DP2015-34

Mate Choice Mechanis m for
Solving a Quas i-Dilemm aTatsuyoshi SAIJO
Junyi SHEN
July 27, 2015

Mate Choice Mechanism for Solving a Quasi-Dilemma
Tatsuyoshi Saijo* and Junyi Shen*** Research Center for Social Design Engineering, Kochi University of Technology, Japan** Research Institute for Economics and Business Administration, Kobe University, Japan, andSchool of Economics, Shanghai University, China

Abstract
Saijo, Okano, and Yamakawa (2014) showed that the mate choice mechanism for a symmetricprisoner’s dilemma (PD) game implements cooperation in backward elimination of weakly
dominated strategies (BEWDS), and it attained almost full cooperation in their experiment.
First, this study shows that the mechanism works well in the class of quasi-dilemma (QD)
games such as asymmetric PD games and coordination games. Second, the class of BEWDSimplementable games is exactly the same as the class of QD games. Third, the mechanism
cannot implement cooperation in a subgame perfect equilibrium. Finally, we confirm that themate choice mechanism works well experimentally for an asymmetric PD game.
1. Introduction
This study is part of our endeavor to find one of the simplest possible mechanisms tosolve social dilemmas, including the prisoner’s dilemma (PD), public good provision, andcoordination, both experimentally and theoretically. 1 Experimentally, we aim to design
mechanisms that can attain a Pareto efficient outcome in a few rounds, 2 because we cannot
repeat the same mechanism many times in real-life settings. Theoretically, we seek natural
behaviors among subjects in experiments rather than stick to Nash or Nash-type equilibrium
See, for example, Huang, Masuda, and Saijo (2014), Masuda, Okano, and Saijo (2014), and Saijo, Okano, andYamakawa (2014). Masuda et al. (2014) constructed a mechanism that implements a Pareto efficient allocation
in a public good economy when the number of players is two and their preferences are linear, and then, they
conducted experiments in which the rate of contribution is 94.9%. The mechanism, called the minimum
approval mechanism, is a version of the mate choice mechanism. Huang et al. (2014) constructed a mechanismthat implements cooperation when the number of players is at least two. They conducted experiments inwhich the rate of cooperation is more than 90% after period 4 with three players. The mechanism is called thesimplified approval mechanism, and it is also based upon the mate choice mechanism.
2 Chen (2005), for example, found that the stability property of mechanisms depends on their supermodularity.
Supermodular mechanisms may require many periods to converge to a desired outcome. The goal of theendeavor is not to find such mechanisms but to find mechanisms that can attain a desired outcome in a few
periods.
1

1

concepts. In addition, we do not use punishment or reward to balance the budget in themechanism design. 3
The study by Saijo, Okano, and Yamakawa (2014) is one of the first attempts to design
such a mechanism for the PD. They proposed the mate choice mechanism after a symmetric PD
game. After observing the choice of cooperation (C) or defection (D) in the PD game, eachplayer approves or disapproves the other’s choice: if both approve it, the outcome is what theychose in the PD game, and if at least one player disapproves the other choice, the outcome isthat when both defect, which is called the mate choice (MC) mechanism. Because the MCmechanism does not have devices such as punishment or reward, it is budget balanced.
Experimentally, they observed that the cooperation rate, that is, the ratio of subjects
who chose cooperation, with the mechanism was 95.0% in round 1 and 96.9% through 19
rounds, when each subject was never matched with the same subject again in all rounds. 4 The(C,C) share, that is, the ratio of pairs in which both chose C, was 90.0% in round 1 and 94.0%
through 19 rounds. They also found that subjects’ behavior was consistent with backwardelimination of weakly dominated strategies (BEWDS) rather than Nash equilibrium (NE) or
subgame perfect equilibrium (SPE) behavior. BEWDS is a procedure to eliminate weakly
dominated strategies in each subgame backwardly. We also call the strategies that survive
through the procedure BEWDS strategies. Theoretically, the MC mechanism implements
cooperation in BEWDS for symmetric PD games. 5
Our paper expands the domain of this mechanism from symmetric PD games toasymmetric games that are not necessarily PD games. We find that the MC mechanism
implements cooperation in BEWDS for the class of quasi-dilemma (QD) games, which contains
coordination games, including the stag hunt game and PD games. Furthermore, under several
mild conditions, we show that the class of games implementing cooperation in BEWDS isexactly the same as the class of QD games. On the other hand, the MC mechanism cannot
implement cooperation in SPE.
In order to test the performance of the MC mechanism in an asymmetric environment,

3 According to Guala (2013), strong reciprocity, in which a player punishes other players using the player’s own
resource, is rare in human history.
4 This is called complete stranger matching, and only a few experiments employ this matching. Saijo et al.
(2014) chose this matching since it is the least favorable design for cooperation with respect to matching.
5 The MC mechanism uses unanimity. Banks, Plott, and Porter (1988) introduced a voting stage after a public
good provision stage and observed that unanimity reduced efficiency. Researchers stopped pursuing thisavenue after Banks et al. (1988) presented their findings. Furthermore, Masuda et al. (2014) found that the MCmechanism cannot implement a Pareto-efficient allocation in BEWDS for an economy with a public good.

2

we choose an asymmetric PD game in which cooperation cannot be attained by thecompensation mechanism in SPE that is proposed by Charness, Fréchette, and Qin (2007). The
compensation mechanism asks players to transfer money to the other player in the first stage,
and then, they both play a PD game. The monetary transfer must be done when the otherplayer chooses cooperation in the PD stage. The compensation mechanism does not cover all
PD games although the mate choice mechanism covers all PD games, and moreover, it covers
non-PD games. That is, there is a class of PD games in which the compensation mechanism
cannot implement cooperation in SPE.
Experimentally, we observed that the cooperation rate with the mechanism is 76.7%
in round 1, 86.7% in round 2, 93.3% in round 3, and 96.7% through 19 rounds. The (C,C) shareis 56.7% in round 1, 73.3% in round 2, 86.7% in round 3, and 93.5% through 19 rounds. That is,
the MC mechanism works reasonably well although it took a few rounds to achieve high (C,C)
shares in an asymmetric PD game with the MC mechanism.
Section 2 describes the MC mechanism applied to QD games. Section 3 proves that
BEWDS implementable games are QD games and shows that the MC mechanism cannot
implement cooperation in SPE. Section 4 presents the experimental design, and section 5, theresults. Section 6 provides suggestions for further research.
2. The mate choice mechanism and quasi-dilemma games
Consider a 2 x 2 game that has two strategies: cooperation (C) and defection (D).

C
DC(a,v)=V(c,x)=XD(b,w)=W(d,z)=Z

Figure 1. A QD game.
Define function p as follows: p(C,C) = (a,v) = V, p(D,C) = (c,x) = X, p(C,D) = (b,w) = W,
and p(D,D) = (d,z) = Z. If p satisfies V > Z (a > d and v > z), X ≥/ Z (d > c or z > x), and W ≥/ Z (d >b or z > w), then p is a QD game, and if p satisfies V > Z, ( c , d ) > ( a , b) and ( w , z ) > ( v , x) , then p
is a PD game. 6 Coordination games, including the stag hunt game, are QD games.

“>” shows that each element of the left-hand-side vector is “strictly greater than” each element of theright-hand-side vector, and “≥” shows that each element of the left-hand-side vector is “greater than or equalto” each element of the right-hand-side vector.
63Property 1. A prisoner’s dilemma game is a quasi-dilemma game.
Proof. Let p be a PD game. Suppose that (a,v) > (d,z), ( c , d ) > ( a , b) , and ( w , z ) > ( v , x) . Given V =(a,v) and Z = (d,z), let X = (c,x) and W = (b,w) satisfy the conditions. Then, c > a, z > x, d > b, andw > v. This implies that z > x and d > b, showing that (d > c or z > x) and (d > b or z > w). That is,
X ≥/ Z and W ≥/ Z. █Player 2's
parametersvz

PD Region
R1 for W



QD Region
for X & W

V

R2

e

Z
R3

0

PD Region
R5 for X ' Player 1's

R4
d

a

parameters

Figure 2. Possible locations of X and W in QD and PD games.
Figure 2 shows the possible locations of X and W in QD and PD games. The region for
QD games is the dark L-shaped area (i.e., R1, R2, R3, R4, and R5), and the regions for PD
games are R1 for W and R5 for X. Line  −  ' shows that the sum of benefits of both players isequal to a + v. If V must be Pareto efficient, X and W must be located under  −  ' . We do not
impose this condition hereafter.
Let us define the MC mechanism. Players first choose either C or D in Figure 1. After
this stage, each subject can approve (y) or disapprove (n), the other player’s choice in thesecond stage. If both approve the other player’s choice, the outcome is what they choose, and ifeither one disapproves the other player’s choice, the outcome is that both defect. Let Mi , mi ,and ui be player i’s choice between C and D, i’s choice between y and n, and i’s payoff,
respectively. Then, the MC mechanism is defined by the following rule: ifm=m=y , then (u1 , u2 ) = p( M1 , M 2 ) ; otherwise, (u1 , u2 ) = p(D, D). In general, we call the12
two-stage game without rule specification an approval mechanism.

4

Saijo et al. (2014) prepared five possible behavioral principles in their experiments for
the MC mechanism: the NE, the SPE, evolutionarily stable strategies (ESS), neutrally stablestrategies (NSS), and BEWDS. Of the five behavioral principles, the data of their experiments
are most compatible with BEWDS.
Since we focus upon SPE and BEWDS, let us define them. As we show later, since
there is no need to consider mixed strategies in our framework, a profile of strategies indicatesan assignment of a pure strategy for each information set. A profile of strategies is a SPE if therestriction of the profile at each subgame is a Nash equilibrium. Let us fix a subgame, and wesay that a strategy at an information set in the game survives the elimination of weakly dominatedstrategies if the strategy is not weakly dominated by any other strategies in the set. 7 A profileof strategies is a BEWDS if the restriction of the profile at each subgame survives theelimination of weakly dominated strategies. A mechanism implements cooperation in SPE (or
BEWDS) if all players choose cooperation in the first stage under SPE (or BEWDS). Next, weshow that the MC mechanism with a QD game implements cooperation under BEWDS.
Property 2. The mate choice mechanism with a QD game implements the cooperative outcome under
BEWDS.
Proof and Interpretation. The MC mechanism with a QD game has four subgames in Figure 3.
Because of the definition of the MC mechanism, we have (u1 , u2 ) = ( d , z) at (y,n), (n,y), and(n,n) for each subgame. This was termed the MC flat by Saijo et al. (2014), because the threecells other than (y,y) have the same payoff vector.
Consider subgame CC where both choose C. Player 1 must compare (a,d) and (d,d).
Because a > d, player 1 chooses y based upon the elimination of weakly dominated strategies.
However, player 1 does not necessarily compare two vectors because of the MC flat. Player 1
should compare a and d to understand this domination.
Although player 1 can choose either y or n based only on the elimination of weakly
dominated strategies, player 1 must also consider player 2’s choice. Player 1 compares v and z,
and hence, player 1 understands that player 2 chooses y. Thus, player 1 understands that the
choice at (C,C) is (y,y), which is shown by the bold square in Figure 3 for subgame CC.
Therefore, player 1 can fill the (C,C) part of the reduced normal form game with (a,v), which is
7 Let X and Y be strategies of a player, and let (a,b) and (c,d) be payoff vectors when the player chooses X andY, respectively. X weakly dominates Y if and only if a ≥ c , b ≥ d , and there is at least one strict inequality. Noticethat no strategies could survive if we use strong domination with strict inequality of each element.

5located above the four subgames in Figure 3.
Consider subgame DC. Because X ≥/ Z, d > c or z > x. Suppose that z > x. Then, player 2chooses n and understands that the outcome is (d,z) regardless of the choice of player 1 in thissubgame. However, as a thought experiment, player 1 must consider player 2’s choice if itwere c > d. That is, although player 1 would choose y, player 1 could not identify which of (c,x)
and (d,z) would be realized without knowing player 2’s choice. Suppose d > c. Player 1 chooses
n and understands that the outcome is (d,z) without considering player 2’s choice.
Repeating the same procedure at each of the subgames CD and DD, player 1 canconstruct the reduced normal form game above the four subgames in Figure 3. Because thegame also has the MC flat and a > d, player 1 chooses C.
If player 1 understands that player 2’s position is the same as that of player 1 using
the same procedure, player 1 is convinced that player 2 also chooses (C,y). If so, theequilibrium path under BEWDS is ((C,C),(y,y)). Thus, we simply write (C,C,y,y) hereafter. █

2 C

1

1
y
n

2 y

a

v

d

z
z
d
d
Subgame
(C,C)

a

D

d

y

n

z

C

y
n

c

x

v

z

D

d
d

z

z

n

d

y

z

y

z
z
d
d
Subgame
(D,C)

n

b

w

y

n

d

z

z
z
d
d
Subgame
(C,D)

y

d

n

d

z

z

n

d
d

z

z

Subgame(D,D)

Figure 3. Construction of the reduced normal form game.
Although this proof is mathematically almost the same as that of Saijo et al. (2014),
there are several differences between the two because of possible asymmetry. First, because
they used the symmetric payoff table, d = z, all components of the three cells have the same
number. In this sense, these three cells have the completely flat property. Therefore, the change
from d = z to d ≠ z might increase the player’s burden to understand the game. Second, (a,d) isdifferent from (v,z), although (a,d) = (v,z) in Saijo et al. (2014). This change also increases theplayers’ burden. The same argument can be applied to all subgames. Third, players who
understand the strategic implication of vectors V, W, X, and Z in the symmetric case might not

6be able to understand it in the asymmetric case under BEWDS. Finally, even though players
may understand these points, the difference between a and v may evoke an “equity” module intheir brains, which could trigger “non-rational” motivation.
An important property of the MC mechanism is that it is onto: the set of possible
outcomes of a QD game is equal to those of the MC mechanism. The former is {V,W,X,Z}, andthe latter must be the same. For example, although player 1 does not want to choose y at (C,D),
W is the outcome if both choose y. This condition excludes any payoff flow from or to the game.
In other words, no outside penalty or reward is given in order to maintain a balanced budget. 8
3. Backward elimination of weakly dominated strategies implementable games are
quasi-dilemma games
We consider the tightness of the parameter space of QD games and the implementableparameter space of BEWDS. If the latter is larger, the idea of BEWDS implementation can beapplied to many other 2 x 2 games. However, we show that these two spaces are identicalunder several assumptions. 9First, the approval mechanism satisfies forthrightness: If both choose y in the second
stage after the choice of a strategy pair in the first stage, the outcome must be that strategy
pair. 10 That is, the outcome must be what they choose whenever both choose y. Second, theapproval mechanism has a flat: the outcome of the second stage, except for (y,y), and that of the
reduced game, except for (C,C), are the same. Thus, we have the following property.
Property 3. Suppose that V > Z and that an approval mechanism satisfies forthrightness with the flat,
Z. Then, the class of games implementing cooperation in BEWDS is exactly the same as that of QD
games.
Proof. Consider any approval mechanism implementing outcome V in BEWDS. Because
forthrightness is satisfied, if both choose y, the outcomes of subgames CC, CD, DC, and DD
must be V, W, X, and Z, respectively. Because Z is the flat and the mechanism implements
8 The money-back-guarantee mechanism introduced by Dawes, Orbell, Simmons, and van de Kragt (1986) andIsaac, Schmidtz, and Walker (1989) is not onto. Let C be a fixed amount of contribution for public good
provision, and let D be no contribution. The money-back-guarantee mechanism returns the contribution to aplayer if both do not contribute, and hence, X and W cannot be the outcomes of the mechanism. If player 1 is autilitarian, and hence prefers X to Z, the MC mechanism can choose X unlike the money-back-guarantee
mechanism.
9 These assumptions are essentially the same as those introduced by Saijo et al. (2014).
10 Saijo, Tatamitani, and Yamato (1996) introduced forthrightness in the natural mechanism design.

7

cooperation, the outcome should be Z in subgames CD and DC. That is, (d > b or z > w) and (d> c or z > x) for subgames CD and DC, respectively. Because ( d > b or z > w) =( d ≤ b and z ≤ w) =( W ≥ Z ), (d > b or z > w) is equivalent to W ≥/ Z. Similarly, we obtain X ≥/ Z.

That is, the class of games implementing cooperation in BEWDS is exactly the same as the classof QD games. █Let us now consider the SPEs of QD games. Since the Nash equilibrium outcomes are(a,v) and (d,z) at subgame CC and (d,z) at subgames DC, CD, and DD, two cases exist in the
reduced normal form games shown in Figure 4. That is, because all combinations, (C,C), (D,C),(C,D), and (D,D), are Nash equilibria of the games, they are also the outcomes of SPEs.
Therefore, the MC mechanism cannot implement cooperation in SPE.

Player 2C
Player 1

Player 2

D

C

a,v

d,z

D

d,z

d,z

Player 1(i) (y,y) in subgame CC

C

D

C

d,z

d,z

D

d,z

d,z

(ii) (n,n) in subgame CC

Bold italic cells indicate Nash Equilibria in the reduced normal form games.
Figure 4. Two reduced normal form games.
As the above discussion shows, multiple Nash equilibria in a subgame generate
multiple SPEs due to the MC flat. Furthermore, players in SPE must consider how the otherplayer behaves in each subgame. In this sense, SPE might place a heavy information burden on
players. On the other hand, players in BEWDS do not need information about how the otherplayer behaves since they care about the comparison of their own strategies in each subgame
as far as the BEWDS strategy is unique. However, this fact does not necessarily support
BEWDS in general since the facts are specific to the MC mechanism. The next two sections
describe the experiment and its results.
4. Experimental design
Our experimental focus is payoff asymmetry of the PD game. We conducted
experiments of an asymmetric PD game with the MC mechanism (AsymPDMC). For
comparison with the data of the AsymPDMC, we borrowed the data regarding a symmetric PD
game with the MC mechanism (SymPDMC) and a symmetric PD game without the MC8mechanism (SymPD) from Saijo et al. (2014).
We chose an asymmetric payoff table in which cooperation cannot be implemented
by the compensation mechanism in SPE, but it can be implemented by the MC mechanism in
BEWDS. The compensation mechanism also has two stages. It asks players to transfer money
to the other player in the first stage, and then, both play a PD game. The monetary transfer
must be done when the other player chooses cooperation in the PD stage. Varian (1994)
designed the compensation mechanism in a general setting, and then, Andreoni and Varian(1999) and Charness et al. (2007) conducted the experiments using the PD games. As Charness
et al. (2007) showed, the compensation mechanism does not cover the entire class of PD games.
In this sense, we chose one of the least favorable matrices to achieve cooperation in our
experimental design. Figure 5 shows the symmetric and asymmetric payoff matrices. 11 The
symmetric payoff table comes from Saijo et al. (2014) and the asymmetric payoff matrix comesfrom Charness et al. (2007). The average cooperation rates in Charness et al. (2007) are 43-68%
with the compensation mechanism.

Player 2C
Player 1

Player 2
D

C

14,14

7,17

D

17,7

10,10

C
Player 1(i) Symmetric payoff matrix

D

C

44,36

8,44

D

52,0

32,28

(ii) Asymmetric payoff matrix

Figure 5. Symmetric and asymmetric payoff matrices.
All the above-mentioned experiments were carried out in Osaka University during
the period from November 2009 to December 2011. The AsymPDMC and SymPDMC
experiments each had three sessions, and the SymPD experiment had one session. Twenty
subjects participated in each session and no subject attended more than one session. We
recruited these 140 undergraduate subjects with different majors through campus-wide
advertisements. They were told that there would be an opportunity to earn money in a
research experiment.
At the beginning of the experiment, each subject had a set of printed instructions anda record sheet. Instructions were read aloud by an experimenter. After that, subjects were

In the experiment, we used payoff numbers that are 100 times the numbers in Figure 5 due to the exchange
rate.
11

9

given five minutes to ask private questions. Communication among subjects was prohibited,
and we declared that the experiment would be stopped if it was observed. This never
happened. There was no practice period. We used the z-Tree software (Fischbacher, 2007) forthe experiment.
The experimental procedure was as follows. We formulated 10 pairs out of the 20
subjects seated at computer terminals in each session. These pairings were anonymous anddetermined in advance in order not to pair the same two subjects more than once. Since mostprevious studies, such as Andreoni and Varian (1999) and Charness et al. (2007), have
employed random matching among four to eight subjects (two to four groups), 12 such
repetition necessarily entails pairings of the same two subjects. Therefore, compared withprevious experiments, this “complete” strangers design might reduce the possibility ofcooperation among subjects. 13 Each subject received an instruction sheet and a record sheet.
Let us explain the PDMC experiment. When the period started, each subject selected
either A (defection) or B (cooperation) in the choice (or PD) stage and then inputted the choice
into a computer and also noted it on the record sheet. After that, each subject explained thereason behind this choice in a small box on the record sheet. The next step was the decision (orapproval) stage. Based on the knowledge of the other subject’s choice, each subject chose toeither “accept” or “reject” it and then inputted the decision into a computer, noted it on therecord sheet, and explained his or her reasoning as before. Once subjects had finished the task,each could see “your decision,” “the other’s decision,” “your choice,” “the other’s choice,”“your points,” and “the other’s points” on the computer screen. However, neither the choices
nor the decisions in pairs other than “your” own were shown on the computer screen. This
ended one period. The experiment without the approval stage became the PD experiment.
After finishing all 19 periods, every subject filled in questionnaire sheets.
Each session lasted approximately 90 minutes including the time spent on answering
the post-experiment questionnaires and payment. Subjects earned, on average, 5233 JPY (about
43.61 USD, using 1 USD=120JPY), 4873 JPY (about 40.61 USD), and 3920 JPY (about 32.67USD)
in AsymPDMC, SymPDMC, and SymPD sessions, respectively.
5. Experimental results
Figure 6 shows the time path of cooperation rates over the 19 periods. The
12 Charness et al. (2007) divided 16 subjects in one session into four separate groups, with four subjects in eachgroup interacting only with each other over the course of the session.
13 An exception is Cooper, DeJong, Forsythe, and Ross (1996), who employed complete stranger matching.

10

cooperation rate in each period is defined as the ratio of number of subjects choosing C to thetotal number of subjects. As shown, the cooperation rate of PD started at 15% in the first three
periods, and then ranged between 5% and 10% in the next 16 periods. In contrast, thecooperation rate of SymPDMC was always above 90% in each period, while that of AsymPDMC
started at about 76.7% in the first period, rose to 86.7% in the second period, and then stayed
above 90% in the remaining 17 periods.
The large gap in cooperation rates between PD and PDMC (either symmetric or
asymmetric) was statistically supported by the proportion test. All the p values for comparing
the cooperation rate of AsymPDMC or SymPDMC with that of PD are smaller than 0.001 ineach period, which suggests that introducing the second stage after the PD game dramaticallyincreases the cooperation rate in both symmetric and asymmetric PD games.
For the comparison of AsymPDMC with SymPDMC, we ran the proportion test byusing both the data of each period and the data pooled over all periods. The two-tailed pvalues are reported in Table 1 (see the fourth column from the left). Generally, there is no
significant difference in the cooperation rates between an asymmetric PD and a symmetric PD
games when the data were pooled over all periods (p = 0.7212). In the first two periods, thecooperation rate is significantly lower under the asymmetric environment. Thereafter, thecooperation rate in the asymmetric mate choice mechanism no longer remains statisticallysignificantly lower than its symmetric counterpart. In fact, a significantly higher cooperation
rate in AsymPDMC is even observed in 3 periods (i.e., periods 7, 14, and 15) out of theremaining 17 periods.
With regard to the share of the (C,C) combination, Figure 7 shows its time path overthe 19 periods. Applying the proportion test, we found that the (C,C) share is significantly
higher in either AsymPDMC or SymPDMC than in PD in each period (all p values < 0.001).
Additionally, as indicated in the last column of Table 1, there is no significant difference in theshare of the (C,C) combination between AsymPDMC and SymPDMC (p = 0.6031) when the data
were pooled over all periods. However, if we look at the p values by period, we find that the(C,C) share is significantly lower in the asymmetric environment than in the symmetricenvironment in the first four periods. Thereafter, similar to the case of the cooperation rate, theshare of AsymPDMC is once again always statistically higher than or equal to its symmetriccounterpart.

11

Cooperation rate

100.0%
90.0%
80.0%
70.0%
60.0%
50.0%
40.0%
30.0%
20.0%
10.0%
0.0%
1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19

PeriodAsymPDMC (96.7%)

PD (7.9%)

SymPDMC (96.9%)

Share of (C,C)

Figure 6. Cooperation rates by periods.

100.0%
90.0%
80.0%
70.0%
60.0%
50.0%
40.0%
30.0%
20.0%
10.0%
0.0%
1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19

PeriodAsymPDMC (93.5%)

PD (0.0%)

Figure 7. Share of (C,C) by periods.

12

SymPDMC (94.0%)

Table 1. Proportion test results for AsymPDMC vs. SymPDMC.
PeriodCooperation rate

Share of (C,C)

AsymPDMC

SymPDMCp value

AsymPDMC

SymPDMCp value

1

76.7%

95.0%

0.0040

56.7%

90.0%

0.0000

2

86.7%

96.7%

0.0475

73.3%

93.3%

0.0033

3

93.3%

98.3%

0.1705

86.7%

96.7%

0.0475

4

93.3%

98.3%

0.1705

86.7%

96.7%

0.0475

5

96.7%

96.7%

1.0000

93.3%

93.3%

1.0000

6

98.3%

95.0%

0.3091

96.7%

90.0%

0.14327100.0%

95.0%

0.0794100.0%

90.0%

0.0120

8

98.3%

98.3%

1.0000

96.7%

96.7%

1.0000

9

98.3%

98.3%

1.0000

96.7%

96.7%

1.0000

10

98.3%

98.3%

1.0000

96.7%

96.7%

1.000011

100.0%

98.3%

0.3153100.0%

96.7%

0.153812100.0%

98.3%

0.3153100.0%

96.7%

0.153813100.0%

98.3%

0.3153100.0%

96.7%

0.153814100.0%

93.3%

0.0419100.0%

86.7%

0.003415

100.0%

95.0%

0.0794100.0%

90.0%

0.0120

16

98.3%

95.0%

0.3091

96.7%

93.3%

0.402217100.0%

96.7%

0.1538100.0%

93.3%

0.0419

18

98.3%

98.3%

1.0000

96.7%

96.7%

1.000019

100.0%

98.3%

0.3153100.0%

96.7%

0.1538All periods

96.7%

96.9%

0.7212

93.5%

94.0%

0.6031

Notes: The reported p values are based on the two-tailed proportion test.
6. Concluding remarks
We showed that the MC mechanism implements cooperation in BEWDS for QD
games and that BEWDS-implementable games are QD games. QD games include not only PD
games but also coordination games. Furthermore, the mechanism cannot implement
cooperation in SPE.
The MC mechanism is essentially a unanimous voting rule between two players, andit can be interpreted as a “minimum” communication device to achieve cooperation. In the firststage, each player reveals the choice of C or D. Then knowing the other’s choice, each chooses y13

or n. If both choose y, the outcome is what they choose in the first stage; otherwise, theoutcome is (D,D). This mechanism implicitly presupposes that the status quo is (D,D) anddisagreement results in the status quo. This procedure can be a natural way to avoid conflict,such as PD, or a coordination situation in daily life. Using functional near-infrared
spectroscopy (fNIRS), Nagatsuka, Shinagawa, Okano, Kitamura, and Saijo (2013) found thatcompared with the PD game, subjects made their choices with less stress using only the MCmechanism. We also found that except for first few periods, the MC mechanism works wellunder an asymmetric environment.
Masuda, Okano, and Saijo (2014) expanded the idea of the MC mechanism to public
good provision and showed that the minimum approval mechanism implements an efficient
allocation in BEWDS theoretically and experimentally. Furthermore, Huang, Masuda, Okano,and Saijo (2014) designed a simplified approval mechanism in the spirit of the MC mechanism
in a social dilemma and showed theoretically and experimentally that it implements
cooperation in BEWDS when there are at least two players. However, designing a reasonableapproval mechanism to implement cooperation in multiple choices and players is an open
question.
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