Chun Lei Yang

8/8/2019 Chun Lei Yang

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The Rise of Cooperation in Correlated

Matching Prisoners Dilemma: An

Experiment

by

Chun-Lei Yang

Institute for Social Sciences and Philosophy, Academia Sinica

128 Academia Rd. Sec. 2, Taipei 115, Taiwan, ROC

[email protected], phone: 886-2-27898161, fax: 886-2-27864160

and

Ching-Syang Jack Yue

Department of Statistics, National Chengchi University,

64 Chi-Nan Rd. Sec. 2, Taipei 11623, Taiwan, ROC

and

I-Tang Yu

Department of Statistics, National Chengchi University,

64 Chi-Nan Rd. Sec. 2 Taipei 11623, Taiwan, ROC

Keywords: Prisoners dilemma, cooperation, experiment, correlated matching,

multi-level selection

JET class. No.: B52, C91, D74

Corresponding author.

1
mailto:[email protected]:[email protected]


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The Rise of Cooperation in Correlated Matching

Prisoners Dilemma: An Experiment

March 2004

Abstract

Recently, there has been a Renaissance for multi-level selection models to explain the

persistence of unselfish behavior in social dilemmas, in which assortative/correlated matching

plays an important role. In the current study of a multi-round prisoners dilemma experiment, we

introduce two correlated matching procedures that match subjects with similar action histories

together. We are interested in the question of whether and how more cooperation results or

subject behavior changes, compared to the control procedure of random matching. We discover

significant treatment effects for both procedures. Particularly with the weighted history

matching procedure we find bifurcations regarding group outcomes. Some groups converge to

the all-defection equilibrium even more pronouncedly than the control groups do, while others

are much more cooperative, with higher reward for a typical cooperative action. Moreover, the

reward ratio between cooperation and defection is significantly higher if we only focus on the

later rounds, close to or higher than 1.0 in all the more cooperative groups. All in all, the data

show that cooperation does have a much better chance to persist in a

correlated/assortative-matching environment, as predicted in the literature.

Keywords: Prisoners dilemma, cooperation, experiment, unselfish behavior, evolution,

assortative matching, correlated matching, multi-level selection

JET class. No.: B52, C91, D74

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1. Introduction

Cooperation in social dilemma situations like the prisoners dilemma (PD), both in human

societies and primitive species, has been a puzzle for philosophers, social scientists, and

biologists alike. Assuming rationality and sufficient sophistication, cooperation can be

sustained as a result of reciprocal actions in the context of 2-person repeated PD games,

as argued in Trivers (1971), Axelrod and Hamilton (1981), Kreps et al. (1982), which in

some cases can be evolutionarily stable as shown by Fudenberg and Maskin (1990) and

Binmore and Samuelson (1992). The same idea can be extended so as to sustain

cooperation with various sort of contagious punishments even if a finite number of

individuals are randomly matched to play the game, as discussed in Milgrom et al.

(1990), Kandori (1992), Ellison (1994). The focus on ostracism by Hirshleifer and

Rasmussen (1989) and that on opting-out with a pool of myopic defectors by Ghosh and

Ray (1996) are further extensions of the idea that reciprocity sustains cooperation.

However, reciprocity cannot explain why people do things that are good for someone

else, the group, or society without expecting direct personal reward from their actions.

To explain this seemingly unselfish behavior, recent theoretical development has

focused on the general principle of correlated, or assortative, matching within the

population. The basic idea is that such unselfish behavior can only have a survival

chance if its adoption entails advantages for its hosts over the selfish behavior at least in

some contexts. These advantages can be subsumed under the notion of

higher-than-chance likelihood to be matched with another unselfish type. Eshel and

Cavalli-Sforza (1982) and Bergstrom (2001) among others have abstract models to

explicitly calculate the fitness implication of such assortative meeting. Sober and

Wilson (1998) discuss all those models that are based on the general principle of

group/multilevel selection: more or less localized or isolated interactions for more or

less extended generations before dispersion can generate assortative matching effects.

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They show how this could sustain a positive level of unselfish genes with scores of

examples from nature. Bowles and Gintis (2002) survey some recent work in support of

the multi-level selection model. Also, viscosity models by Pollock (1989) and Nowak

and May (1992) can be reinterpreted as models of multi-level selection.

In the economics literature, Frank (1988) for example echoes the correlated matching

theme to explain cooperation in PD of one-shot nature. He argues that the unselfish and

cooperative act as such leaves traces on its hosts that can be observed by potential

trading partners over ones lifespan. This signal of honesty must not be easily imitated.

An assortative procedure that matches people similar in this signal to play one-shot PD

will have the cooperative people more likely to be matched together, and similarly for

the selfish ones. In other words, acting is also investing in the chance of encountering

people of similar action histories. (See Frank, 1994, for a discussion of its relationship

with the multilevel selection idea.) Robson (1990), Amann and Yang (1998), and Vogt

(2000) have extended this theoretical approach by introducing mutant types into the

problem who are more sophisticated at recognizing others types. This can

endogenously induce correlated matching among those who play cooperation. Also,

Bergstrom and Stark (1993) discuss several models of multilevel selection, i.e.

assortative matching, based on family interactions.

Can people who cooperate really recognize other cooperators? Are there other

mechanisms that achieve correlated matching in a similar manner? Can we expect more

cooperation consequently, given those mechanisms? Can experiments shed light on

these questions?

There are some experimental studies related to correlated matching in a PD environment.

Camerer and Knez (2000) show that a successful group experience of solving a

coordination problem can affect the willingness to cooperate in the one-shot PD. Despite

their strong presentation effect, the experience can be interpreted as a catalyst to identify

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same-minded peers. Frank et al. (1993), as confirmed and refined by Brosig (2001),

show that pre-play communication leads to better assessment of the cooperative types

and a very high, yet difficult to explain, rate of cooperation even in one-shot PD.

Yamagishi et al. (2003) show evidence that humans better recognize faces of cheaters

than those of cooperators. Orbell and Dawes (1993), featuring the option of autarky and

the false consensus effect, show that more cooperation may result among those who

ventured to enter the PD game with another partner in an open-room six-people group

without communication.

Actually, for experimental purposes, the problem of cooperation via correlated matching

is more complex than it appears in a theoretical model. The reason is that theoretical

models assume either extremely sophisticated or extremely simple individuals, while

neither is the case with real subjects. In other words, real subjects are more sophisticated

than the basic evolutionary models assume. This makes it a nontrivial issue as to the

characterization ofsimilarindividuals. As the working assumption, we identify the type

of an individual with his history of actions, as if saying, a man is what he does.

The novelty of the current experimental study is that we separate recognition from

action in subjects behavior. More precisely, unlike in the studies cited above, subjects

are not given the opportunity, which is uncontrollable for the purpose of the experiment,

to assess the type of their matched partners before making the trade decision. In contrast,

we set objective criteria by which they are endogenously matched with one another,

according to the similarity of their action histories within the cohort. We introduce two

such correlated matching procedures in a multi-round PD experiment. We are interested

in the questions of whether subjects respond to the matching procedure and whether and

how more cooperation results or subject behavior changes, compared to the control

procedure of random matching. If indeed acting is investing as envisioned by Frank

(1988), can those who sow also reap the fruit, i.e. those who cooperate more encounter

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more cooperative actions in general?

It turns out that the answers depend both on the specific correlated-matching procedure

and on the group composition. If the matching-relevant history is limited to only one

round, we have some treatment effect. In the weighted-history matching procedure

involving history up to five rounds, some groups achieve high levels of convergent

behavior with or without positive numbers of subjects sticking to cooperation. In some

other groups, high volatility persists even late in the game. In any case, the total amount

of cooperation as well as the likelihood for cooperators to meet one another shows a

more extreme distribution here, compared both to the control and the one-period

correlation treatments. This bifurcation result also extends to the reward for a typical

cooperative action. Moreover, even the reward ratio between cooperation and defection

is significantly higher if we only focus on the later rounds, close to or higher than 1.0 in

all the more cooperative groups. All in all, the data show that cooperation does have a

much better chance to persist in a correlated/assortative-matching environment, as

predicted by the multilevel selection theory in the literature.

Before presenting the details of the paper, it is important to note that there are other

kinds of experiments on cooperative behavior without the option of strategic reciprocity

that support the same multilevel selection idea. For example, Bolton et al. (2001) show

in the Nowak and Sigmund (1998) framework of one-way cooperative giving that mere

information about the receivers last-round action alone suffices to induce a significant

increase in cooperation, which is also the object of Wedekind and Milinski (2000), and

Seinen and Schram (2000). Although their matching procedure is random, the individual

value of reputation is close to the basic idea behind Frank (1988) that acting

(cooperatively) is investing (into social reputation or score for a good matching position).

Fehr and Gaechter (2000, 2002) show, in public goods experiments, how cooperation

can proliferate even in complete-stranger settings when costly punishment is feasible.

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Gunnthorsdottir et al. (2001) show that sorting the groups by past contributions can in

fact sustain high-level contributions among the cooperators, even though they are

unaware of this sorting mechanism. Page et al. (2002) also obtain higher levels of

contribution by allowing subjects to determine the new group formation based on past

actions.

2. Game, Matching Procedures, and Design of Experiment

The Payoffs In this study, we consider the following basic prisoners dilemma under

different matching schemes. Table 1 shows the payoffs in NT (1 USD = ca. 34NT) for

the row players. C and D stand for the strategies of cooperation and defection

respectively. The payoff numbers are chosen so that the incentive to play defection is

strong.

Table 1: Payoff Matrix

C D

C 8 1

D 12 3

The Matching Procedures Besides the common procedure of random matching (RM),

we introduce two novel procedures, the one-period correlated matching (OP) and the

weighted-history correlated matching (WH). In the OP procedure, subjects in each

period are randomly paired with someone who uses the same action in the previous

period, as long as there is such a person available. The WH procedure extends the OP

one so as to pair subjects according to some well-defined weighted history of previous

actions. For the current study, we track only five rounds back and the history is weighted

using the Fibonacci numbers1. Each subject starts with a sorting score T(t)=0 at the start

of the game, for all t1. At each particular round t, his sorting score is defined as T(t) =

5a(t-1) + 3a(t-2) + 2a(t-3) + 1a(t-4) + 1a(t-5), where a(s) is 0 if his action in period s is

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defection, and 1 otherwise. In each round, all subjects are sorted after their current

T-values. The relative order among those with the same T-value is randomly determined.

And starting from the bottom, they are matched pair-wise in that order. An additional

test ensures that subjects understand the matching rule correctly. See the corresponding

instruction in the appendix. Subjects do not know their match-partners T-values, nor are

they informed about the distribution of T-value in the group. This uncertainty reflects the

feature in Franks model of imperfect signal recognition. Note that we chose not to use

the simpler linear weighting of history, because discounting makes the recent actions

more salient, which in turn may help get people to stick to a stable course of action.

Theoretical and Experimental Considerations Assuming perfect rationality, any finite

repetition of the PD game with any of the above three matching procedures entails the

unique subgame perfect equilibrium (SPE) of defection throughout. With infinite

repetition, always-defection is still an SPE for all matching procedures. Kandori (1992)

and Ellison (1994) show that the so-called contagious punishment scheme can sustain

always-cooperation as an SPE under some conditions in the RM procedure. However,

all-cooperation can never be an SPE in the OP and WH matching situations. In that case,

each subject would be tempted to deviate to defection unilaterally, since contagious

punishment would not be sustainable in equilibrium. The one he exploited would have

no incentive to play defection, as required by contagious punishment. At least with the

WH procedure, punishing the next partner would imply a rematch with the previous

defector for many more rounds for sure, in exchange for the high likelihood of meeting

someone in the previous all-cooperation class of players. In fact, with the correlated

matching procedures, there cannot even exist a stationary equilibrium with positive

numbers of cooperators if subjects are aware of the stationary distribution. To be more

1 Note that Fibonacci numbers F(n) has the property that F(n)/F(n+1) Golden ratio.

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precise, in any stationary equilibrium, the payoffs for the groups of always-cooperators

and always-defectors must be close, so that nobody can profit by unilaterally changing

his action(s). Yet, due to the correlated matching scheme, cooperators roughly expect the

symmetric cooperation payoff while defectors get the symmetric defection payoff, if the

population size is not too small. It is conceivable that a fleshed-out behavior model may

lead to spatial chaos in the mode of Novak and May (1992).

For experimental purposes, there are other more salient issues at hand. First, we have to

have a finite horizon in any experiment, but we still can find significant and persistent

cooperation even late in the game (15-20%), as discussed by Andreoni and Miller (1993),

Cooper et al. (1996), Friedman (1996) among many others for the RM procedure.

Second, subjects ability of backwards induction has proven to be very limited (e.g.

Selten and Stoecker, 1986), rendering the SPE based on complicated backwards payoff

calculations unfit as the prediction to be tested in experiment. However, the lack of

refined sophistication is not necessarily an obstacle for the rise of cooperation in human

society. As suggested by recent literature on the evolution of cooperative behavior via

multi-level selection, the correlated matching designs here are meant to create a device

that can potentially increase the inherent value of cooperation. However, as all-defection

is always equilibrium, it is still a big question whether and how this potential can be

materialized.

As for our specific correlated matching procedures, since subjects are not informed

about the current action distribution during the game, it is indeed conceivable that the

play can converge to some stationary state with positive numbers of cooperators present.

And we may speculate that WH is better at enabling such a stationary state than OP,

since it takes a person only one round to get back to the elite class of cooperators as

identified by the matching method in the OP procedure, while five rounds are needed for

the same task under the WH procedure.

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Working Hypothesis. In the correlated matching treatments, we expect a higher chance

that cooperative actions meet one another, and thus higher reward ratios. At least in

some groups we expect a higher total rate of cooperation.

The Experiment Design We have one treatment for each of the matching procedures RM,

OP, and WH. Each treatment has been conducted with 5 groups of 14 subjects each. So,

a total of 210 students of various majors from Chengchi University in Taiwan

participated, recruited via announcement on bulletin boards and flyers with standard

slogans to appeal for participation. There is a show-up fee of 50 NT.

General instructions are handed out and read, and questions answered. Each subject is

randomly assigned to a terminal with an ID card in the spacious computer room of the

statistics department. At the beginning of each game, subjects get the specific instruction

about the game to be played, i.e. about both the payoffs and the matching methods as

well as the number of rounds. After nobody has any further question, the game is played.

The total duration of the session is less than 90 minutes. Subjects are then separately

paid off and dispatched. Anonymity is ensured throughout. The sequences of games

played in the specific treatments are summarized below. Subjects have no information

about what is to come after the current game they are in.

Table 2: Treatment summary

Treatment Game 1 Game 2 Game 3

1: OP RM-5 OP-25 RM-5

2: RM RM-5 RM-25 RM-5

3: WH RM-5 WH-25 RM-5

The number indicates the number of rounds played in that game. Game 1 and 3 are

5-round random matching PD games with no feedback. Only at the very end of the

experiment will subjects get informed of the results in those games and their total

account balances will be adjusted accordingly. With this design, we want to test whether

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all groups are statistically from the same sample pool initially. Moreover, we avoid any

dynamic effect before game 2 by the no-feedback feature. Game 2 is a 25-round PD

game with feedback following the matching procedures of OP, RM, and WH in

treatments 1, 2, and 3 accordingly. In game 2, each subjects view window carries his

own play and payoff history as well as the current balance of account. Note that subjects

in each session played two more games after this, which does not affect the subsequent

data analysis. Average payoff for participants during the whole session is about 350 NT

while the hourly wage for student jobs is around 120 NT in general.

3. Data Analysis

Game 1 is designed to check the initial inclination to cooperation in the population and

to make sure that the groups are indeed random samples from the same population.

Game 3 is designed to check what changes with added experience during the

experiment.

Observation 1. The Kruskal-Wallis rank sum test confirms that there is no significant

sampling bias regarding the aggregate rate of cooperation p1(c) across all three

treatments, p = 0.221, in game 1. Also, cooperation in game 3 is significantly lower,

p=0.000, than that in game 1. In fact, the 36% average rate of cooperation in game 1

from subjects without experience and 18% in game 3 are compatible with data in the

literature for RM prisoners dilemma experiments.

Note that, because the results of using Mann-Whitney and Wilcoxon rank sum tests in

this paper coincide with those of the Kruskal-Wallis (KW) test, all equal median tests in

this paper refer to the KW test, unless stated otherwise. Table A.1, Appendix 2,

summarizes the group data for games 1 and 3. A logistic regression shows that

individual inclination to play cooperation in game 3 is positively affected by his/her

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game 1 and game 2 cooperative actions and how often he/she switches actions in game

2.

We now turn to game 2 to investigate the treatment effects. Throughout the paper, we

will use data from rounds 6-23 only, to avoid any potential effects of the initial learning

and end-game behavior often reported in the literature. Note that, in treatments 1 and 3,

there indeed seems to be a decrease in cooperation in the last two rounds, indicating

likely strategic intention associated with action c over the course.

Table 3: Group level summary over rounds 6-23

Treatment Gr. p2(c) P2(cc|c) p2(dd|d) av-r(c) av-r r-ratio

1 0.183 0.261 0.835 2.826 4.183 0.630

2 0.230 0.345 0.804 3.414 4.452 0.717

OP 3 0.246 0.355 0.789 3.484 4.548 0.718

4 0.190 0.292 0.833 3.042 4.222 0.676

5 0.306 0.468 0.766 4.273 4.853 0.836

avg. 0.231 0.344 0.806 3.408 4.452 0.714

std 0.049 0.079 0.030 0.554 0.272 0.077

6 0.214 0.074 0.747 1.518 4.468 0.288

7 0.258 0.400 0.791 3.800 4.599 0.779

RM 8 0.139 0.171 0.866 2.200 3.925 0.523

9 0.179 0.222 0.831 2.556 4.171 0.565

10 0.218 0.218 0.782 2.527 4.433 0.509

avg. 0.202 0.217 0.804 2.520 4.319 0.533

std 0.045 0.118 0.046 0.828 0.270 0.175

11 0.409 0.816 0.872 6.709 5.194 1.617

12 0.083 0.095 0.918 1.667 3.567 0.446

WH 13 0.333 0.476 0.738 4.333 5.016 0.809

14 0.405 0.588 0.720 5.118 5.357 0.927

15 0.135 0.176 0.872 2.235 3.897 0.538

avg. 0.273 0.430 0.824 4.012 4.606 0.867

std 0.154 0.297 0.089 2.077 0.815 0.463

KW test p-value 0.677 0.228 0.811 0.228 0.733 0.185

Note: Total number of actions in each group: 252. (p: proportion; c, d: cooperation,

defection; cc, dd: outcomes where both subjects play the same action; cc|c: cc

occurs conditional on playing c oneself; av-r: average reward)

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Table 3 summarizes the most relevant group aggregate data. First, the rate of

cooperation p2(c), though slightly higher in WH, is not significantly different across

treatments, p=0.677. Exactly the same is true for the likelihood p2(cc|c) that a

cooperator expects to meet another cooperator in the same round (p=.228), and for the

average payoff a typical cooperative action expects av-r(c) (p=.228), the average payoff

av-r for the whole group (p=.733), and the empirical reward ratio, r-ratio, between

cooperation and defection.

Conspicuously, however, the WH treatment displays much higher standard deviations

with respect to all variables in Table 3. In fact, groups 11, 13 and 14 display the highest

rates of cooperation, the highest chances that a cooperative action encounters another

cooperative one, and the highest average payoffs for the group and for cooperation

within the group, among all 15 groups, while groups 12 and 15 rank among the lowest

three groups regarding all the above variables except for average payoff for cooperation.

In fact, this bifurcation effect of WH is also evident when we look at the data in rounds

6-14 and 15-23 separately, as summarized in Tables A.2 and A.3 in Appendix 2. Table 4

below summarizes results of all equal median and equal dispersion tests relevant.

Observation 2.All rounds together (6~23), WH has a significantly greater dispersion

than RM in p2(c), p2(dd|d), andav-r.

Observation 3.Within rounds 15~23, WH has a significantly higher median than RM,

in av-r(c) and r-ratio with p-values 0.047 and 0.016 respectively, and a greater

dispersion in p2(c), p2(dd|d) , andav-r. Within rounds 6~14, WH has a greater

dispersion than RM in all variables except forp2(dd|d).

These also indicate that the advantage of playing c in WH compared to RM can increase

over time, after an extended period of learning and adaptation to the environment. In fact,

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comparing data from 6~14 with that from 15~23 as shown in Table A.4 in the appendix,

we see that the average payoff for cooperation and the r-ratio greatly increase even in

groups 12 and 15 where the total number of cooperation is lower than in the worst

groups in RM.

Table 4. Tests of treatment effect for group-aggregate variables

Test of Equal DispersionTest of Equal

Median OP vs. RM OP vs. WH RM vs. WH

6~23 6~14 15~23 6~23 6~14 15~23 6~23 6~14 15~23 6~23 6~14 15~23

p2(c) .677 .577 .706 1 .828 .671 .011 .011 .034 .011 .011 .034

p2(cc|c) .228 .632 .063 .203 .519 .396 .011 .011 .090 .203 .053 1

p2(dd|d) .811 .810 .733 .671 .384 .203 .011 .034 .011 .011 .203 .011av-r(c) .228 .632 .063 .203 .519 .396 .011 .011 .090 .203 .053 1

av-r .733 .645 .733 .671 .830 .671 .011 .011 .034 .011 .011 .034

r-ratio .185 .590 .027 .203 .519 .396 .034 .011 1 .396 .090 1

Note: The tests of equal median and dispersion are Kruskal-Wallis and Ansari-Bradley2

tests,

while and indicate 10% and 5% significance, respectively.

Observation 4. WH has significantly greater dispersion than OP in all relevant

variables, which, however, fades away in later rounds 15~23 forr-ratio.

Observation 5. The only differences between OP and the control treatment RM are

regarding the medians for p2(cc|c), av-r(c), and the r-ratio in rounds 15~23, with

p-values 0.028, 0.047, and 0.047, respectively.

These indicate that OP does induce some behavior difference compared to RM, in the

later rounds. But this effect is by far not as strong as WHs.

Observation 6. In both OP and WH, p2(cc|c), av-r(c), andr-ratio display increase from

6~14 to 15~23, to the significance level 10% two-way, or 5% one-way, with the

Wilcoxon signed rank test. (Table A.4, Appendix.)

2

Ansari-Bradley test and Siegel-Tukey test are two frequently used non-parametric tests for dispersion.Since the results of Ansari-Bradley and Siegel-Tukey tests are very similar, we only show the p-values ofthe Anasari-Bradley test. For detailed discussion of nonparametric tests, see for example, Daniel (1990).

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This shows that the group aggregate behavior is still evolving in the OP and WH

treatment, but not so much in the control treatment RM.

On the group-aggregate level, we have found significant differences across treatments.

The legitimate question is whether we can observe consistent and compatible treatment

effects if the data are aggregated differently. Next, we will show that the observed

treatment effects are indeed robust when we focus on distributions of individual

characteristics, which also offers another angle to understand the complex structure of

subject behavior in the experiment.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.0

0.5

1.0

Group

%o

fC

OP RM WH

Figure 1: Boxplots of individual c-frequency in Game 2

Figure 1 is a boxplot presentation of the individual frequency of cooperation. The

medians are rather indistinguishable across treatments. However, a complete display of

this distribution reveals a lot of interesting insights.

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Table 5: Distribution of individual c-frequency in Game 2 (rounds 15~23)

OP RM WH

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 1 0 0 0 0 0 0 1 0

0 0 0 0 1 0 1 0 0 0 0 0 0 2 0

0 0 1 0 2 0 1 0 0 0 1 0 1 3 0

1 1 1 0 2 0 1 0 0 1 1 0 2 3 0

1 1 1 0 3 1 1 0 0 1 1 0 2 3 0

1 1 2 0 3 1 2 0 0 2 1 0 2 4 0

2 2 3 1 3 2 2 1 1 3 8 0 2 4 1

2 3 3 2 4 2 3 1 1 3 9 1 4 4 2

2 4 4 2 5 3 3 1 3 3 9 1 5 6 2

3 4 5 3 5 4 4 2 3 4 9 1 6 6 2

4 6 6 7 7 7 6 3 6 4 9 3 9 9 3

4 7 9 9 9 9 9 3 9 9 9 3 9 9 3

Note: Each column has 14 entries for the 14 players of each group. Max.= 9.

Table 5, for example, reveals that group 11 has apparently converged to a stationary state

of 6 cooperators and 8 defectors, with nobody deviating more than once in a total of 9

rounds (15~23). From Table 5 we can also see that both groups 13 and 14 have exactly

one pair locked-in in the always-cooperation mode. No other groups come close, though

the presence of four always-cooperators in RM suggests that there are always some

people who might not have understood the game or are just unconditional cooperators

for whatever reason. Conversely, no players in groups 12 and 15 chose cooperation more

than 3 times in the last 9 rounds, indicating a convergence to the all-defection

equilibrium. They also display less overall inclination to cooperation than even groups

in the random matching case.

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Table 6: Derived variables for Game 2

OP RM WH

Group 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

H6

mean

5.0 7.17 7.17 7.0 9.83 8.5 8.17 5.33 7.5 7.33 15.67 3.33 10.5 11.5 4.5

6-14 2.5 3.67 3.5 3.17 4.33 4 4.17 3.67 3.67 3.67 6.83 2 5.83 5.83 2.67

15-23 2.83 4.33 5 4 5.5 4.5 4.5 1.83 3.83 4.33 8.33 1.5 5.83 6.33 2.17

L6

mean

1.5 1.17 1.5 0.17 1.5 0 1.5 0 0 0.5 0.5 0 1.83 3.17 0.5

6-14 1.17 0.33 0.17 0.17 0.67 0 0.67 0 0 0.17 0.17 0 0.5 0.83 0.17

15-23 0.17 0.17 0.33 0 0.83 0 0.67 0 0 0.17 0.33 0 0.5 1.67 0

H6L6 3.5 6.0 5.67 6.83 8.33 8.5 6.67 5.33 7.5 6.83 15.17 3.33 8.67 8.33 4.06-14 1.33 3.33 3.33 3 3.67 4 3.5 3.67 3.67 3.5 6.67 2 5.33 5 2.5

15-23 2.67 4.17 4.67 4 4.67 4.5 3.83 1.83 3.83 4.17 8.5 1.5 5.33 4.67 2.17

#c50% 0 1 2 2 4 2 2 1 2 2 6 0 3 6 06-14 0 1 0 1 2 2 2 3 2 2 5 0 3 4 0

15-23 0 2 3 2 4 2 2 0 2 1 6 0 4 4 0

#c=0 0 2 3 5 1 6 0 6 8 3 4 7 1 0 4

6-14 1 4 5 5 2 7 2 6 9 5 5 8 4 3 5

15-23 5 5 4 8 3 6 2 8 8 5 4 9 4 1 8

Changes

= 01 3 4 6 2 9 6 9 11 4 8 9 5 0 5

Note: Definitions of variables in Table 6 can be found in the following observations.

To illustrate where the above observed aggregate treatment effects may have originated,

we define some further statistical variables. Their values for all groups and all three

time-intervals we discuss are summarized in Table 6, and the corresponding tests are

summarized in Table 7.

Table 7. Tests of treatment effect for derived variables

Test of Equal DispersionTest of Equal

Median OP vs. RM OP vs. WH RM vs. WH

6~23 6~14 15~23 6~23 6~14 15~23 6~23 6~14 15~23 6~23 6~14 15~23

H6 .677 .493 .674 .667 .128 .667 .011 .032 .011 .011 .007 .032

L6 .210 .201 .578 .430 .397 .676 .070 .624 .250 .933 .676 .549

H6L6 .686 .228 .580 .830 .364 .825 .053 .090 .025 .034 .010 .032

50% of c .808 .165 .544 .397 .621 .196 .065 .027 .099 .038 .056 .018

# of 0s .381 .206 .815 .919 .770 .768 .210 .435 .432 .728 .712 .708

Changes=0 .075 .787 .048 .247

Note: The tests of equal median are Kruskal-Wallis tests. The tests of dispersion are

Ansari-Bradley tests for the continuous variables and bootstrap simulation tests for the discretevariables. Also, and indicate 10% and 5% significance, respectively.

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Observation 7. With respect to the mean of the 6 highest-ranking cooperators

(H6-mean), that of the 6 lowest ranking cooperators (L6-mean), and their difference

H6-L6, there is no significant difference in the treatment median.

Observation 8. WH has significantly higher dispersion than OP and RM in H6-mean

andH6-L6.

Observation 9. With the exception of OP vs. WH for rounds 6~23, there is no significant

difference in dispersion forL6-mean across the treatments.

Observation 10. Regarding the statistical variable of the number of subjects in a group

who played cooperation at least 50% of the time, #c 50%, WH displays significantly

higher dispersion than both OP and RM in all three time frames.

Observation 11. Regarding the number of subjects in a group who consistently chose

defection, #c=0, there is no significant difference across treatments whatsoever.

Observations 7 through 11 combined indicate that the afore-mentioned bifurcation effect

of the WH treatment, groups 11, 13, 14 vs. groups 12, 15, mainly stems from the

bifurcation of the number of most cooperative players. Regarding the number of least

cooperative subjects, there seems to be parity across treatments, except for OP vs. WH

for timeframe 6~23. In the latter case, groups 13, 14 show higher means for their least

cooperative subjects than the groups in OP, while groups 11, 12, 15 show lower means

that are similar to the groups in RM. However, this significance does not hold if we

consider rounds 6~14 and 15~23 separately. In any case, this oddity may be from the

fact that groups 13 and 14, like most groups in OP, still display high volatility in subject

behavior, while subject behavior is more settled in groups 11, 12, 15 and the RM groups,

especially among those who have decided to stick to defection. This is also evident from

the following observation.

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Observation 12. The number of subjects without a change in cooperation attitude from

rounds 6~14 to 15~23 (Changes =0) is significantly higher in RM than in OP, p= 0.027.

Besides, WH displays a significantly higher dispersion than OP.

Table A.7 in the appendix ranks subjects within each group according to their changes in

total number of cooperation choices from rounds 6~14 to 15~23, from which the

variable Changes = 0 is derived straightforwardly.

So far, we have discussed treatment effects based on various kinds of group-level

variables. Now we will show more evidence to further confirm that cooperation has a

much better chance to survive in correlated matching environments. Figures 2, 3, and 4

show the scatter plots for the joint distribution of average individual T-value and payoff

in the RM, OP, and WH treatments, respectively. Each graph contains 70 dots for the

same number of subjects in each treatment. The correlation coefficient between T-value

and payoff in OP is 0.1412 and not significantly different from 0. It is negatively

correlated in the RM treatment (r=0.5717), but positively correlated in WH (r=0.7705),

both significantly differently from 0.

Tvalue

payoff

0 2 4 6 8 10 12

2

3

4

5

6

7

Figure 2. T vs. payoff for RM (r=0.5717)

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Tvalue

payoff

0 2 4 6 8

3

4

5

6

Figure 3. T vs. payoff for OP (r=0.1412)

Tvalue

payoff

0 2 4 6 8 10 12

3

4

5

6

7

8

Figure 4. T vs. payoff for WH (r=0.7705)

It is evident that the negative correlation of RM, which reflects the very nature of the

predicted failure of cooperation in PD, disappears with the introduction of both the OP

and the WH treatments. This observation accentuates the insight we gained with the

variable r-ratio in Table 3 and Table A.3 with statistical significance stated in

Observation 3. It implies that, particularly with the WH matching device, cooperation

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has a good chance to prevail. Note that the qualitative results remain exactly the same if

we replace T-value with individual 5-round average c-frequency, i.e. the linear

weighting, as shown in Figures A.1, A.2, and A.3 in the appendix.

The final part of the current analysis focuses on the likelihood of cooperation aspect of

the T-related behavior patterns. Figure 5 shows that the rate of cooperation displays an

increasing tendency with T for all treatments.

T-value

Prob.ofCooperation

0 2 4 6 8 10 120.0

0.2

0.4

0.6

0.8

OPRMWH

Figure 5: T-dependent probability of cooperation

Surprisingly, it appears that there is not much difference in this regard across the

treatments. This makes the variable T-value seem to be compatible with the

interpretation as an inertia variable: subjects inclination towards cooperation is similar,

i.e. positively correlated, with their average behavior in the recent history of play.

Starting with 15 group dummies and iteratively eliminating dummies of p-values greater

than 0.05, it turns out that the following logistic regression outcome fits the data quite

well.

15&12Gr7458.32766.0096.1520.8277.3999.2)(logit 32 +++= TrtTTTp (1)

(.0889) (.0855) (.0201) (.0012) (.1090) (.1762)

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(-2LL = -1585.92, Concordance = 76.3%, Hosmer-Lemeshow p-value = .587)

Note that we initially ended up with significant dummies left for groups 12, 14, and 15 in

this manner. The Hosmer-Lemeshow goodness-of-fit test3

of this particular model failed,

however. Note also that we unsuccessfully experimented with the model conditioned on

whether subject behavior was C or D for the same T-value. After trial and error, model (1)

ended up as the only one with all-around satisfactory statistical properties. So, the

bifurcation effect of the WH treatment resurfaces here. Figure 6 displays the fitted

curves.

T-Value

P

rob.ofCooperation

0 2 4 6 8 10 12

0.2

0.4

0.6

0.8

OP & RMWH (Groups 11, 13, 14)WH (Groups 12, 15)

Figure 6. Fitted curves for the joint logistic model

Summary. Piecing together the evidence, we see that the OP matching procedure has a

slight effect of increasing cooperation, which can even get more pronounced with time.

But subjects seem to be still very actively changing their actions, so our current design is

3 Two frequently used goodness-of-fit are Pearson and Deviance tests. However, as the number of groups

increases, these two tests are more likely to falsely reject the null hypothesis. Another (Pearson-like)

goodness-of-fit test, proposed by Hosmer and Lemeshow (1980), is used more often in practice, as in this

paper. Note that the Hosmer-Lemeshow (H-L) test is to group residuals based on the values of the

estimated probabilities. The default number of groups in H-L test is 10, as in this paper.

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not particularly conducive to founded speculation as to where all this is leading.

However, the WH procedure shows much clearer effect compared to the control

treatment. There is a clear bifurcation of groups under WH. In some groups, 12 and 15,

the behavior seems to converge to the all-defection equilibrium, even more accentuated

than in the RM groups. In others, 11, 13, and 14, we observe strong performance for

cooperative actions, particularly evident in the later rounds (15~23). Group 11 is notable

for an extreme, within-group bifurcation of persistent defectors and cooperators. On the

other hand, groups 13 and 14 are similar to the OP groups with still a lot of subjects

changing behavior in the later rounds, but with apparently better reward overall. Most

notably, with an average of 0.8 and 1.07, both OP and WH show a significantly better

reward ratio between cooperation and defection than RM, in the later rounds. Even in

the defective groups 12 and 15, the r-ratios are very high compared to the RM groups.

These are all evidence that the assortative matching devices we investigate in this study

are indeed effective as the theory predicted.

4. Concluding Remarks

Think of the groups as different tribes in competition for hunting territory. And, think of

their military power as proportionate to the groups total reward accruing from those

repeated PD games. Those tribes that manage to end up being more cooperative within

the tribe, like our groups 11, 13, and 14, will have a better chance to be the ones left in,

say, a war of attrition. Alternatively, if the reward is interpreted as representative of the

reproductive power common to biology models, these groups also will have more

offspring to participate in the next generation group formations and we will have higher

chances of having similarly composed groups with the hope of similarly better group

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total reward.4

In any case, with the correlated, particularly the WH, matching device,

society as a whole has a better chance to have more cooperation persistent in the long

run. Figure 4 in contrast to Figure 2 very well illustrates this phenomenon, that it often

pays to be consistently cooperative in WH.

Now that we have confirmed the theoretical prediction that correlated/assortative

matching can indeed break the curse of defection in RM prisoners dilemma, the next

step is to characterize the process and outcomes in this new environment. Is that just

chaos now? Our data indicate that we may expect a bifurcation in aggregate group

outcomes: Some groups may converge to the all-defection equilibrium even faster than

in the RM environment, while cooperation can thrive among a sub-group of subjects in

the other groups. The latter can be further divided into two types. Some groups have the

more cooperative subjects sticking to cooperation quite soon in the experiment, while

most of them in the other groups are still not fully committed to cooperation even late in

the game, quite similar to the typical behavior in the OP treatment. It is not clear whether

this phenomenon is only transient, i.e. reflecting the natural learning process towards a

more stable behavior pattern, or whether it already reflects some stable pattern of the

subjects who explicitly want to exploit fellow cooperators every now and then.

However, the potential danger of fatigue or boredom due to long experimental sessions

limits our options to directly answer the last question by experiment. We hope to

develop learning models in the future to be able to simulate the long-run outcomes with

data from sessions of limited duration. Also, it is relevant to manipulate the information

the subjects receive, in order to check the robustness of the results in this study.

Note that this bifurcation result is also consistent with the conditional cooperation

hypothesis by e.g. Keser and van Winden (2000) that many cooperators are only willing

4 See e.g. Sobel and Wilson (1998, pp.63, pp.65) for this line of argument and for further references.

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to cooperate if they expect others to do the same, an idea also to be found in Rabin

(1993). Amann and Yang, 1998, call them cautious cooperators in an evolutionary

model. Our study shows the experimental evidence that correlated matching devices can

induce those cautious cooperators to actually cooperate in the end.

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Appendix 1: Instructions

General Instruction

You are about to participate in an economic experiment on multi-person interactive decisions.

The study is funded by the NSC and Academia Sinica.

Basic experimental rules

1. Please read these instructions completely and carefully. Keep quiet during the experiment

and do not contact fellow subjects under any circumstances. Any question shall be raised to

the experimenters individually.

2. The experiment is anonymous and you will get an ID card before the start.

3. Please do not do anything on the PC that is irrelevant to the experiment.

4. You will be paid off anonymously after the experiment ends, in exchange for the ID card you

hold.

5. Please return all instructions you have received at the end of the experiment.

6. To help us collect reliable data, please do not talk about this experiment in the next two

weeks with those who have not participated. Thank you for your cooperation.

Generic decision rules

There are 14 subjects who participate in this session. Each session contains several separate

2-person games of length from 5 to 25 rounds. Below is a demo window for what you will see on

the PC in the experiment. (Correct payoffs numbers will be given during the experiment.)

In each round, when the decision bars become highlighted, you can start to make a decision

between A and B. An example is shown in these printed instructions to tell you how to find out

what you and your counterpart will receive as a consequence of your decisions.

At the end, you will be paid off the sum of the payoffs you get in each round, plus a 50NT

show-up fee. For experimental purposes, in some of the games you will not be informed

immediately of the action of the other person and thus of your own payoff there. On the upper

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right of the window, you will see your account balance for all the other games/rounds played,

including the 50 NT show-up fee. At the end, your final balance will be updated, with all the

payoff and action information previously withheld.

Note that the computer rematches the participants into pairs after each round. The matching

procedure changes during the session. You will be informed about the details in special written

instructions accordingly. Wait for the experimenters further instruction.

During the session, you will find the following information on the right of the window: (1) Your

own choices so far; (2) Your counterparts choices; (3) Your own payoffs in the previous rounds;

(4) Other information and instructions. For part of the experiment, you will not get information

about (2) and (3) right away, but you will find ? instead. This information will be given at the

end of the session.

Special Instructions

Random Matching: 5 rounds no feedback (Games 1 and 3)

In this part, there will be five rounds. At the beginning of each round, subjects will be randomly

paired to play the game you will find on the window. This means that you have the same chance

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to meet any of the other 13 subjects in each round, independent of what has happened so far. For

experimental purposes, you will be informed neither of your counterparts action nor of your

own payoff, at this time. You will receive the relevant information at the end of the session and

your total final payoff will be updated accordingly.

Random Matching: 25 rounds (Game 2)

In this part, there will be 25 rounds. At the beginning of each round, subjects will be randomly

paired to play the game you will find on the window. This means that you have the same chance

to meet any of the other 13 subjects in each round, independent of what has happened so far.

One-round Correlated Matching (Game 2)

In this part, there will be 25 rounds. At the beginning of the first round, subjects will be

randomly paired to play the game you will find on the window. For the other rounds, the

matching procedure has the following form. All subjects who have made the same decision in

the previous round will be randomly paired with one another. Only in case of odd numbers will

there be one pair in which the subjects have made different decisions in the previous round.

Weighted-history Correlated Matching (Game 2)

In this part, there will be 25 rounds. At the beginning of the first round, subjects will be

randomly paired to play the game you will find on the window. For the other rounds, the

matching procedure takes into account what the subjects have done in the previous five rounds

in the following form.

Step 1: Calculation of the sorting score (T) Rd Last 5 decisions T= T1 + T2 + T3 + T4+ T5

1 none 0

2 A 0 = 0

3 BA 5 + 0 = 0

4 BBA 5 + 3 +0 = 8

5 ABBA 0 + 3 + 2 +0 = 5

6 BABBA 5 + 0 + 2 + 1 + 0 = 8

7 ABABB 0 + 3 + 0 + 1 + 1 =5

At each round, if your choice in the previous

round is B you will get a score T1=5. If B is your

choice in the 2nd

previous round, you get a score

T2=3. If B is your choice in the 3rd

previous round,

you get a score T3=2. If B is your choice in the 4th

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previous round, you get a score T4=1. If B is your choice in the 5th

previous round, you get a

score T5=1. Otherwise, i.e. if your choice is A in the n-th previous round, your score is Tn=0.

The total sorting score T = T1+T2+T3+T4+T5. For the very first 5 rounds of this part, T is

calculated with the previous rounds actions only. By definition, then, all subjects start this part

with the same T=0.

For illustration, assume that somebodys choices in round 1-6 are ABBAABA.

For example, in round 7, T=5 is calculated in the following way. Since A is the choice in the

previous round (round 6), T1=0. B in the 2nd

previous one (round 5), T2=3. Etc.

Step 2: Matching according to the sorting scores

At the beginning of each round, all subjects will be ranked according to their ranking scores T

and be paired with their neighbors, subsequently from top to bottom. Subjects with the same

ranking score T will be ranked randomly among themselves.

For example, if there were four subjects a, b, c and d in the experiment (in reality 14) with the

ranking scores 5, 5, 0 and 8 accordingly, then they would be ranked either as {d,a,b,c} or {d,b,a,c}

with equal chance. Then, we would end up with the matching result of either {(d plays with a), (b

with c)} or {(d with b), (a with c)} with equal chance accordingly.

Note that this matching procedure ignores the actions earlier than five rounds ago. Also you can

at any time find out in the window on the right side of your monitor about your own previous

choices, your previous counterparts choices, your payoffs, your account balance, and your

current ranking score T.

Test

1. If subject as choices in the first 4 rounds are ABBA, then as ranking score in the 5th round is:

(1) 0, (2) 1, (3) 3, (4) 5.

2. If as choices from round 3 to round 7 are ABABB, then his ranking score in the 8th round is:

(1) 0, (2) 1, (3) 5, (4) 9.

3. If six subjects a, b, c, d, e, f participate in the experiment and, at some round, have the

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ranking scores of 8, 12, 10, 12, 5, 0, then a can be potentially matched with: (1) only b, (2)

only c, (3) b or c, (4) b or d.

4. If six subjects a, b, c, d, e, f participate in the experiment and, at some round, have the

ranking scores of 8, 12, 10, 10, 5, 0, then a can be potentially matched with: (1) only b, (2)

only c, (3) c or d, (4) b or d.

Answers: 1. (3); 2. (4); 3. (2); 4. (3).

Appendix 2: Further Data

Table A.1: Group data on the 5-round, no-feedback RM games

Trt1(OP) Trt2(RM) Trt3(WH)

group p1(d) p3(d) group p1(d) p3(d) group p1(d) p3(d)

1 0.600 0.843 6 0.714 0.786 11 0.629 0.7432 0.757 0.900 7 0.557 0.686 12 0.743 0.9143 0.800 0.886 8 0.657 0.900 13 0.686 0.8144 0.657 0.929 9 0.629 0.914 14 0.529 0.7435 0.729 0.657 10 0.643 0.771 15 0.571 0.829

avg. 0.709 0.843 avg. 0.640 0.811 avg. 0.631 0.809std

0.080

0.108 std

0.057

0.096std

0.095

0.075

Defection frequency in game 3, p3(d), proves to be significantly higher than in game 1

(KW test, p=0.000), i.e. the share of cooperation decreases from ca. 36% to only ca. 18%.

A logistic regression shows that individual behavior in game 2, along with his game 1

behavior, has a significant effect. The regression has the following form, with the

estimated coefficients and standard deviations.

(0.0204)(0.3653)(0.3437)(0.3034)

noise)(21003.0)(23695.2)(16213.14809.0)(3logit +++= swpdpdpdp (A-1)(Log-Likelihood = -426.071; Concordant = 74.8%; Hosmer-Lemeshow (H-L)

goodness-of-fit test p =0.493; p2(sw) denotes the individual frequency of action

switches in game 2.) Note that we have experimented with different models with

different combinations of potentially relevant variables. Yet, the associated H-L

goodness-of-fit test failed for other models.

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Table A.2: Group level summary over rounds 6-14 (Game 2)

Treatment Gr. p2(c) p2(cc|c) p2(dd|d) av-r(c) av-r r-ratio

1 0.206 0.190 0.813 2.615 4.349 0.545

2 0.230 0.308 0.791 3.413 4.452 0.717

OP 3 0.214 0.182 0.800 2.556 4.405 0.521

4 0.190 0.200 0.826 2.750 4.238 0.599

5 0.262 0.345 0.771 3.545 4.643 0.705

avg. 0.221 0.245 0.800 2.976 4.417 0.617

std 0.027 0.076 0.021 0.467 0.149 0.090

6 0.198 0.000 0.769 1.000 4.389 0.191

7 0.246 0.414 0.795 3.710 4.532 0.773

RM 8 0.190 0.250 0.839 2.750 4.238 0.599

9 0.175 0.211 0.910 2.273 4.159 0.499

10 0.198 0.261 0.809 2.680 4.294 0.571

avg. 0.202 0.227 0.802 2.482 4.322 0.527

std 0.027 0.149 0.025 0.982 0.144 0.213

11 0.365 0.700 0.833 5.870 5.048 1.283

12 0.095 0.000 0.891 1.000 3.667 0.253

WH 13 0.333 0.474 0.730 4.333 5.016 0.809

14 0.373 0.585 0.761 4.872 5.198 0.904

15 0.167 0.100 0.804 1.667 4.135 0.360

avg. 0.267 0.372 0.804 3.548 4.613 0.722

std 0.127 0.307 0.063 2.109 0.674 0.420

p-value 0.578 0.724 1.000 0.633 0.646 0.590

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Table A.3: Group level summary over rounds 15-23 (Game 2)

Treatment Gr. p2(c) p2(cc|c) p2(dd|d) av-r(c) Av-r r-ratio

1 0.159 0.333 0.872 3.100 4.016 0.740

2 0.230 0.385 0.814 3.414 4.452 0.717

OP 3 0.278 0.516 0.815 4.200 4.690 0.861

4 0.190 0.300 0.848 3.333 4.206 0.756

5 0.349 0.513 0.740 4.818 5.063 0.927

avg. 0.241 0.409 0.818 3.773 4.486 0.800

std 0.075 0.100 0.050 0.716 0.411 0.090

6 0.230 0.148 0.729 1.966 4.548 0.369

7 0.270 0.370 0.800 3.882 4.667 0.783

RM 8 0.087 0.000 0.891 1.000 3.611 0.259

9 0.183 0.286 0.835 2.826 4.183 0.630

10 0.238 0.214 0.738 2.400 4.571 0.457

avg. 0.202 0.204 0.799 2.415 4.316 0.500

std 0.071 0.141 0.068 1.064 0.435 0.207

11 0.452 0.902 0.918 7.386 5.341 2.022

12 0.071 0.000 0.933 2.556 3.468 0.722

WH 13 0.333 0.486 0.747 4.333 5.016 0.809

14 0.437 0.600 0.677 5.327 5.516 0.941

15 0.103 0.333 0.920 3.154 3.659 0.849

avg. 0.279 0.464 0.839 4.551 4.600 1.069

std 0.182 0.333 0.119 1.912 0.965 0.539

p-value 0.706 0.100 0.527 0.063 0.733 0.027

Notes: (1) OP and RM have significant difference in p2(CC|C) with p-value=0.028.

(2) WH and RM have significant difference in av-r(c) with p-value=0.047.

(3) OP and RM have significant difference in r-ratio with p-value=0.047;

WH and RM have significant difference in r-ratio with p-value=0.016.

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Table A.4: Group level summary for change from rounds 6~14 to 15-23

Treatment Gr. p2(c) p2(cc|c) p2(dd|d) av-r(c) Av-r r-ratio

1 -0.047 0.143 0.059 0.485 -0.333 0.195

2 0 0.077 0.023 0.001 0 0

OP 3 0.063 0.334 0.015 1.644 0.285 0.340

4 0 0.100 0.022 0.583 -0.032 0.157

5 0.087 0.168 -0.031 1.273 0.420 0.222

avg. 0.021 0.164 0.018 0.797 0.680 0.183

std 0.054 0.101 0.032 0.656 0.294 0.123

Wilcoxon p-value 0.423 0.059 0.418 0.059 0.855 0.100

6 0.032 0.148 -0.040 0.966 0.159 0.178

7 0.024 -0.044 0.005 0.172 0.135 0.010

RM 8 -0.103 -0.250 0.052 -1.750 -0.627 -0.340

9 0.008 0.075 -0.075 0.553 0.024 0.131

10 0.040 -0.047 -0.071 -0.280 0.277 -0.114

avg. 0 -0.024 -0.026 -0.068 -0.006 -0.027

std 0.059 0.151 0.054 1.047 0.358 0.209

Wilcoxon p-value 0.590 1.000 0.418 1.000 0.590 1.000

11 0.087 0.202 0.085 1.516 0.293 0.739

12 -0.024 0 0.042 1.556 -0.199 0.469

WH 13 0 0.012 0.017 0 0 0

14 0.063 0.015 -0.084 0.455 0.318 0.037

15 -0.063 0.233 0.116 1.487 -0.476 0.489

avg. 0.013 0.092 0.035 1.003 -0.013 0.347

std 0.062 0.115 0.077 0.726 0.336 0.318

Wilcoxon p-value 0.715 0.100 0.281 0.100 1.000 0.100

KW test p-value 0.970 0.102 0.265 0.228 0.983 0.129

Note: The Wilcoxon signed rank test is to test if the median is zero. The alternative

hypothesis is that the median is not equal to zero. If we let the alternative hypothesis be

that the median is greater than zero, then the p-value will be 0.050 and 0.030,

corresponding to the original 0.100 and 0.059, respectively.

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Table A.5: Distribution of individual c-frequency in Game 2 (rounds 6~14)

OP RM WH

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 0 0 1 0 1 0 0 0 0 0 0 0 0

1 0 0 0 1 0 1 0 0 0 0 0 0 1 0

2 1 0 0 1 0 1 0 0 0 0 0 1 2 0

2 1 1 1 1 0 1 0 0 1 1 0 2 2 1

2 2 2 2 1 0 1 1 0 1 2 0 2 3 2

2 3 3 2 2 1 1 1 0 1 2 0 2 4 2

2 3 3 2 2 1 2 1 0 2 2 1 3 4 2

2 3 3 2 4 3 2 2 1 2 5 1 4 4 2

2 3 3 3 4 3 3 2 3 3 7 2 4 6 23 4 4 3 4 3 4 5 4 3 9 2 6 6 3

3 4 4 4 6 5 6 6 5 5 9 3 9 7 3

3 5 4 5 6 9 8 6 9 7 9 3 9 8 4

Table A.6: Distribution of individual c-frequency in Game 2 (rounds 6~23)

OP RM WH

1 2 3 4 5 6 7 8 9 10 11 12 13 14 151 0 0 0 0 0 1 0 0 0 0 0 0 1 0

1 0 0 0 1 0 1 0 0 0 0 0 0 1 0

1 1 0 0 1 0 1 0 0 0 0 0 2 3 0

2 1 1 0 2 0 2 0 0 1 0 0 2 4 0

2 2 4 0 2 0 2 0 0 1 1 0 3 4 1

2 3 4 1 3 0 2 0 0 1 2 0 4 6 2

3 4 5 2 4 1 3 1 0 4 3 0 5 7 2

4 4 5 3 5 2 4 2 0 4 3 1 5 7 2

4 5 5 4 7 3 4 2 1 4 10 2 6 9 3

5 6 5 5 7 5 4 3 2 4 14 2 6 9 3

5 7 6 5 9 6 5 4 6 5 16 3 6 10 4

5 7 7 5 10 7 7 6 8 7 18 3 9 11 55 8 10 11 13 12 12 8 10 10 18 4 18 13 5

6 10 10 12 13 18 17 9 18 14 18 6 18 17 7

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Table A.7: Ranked changes in individual number of c from rounds 6~14 to 15~23

OP RM WH

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

-2 -4 -4 -5 -2 -1 -2 -6 -2 -4 -1 -2 -6 -3 -3-2 -3 -3 -2 -1 0 -1 -3 0 -2 -1 -2 -3 -2 -2

-2 -2 -2 -1 -1 0 -1 -2 0 -1 0 -1 -2 -2 -2

-2 -2 -1 -1 -1 0 -1 -1 0 -1 0 -1 -1 -1 -2

-1 -1 0 -1 0 0 0 -1 0 0 0 0 0 -1 -1

-1 -1 0 0 0 0 0 0 0 0 0 0 0 -1 -1

-1 0 0 0 1 0 0 0 0 0 0 0 0 -1 0

-1 0 0 0 1 0 0 0 0 0 0 0 0 -1 0

-1 0 1 0 1 0 0 0 0 1 0 0 0 1 0

0 1 1 0 1 0 0 0 0 1 0 0 1 1 0

1 1 1 0 2 1 1 0 0 1 1 0 1 3 01 2 2 1 2 1 1 0 0 2 2 0 2 4 1

2 2 5 3 3 1 3 0 1 4 4 0 2 4 1

3 7 8 6 5 2 3 0 2 4 6 3 6 7 1

Note: Positive numbers indicate an increase in cooperation moves.

Graph of Linear T vs. Payoff (OP, r=0.1330)

Tvalue

payoff

0 1 2 3

3

4

5

6

Figure A-1. Linear T vs. payoff for OP

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Graph of Linear T vs. Payoff (RM, r=-0.5797)

Tvalue

payoff

0 1 2 3 4 5

2

3

4

5

6

7

Figure A-2. Linear T vs. payoff for RM

Graph of Linear T vs. Payoff (WH, r=0.7599)

Tvalue

payoff

0 1 2 3 4 5

3

4

5

6

7

8

Figure A-3. Linear T vs. payoff for WH

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Chun Lei Yang