to assign the object without payment to the player whose valuationis the highest. We say that such an allocation is “first-best.” We experimentallycompare the relative performance of the mechanism of Mihara (Japanese Economic
Review, 63(3), 420 - 429, 2012) and a mechanism which we modify Mihara’s mechanism. We find that a modified Mihara’s mechanism relatively works better than
Mihara’s mechanism from the following five view points: (1) the proportion of thefirst-best allocations; (2) the proportion of the right-player allocations; (3) resource
inefficiency and wrong-player inefficiency; (4) net mean efficiency; and (5) players’
behavior.
JEL Classification: C92; D44; D78
Keywords: Implementation in iterative elimination of obviously dominated strategies; King Solomon’s Dilemma; Mihara’s mechanism; Ascending clock auctions;
Laboratory experiment∗

The authors are grateful to Emiko Fukuda, Ryo Kawasaki, Akiyoshi Shioura, Hirofumi Yamamura,and Takehiko Yamato for their helpful comments. Hagiwara was partially supported by JSPS KAKENHI
Grant Number JP17J01520.
†
Corresponding author: Faculty of Economics, Osaka University of Economics, 2-2-8, Osumi,
Higashiyodogawa-ku, Osaka, 533-8533, Japan; E-mail: m.hagiwara@osaka-ue.ac.jp‡
Department of Industrial Engineering and Economics, School of Engineering, Tokyo Institute ofTechnology, 2-12-1 Ookayama, Meguro-ku, Tokyo, 152-8552, Japan; E-mail: fumi314@yahoo.ne.jp11

Introduction“King Solomon’s Dilemma” is based on the biblical story in which King Solomon must
give a child to one of two women who claim to be the true mother. The King and both
women know that only one of the two women is the true mother. The King does notknow which the right mother is. On the other hand, both women know who the true
mother is, but the false mother does not want to disclose her true identity. The King
wants to give the child to the true mother without payment. This problem is considered
as an allocation problem for an identical and indivisible object among two players, andthere are several examples of which this model is a realistic description (see Glazer andMa, 1989; Footnote 1).1 A social planner wants to assign the good without payment to
the player whose valuation is the highest. We say that such an allocation is “first-best.”
Since the problem involves a transfer of money, an auction itself does not solve it (seeGlazer and Ma, 1989; Moore, 1992; Perry and Reny, 1999). Several mechanisms havebeen proposed in order to solve the allocation problem.
Glazer and Ma (1989) and Moore (1992) assume that each player knows the otherplayer’s valuation. Under this information setting, they construct multi-stage mechanismsthat implement the first-best allocation in subgame perfect equilibrium. Although thosemechanisms consist of more than two stages, they have an appealing feature that only
one player moves at each stage.
On the other hand, the four studies of Perry and Reny (1999), Olszewski (2003), Qin
and Yang (2009), and Mihara (2012) assume that each player does not know the otherplayer’s valuation but knows her own valuation and whether she is the top valuation
player. Perry and Reny (1999) and Olszewski (2003) construct mechanisms that implement the first-best allocation in iterative elimination of weakly dominated strategies.
Qin and Yang (2009) design a mechanism that implement the first-best allocation in one
round of elimination of weakly dominated strategies, followed by at most four rounds ofiterative elimination of strictly dominated strategies. In contrast to these mechanism,
Mihara (2012) provides a mechanism whose advantage is that only the top valuation
player chooses to participate in the auction, which rules out the need to hold the auction.
Mihara’s mechanism consists of the following two stages: At the first stage, each playerchooses whether she participates in the second-price auction or not: At the second stage,if both players choose to participate in the auction, all players pay a fee and partici1

King Solomon’s Dilemma is generalized by Qin and Yang (2009) as the allocation problem fork identical and indivisible objects among at least two players. Mihara (2012) also investigates thisgeneralized problem.

2

pate the second-price auction. Mihara (2012) assumes that the players and the plannerknow that there is a gap between the two players’ valuations which is greater than M .
Mihara’s mechanism implements the first-best allocation in one round of elimination of
weakly dominated strategies, followed by two rounds of iterative elimination of strictly
dominated strategies.
For the result of Mihara (2012), an important feature of the second-price auction is“strategy-proofness.” This axiom states that in the direct revelation mechanism, truthtelling is a weakly dominant strategy for each player. Even though choosing when to quit
in an ascending clock auction is the same as choosing a bid in the second-price auction,
ascending clock auctions are simpler for real people to understand than the second-priceauction. As evidence from a laboratory experiment, subjects are substantially more
likely to play the dominant strategy under an ascending clock auction than under thesecond-price auction (Kagel, Harstad, and Levin, 1987). Inspired by this observation,
Li (2017) models what it means by “obvious dominance.” This definition distinguishes
ascending clock auctions and the second-price auction. Ascending clock auctions are
obviously strategy-proof. On the other hand, the second-price auction is strategy-proof, but
not obviously strategy-proof. Therefore, in iterative elimination of obviously dominatedstrategies, Mihara’s mechanism does not implement the first-best allocation.
We modify Mihara’s mechanism as changing the second-price auction into an ascending clock auction. This modified mechanism implements the first-best allocationin iterative elimination of obviously dominated strategies (Proposition 2). This resultis shown, together with Table 1 of Mihara (2012), by the reason that in an ascending
clock auction, for each player, quitting when the price is her own valuation is the uniqueobviously dominant strategy.
A bunch of experimental studies suggest that players perform only a few number ofsteps of iterative elimination of dominated strategies (Nagel, 1995; Sefton and Yavas,
1996; Costa-Gomes, Crawford, and Broseta, 2001; Katok, Sefton, and Yavas, 2002;
Rapoport and Amaldoss, 2000; Kneeland, 2015). While there is evidence that it is hard
to apply the notion of implementation in iterative elimination of dominated strategies
to real people, the result of Masuda, Okano, and Saijo (2014) suggests that the notionof implementation in iterative elimination of obviously dominated strategies is not thecase.2 They report that in the public good provision, the mechanism which implements
the Pareto-efficient allocation in iterative elimination of obviously dominated strategies
2

Note that Masuda, Okano, and Saijo (2014) do not state the feature of implementation of iterative
elimination of obviously dominated strategies in their mechanism.

3

works well. In their laboratory experiment, players simultaneously and independently
choose their contributions to the public good from 0 to 24 as integers. Moreover, in eachmechanism designed by them, players simultaneously and independently decide whether
to approve the other’s choice or not. Then, the number of strategies in their experiment
for each player is 25×2 = 50. On the other hand, as in Section 3, the number of strategies
in our experiment for each player is 2 × 271 = 542. Even though the number of strategies
for each player is greater, our mechanism works well.
We experimentally compare the relative performance of Mihara’s mechanism and ourmechanism. Our experiment find the following five observations: (1) Our mechanism
attains the first-best allocation more frequently than Mihara’s mechanism (Result 1); (2)Our mechanism allocates the object to the right player more frequently than Mihara’s
mechanism (Result 2)3 ; (3) Mihara’s mechanism is more inefficient than our mechanism
from the view point of resource inefficiency and right-player inefficiency (Results 3 and 4);(4) Net mean efficiency under our mechanism was 91.3%, while net mean efficiency under
Mihara’s mechanism was 71.8% (see Table 2); and (5) The equilibrium strategies are frequently taken under our mechanism than under Mihara’s mechanism (see Subsection 4.3).
In addition, we compare the results of Elbittar and Di Giannatale (2017) with our results
under the same setting as ours.4 Our mechanism works better than any mechanisms
investigated by Elbittar and Di Giannatale (2017).
The rest of the paper is organized as follows. Section 2 provides the model and two
mechanisms. Section 3 describes our experimental design. Section 4 reports our results
and provides discussion. Section 5 summarizes our results and further researches as aconclusion.

2

The Model and Two Mechanisms

Suppose that there are only two players, 1 and 2. A single, identical, and indivisible object
is to be allocated to the player whose valuation is the highest. It is common knowledge
that one player’s valuation exceeds the other’s. Each player knows her own valuation,
and whose valuation is higher. A social planner knows neither players’ valuations nor
3

For both mechanisms, the first-best allocations and the right-player allocations are more frequently
achieved in the latter half than in the first half.
4
Ponti, Gantner, López-Pintado, and Montgomery (2003) also conducted an experiment for King
Solomon’s Dilemma. While they considered the only five cases of payoff vectors that the two players
have, both this paper and Elbittar and Di Giannatale (2017) investigate much more cases of payoff
vectors and each case is randomly chosen by computer. Then, it is hard to compare directly the resultsof Ponti, Gantner, López-Pintado, and Montgomery (2003) with ours nor the results of Elbittar and
Di Giannatale (2017).

4

whose valuation is higher. Each player’s payoff of obtaining the object and payment p is
v + p, where v is the player’s valuation of the object. Each player’s payoff of obtaining noobject and payment p is p. We assume that the two players and the planner know that
there is a gap between the highest and the second highest valuations which is greaterthan a positive real number M > 0. The planner wants to allocate the object without
payment to the player whose valuation is the highest. Hereafter, we say that such an
allocation is first-best.
For the allocation problem, the following mechanism is designed by Mihara (2012):
The Second-Price Auction with the Participation Stage (hereafter, SPAPS)
The participation stage: Both two players simultaneously saying either “auction” or “noauction.”
If only one player says “auction,” then this player is eligible to acquire the object
without payment and the other player gets and pays nothing, and the game ends. If no
players say “auction,” then the object is assigned randomly, and the game ends. If both
two players say “auction,” then we move to the second-price auction after each player
pays the participation fee, M .
The auction stage: Both two players simultaneously and independently offer bids for theobject. The player whose bid is the highest is eligible to acquire the object at the pricethat the other player offers a bid and the other player gets and pays nothing except forthe participation fee, and the game ends.
Regarding SPAPS, Mihara (2012) provides the following result:
Proposition 1. (Mihara, 2012) SPAPS implements the first-best allocation in one roundof elimination of weakly dominated strategies, followed by two rounds of iterative elimination of strictly dominated strategies
Strategy-proofness requires that in the direct revelation mechanism, truth-telling is aweakly dominant strategy for each agent.5 While the second-price auction is strategyproof, it is not obviously strategy-proof.6 Hence, SPAPS does not implement the first-best
allocation in iterative elimination of obviously dominated strategies. Then, we modify
SPAPS as follows:5

For the definition of strategy-proofness, see for example Barberà (2011).
A strategy is obviously dominant if, for any deviation, at any information set where both strategies
first diverge, the best outcome under the deviation is no better than the worst outcome under thedominant strategy. A mechanism is obviously strategy-proof if it has an equilibrium in obviously dominant
strategies. For the formal definitions of obviously dominant strategies and obvious strategy-proofness,see Li (2017).
6

5

An Ascending Clock Auction with the Participation Stage (hereafter, ACAPS)
The participation stage: This stage is the same as Mihara’s mechanism.
The auction stage: The price of the object begins at 0 and increases the price for theobject. Each participant will be regarded as an active bidder for the object until theplayer cease to bid on the object. Each player can drop out of the auction. Exit from theauction is not reversible i.e., no player can re-enter once she is out. The only one player
who is still an active bidder, i.e. did not drop out, is eligible to acquire the object at theprice at which the other bidder exited.
Since an ascending clock auction is obviously strategy-proof, together with Table 1 ofMihara (2012), we have the following result:
Proposition 2. ACAPS implements the first-best allocation in iterative elimination of
obviously dominated strategies.
The idea to have Proposition 2 is not difficult. Suppose that at the participation stage,
each player says “auction.” In the auction stage, for each player, quitting when the priceis her own valuation is the unique obviously dominant strategy. Given this remaining
strategy profile, saying “auction” is the unique obviously dominant strategy for the player
whose valuation is the highest. Given this remaining strategy profile in the auction stage
and this remaining strategy of the player whose valuation is the highest, saying “noauction” is the unique obviously dominant strategy for the player whose valuation is
the lowest. Therefore, the unique remaining strategy profile from iterative eliminationof obviously dominated strategies is that the player whose valuation is the highest says
“auction” and the player whose valuation is the lowest says “no auction.” Therefore, by
these three steps, we have the statement of Proposition 2.

3

Experimental Design

Our experiment compares the relative performance of SPAPS and ACAPS. All settings
regarding our experimental design are the same as those of Elbittar and Di Giannatale(2017) except the amount of the entry payment for our experiment and the number ofthe practice and real rounds. To compare our results with theirs, we use the almost samesettings and the reasons for three different settings are discussed in Footnotes 8 and 10.
The details in the experimental design are as follow.
Subjects. For each session, the subjects were drawn from a wide of undergraduate
6and graduate students at Tokyo Institute of Technology. The subjects were told that
there would be an opportunity to earn money in our experiment. None of them had
prior experience in a second-price-auction experiment nor an ascending-clock-auction
experiment nor a King-Solomon-Dilemma experiment.
Two sessions were conducted under each of SPAPS and ACAPS, and 20 subjects
participated in each session (80 separate subjects in total). No subject attended morethan one session. Our experiment was conducted at Tokyo Institute of Technology during
December of 2019 using computers with the experimental software, z-Tree (Fischbacher,2007). In each session, 20 subjects were seated at computer stations that were separated
with visual partitions in the Experimental Economics Laboratory at Tokyo Institute ofTechnology.
Flow of the Experiment. Upon subjects’ arrival, they were randomly assigned to acomputer terminal. Each subject received an instruction, a post-experimental questionnaire, and a record sheet which all are in Japanese, together with writing materials such as
two pencils.7 Each instruction explains the summary of our experiment, the details of ourexperiment, the procedures of our experiment, and how to decide points and payments.
After all subjects were seated and the experimenter confirmed that subjects received all
experimental materials, the instruction was read aloud by the experimenter for subjects
to understand the instruction equally. Subjects were allowed to ask any question they
had during the whole process. After reading the instruction, our experiment was initiatedby computer. At the end of our experiment, each participant answered a questionnaire
and then the payments obtained by subjects were made by cash.
Practice and Real Periods. In order to familiarize the subjects with the procedures,
3 practice periods and 20 real periods.8
Matching Procedure. After the three practice periods, each subject was designatedeither as a High Valuation Player (hereafter, HVP) or a Low Valuation Player (hereafter,
LVP). These roles remained fixed for the whole session, so that each type of a player identifies herself with her role. In each period, an HVP was paired with an LVP, and eachpairing was randomized. They were never paired with the same subject more than twice,7

In the following website, see the instructions, the post-experimental questionnaire, and the record
sheets in English that are translated from Japanese and in Japanese that are used in our experiment:
https://sites.google.com/view/makotohagiwara-economics/research8
In contrast to ours, Elbittar and Di Giannatale (2017) conducted the 2 practice periods and the
10 real periods. We changed the settings because we would like for subjects to understand all cases of
SPAPS or ACAPS and also find learning effect.

7and were never paired with the same subject in two consecutive periods. Furthermore,they did not know who they were paired with in any given period.
Valuations. Let θHV P be the valuation of an HVP and θLV P be the valuation of an
LVP. Players’ valuations for each period were drawn randomly from the interval [0, 200]
as integers, with the restriction that the difference between θHV P and θLV P would be
more than M ≡ 50:
θHV P − θLV P > M ≡ 50.
Information Setting. We consider the following information setting: both players were
informed of the ranking of their valuations i.e., whether they had the higher or lower valuation, but they were not informed of the exact amount of the other player’s valuation.
Under this information setting, each player’s screen showed her own valuation, but not
the valuation of the player with whom was paired. However, the players knew that thedifference between the players’ valuations was more than 50 francs.
Initial Capital. All players were endowed with an initial capital balance in according to
the role in the conducted period. The initial capital balances were 30 francs for the HVPs
and 70 francs for the LVPs.9 The reason for the difference between the initial capital of
the LVPs and that of the HVPs was to compensate for the asymmetry in their valuations
and deter envy-driven actions, as in Elbittar and Di Giannatale (2017).
Information Feedback. During the decision process, some of the calculations about
possible payoffs were given privately by the computer to each subject at different decisionnodes. After participating an auction and the bids were submitted for the second-priceauction, or the first drop-out occurred for an ascending clock auction, the payoffs conditional on keeping or giving up the object were calculated for the winner. Finally, at theend of each round, each subject received complete feedback about her own payoff.
Payoffs and Final Payments. The final payoff was determined selecting one roundrandomly out of the 20 real periods played for money. Subjects were also informed thatany winnings would be added to the initial capital balance, and any losses would be
subtracted from it. The initial capital balance, plus gains or minus losses, was considered
their possible payoff for each period. The equilibrium expected payoffs were 193 francs
for the HVPs and 70 francs for the LVPs, as in Elbittar and Di Giannatale (2017).
Under SPAPS, the final payment on average was 3054 Japanese yen, and the experimental time was about 2 hours and a half. Under ACAPS, the final payment on average
9

Regarding the initial capital balances, we will discuss those in Section 5.

8was 3719 Japanese yen, and the experimental time was about 3 hours.10 Note that theexperimental time was different between SPAPS and ACAPS due to the waiting time inan ascending-clock auction.
Bidding Restrictions. For the ascending clock auction, subjects were informed thatthe clock price had reached the maximum level of 270 without a subject dropping out, thesale price for the item would be 270 and the item would be sold to one of the claimants(chosen randomly by computer) at this price, without the option to withdraw, and theother bidder would pay nothing. In order to compare an ascending-clock auction withthe second-price auction, subjects were not allowed to bid above 270. This restriction
was imposed to avoid allowing the subjects to incur losses.

4

Results and Discussion

4.1

Efficiency

To compare the results of our experiment with the results of Elbittar and Di Giannatale(2017), we consider the two different measures of “efficiency:” (1) the proportion of
first-best allocations, and (2) the proportion of right-player allocations. The
first is obtained by dividing the number of objects given to HVPs without payment by
the total number of allocations. The second is obtained by dividing the number of objects
given to HVPs with or without payment by the total number of allocations.
Figure 1 (resp. Figure 2) reports the transition of the proportions of each of the firstbest allocations and the right-player allocations under SPAPS (resp. ACAPS). Figure 3
reports the aggregate proportions of the first-best allocations and right-player allocations
under SPAPS and ACAPS. From these figures, the performance regarding each of thefirst-best allocations and the right-player allocations under ACAPS is better than under
SPAPS.
To investigate these observation statistically, we conduct pairwise z-tests between
SPAPS and ACAPS (see Table 1).11
The following are our two results:
Result 1. (First-best allocations) The ratio of the first-best allocations under ACAPS
10

While Elbittar and Di Giannatale (2017) used the payment of 50 francs for each subject as an entry
payment in their experiment, we payed 1013 Japanese yen (or 506.5 francs) hourly. The amount of 1013
Japanese yen is decided for the compensation of the minimum wage in Tokyo, Japan in 2019 which isirrelevant to the performances of subjects. See the following URL:
https://pc.saiteichingin.info/kouho/pdf/leaflet/english/13_mw2019_A4_english_toukyou.pdf
11

For pairwise z-tests, we essentially follow Ponti, Gantner, López-Pintado, and Montgomery (2003)(see Footnote 19 on page 225 of their paper).

9

Figure 1: Transition of the proportion of each of the first-best and the right-player allocations under SPAPS.

Figure 2: Transition of the proportion of each of the first-best and the right-player allocations under ACAPS.

Figure 3: Aggregate proportions of each of the first-best allocations and right-player
allocations under SPAPS and ACAPS.

10

Table 1: Aggregate proportions of allocations for SPAPS and ACAPS.
1-20 periods

1-10 periods

11-20 periods

SPAPS

ACAPS

SPAPS

ACAPS

SPAPS

ACAPS

(1) HVPs get the object

0.88

0.97∗∗∗

0.84

0.96∗∗∗

0.93

0.99∗∗(1a) without payment

0.74

0.87∗∗∗

0.65

0.81∗∗∗

0.83

0.92∗∗(1b) with payment

0.14

0.11

0.19

0.15

0.10

0.07

(2) LVPs get the object

0.10

0.02∗∗∗

0.14

0.04∗∗∗

0.07

0.01∗∗(2a) without payment

0.02

0.00∗∗

0.03

0.00∗

0.01

0.00

(2b) with payment

0.08

0.02∗∗∗

0.11

0.04∗∗

0.06

0.01∗∗(3) No one gets the object

0.02

0.01

0.03

0.01

0.01

0.01

Observations

400

400

200

200

200

200

Note: ∗ p <0.05; ∗∗ p <0.01; ∗∗∗ p <0.001

11

is significantly larger than under SPAPS for total 20 periods, 1 − 10 periods, and 11 − 20
periods, respectively.
Support. From (1a) in Table. 1, the aggregate proportion of allocations for total 20 periods(resp. 1 − 10 periods, 11 − 20 periods) when HVPs get the object under SPAPS without
payment is 0.74 = 296/400 (resp. 0.65 = 130/200, 0.83 = 166/200), and that for ACAPS
is 0.87 = 346/400 (resp. 0.81 = 162/200, 0.92 = 184/200). For each of total 20 periods,
1 − 10 periods, and 11 − 20 periods, since the latter proportion is greater than theformer proportion, ACAPS performs better than SPAPS. In addition, the difference isstatistically significant at the 0.1%−level (resp. 0.1%, 1%).
Result 2. (Right-player allocations) The ratio of the right-player allocations under
ACAPS is significantly larger than under SPAPS for total 20 periods, 1 − 10 periods, and11 − 20 periods, respectively.
Support. From (1) in Table. 1, the aggregate proportion of allocations for total 20 periods(resp. 1 − 10 periods, 11 − 20 periods) when HVPs get the object under SPAPS with orwithout payment is 0.88 = 353/400 (resp. 0.84 = 167/200, 0.93 = 186/200), and that for
ACAPS is 0.97 = 388/400 (resp. 0.96 = 191/200, 0.99 = 197/200). For each of total 20
periods, 1 − 10 periods, and 11 − 20 periods, since the latter proportion is greater than
the former proportion, ACAPS performs better than SPAPS. In addition, the differenceis statistically significant at the 0.1%−level (resp. 0.1%, 1%).

From these results, we observed that ACAPS achieved both the first-best allocations
and the right-player allocations more frequently than SPAPS for total 20 periods, 1 − 10
periods, and 11−20 periods, respectively. In addition, for both mechanisms, the first-best
allocations and the right-player allocations were more frequently achieved in the latter
half than in the first half.
We compare our experimental results under SPAPS and ACAPS and those of Elbittarand Di Giannatale (2017) regarding the following three mechanism: Perry and Reny
(1999) mechanism (hereafter, PRM), PRM with an ascending clock auction (hereafter,
PRMACA), and PRM with a slow ascending clock (hereafter, PRMSACA).12
Figure 4 provides the comparison of the experimental results under these five mechanisms. Note that since Elbittar and Di Giannatale (2017) conducted their experiments
12
Note that the experiment of Elbittar and Di Giannatale (2017) was conducted in Mexico City, while
our experiment was run in Tokyo, Japan.

12Figure 4: Comparison with SPAPS, ACAPS, PRM, PRMACA and PRMSACA.

for only 1 − 10 periods, we also focus on 1 − 10 periods. As in Figure 4, ACAPS attains both the first-best allocations and the right-player allocations more frequently than
PRM, PRMACA and PRMSACA. The aggregate proportions of the first-best allocations under SPAPS, ACAPS, PRM, PRMACA and PRMSACA are 65%, 81%, 40%, 39%
and 51%, respectively. The aggregate proportions of the right-player allocations under
SPAPS, ACAPS, PRM, PRMACA and PRMSACA are 84%, 96%, 82%, 80% and 87%,
respectively.

4.2

Sources of Inefficiency

To compare our experimental results with those of Elbittar and Di Giannatale (2017),
we consider two possible sources of inefficiency: (1) wrong-player inefficiency, and(2) resource inefficiency. The first is obtained by dividing the high valuation minus
winner’s valuation by the summation of the high valuation and the initial capitals of an
HVP and an LVP. The second is obtained by dividing the summation of the two players’
payments by the summation of the high valuation and the two players’ initial capitals.
In other words, the proportion of wrong-player inefficiency tells us the inefficiency fromallocating the object to an LVP, and the proportion of resource inefficiency tells us theinefficiency of monetary resources from paying the participation fee and the winning bid.
Before analyzing these two sources of inefficiency, let us introduce the notion of net
mean efficiency as the total amount of players’ net gains as a proportion of the total
surplus. At equilibrium, the net mean efficiency should be a hundred percent of the total surplus, otherwise there would be two types of inefficiency: wrong-player inefficiency
and resource inefficiency. Table 2 reports the aggregate proportions of net mean effi-

13Table 2: Aggregate proportions of Net Mean Efficiency (NME), Resource Inefficiency(RI), and Wrong-Player Inefficiency (WPI).
Net Mean Efficiencya

Resource Inefficiencyb

Wrong-Player Inefficiencyc

SPAPSd

71.8%

23.0%

5.2%

ACAPSd

91.3%

7.2%

1.5%

PRM

57.4%

35.9%

6.7%

PRMACA

70.1%

23.4%

6.5%

PRMSACA

75.0%

21.2%

3.8%

a

NME=(Winner’s valuation−Players’ payments+Players’ initial capital)/(Highest valuation+Players’ initial capital)b
RI=Players’ payments/(Highest valuation+Players’ initial capital)c
WPI=(Highest valuation−Winner’s valuation)/(Highest valuation+Players’ initial capital)d
In order to compare with Elbittar and Di Giannatale (2017), the aggregate proportion isfor 1 − 10 periods.

ciency, wrong-player inefficiency and resource inefficiency under SPAPS, ACAPS, PRM,
PRMACA, and PRMSACA.13 From Table 2, resource inefficiency is the main source of
inefficiency under each of all five mechanisms, and ACAPS achieves higher net mean
efficiency than the other four mechanisms.
We measure how players’ valuations and a different mechanism affect our sources of
inefficiency. In the following Ordinary Least Squares (OLS) regression regarding each ofwrong-player inefficiency and resource inefficiency, SPAPS is the baseline model and iscompared to ACAPS. The specifications can be formulated, in general, as follows:
yit = intercept + β1 θHV P,it + β2 θLV P,it + β3 per + γdACAP S + vi + it ,where yit represents the proportion of either wrong-player inefficiency or resource inefficiency for player i at period t; θHV P,it represents the valuation of the HVP for player
i at period t; θLV P,it represents the valuation of the LVP for player i at period t; perrepresents the period treating time as a continuous variable; dACAP S represents a dummy
variable that takes the value of one in the case of ACAPS; vi is the individual-specificerror term and it is the usual error term assumed to be i.i.d. for player i at period t.
Table 3 provides the regression results for sources of inefficiency.
13

The aggregate proportions of net mean efficiency, wrong-player inefficiency and resource inefficiency

14

Table 3: Regression results for sources of inefficiency.
Resource Inefficiency

Wrong-Player Inefficiency

θHV P−0.0001(0.0002)−0.0004∗∗∗(0.0001)

θLV P

0.003∗∗∗(0.0002)0.0002∗∗∗(0.0001)

per−0.007∗∗∗(0.001)−0.002∗∗∗(0.0004)dACAP S

−0.082∗∗∗(0.026)−0.033∗∗∗(0.007)

Intercept

0.136∗∗∗(0.037)0.111∗∗∗(0.012)

Observations

1,600

1,600

Note: Numbers in parentheses below each coefficient represent the coefficient’s standard
error.
∗
p <0.05; ∗∗ p <0.01; ∗∗∗ p <0.001

15

The following are our two findings regarding sources of inefficiency:
Result 3. (Resource inefficiency)
1. The proportion of the inefficiency due to the use of monetary resources in the allocation process in ACAPS is lower than in SPAPS.
2. When LVPs’ valuations increase, the proportion of the inefficiency of the monetary
resources increases.
3. As the period progresses, the proportions of the inefficiency of the monetary resources under SPAPS and ACAPS decrease.
Support.

1. In Table 3, the coefficient for dACAP S is negative (−0.082) and the differ-ence is statistically significant at 0.1%−level, which indicates a reduction in the useof monetary resources under ACAPS.
2. The coefficient for θLV P is positive (0.003) and the difference is statistically significant at 0.1%−level.
3. The coefficient for per is negative (−0.007) and the difference is statistically significant at 0.1%−level.

Result 4. (Wrong-player inefficiency)
1. The proportion of the inefficiency for allocating the object to the wrong player under
ACAPS is lower than under SPAPS.
2. When HVPs’ valuations decrease or LVPs’ valuations increase, the proportions of
the inefficiency for allocating the object to the wrong player decrease.
3. As the period progresses, the proportion of the inefficiency from allocating the object
to the wrong player decreases.
Support.

1. In Table 3, the coefficient for dACAP S is negative (−0.033). In addition,

the difference is statistically significant at 0.1%−level, which indicates a reductionof cases that the object is allocated to the wrong player under ACAPS.
2. The coefficient for θHV P is negative (−0.0004) and that for θLV P is positive (0.0002).
For each of these coefficients, the difference is statistically significant at 0.1%−level.
under PRM, PRMACA, and PRMSACA are come from Table 4 in Elbittar and Di Giannatale (2017).

16

Table 4: Players’ behavior at the participation stage.
SPAPS

ACAPS

(1a) “auction”

96.5% = 772/800

99.3% = 794/800(1b) “no auction”

3.5% = 28/800

0.7% = 6/800(2a) “auction”

24.3% = 194/800

12.8% = 102/800(2b) “no auction”

75.8% = 606/800

87.3% = 698/800(1) HVPs’ decision(2) LVPs’ decision

3. The coefficient for per is negative (−0.002), and the difference is statistically significant at 0.1%−level.

4.3

Players’ Behavior

4.3.1

The Participation Stage

Table 4 describes the aggregate proportions of the players who say “auction” or “no auction” at the participation stage for total 20 periods under SPAPS and ACAPS. Under
SPAPS, HVPs say “auction” at 96.5% and LVPs say “no auction” at 75.8%. Under
ACAPS, HVPs say “auction” at 99.3% and LVPs say “no auction” at 87.3%. Therefore,
the performance under ACAPS is better than under SPAPS at the participation stage.
Figure 5 (resp. Figure 6) describes the transition regarding the aggregate proportion of
the players who say “auction” at the participation stage for each period under SPAPS(resp. ACAPS). These figures seem to represent the tendency that as the period progresses, the equilibrium action at the participation stage for HVPs and LVPs under eachmechanism is more frequently taken.
Regarding the above observations, we statistically measure how players’ valuations
and a different mechanism affect players’ behavior regarding whether they participate in
the auction or not. In the following logit regression regarding each of saying “auction”
and saying “no auction,” SPAPS is the baseline model and is compared to ACAPS. The
specifications can be formulated, in general, as follows: For HVPs,
yit0 = intercept + β10 θHV P,it + β20 per + γ 0 dACAP S + vi0 + 0it ,17

Figure 5: Participation rate for each period at the participation stage under SPAPS

Figure 6: Participation rate for each period at the participation stage under ACAPS

18

Table 5: Regression results for HVPs’ behavior and LVPs’ behavior at the participation
stage.
HVPs

LVPs

θHV P

0.033∗∗∗(0.005)

-

θLV P

-

0.024∗∗∗(0.002)per

0.095∗∗(0.034)−0.098∗∗∗(0.014)dACAP S1.724∗∗∗(0.464)−1.098∗∗∗(0.292)

Intercept−1.896∗∗(0.689)−1.677∗∗∗(0.257)800

800

Observations

Note: Numbers in parentheses below each coefficient represent the coefficient’s standard
error.
∗
p <0.05; ∗∗ p <0.01; ∗∗∗ p <0.001

and for LVPs,
yit00 = intercept + β100 θLV P,it + β200 per + γ 00 dACAP S + vi00 + 00it ,where yit0 (resp. yit00 ) represents either “auction” or “no auction” chosen by HPV (resp.
LVP) i at period t and both yit0 = 1 and yit00 = 1 (resp. yit0 = 0 and yit00 = 0) mean that atperiod t, player i says “auction” (resp. “no auction”); vi0 and vi00 are the individual-specificerror terms and 0it and 00it are the usual error terms assumed to be i.i.d. for player i atperiod t. Table 5 provides the regression results for HVPs’ behavior and LVPs’ behavior
at the participation stage.
The following are our two findings regarding players’ behavior at the participation
stage:
Result 5. (HVPs’ behavior at the participation stage)
1. Saying “auction” at the participation stage for HVPs under ACAPS is more frequently taken than under SPAPS.
19

2. When HVPs’ valuations increase, saying “auction” at the participation stage ismore frequently taken.
3. As the period progresses, saying “auction” at the participation stage for HVPs ismore frequently taken.
Support.

1. In Table 5, the coefficient for dACAP S is positive (1.724) and the differenceis statistically significant at 0.1%−level, which indicates the increase that HVPs
say “auction” under ACAPS.
2. The coefficient for θHV P is positive (0.033) and the difference is statistically significant at 0.1%−level.
3. The coefficient for per is positive (0.095) and the difference is statistically significant
at 1%−level.

Result 6. (LVPs’ behavior at the participation stage)
1. Saying “no auction” at the participation stage for LVPs under ACAPS is more
frequently taken than under SPAPS.
2. When LVPs’ valuations increase, saying “auction” at the participation stage is more
frequently taken.
3. As the period progresses, saying “no auction” at the participation stage for LVPs ismore frequently taken.
Support.

1. In Table 5, the coefficient for dACAP S is negative (−1.098) and the dif-ference is statistically significant at 0.1%−level, which indicates the reduction that
LVPs say “auction” under ACAPS.
2. The coefficient for θLV P is positive (0.024) and the difference is statistically significant at 0.1%−level.
3. The coefficient for per is negative (−0.098) and the difference is statistically significant at 0.1%−level.

20

Table 6: Players’ behavior at the auction stage.
SPAPS

ACAPS

Auction stage

22.5% = 180/800

12.8% = 102/800(1) HVPs win the auction

63.3% = 114/180

82.4% = 84/102(1a) and bSHB ∈ [0, θLV P − 2)a

28.1% = 32/114

21.4% = 18/84(1b) and bSHB ∈ [θLV P − 2, θLV P + 2]

19.3% = 22/114

45.2% = 38/84(1c) and bSHB ∈ (θLV P + 2, 270]

52.6% = 60/114

33.3% = 28/84

36.7% = 66/180

17.6% = 18/102(2a) and bSHB ∈ [0, θHV P − 2)

69.7% = 46/66

88.9% = 16/18(2b) and bSHB ∈ [θHV P − 2, θHV P + 2]

18.2% = 12/66

0.0% = 0/18

(2c) and bSHB ∈ (θHV P + 2, 270]

12.1% = 8/66

11.1% = 2/18

(2) LVPs win the auction

a

4.3.2

bSHB : the second highest bid (or the first drop-out price)
The Auction Stage

Table 6 provides the distributions of players’ bids under SPAPS and ACAPS. In orderto analyze whether HVPs and LVPs bid rationally, we divide the regions of the secondhighest bid under SPAPS (or the first drop-out price under ACAPS), bSHB , into threecases: (1) bSHB is less than loser’s valuation minus 2 francs, (2) bSHB is in the interval
between loser’s valuation plus 2 francs and minus 2 francs, and (3) bSHB is more thanloser’s valuation plus 2 francs. Notice that we would like to analyze whether loser’s bid is
her own valuation (or quit when the price is her own valuation) of the equilibrium pricewithin 2 francs as in Elbittar and Di Giannatale (2017) and Li (2017).
First, we investigate the behavior of LVPs. HVPs win the auction at 63.3% = 114/180
under SPAPS and at 82.4% = 84/102 under ACAPS. When we focus on the cases where
the auction stage is proceeded and HVPs win the auction, bSHB is less than loser’s
valuation minus 2 francs at 28.1% = 32/114 under SPAPS and at 21.4% = 18/84 under
ACAPS; bSHB is between loser’s valuation plus 2 francs and minus 2 francs at 19.3% =22/114 under SPAPS and at 45.2% = 38/84 under ACAPS; bSHB is more than loser’s
valuation plus 2 francs at 52.6% = 60/114 under SPAPS and at 33.3% = 28/84 under
ACAPS. Note that the number of the players who proceed to the auction under ACAPS

21

(102/800) is much less that under SPAPS (180/800). Even though, not only the aggregate
proportion of the LVPs who bid between loser’s valuation plus 2 francs and minus 2 francs
under ACAPS (45.2%=38/84) but also the number of the LVPs itself are greater thanunder SPAPS (19.3%=22/114). In addition, not only the aggregate proportion of the
LVPs who bid more than loser’s valuation plus 2 francs under ACAPS (33.3%=28/84)
but also the number of the LVPs itself are less than under SPAPS (52.6%=60/114).
Next, we investigate the behavior of HVPs. LVPs win the auction at 63.3% = 114/180
under SPAPS and at 82.4% = 84/102 under ACAPS. When we focus on the cases where
the auction stage is proceeded and LVPs win the auction, they win the auction at 36.7% =66/180 under SPAPS and at 17.6% = 18/102 under ACAPS; bSHB is less than loser’s
valuation minus 2 francs at 69.7% = 46/66 under SPAPS and at 88.9% = 16/18 under
ACAPS; bSHB is between loser’s valuation plus 2 francs and minus 2 francs at 18.2% =12/66 under SPAPS and at 0.0% = 0/18 under ACAPS; bSHB is more than loser’s
valuation plus 2 francs at 12.1% = 8/66 under SPAPS and at 11.1% = 2/18 under
ACAPS. In contrast to the results for LVPs, the aggregation proportions of the HVPs who
bid less than loser’s valuation minus 2 francs under SPAPS (69.7%=46/66) and ACAPS(88.9%=16/18) are large. Note that even though, most HVPs do not say “auction” at theparticipation stage under both mechanisms as in Section 4.3.1. This tendency is different
from the results of experiments regarding the second-price auction and an ascending
auction (e.g., Li, 2017). Then, this tendency seems to be come from the reason that two
types of high values and low values are assigned and players know their types.

5

Conclusion

We first investigated a mechanism for King Solomon’s Dilemma theoretically. While themechanism of Mihara (2012) includes the second-price auction, we changed it into an
ascending clock auction. By this modification, we had the result that a modified Mihara’s mechanism implements the fist-best allocation in iterative elimination of obviously
dominated strategies.
We experimentally compared the performance of Mihara’s mechanism and a modified Mihara’s mechanism for King Solomon’s Dilemma. Our findings were as follows:(1) ACAPS attained the first-best allocation more frequently than SPAPS (Result 1);14(2) ACAPS allocated the object to the right player more frequently than SPAPS (Result 2); (3) SPAPS was more inefficient than ACAPS from the view point of resource
14

For both mechanisms, the first-best allocations and the right-player allocations were more frequently
achieved in the latter half than in the first half.

22inefficiency and right-player inefficiency (Results 3 and 4); (4) Net mean efficiency under
ACAPS was 91.3%, while net mean efficiency under SPAPS was 71.8% (Table 2); and(5) The equilibrium strategies were frequently taken under ACAPS than under SPAPS(Section 4.3).
Although the performance of ACAPS was relatively better than that of SPAPS in
our experiment, there are still open questions. The following are three instances; theparticipation fee, the number of players, and the types of players.
First, in our experiment, when LVPs’ valuations increase, the proportions of theinefficiency of the monetary resources and the inefficiency regarding the wrong-player
allocations increases, as in Subsection 4.2. A possible explanation for it is that if an
LVP’s valuation for the object is higher than the participation fee, he considers the casethat he can get the object with payment which is lower than her valuation in the auction
stage. Thus, we expect that increasing the participation fee may lower the rate of the
choices of “auction” selected by LVPs at the participation stage. Conversely, we would
like to investigate whether decreasing the participation fee may lower the rate of the
choices of “no auction” selected by LVPs at the participation stage.
Second, in our experiment, there are only two players. If the number of players
increases, then it may be harder for players to find the equilibrium strategies. Even inthis case, we expect that the relative performance of our mechanism may be better thanthat of Mihara’s mechanism.
Third, in our experiment, the two types of players were fixed, as in Elbittar and
Di Giannatale (2017). The reason for this setting is that we would like to find whether
there is learning effect or not. While we found learning effect as in Sections 4.2 and 4.3, a
few subjects with low valuations (two subjects out of total 20 subjects with low valuations
under each of SPAPS and ACAPS) participated the auctions to spite their counterparts
with high valuations.15 This is because subjects’ types were fixed so that subjects withlow valuations envied their counterparts.16 On the other hand, in the experiment ofPonti, Gantner, López-Pintado, and Montgomery (2003), the two types of players were
15

This observation is come from the post-experimental questionnaires. While the subjects who spited
their counterparts understood that saying “no auction” is the best response for theirselves, they participated the auctions as their counterparts had to pay money including the participation fee.
16
The subjects who spited their counterparts might not very much consider their own payoffs. As in
our instructions, if subjects got negative points, the minimum hourly payment of 1013 Japanese yen was
guaranteed. Even though, if they did not participate in the auction, they could get the larger amountof money. Then, our setting regarding how to calculate the final payment from subjects’ payoffs gave
the incentives to get the much larger amount of payoffs to the subjects. In short, while they had thechance to get larger amount of money, they might have decided to spite their counterparts since the lossof money by this behavior is not large.

23

not fixed. If we follow this setting, each subject may change her strategy because he canbe either an HVP or an LVP.

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