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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
cly@gate.sinica.edu.tw, 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.
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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