Social Design Engineering Series

SDES-2017-22

Mate Choice Mechanism for Solving a Quasi-Dilemma

Tatsuyoshi Saijo
School of Economics and Management, Kochi University of Technology
Research Center for Future Design, Kochi University of Technology

Junyi Shen
Research Institute for Economics and Business Administration, Kobe University

30th October, 2017

School of Economics and Management
Research Center for Future Design
Kochi University of Technology
KUT-SDE working papers are preliminary research documents published by the School of Economics and Management jointly with the ResearchCenter for Social Design Engineering at Kochi University of Technology. To facilitate prompt distribution, they have not been formally reviewed
and edited. They are circulated in order to stimulate discussion and critical comment and may be revised. The views and interpretations expressedin these papers are those of the author(s). It is expected that most working papers will be published in some other form.

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

Abstract
Saijo, Okano, and Yamakawa (2015) 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.
This study theoretically shows, first, that this mechanism works well in the class ofquasi-dilemma (QD) games, such as asymmetric PD games and coordination games. Second,the class of BEWDS-implementable 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 the mate choice mechanism works well experimentally for an asymmetric PD
game.
Keywords: Asymmetric prisoner’s dilemma; Quasi-dilemma; Mate choice mechanism;
Cooperation; Experiment
JEL codes: C72, C92, D74, P43* Corresponding author: Junyi Shen. 2-1 Rokkodai, Kobe, Hyogo 657-8501, Research Institutefor Economics and Business Administration, Kobe University, Japan. Tel: +81 78 803 7013.
Email: shen@rieb.kobe-u.ac.jp

11. Introduction
Over the past six decades, the prisoner’s dilemma (PD) game has been extensively
discussed in both the public and academic press (see Poundstone (2011) for a historicalperspective). The PD game presents a scenario in which the outcome of one person’s decision
is determined by the simultaneous decisions of the other participants, resulting in a bad
outcome for all of them (a Pareto-inefficient Nash equilibrium) if all act in their own
self-interest. The key characteristic of this game is that while there are substantial gains that
could be attained through cooperation, non-cooperation is dominant for each player (see Kuhn
(2014) for an overview of PD literature).
Since participants in laboratory experiments consider non-monetary factors such associal norms and anonymity as well as monetary stakes, the cooperation rates, that is, the ratioof participants who chose cooperation, in PD game experiments are well above zero (seeChaudhuri (2011) for a survey), but not close to one. In order to increase the cooperation rates
in PD games, researchers, such as Yamagishi (1986), Banks, Plott, and Porter (1988), Varian(1994), Fehr and Gächter (1999), Andreoni and Varian (1999), and Charness, Fréchette, and Qin(2007), started adding a stage before or after the PD games.1 Adding a stage after the PD gameallows participants to punish non-cooperation. Although participants in such games do nothave a monetary incentive to punish other players, cooperation rates in experiments withpunishment increase, but still do not become close to one (see Yamagichi (1986) and Fehr and
Gächter (1999)). Varian (1994) introduced a compensation mechanism before the dilemma game.
Under this mechanism each player is asked in the first stage to choose how much to pay his or
her counterpart for cooperating. After learning the payments offered in the first stage, theplayers then play a normal PD game. Andreoni and Varian (1999) conducted experiments withthe compensation mechanism and observed that cooperation rates increased from 25.8% whentransfer payments were not feasible to 50.5% when transfer payments were permitted, and thecooperation rate was around 20% in the first two rounds. Charness, Fréchette, and Qin (2007)
conducted experiments and observed that cooperation rates increased from 11–18% whentransfer payments were not feasible to 43–68% when transfer payments were permitted.
Our aim in this study is to find one of the simplest possible mechanisms to solve social
dilemmas, including the prisoner’s dilemma, both experimentally and theoretically.
Experimentally, we focus on designing a mechanism that can attain a Pareto efficient outcome

1

The choice of a participant in the first stage is scrutinized by the other participant before deciding the choicein the second stage. As Levitt and List (2007) suggested, this might increase cooperation.

2

in a few rounds,2 because we cannot repeat the same mechanism many times in real-life
settings. Theoretically, we do not stick to Nash or Nash-type equilibrium concepts, but search
for a behavioral principle among subjects in experiments that implements cooperation. In
addition, we do not use punishment or reward to balance the budget, that is, there is nomonetary inflow or outflow in the mechanism design.3
The study by Saijo, Okano, and Yamakawa (2015) is one of the first attempts to design
such a mechanism for the PD.4 They proposed the mate choice (MC) mechanism, which occurs
after a symmetric PD game. After observing the choice of cooperation (C) or defection (D) in the
PD game, each player is asked to approve or disapprove of the other’s choice. If both approveit, the outcome is what they chose in the PD game; if at least one player disapproves of theother’s choice, the outcome is the same as when both defect in the PD game.5 Experimentally,
Saijo et al. (2015) observed that the cooperation rate with the mechanism was 95.0% in round 1
and 96.9% over 19 rounds, when each subject was never matched with the same subject again
in all rounds.6 The (C, C) share, that is, the ratio of pairs in which both chose cooperation, was
90.0% in round 1 and 94.0% over 19 rounds. They also found that subjects’ behavior wasconsistent with backward elimination of weakly dominated strategies (BEWDS) rather thanNash equilibrium (NE) or subgame perfect equilibrium (SPE) behavior. BEWDS is a procedurethat eliminates weakly dominated strategies in each subgame, backwardly. The strategies thatsurvive through the procedure are called BEWDS strategies. Theoretically, Saijo et al. (2015)
proved that the MC mechanism implements cooperation in BEWDS for symmetric PD games.7
In two related studies, Masuda, Okano, and Saijo (2014) constructed a minimum
approval mechanism, which is a version of the MC mechanism, in a public good economy
when the number of players is two and their preferences are linear. They showed
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.
3 According to Guala (2013), strong reciprocity, in which a player punishes other players using the player’s own
resources, is rare in human history.
4 Saijo, Masuda, Okano and Yamakawa (2017) is a simplified version of Saijo et al. (2015).
5 Because the MC mechanism does not have devices such as punishment or reward, it is budget balanced.
6 This is called complete stranger matching, and only a few experiments employ this matching. Saijo et al.
(2015) chose this matching since it is the least favorable design for cooperation with respect to matching.
7 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, Okano, and Saijo (2014) foundthat the MC mechanism cannot implement a Pareto-efficient allocation in BEWDS for an economy with apublic good.
2

3

experimentally that the mean contributions ranged from 76.9% to almost 100.0%, with anaverage of 94.9%. Huang, Masuda, and Saijo (2014) constructed a simplified approvalmechanism, which is also based on the MC mechanism, for a symmetric PD game in which thenumber of players was expanded from two to three. They observed experimentally that themean cooperation rate increased from 44.4% in round 1 to above 90.0% in round 5 andmaintained that level in the remaining 10 rounds.
From the above-mentioned studies, it seems that the MC mechanism performs very
well in stimulating the players to cooperate in a symmetric environment. Consequently, thequestion of whether the MC mechanism is also effective in an asymmetric environment isnatural. As Andreoni and Varian (1999) found, a participant with a relatively low payoff tends
not to cooperate and this might influence the cooperation rate. Hence, in this paper we expand
the domain of this mechanism from symmetric PD games to asymmetric games that are notnecessarily PD games. We find theoretically that the MC mechanism implements cooperationin 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, weshow that the class of games implementing cooperation in BEWDS is exactly the same as theclass of QD games, and that the MC mechanism cannot implement cooperation in SPE.
In order to test the performance of the MC mechanism experimentally in an
asymmetric environment, we choose an asymmetric parameterization of the PD game (“Game3” in Charness et al. (2007)) because the cooperation rate of this parameterization (42.9%) was
worse than those for the other two asymmetric parameterizations (53.9% in “Game 1” and68.1% in “Game 2”). This fact also motivates us to investigate whether the MC mechanism
performs better than the compensation mechanism does. It should be noted that thecompensation mechanism does not cover all PD games; in contrast, the MC mechanism covers
all PD games and non-PD games. That is, there is a class of PD games in which thecompensation mechanism cannot implement cooperation in SPE. Game 3 belongs to this class.
Experimentally, we observed that the cooperation rate with the MC mechanism in an
asymmetric environment started at about 76.7% in round 1, rose to 86.7% in round 2, 93.3% inrounds 3 and 4, 96.7% in round 5, and then stayed above 98.0% in the remaining 14 rounds.
The overall average cooperation rate over 19 rounds was 96.7%. The (C,C) share started at
56.7% in round 1, rose to 73.3% in round 2, to 86.7% in rounds 3 and 4, to 93.3% in round 5, andthen stayed above 96.0% in the remaining 14 rounds. The overall average (C,C) share over 19
rounds was 93.5%. That is, the MC mechanism works reasonably well, although it took a few

4rounds to achieve high (C,C) share in an asymmetric PD game.
A good example of the mate choice mechanism in an asymmetric environment is
so-called MAD (Mutually Assured Destruction) that led the earth to the avoidance of nuclear
disaster around the last half of the twentieth century. Even though superpower A attacks theother superpower S using nuclear weapons, superpower S can monitor the attack and then has
enough time to mount the counterattack. In other words, this is a two-stage game where thefirst stage is a PD game, and the second stage is a special case of the approval stage. The
approval in the second stage is “No (Further) Attack” and the non-approval is “(Counter)
Attack.” If a superpower decides to choose “Attack” in the PD game, she must choose “No(Further) Attack” automatically in the second stage since she has already chosen “Attack” inthe first stage. Then each chooses “No Attack” or “No Action” in the first stage, and thenchooses “No (Further) Attack” in the second stage is the unique BEWDS path.8 Notice that thesecond stage mechanism is not by man made one such as convention, but by an evolving
mechanism due to technological constraints including the monitoring accuracy and the timelag between the discharge and explosion that are called second-strike capability by Russett,Starr and Kinsella (page 237, 2009). The technological progresses were due to the battle ofholding hegemony over the other superpower.
There are many other examples of the mate choice mechanism. Consider a merger ora joint project of two companies. They must propose plans (the contents of cooperation) in thefirst stage, and then each faces the approval decision in the second stage. In order to resolve
the conflicts such as prisoner's dilemma, interested parties usually form a committee consistingof representatives of the parties. Consider two companies facing confrontation on thestandardizations of some product. Each company chooses cooperation (or compromise) or
defection (or advocating of the own standard), and then the committee consisting of two
company members and/or bureaucrats gives the approval. Another example is the two party
system. Each party chooses either cooperation (or compromise) or defection (or insistence ofpolicy for the own party), and then diet (or national assembly) plays a role of approval. The
bicameral system also has two stages. One chamber decides a policy (or compromise) and theother chamber plays a role of approval. The negotiation process at United Nations also has thisstructure. Negotiators among relevant countries get together to find compromise, i.e., theRobert J. Aumann (2006) in his Nobel Lecture described MAD as an outcome of infinitely repeated games inorder to maintain cooperation. The idea of approval mechanism is not to use infinite periods but to consider
the game in two stages. Notice also that (Attack, Attack) is a part of SPE, but not a part of the BEWDS path. Thatis, it was fortunate that the decision makers of the superpowers did not follow this path.
85

content of cooperation in the first week and then high ranked officials such as presidents andprime ministers get together to approve or disapprove it in the second week. Adding thesecond stage in resolving conflicts has been used widely in our societies.
The paper is organized as follows. 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 presents the results. Section 6 provides suggestions for further research.
2. The mate choice mechanism and quasi-dilemma games
Consider a 2 × 2 game that has two strategies: cooperation (C) and defection (D).

Player 1

C
D

Player 2
CD(a,v)=V (b,w)=W(c,x)=X (d,z)=Z

Figure 1. A QD game.
Define a payoff function p for the game 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; if p satisfies V > Z, (c , d)  ( a, b) and( w, z)  ( v, x) , then p is a PD game.9 Coordination games, including the stag hunt game, are QD
games.
Property 1. A prisoner’s dilemma game is a quasi-dilemma game.
Proof. Let p be the payoff function of a PD game. Then (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.
Thus, c > a, z > x, d > b, and w > v. Since z > x and d > b, it follows that (d > c or z > x) and (d > bor z > w). That is, X  Z and W  Z. █“>” indicates that each element of the left-hand-side vector is “strictly greater than” each element of theright-hand-side vector, and “≥” indicates that each element of the left-hand-side vector is “greater than orequal to” each element of the right-hand-side vector.
96

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
a

d

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. The line  ' shows points where the sum of benefits toboth players is equal to a + v. If V must be Pareto efficient, then X and W must lie 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) of the other player’s choice in thesecond stage. If both approve the other player’s choice, the outcome is what they chose; ifeither one disapproves of 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: if

m1  m2  y , then (u1 , u2 )  p( M1 , M2 ) ; otherwise, (u1 , u2 )  p(D, D). In general, the two-stage
game without any rule specification is called an approval mechanism.
Saijo et al. (2015) considered five possible behavioral principles for the MCmechanism in their experiments: the NE, the SPE, evolutionarily stable strategies (ESS),
neutrally stable strategies (NSS), and BEWDS. Of the five behavioral principles, the data fromtheir 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 indicates7

an assignment of a pure strategy for each information set. A profile of strategies is an 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 available at that
information set.10 A profile of strategies is a BEWDS if the restriction of the profile at eachsubgame survives the elimination 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, we show 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 the four subgames shown in
Figure 3. Because of the definition of the MC mechanism, we have (u1 , u2 )  (d , z) for (y,n),(n,y), and (n,n) in each subgame. This was termed the MC flat by Saijo et al. (2015), because thethree cells other than (y,y) have the same payoff vector.
Consider subgame CC, where both players have chosen C. Player 1 must compare(a,d) and (d,d). Because a > d, player 1 chooses y because of 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, basing this on the elimination of weakly
dominated strategies alone, player 1 must additionally 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 forsubgame CC. Therefore, player 1 can fill the (C,C) part of the reduced normal form game with(a,v), which is located above the four subgames in Figure 3.
Consider subgame DC. Because X  Z, it must be that d > c or z > x. Suppose that z > x.
Then, player 2 chooses n and understands that the outcome is (d,z) regardless of the choice of
player 1 in this subgame. However, as a thought experiment, player 1 must consider player 2’s
choice if it were the case that c > d. That is, although player 1 would choose y, player 1 could
For the game shown in Figure 1, C weakly dominates D for player 1 if and only if a  c , b  d , and there is atleast one strict inequality. Notice that no strategies could survive if we use strong domination with strict
inequality of each element.
108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

C

a

D

d

v

D

d

z

d

z
z

2 y

n

y

n

y

n

y

v

z

x

z

w

z

z

a

d

z
z
d
d
Subgame
(C,C)

y
n

c

d

y

z
z
d
d
Subgame
(D,C)

n

b

d

z
z
d
d
Subgame
(C,D)

y

d

n

d

n

d
z

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. (2015),
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), whereas (a,d) = (v,z) in Saijo et al. (2015). This change also increases theplayers’ burden. The same argument can be applied to all subgames. Third, players who
understand the strategic implications of vectors V, W, X, and Z in the symmetric case might
not be able to understand them 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 in their brains, which could trigger “non-rational” motivation.
An important property of the MC mechanism is that it is onto: the set of possible

9

payoffs of a QD game is equal to the set of those of the MC mechanism. The former is{V,W,X,Z}, and the latter must be the same. For example, although player 1 does not want tochoose y at (C,D), W is the payoff pair if both choose y. This condition excludes any payoff flowfrom or to the game. In other words, no outside penalty or reward is given in order to maintaina balanced budget.11
3. Games that are implementable by BEWDS 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 × 2 games. However, we show that these two spaces are identicalunder several assumptions.12
First, 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.13 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 payoffs of subgames CC, CD, DC, and DD mustbe V, W, X, and Z, respectively. Because Z is the flat and the mechanism implements
cooperation, the payoff should be Z in subgames CD and DC. That is, (d > b or z > w) and (d > cor 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 class
The money-back-guarantee mechanism introduced by Dawes, Orbell, Simmons, and van de Kragt (1986)
and Isaac, 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 payoffs produced by the mechanism. If player
1 is a utilitarian, and hence prefers X to Z, the MC mechanism can lead to X unlike the money-back-guarantee
mechanism.
12 These assumptions are essentially the same as those introduced by Saijo et al. (2015).
13 Saijo, Tatamitani, and Yamato (1996) introduced forthrightness in natural mechanism design.
1110

of QD games. █Let us now consider the SPEs of QD games. Since the Nash equilibrium payoffs 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. 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, theMC mechanism cannot implement cooperation in SPE. That is, multiple Nash equilibria in a
subgame generate multiple SPEs due to the MC flat. Furthermore, players in an SPE must
consider how the other player behaves in each subgame.

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.
Consider now informational burden to a player using BEWDS. Since subjects caninstantaneously understand that each subgame in the second stage has the MC flat, and canidentify (d,z) as the outcome of subgame DD in Figure 3, the cells or triangles that subjects
must see or consider are dark parts in subgames CC, CD and DC in the second stage under
BEWDS. Subject 1's own possible outcomes to be compared are the two lower left triangles ofthe left column. Hence, the number of triangles that each subject must see is four in each ofsubgames CC, CD and DC. That is, the remaining triangles are unnecessary for their decision
making including the lower right cell.
Let us consider the minimum informational or intellectual requirement to achieve
CCyy under BEWDS. Consider subject 1. The information of the two lower left triangles of theleft column in each of subgames CC, CD, and DD is enough to solve or choose either y or n ineach subgame, but this is not enough to solve the two stage game since subject 1 cannot
identify which cell would be realized without having the information of the two upper right
triangles of the upper row at subgames CC and DC. In other words, subject 1 must use theory ofmind to understand which strategy subject 2 chooses. If this is successful, subject 1 canconstruct the reduced normal form game out of two stages shown at the top in Figure 3.
11During this construction, subject 1 understands that (s)he really needs to know for the decision
making is the two lower left triangles of the left column in the reduced normal form game, i.e.,
subject 1's outcomes of subgames CC and DC. Using this information, subject 1 chooses C. In
this sense, subject 1 must have backwardability that identifies chosen cells in subgames CC, CD
and DD, and then finds his or her own two triangles corresponding to subgames CC and DC in
the reduced normal form game. Finally, subject 1 also uses a simple heuristic: "the other subject
thinks the same way as I think." For example, subject 1 who understands the outcome of subgame
DC can find the outcome of subgame CD using this heuristic. These two simplified methods
mitigate subjects' burden considerably.14,15 Of course, these facts do not necessarily support
BEWDS in general since the facts are specific to the MC mechanism.
In the next two sections, we describe our experimental investigation of the MCmechanism and its results.
4. Experimental design
Our experimental focus is payoff asymmetry in the PD game. We conducted
experiments using an asymmetric PD game with the MC mechanism (AsymPDMC). For
comparison with the data for AsymPDMC, we borrowed the data for a symmetric PD gamewith the MC mechanism (SymPDMC) and a symmetric PD game without the MC mechanism(SymPD) from Saijo et al. (2015).
We chose an asymmetric payoff table in which cooperation cannot be implemented
by the compensation mechanism in SPE, but 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

Player 1 must compare two numbers six times in order to decide (C,y) in Figure 3: two comparisons (i.e., myown a and d, and the other’s v and z) in subgame CC, one comparison (i.e., my own d and c, and the other’s
choice does not matter since my own outcome is d regardless the choice of the other) in subgame CD, twocomparisons in subgame DC, and one comparison between a and d in the reduced normal form game. This isquite a contrast when we find Nash equilibria of the two stage game. Since the number of information set is 5,5each player has 2 strategies. Player 1 must compare (2 5  1) numbers to find best responses for any given
strategies of player 2. This indicates that player 1 must compare two numbers (2 5  1) 2 5 times. In order to findthe Nash equilibria, player 1 must find the best responses of player 2, and hence must compare two
numbers (2 5  1) 2 5 times. That is, the number of comparisons is 2  (2  1)  2  1984 which is more than 300
times of 6, which might trigger qualitative difference between NE and BEWDS. See Saijo et al. (2015) for
detailed comparison of several equilibrium concepts such as Nash equilibria, subgame perfect equilibria,
neutrally stable strategies, and evolutionarily stable strategies for symmetric games.
15 A reviewer suggested to introduce trembling-hand SPE or conditional cooperator in order to eliminate“bad” outcomes, but these are our future agenda.
14

5

12

5

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 experiments using 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 the least favorable matrix (i.e., the Game 3 matrix in Charness et al. (2007))
to examine cooperation levels in our experimental design. Figure 5 shows the symmetric andasymmetric payoff matrices.16 The symmetric payoff table comes from Saijo et al. (2015) andthe asymmetric payoff matrix comes from Charness et al. (2007).

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 was given a set of printed
instructions and a record sheet. Instructions were read aloud by an experimenter. After that,
subjects were given five minutes to ask private questions. Communication among subjects wasprohibited, and we declared that the experiment would be stopped if it was observed. Thisnever happened. There was no practice period. We used the z-Tree software (Fischbacher,2007) for the experiment.
The experimental procedure was as follows. We assigned the 20 subjects seated atcomputer terminals in each session to 10 pairs. These pairings were anonymous andIn the experiment, we used payoff numbers that are 100 times the numbers in Figure 5 due to the exchange
rate between the Japanese yen and the US dollar.
16

13

determined in advance in order not to pair the same two subjects more than once. Since mostprevious studies, such as those of Andreoni and Varian (1999) and Charness et al. (2007), have
employed random matching among four to eight subjects (two to four groups),17 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.
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. 18 The next step was the decision(or approval) 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 PD experiment omitted the approval stage. After finishing all 19
periods, every subject filled in a questionnaire.
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 paths of cooperation rates over the 19 periods. The
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%, while that of AsymPDMC started atabout 76.7% in the first period, rose to 86.7% in the second period, and then stayed above 90%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.
18 We had wished that we could analyze these reasons. However, most of the subjects did not write down
reasons or just wrote “in order to earn more money”, which leaves analyses of this information uninformative.
Hence, we do not report these results in the paper.
17

14

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 using
both the data from each period and the data pooled over all periods. The two-tailed p values
are reported in Table 1 (see the fourth column from the left). Generally, there is no significant
difference in cooperation rates between asymmetric PD and symmetric PD games when thedata were pooled over all periods (p = 0.7212). In the first two periods, the cooperation rate issignificantly lower in the asymmetric environment. Thereafter, the cooperation rate in theasymmetric MC mechanism is not significantly lower than its symmetric counterpart. In fact, asignificantly higher cooperation rate in AsymPDMC is observed in 3 periods out of theremaining 17 periods (specifically, periods 7, 14, and 15). In addition, Andreoni and Varian(1999) found a significant difference in cooperation rates between subjects with a relatively low
payoff (41.8%) and their counterparts with a relatively high payoff (59.2%). However, as
shown in Table 2 we did not find such significant differences in cooperation rates between
these two types of subject. Moreover, compared to the cooperation rate with the compensation
mechanism in Charness et al. (2007) the one in our AsymPDMC is obviously higher, whichsuggests the superiority of an MC mechanism over a compensation mechanism.
With regard to the share of the (C,C) combination, Figure 7 shows its time paths 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, as with the cooperation rate, the (C,C) shareof AsymPDMC is once again always statistically higher than or equal to that of its symmetriccounterpart.

15

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 period.

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 period.

16

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.

17

Table 2. Cooperation rates of subjects with low payoff and with high payoff in AsymPDMC.
Subjects with relative low payoff

Subjects with relative high payoff

p value

1

80.0%

73.3%

0.5416

2

83.3%

90.0%

0.4475

3

93.3%

93.3%

1.0000

4

93.3%

93.3%

1.00005

100.0%

93.3%

0.15036

100.0%

96.7%

0.31327100.0%

100.0%

–

8

96.7%

100.0%

0.3132996.7%

100.0%

0.31321096.7%

100.0%

0.313211100.0%

100.0%–

12

100.0%

100.0%–

13

100.0%

100.0%–

14

100.0%

100.0%–

15

100.0%

100.0%

–

16

96.7%

100.0%

0.313217100.0%

100.0%–

18

100.0%

96.7%

0.313219100.0%

100.0%–All periods

96.7%

96.7%

1.0000PeriodNotes: The reported p values are based on the two-tailed proportion test. A “–“ means the
proportion test cannot be performed because the cooperation rates in both conditions are
100.0%

6. Concluding remarks
We have shown theoretically 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 for two players, and it can

18

be interpreted as a “minimum” communication device to achieve cooperation. In the first stage,
each player reveals their choice of C or D. Then knowing the other’s choice, each chooses y or n.
If both choose y, the outcome is what they chose in the first stage; otherwise, the outcome is(D,D). This procedure can be a natural way to avoid conflict, such as a PD or a coordination
situation, in daily life. Using functional near-infrared spectroscopy (fNIRS), Nagatsuka,
Shinagawa, Okano, Kitamura, and Saijo (2013) found that compared with the PD game,subjects made their choices with less stress when using the MC mechanism. We also foundexperimentally that except for first few periods, the MC mechanism works well in an
asymmetric environment.
Masuda et al. (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 both theoretically and experimentally. Furthermore, Huang et al. (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 reasonable approvalmechanism to implement cooperation with more than two choices and players is still an open
question that needs to be answered.
References
Andreoni, J., and Varian, H. 1999. “Preplay Contracting in the Prisoners’ Dilemma.”
Proceedings of the National Academy of Sciences, 96(19), 10933–10938.
Aumann, Robert J. 2006. “War and Peace.” Proceedings of the National Academy of Sciences,103(46): 17075-78.
Banks, J.S., C.R. Plott, and D.P. Porter. 1988. “An Experimental Analysis of Unanimity in Public
Goods Provision Mechanisms.” Review of Economic Studies, 55(2): 301–322.
Charness, G., G.R. Fréchette, and C.-Z. Qin. 2007. “Endogenous Transfers in the Prisoner’s
Dilemma Game: An Experimental Test of Cooperation and Coordination.” Games andEconomic Behavior, 60(2): 287–306.
Chaudhuri, A. (2011). Sustaining cooperation in laboratory public goods experiments: aselective survey of the literature. Experimental Economics, 14(1), 47-83.
Chen, Y. 2005. “Dynamic Stability of Nash-efficient Public Goods Mechanisms: Reconciling
Theory and Experiments.” In A. Rapoport and R. Zwick (eds.). Experimental Business Research.
Vol II, 185–200.
Cooper, R., D.V. DeJong, R. Forsythe, and T.W. Ross. 1996. “Cooperation Without Reputation:

19

Experimental Evidence from Prisoner’s Dilemma Games.” Games and Economic Behavior,
12(2), 187–218.
Dawes, R., J. Orbell, R. Simmons, and A. van de Kragt. 1986. “Organizing Groups forCollective Action.” American Political Science Review, 80(4), 1171–1185.
Fehr, E., & Gächter, S. (1999). Cooperation and punishment in public goods experiments.
Institute for Empirical Research in Economics working paper, (10).
Fischbacher, U. 2007. “Z-Tree: Zurich Toolbox for Ready-made Economic Experiments.”
Experimental Economics, 10(2), 171–178.
Guala, F. 2013. “Reciprocity: Weak or Strong? What Punishment Experiments Do (and Do Not)
Demonstrate.” Behavioral and Brain Sciences, 35, 1–59.
Huang, X., T. Masuda, Y. Okano, and T. Saijo. 2014. “Cooperation among Behaviorally
Heterogeneous Players in Social Dilemma with Stay or Leave Decisions.” Working Paper in
Social Design Engineering Series. SDES-2014-7, Research Center for Future Design, Kochi
University of Technology.
Isaac, R.M., D. Schmidtz, and J.M. Walker. 1989. “The Assurance Problem in a Laboratory Market.”Public Choice, 62 (3), 217–236.

Kuhn, S. 2014. “Prisoner's Dilemma”, The Stanford Encyclopedia of Philosophy (Fall 2014 Edition),
Edward N. Zalta (ed.), URL =<http://plato.stanford.edu/archives/fall2014/entries/prisoner-dilemma/>.
Levitt, S. D., & List, J. A. (2007). “What do Laboratory Experiments Measuring Social
Preferences Reveal about the Real World?” Journal of Economic Perspectives, 21(2), 153-174.
Masuda, T., Y. Okano, and T. Saijo. 2014. “The Minimum Approval Mechanism Implements
the Efficient Public Good Allocation Theoretically and Experimentally.” Games and Economic
Behavior, 83, 73–85.
Nagatsuka, M., H. Shinagawa, Y. Okano, Y. Kitamura, and T. Saijo. 2013. “Using Economic
Games to Investigate the Neural Substrates of Cognitive Processes.” American Journal of
Clinical Medicine Research, 1(4), 71–74.
Poundstone, W. (2011). Prisoner's dilemma. Anchor.
Russett, B., H. Starr and D. Kinsella. 2009. World Politics: The Menu for Choice, Wadsworth
Publishing. 9th edition.
Saijo, T., T. Masuda, Y. Okano, and T. Yamakawa. 2017. “Approval Mechanism to Solve
Prisoner’s Dilemma,” submitted to Social Choice and Welfare, revision requested.
Saijo, T., Y. Okano, and T. Yamakawa. 2015. “The Approval Mechanism Experiment: A
Solution to the Prisoner’s Dilemma.” Working Paper in Social Design Engineering Series.
SDES-2015-12, Research Center for Future Design, Kochi University of Technology.

20

Saijo, T., Y. Tatamitani, and T. Yamato. 1996. “Toward Natural Implementation.” International
Economic Review, 37(4), 949–980.
Varian, H.R. 1994. “A Solution to the Problem of Externalities When Agents are
Well-informed.” American Economic Review, 84, 1278–1293.
Yamagishi, T. (1986). “The Provision of a Sanctioning System as a Public Good.” Journal of
Personality and social Psychology, 51(1), 110-116.

21