Department of Economics, UCSB
UC Santa Barbara
Title:
Endogenous Transfers in the Prisoner's Dilemma Game: An Experimental Test Of Cooperation
And Coordination
Author:
Charness, Gary B , Economics Department, University of California, Santa Barbara
Qin, Cheng-Zhong , University of California, Santa Barbara
Publication Date:
02-06-2005
Series:
Departmental Working Papers
Publication Info:
Departmental Working Papers, Department of Economics, UCSB, UC Santa Barbara
Permalink:
http://escholarship.org/uc/item/9cm846c4
Keywords:
Prisoner's dilemma, Endogenous transfer payments, Compensation mechanism, Coase theorem,
Coordination games, Equilibrium selection
Abstract:
We study experimentally a two-stage compensation mechanism for promoting cooperation in
prisoner's dilemma games. In stage 1, players simultaneously choose binding non-negative
amounts to pay their counterparts for cooperating in a given prisoner's dilemma game, and then
play the prisoner's dilemma game in stage 2 with knowledge of these amounts. For the asymmetric
prisoner's dilemma games we consider, all payment pairs consistent with mutual cooperation in
subgame-perfect equilibrium transform these prisoner's dilemma games into coordination games,
with both mutual cooperation and mutual defection as Nash equilibria in the stage-2 game. We
find considerable empirical support for the mechanism, as cooperation is much more common
when these endogenous transfer payments are feasible. We identify patterns among transfer pairs
that affect the likelihood of cooperation. Mutual cooperation is most likely when the payments
are identical; it is also substantially more likely with payment pairs that bring the payoffs from
mutual cooperation closer together than with payment pairs that cause them to diverge. There
is substantial scope for this compensation mechanism to achieve beneficial social outcomes in
commerce and in international affairs, and reason to be concerned about the ability of firms to
design collusive agreements.
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Endogenous Transfers in the Prisoner's Dilemma Game: An
Experimental Test Of Cooperation And Coordination
Gary Charness, Guillaume Frechette & Cheng-Zhong Qin
February 6, 2005
Abstract: We study experimentally a two-stage compensation mechanism for promoting
cooperation in prisoner's dilemma games. In stage 1, players simultaneously choose binding
non-negative amounts to pay their counterparts for cooperating in a given prisoner's dilemma
game, and then play the prisoner's dilemma game in stage 2 with knowledge of these amounts.
For the asymmetric prisoner's dilemma games we consider, all payment pairs consistent with
mutual cooperation in subgame -perfect equilibrium transform these prisoner's dilemma games
into coordination games, with both mutual cooperation and mutual defection as Nash equilibria
in the stage-2 game. We find considerable empirical support for the mechanism, as cooperation
is much more common when these endogenous transfer payments are feasible. We identify
patterns among transfer pairs that affect the likelihood of cooperation. Mutual cooperation is
most likely when the payments are identical; it is also substantially more likely with payment
pairs that bring the payoffs from mutual cooperation closer together than with payment pairs that
cause them to diverge. There is substantial scope for this compensation mechanism to achieve
beneficial social outcomes in commerce and in international affairs, and reason to be concerned
about the ability of firms to design collusive agreements.
Keywords: Prisoner's dilemma, Endogenous transfer payments, Compensation mechanism,
Coase theorem, Coordination games, Equilibrium selection
JEL Classifications: A13, B49, C72, C78, C91, K12
Charness is at the Department of Economics, University of California Santa Barbara, Santa Barbara, CA 93106-
9210, http://www.econ.ucsb.edu/~charnes s/, email: charness(S),econ.ucsb.edu , fax: 805-893-8830; Frechette is at
New York University, Department of Economics, 269 Mercer Street, 7 th floor, New York, NY 10003,
http://www.people.hbs.edu/gfrechette/html/econ.htm . e-mail: gfrechette@hbs . edu , and Qin is at the Department of
Economics, University of California Santa Barbara, Santa Barbara, CA 93106-9210,
http://www.econ.ucsb.edu/faculty/~qin/ . email: qin@econ.ucsb.edu. Financial support from the UCSB Academic
Senate is gratefully acknowledged.
1. Introduction
The prisoner's dilemma is by far the most famous example of a game with a unique
Pareto-inefficient Nash equilibrium. The chief characteristic of this game is that while there are
substantial gains that could be attained through cooperation, non-cooperation (defection) is
dominant for each player. The theoretical result is that all players defect, even though joint
defection leaves each player with less than he or she could have obtained through mutual
cooperation. A multitude of experiments have been conducted on the prisoners' dilemma (see
Rapaport and Chammah 1965, Dawes 1980, and Roth 1988 for surveys of these experiments).
The central finding in these studies is that mutual cooperation is indeed rather rare in the
prisoner's dilemma. Since players always do better with respect to their individual payoffs by
defecting, few people elect to cooperate in this environment, leading to poor social outcomes. It
is thus desirable to design mechanisms that will implement the efficient outcome.
Coase (1960) presents an example involving a rancher and a farmer, in which the
rancher's cattle stray onto the farmer's property and damage the crops beyond the benefit to the
rancher. Coase argues that even if the rancher's cattle are legally allowed to trespass, the
efficient outcome, as in the case where the rancher's cattle are legally prohibited from
trespassing, will still result because the farmer would then have an incentive to pay the rancher to
cooperate (reducing the number of straying cattle). That is, with well-defined property rights, no
transaction costs, and fully symmetric information, efficiency is neutral to the assignment of
responsibilities for damages; this result has come to be called the Coase theorem.
Varian (1994) presents a general two-stage compensation mechanism that can be seen as
being complementary to the Coasian approach. It implements efficient outcomes through
subgame-perfect equilibria in a wide range of environments with externalities, including
1
prisoner's dilemma games with certain specifications of the payoffs. The mechanism provides a
formalization of bargaining involved in the Coase theorem, and it does not involve a regulator or
central planner mandating taxes or transfer payments; instead it relies upon the parties to design
transfer payments that leads to the efficient outcome. In essence, the prisoner's dilemma can be
seen as an environment with a two-sided externality.
Applying this mechanism to a prisoner's dilemma game, each party would make a
binding pre-play offer to pay the other for cooperating in stage 1 ; upon observing these offers,
each party then chooses to cooperate or to defect in the prisoner's dilemma game in stage 2. A
natural solution concept is subgame-perfect equilibrium (henceforth SPE); while one wishes to
offer enough to induce the other to cooperate, it is best to offer the minimum amount that is
required to achieve this goal. 2 Qin (2002) characterizes the conditions on payment pairs that are
necessary and sufficient to "induce the players to cooperate" (to be defined shortly).
To illustrate the mechanism, consider this example:
Figure 1: A prisoner's dilemma game
Player 2
C
D
c
10, 13
2, 15
D
13,2
7,6
1 Moore and Repullo (1988) show that, given certain assumptions, almost all choice rules can be implemented by
subgame-perfect equilibria. The compensation mechanism seems to be considerably simpler than the examples
provided in Moore and Repullo (1988).
2 See also Ziss (1997) where it is shown that the efficient outcome is not among the set of possible SPE outcomes
for certain prisoner's dilemma games.
3 Jackson and Wilkie (2003) consider more general strategy-dependent transfer payments that players may offer to
each other before playing a game in strategic form. For example, with a prisoner's dilemma game, players can offer
among permissible transfer payments those that will be carried out only when both players defect or only when
player 1 defects while player 2 cooperates, and so on. The main result of their paper is a complete characterization
of supportable equilibrium payoffs rather than transfer payments.
2
Suppose player i (i=l,2) offers to pay an amount H t to player j on the condition that player j
cooperates. Then, if the payments Hi and H 2 are large enough, mutual cooperation will follow in
the subsequent play of the prisoner's dilemma game. For example, if Hi = H 2 = 10, then players'
payoffs in the subsequent play will be:
Figure 2: Transformation of the game in Figure 1 by payment pair (10,10)
Player 2
C
D
c
10, 13
12,5
D
3, 12
7,6
In this case, (C,C) is the unique Nash equilibrium. However, the strategy of choosing a payment
of 10 and then subsequently cooperating cannot survive subgame-perfection. To see this,
suppose that player 2 chooses payment 10. Then, cooperating is dominant for each player in the
subsequent play of the prisoner's dilemma game as long as player 1 chooses a payment greater
than 4. Consequently, player 1 can increase her payoff by lowering her payment to, say 5.
We test the compensation mechanism experimentally, using three parameterizations of
the prisoner's dilemma, and find substantial success: We observe cooperation rates of 43% -
68% when transfer payments are permitted, compared to 1 1% - 18% when transfer payments are
not feasible (almost a fourfold decrease). We also find distinct patterns in cooperation rates;
factors include whether mutual cooperation is a (strict) Nash equilibrium in the subgame induced
by the chosen transfer payments and whether transfer payments make the payoffs from mutual
cooperation closer together or further apart than without the transfers. Cooperation rates are
highest when qualifying transfer payment pairs are identical, while transfer payments that cause
3
the gap in the mutual-cooperation payoffs to grow are less effective than transfer payments that
cause the gap to shrink.
Our study has bearing on issues such as contractual performance and breach, where each
party posts a reward for the other party's performance (or deposits a bond) in an escrow held by
a neutral third party. 4 In the field, this is observed in real estate and construction matters, where
performance bonds and escrows are the rule. Side payments can be seen in international fishing
and international pollution 'contracts'. Alternatively, this could also be relevant for the
provision of public goods, if the parties make pledges conditional on completion of the project.
Agreements to make contributions contingent upon other contributions are seen in many forms
of fundraising, including public television and radio.
In addition, our results have clear legal implications for both positive and negative
interpretations of the prisoner's dilemma. By positive interpretations, we mean situations where
the prisoner's dilemma is a reduced form for cooperation being socially-beneficial, such as in
public-goods games. The success of the mechanism suggests that it may not be necessary for the
legislative authority to attempt to directly implement mutual cooperation. By negative
interpretations, we mean situations where the prisoner's dilemma is a reduced form for a case
where cooperation hurts society, such as with collusion in Cournot quantity competition. In this
case, our results suggest that the legislative authority should be quite careful to ensure that side
contracts are illegal.
2. Background and Theory
There have been several laboratory tests of the Coase theorem, beginning with Hoffman
and Spitzer (1982) and Harrison and McKee (1985). In these studies, there is an optimal choice
4 Williamson (1983) discusses the merits of crafting ex ante incentive structures for the prisoner's dilemma.
4
of lotteries, in terms of total (expected) social payoffs. However, one agent, who has an
individual incentive not to choose the optimal lottery, controls the lottery chosen after the parties
can contract over side payments. These studies generally find that the parties are able to contract
effectively. 5 Nevertheless, as the contracting problem in our case is much more difficult, given
its two-sided nature, in some sense the prisoner's dilemma is a more challenging test.
Andreoni and Varian (1999) were the first to experimentally test the performance of the
compensation mechanism in the prisoner's dilemma. The cooperation rate nearly doubles with
feasible transfers in their game, from 26% to 50%, showing considerable effectiveness for the
mechanism. To some degree, our study follows in their footsteps; nevertheless our design
permits us to go beyond their study in at least two important respects. First, we consider games
where there is a substantial range of payment pairs that induce the players to cooperate in SPE.
While mutual cooperation is predicted with all qualifying payment pairs, it may be that we can
identify factors that behaviorally enhance or inhibit mutual cooperation. In contrast, with integer
payments there is a unique SPE in the game considered in Andreoni and Varian (1999), making
it difficult to discover any such patterns.
Second, the SPE payments in Andreoni and Varian lead to a solution in dominant
strategies, facilitating cooperation given that sufficient transfer values have been chosen. In
comparison, payment pairs required for inducing cooperation transform the experimental
prisoner's dilemma games in the present paper into coordination games between the players, in
which there are two distinct Nash equilibria, (C,C) and (D,D). In our games, mutual defection
can typically be ruled out as part of an equilibrium strategy, as is explained later in the paper.
Nevertheless, this analysis requires fairly sophisticated reasoning unlikely to manifest in an
5 However, Kahneman et al. (1990) find substantially less trading of consumption goods than the level predicted by
the Coase theorem, attributing the gap to the endowment effect.
5
experimental game. This would seem to be a substantially more difficult task than selecting
mutual cooperation when it is the unique Nash equilibrium in the second stage of the game, as in
Andreoni and Varian (1999). Thus, ceterus paribus, one might expect cooperation rates to be
lower in our case.
To describe the compensation mechanism more formally, consider a situation that is
captured by a prisoner's dilemma game in Figure 3 below:
Figure 3: The generic prisoner's dilemma game
Player 2
Player 1
C
D
c
Ri, R2
Si, T 2
D
T U S 2
Pi, Pi
S k , - S t whenever P t - S t > T~ R u (4)
6 In real joint ventures, of course one may not know ex ante the precise values of the exogenous parameters.
However, these values can be considered in expected terms, if we assume that the firms are risk neutral. Similarly,
it may not be feasible to perfectly monitor effort, but we might well be able to observe factors that are correlated
with this effort, such as specific investments or elements of the firm's performance.
7
The set containing all such payment pairs H* often describes a rectangle. 7 For example,
consider our Game 1, where (Si, Pi, Ri, Tj) = (8, 28, 40, 52) and (S 2 , P 2 , R 2 , T 2 ) = (8, 24, 52, 60).
Applying conditions (1) - (4), we see that 8 20. Given that player 1 plays his or her SPE strategy, by offering to pay H 2
> 20 and by cooperating in the second stage conditional on payment pair (Hi*, H 2 ), player 2's
payoff would become 52 - H 2 + Hi * > 60 - H 2 . But this means that by choosing 20 24. Consequently, given player 1 's strategy in that SPE with payment 8 16
or H 2 > 16, but the action pair (D,D) is also consistent with subgame-perfection for other values
in the SPE-region. 10,11
For purposes of statistical analysis, there is a multiple-observation problem, since each
person plays in 25 periods and interacts with other players during the session. While we account
for this in regression analysis, we also perform non-parametric statistical tests across conditions.
To facilitate these tests, we partitioned the 16 participants in each session into four separate
groups, with the four people in each group interacting only with each other over the course of the
12
session. In this way, we obtain four completely independent observations in each session.
The Experiment
We conducted a series of experiments in nine separate sessions at the University of
California at Santa Barbara. We had three sessions for each of three different prisoner's-
dilemma games. For each game, endogenous transfers were permitted in two of these sessions,
while the third session served as a control. There were 16 participants in each session, with
The reasoning goes as follows: Assume (D, D) is played after a payment pair H* in the SPE region, in which case
player 1 gets 32. Notice that player 1 can make C strictly dominant for player 2 in the action subgame by choosing
any Hi > 28. If she does so, then player 2 plays C and player 1 can thus guarantee herself payoff 44 - Hi + H 2 *.
This payoff is greater than 32 if and only if Hi < 12 + H 2 *. Thus player 1 would have an incentive to change her
payment in the pair H* if there exists a payment that simultaneously satisfy 28 < Hi < 12 + H 2 *. Such a payment
exists if and only if H 2 * > 16. Thus H 2 * > 16 is incompatible with the assumption that (D, D) is induced by payment
pair H*. This shows that (D, D) cannot be induced by a payment pair H* in the SPE region if H 2 * > 16. Similar
reasoning shows that (D, D) cannot be induced by a payment pair in the SPE region if Hi* > 16. In summary, (D,
D) cannot be induced by a payment pair H* in the SPE region if either Hi* > 16 or H 2 * > 16.
11 In this case the mechanism has two subgame-perfect equilibria; we are not aware of any study that investigates
what might be expected as an outcome, depending on structural characteristics of the situation, when there are
multiple equilibria for a mechanism.
12 Subjects were told that they were randomly re-matched, but not that this was done in subgroups.
12
average earnings of about $15 (including a $5 show-up payment) for a one-hour session.
Participants were recruited by e-mail from the general student population. 13
We provided instructions on paper, which were discussed at the beginning of the session;
a sample of these instructions is presented in Appendix A. Our computerized experiment was
programmed using the z-Tree software (Fischbacher 1999). After a practice period, participants
played 25 periods; each person was a Row player in some periods and a Column player in others,
with one's role being drawn at random from period to period, and the person with whom one was
matched also being determined at random from the other members of the subgroup.
Players first learned their roles for the period and then (if cooperation-rewards were
feasible) chose amounts to transfer to their counterparts in the event of their cooperation. After
learning the amounts chosen, both players in a pair then simultaneously chose whether to
cooperate or defect in the subgame, and were then informed of the outcome.
Hypotheses
In this section, we formulate several hypotheses based on the predictions of the theory.
We also explore some of the tensions that may stop these predictions from being realized. First,
given that cooperation is a SPE of the game with transfers, but not of the standard prisoner's
dilemma, we have:
Hypothesis 1: There will be more cooperation in the sessions where players can choose
transfer payments.
13 Since part of what we wish to study are decisions conditional on transfers making cooperation a SPE, to improve
our chances of observing SPE transfers potential subjects were told (in a mass e-mail) that we were particularly
interested in students who either had high grade-point averages or who were majoring in mathematics or the
sciences. We then screened the applicants using these criteria; participants typically were either graduate students or
had GPAs above 3.70. As discussed in Section 5, our results in the standard prisoner's dilemma are similar to those
in other experiments.
13
Since cooperation is a Nash equilibrium in the induced subgame for only some transfer
pairs, we have:
Hypothesis 2: There will be more cooperation when mutual cooperation is a Nash
equilibrium in the subsequent subgame led by the chosen transfer payments.
We next consider whether, given transfer pairs consistent with mutual cooperation being
an equilibrium, there are certain characteristics of transfer pairs that are particularly effective in
leading to mutual cooperation. In principle, the theoretical arguments hold regardless of the
location of a point within the mutual-cooperation Nash or SPE regions. Thus, the hypothesis that
emerges from the theory on this point is:
Hypothesis 3: Given that a transfer induces cooperation, the cooperation rate will not
differ according to any characteristics of the transfer pair.
While the standard arguments predict no differences in behavior for qualifying transfer
pairs, the fact that there are multiple equilibria in the subgame leads us to suspect that secondary
factors will influence the choice of play in the subgame, thereby falsifying Hypothesis 3. For
example, reward pairs that are on the Nash 'border' seem less likely to lead to cooperation.
Consider Game 1, with 8 < Hi* and 12 < H 2 *. Suppose the transfer pair is (H]*,H 2 *) = (12,12),
on the southern border of the Nash or SPE regions. The induced subgame, where both (C,C) and
(D,D) are Nash equilibria, is:
14
Figure 8: Game 1 transformed by {H 1 ,H 2 ) = (12,12)
Player 2
C
D
c
40, 52
20, 48
D
40, 20
28,24
If Player 1 thinks Player 2 is going to cooperate, he stands to get 40 with either C or D; however,
C for Player 1 is weakly-dominated by D in the subgame. Furthermore, Player 2 stands to gain a
lot (32) by Player 1 choosing C over D. In this case, Player 1 may feel unhappy that Player 2 has
chosen to give no incremental reward for cooperative play, while hoping or expecting to reap
large rewards from mutual cooperation. In this sense, a border reward is like a zero offer in the
ultimatum game - a rejection doesn't really cost the rejector anything, but punishes the selfish
party. Thus, border reward pairs may be less effective in achieving cooperation.
All else equal, we might also expect players to be more likely to cooperate when transfer
payments (and thus the rewards for cooperation) are higher, even when all transfer pairs
considered are within the Nash or SPE regions. Here risk-dominance considerations (which
choice does better if the other person randomly chooses whether or not to cooperate) might serve
to help select the equilibrium in the induced coordination game.
Finally, it is possible that social preferences come into play, even for 'qualifying' reward
pairs. Since rewards are simply transfers and do not change the total social payoff contingent
upon the players' actions in the subgame, there is no role for 'efficiency' per se. However, the
reward pairs chosen affect the difference between the players' payoffs (or alternatively, the
minimum payoff) that result from mutual cooperation. We might expect the likelihood of
cooperation to increase as the difference in the net payoffs from mutual cooperation decreases, in
15
line with the Fehr and Schmidt (1999), Bolton and Ockenfels (2000), and Charness and Rabin
(2002) utility models.
Our final hypothesis concerns whether behavior is sensitive to identifiable characteristics
of the three different games we test. In principle, since it is possible in each of these games to
choose reward pairs leading to cooperation in equilibrium, we should see mutual cooperation in
every case that such reward pairs are chosen. Even if this is not the case, we should still see no
difference in the effectiveness of transfer payments in achieving cooperation across these games:
Hypothesis 4: Given that a transfer pair is consistent with mutual cooperation either in
the subgame or as part of a SPE, the cooperation rate will not differ across the three
experimental games. Furthermore, the effectiveness of transfer payments in enhancing
cooperation will not vary across games.
On the other hand, we might intuitively expect to see a relationship between risk and
reward. Rapoport and Chammah (1965) presents some ideas on how the relationships between
the entries in the payoff matrix of the prisoner's dilemma might be expected to influence
cooperation rates (without transfers); however, they only consider a 'symmetric' prisoner's
dilemma, where players have identical payoffs in each cell of the 2x2 game. Nevertheless, the
concept of risk and reward may well influence decisions. One simple idea is to compare the size
of the joint (or individual) payoffs with mutual cooperation to those with mutual defection;
bigger gains from mutual cooperation should translate into more cooperation.
4. Experimental Results and Hypothesis Tests
In this section, we first present a summary of our experimental data. We then provide a
regression analysis of the data. Throughout, we relate the data analysis to the hypotheses
elaborated in Section 3.
16
Data Summary
Figure 9 shows the average cooperation rates by game and treatment:
Figure 9 - Cooperation rates, by game
■ No transfers possible
□ Transfers possible
Game 1 Game 2 Game 3
Cooperation rates are clearly higher in treatments where transfers are allowed; the
comparisons are 15.8% vs. 53.9% for Game 1, 17.5% vs. 68.1% for Game 2, and 10.8% vs.
42.9% for Game 3. All of these differences are highly statistically significant (p-value < 0.01,
one-sided Mann- Whitney test), 14 and are also of significant magnitude. Thus, there is clear
support for Hypothesis 1 .
Hypothesis 4 on the other hand finds mixed support. In the conditions without transfer,
the rates of cooperation are not statistically different (p-values of two-sided Mann- Whitney test >
0.1 in all pair-wise comparisons). 15 This is different from previous results ( Rapoport and
Chammah 1965) and might indicate that the differences in the entries of the prisoner's dilemma
14 When performing hypothesis tests of this sort, we will do it for both subject averages and group averages to
eliminate correlation across time. If results are sensitive to the unit of observation, it will be noted. Otherwise, as in
this particular case, the result holds in both cases.
15 The £>-value of the Kruskal-Wallis test which examines the hypothesis that the samples are from the same
population is also greater than 0. 1 .
17
game are not big enough, that these effects are not stable, or that these effects do not generalize
to non-symmetric games. But when transfers are allowed, we do observe differences. All pair-
wise comparisons of rates of cooperation are statistically different (p-values of two-sided Mann-
Whitney test < 0.1 in all cases). 16
In Figure 10, we consider only those sessions in which transfers were possible, and
display cooperation rates as a function of whether mutual cooperation was a Nash equilibrium in
the subgame induced by the transfer pair chosen:
Figure 10 - Cooperation rates, by reward-pair
consistency with equilibrium
100%-
75%-
50%-
25%-
0%-
Game 1 Game 2 Game 3
In games where transfers are allowed, cooperation rates are lowest if the reward pairs
chosen are not consistent with mutual cooperation being a Nash equilibrium in the subgame.
These differences are statistically significant in all treatments (two-sided ^-values of Sign test <
0.01 in all treatments) thus lending support to Hypothesis 2. Cooperation rates are substantially
(10 to 25 percentage points) higher for reward pairs on the 'border' of the NE-region, with a
further substantial (18 to 38 percentage points) increase for reward pairs in the interior of the
16 The p-va\ue of the Kruskal-Wallis test is less than 0.01.
18
NE-region. This difference between the border and the interior is statistically significant (one-
sided /^-values of Sign test < 0.05 in all treatments). This is the first observation against
Hypothesis 3. Namely, some SPE-consistent reward pairs are less likely than others to lead to
cooperation.
In Figure 1 1 we show the proportions of the reward pairs that were variously consistent
with mutual cooperation in a SPE, were such that mutual cooperation was a Nash equilibrium in
the subgame, or were in the region where the transfer pairs make mutual cooperation the unique
Nash equilibrium:
Figure 11 - Proportion of reward-pairs such that
cooperation is consistent with NE and SPE
100%
■ NE
□ SPE
■ Unique NE
Game 1
Game 2
Game 3
The proportion of joint transfers that make mutual cooperation a Nash equilibrium is
rather high, about 68% across the three games, with a lower proportion in Game 1. Further,
more than half of all endogenous reward pairs were consistent with a SPE involving mutual
cooperation.
We now consider how the likelihood of mutual cooperation is affected by how a reward
pair, consistent with a SPE involving mutual cooperation, affects the difference in net payoffs
19
with mutual cooperation: Table 1 reports the rates of mutual cooperation as a function of
whether transfers make final payoffs closer, further or if they remain the same. This is done for
SPE transfers and SPE transfers excluding the NE border (by border we mean the cases where at
least one of the two subjects is indifferent between cooperation and defection given that the other
person cooperates):
Table 1: Percentages of Mutual Cooperation, by Net Transfer Category
Transfers make MC payoffs:
Diverge
Equal
Closer
Total
All SPE Transfers
Game 1
40% (20)
77% (22)
52% (127)
54% (169)
Game 2
47% (30)
67% (45)
61% (122)
60% (197)
Game 3
23% (120)
35% (37)
32% (109)
29% (266)
SPE Transfers, excluding NE Border
Game 1
60% (10)
94% (17)
65% (101)
69% (128)
Game 2
55% (22)
73% (41)
71% (86)
64% (159)
Game 3
25% (81)
40% (25)
40% (86)
33% (192)
(Number of Observations in Parentheses)
We can see the rate of mutual cooperation for SPE-consistent reward pairs is highest (or
tied for highest) in all cases when these transfers are exactly equal. In four of the six cases,
mutual cooperation is nearly as likely when the reward pair brings the players' mutual-
cooperation payoffs closer together, while in two cases it is substantially less likely. We also see
that mutual cooperation is always least likely when the qualifying reward pair makes the mutual-
cooperation payoffs further apart. This is a second observation against Hypothesis 3, and also
doesn't exactly square with what one might have expected; that is, given the evidence on social
preferences (Fehr and Schmidt 1999, Bolton and Ockenfels 2000, and Charness and Rabin 2002)
one might have expected that cooperation is highest when transfers make final payoffs closer.
20
However, this is not the case. Overall, there were many more qualifying reward pairs chosen
that reduce the difference in mutual-cooperation payoffs than the opposite direction (631 to 283).
Regression analysis
Table 2 reports random-effects probit estimates of the determinants of cooperation where
the regressors are what the subject offers to pay (Would Pay), what the subject is offered (Would
Receive) again interacted with a dummy variable for the case where cooperation should result in
equilibrium, a dummy variable taking value 1 if the transfers are such that they are on the border
of the NE region and otherwise, and an indicator variable for when transfers are equal and one
for when they make final payoffs closer. This is estimated on all transfers such that cooperation
is a NE. The data are separated in two cases: closer is better (CB), meaning all the cases where
having transfers that make final payoffs closer imply that the subjects' own payoffs are higher
(type 1 in Games 1 and 2, and Type 2 in Game 3) and closer is worse (CW), which is the
1 7
opposite (Type 2 in Games 1 and 2, and Type 1 in Game 3). The table also reports the
marginal effects for the average subject at the sample mean of the regressors, except for
dichotomous ones where it gives the difference in probabilities when the variable equals 1 and
when it equals 0.
Appendix C provides results by game and role (row or column). Results are similar but two observations should
be made. First, the NE border effect is not as strong, meaning that although it has a negative impact in all games for
both roles, it is statistically significant in only half of them. Second, the effect of equal transfers is statistically
significant in half the cases, and it has the opposite sign in one of the cases where it is not.
We have also tested to see if controlling for periods affected the results. We have done this by including the
period and the period squared as regressors or by including indicator variables for blocks of five periods. In neither
case did it have any qualitative impact on the results. Furthermore, the majority of coefficient estimates of the effect
of period (for both specifications) were statistically insignificant. In the interest of space these are not included but
are available from the authors on request.
21
Table 2: Determinants of Cooperation
Random-effects probit and marginal-effects estimates in NE region
Closer is Better
Closer is Worse
RE Probit
Marginal
Effects
RE Probit
Marginal
Effects
Would Pay
0.001
0.000
-0.013
-0.004
(0.011)
(0.004)
(0.019)
(0.006)
Would Receive
0.093***
0.031***
0.136***
0.042***
(0.018)
(0.006)
(0.023)
(0.008)
NE Border §
-0.637***
-0.231***
-0.746***
-0.255***
(0.166)
(0.063)
(0.186)
(0.067)
Equal Transfers 5
0.535***
0.157***
0.496**
0.131***
(0.194)
(0.050)
(0.219)
(0.051)
Final Payments are Closer 8
0.603***
0.206***
0.077
0.023
(0.156)
(0.055)
(0.172)
(0.053)
Constant
-1.054***
-0.729
(0.346)
(0.485)
Observations
820
820
820
820
Number of Subjects
94
94
96
96
Standard errors in parentheses * significant at 10%; ** significant at 5%; *** significant at 1%
§ Marginal effects report the change in probability when the regressor goes from to 1 .
Clearly the amount one is offered matters. That regressor is always positive and
statistically significant. This is a third observation that is not in line with Hypothesis 3.
However, how much one would pay never has a statistically-significant impact. If the transfers
are exactly on the NE border, then cooperation is less likely, confirming what we had already
noticed from the summary statistics. If the transfers are exactly the same, then cooperation is
more likely. Finally, cooperation is more likely if transfers narrow the payoff difference from
mutual cooperation, but only when the difference of transfers is in the player's favor. These last
observations also do not square with Hypothesis 3, nor with what one might expect given the
current models of social preferences. That is, since the only condition where equality seems to
matter is when it is self-advantageous, people do not seem willing to forego pecuniary benefits in
favor of greater equality of distributions. On the other hand, they do not necessarily prefer to
increase inequality even if it favors them.
22
The marginal effects inform us of each factor's relative importance. Would receive
averages 14 and 13 in CB and CW respectively (with standard deviations of 5 and 7). Thus, in
the case of CB, increasing the amount a subject is offered by one standard deviation has the same
impact as making the transfers equal, increasing the probability of cooperation by 15 percentage
points. Making the transfers such that final payoffs are closer increases cooperation by 21
percentage points while being on the NE border reduces cooperation by 23 percentage points.
The effects are similar in CW except that the effect of what one is offered is greater, while the
effect of having equal transfers is slightly less; having closer final payoffs has no impact.
Thus far we have analyzed determinants of individual behavior. Such behavior has
implications for the groups, and we now turn to a more detailed analysis of the factors affecting
mutual cooperation. We estimate a random-effects probit model of the determinants of mutual
cooperation, reported in Table 3. 19 Besides indicator variables for equal transfers and transfers
that make final payoffs closer, these include a dummy for the NE border, and a dummy for the
sum of transfers. The same specification is estimated pooling all games, with indicator variables
for game 1 and game 2 included and marginal effects reported. The analysis will focus on the
pooled results.
For binary regressors this reports the change in probability when the regressor goes from to 1 (keeping all other
regressors at their sample mean).
19 Of course these possibly imply complicated correlation structures across observations since they are a single
observation for a pair of subjects. The estimates reported use the pair as the random effects. Estimates where
individuals are taken as the random effect are provided in Appendix D. All results are qualitatively unchanged.
23
Table 3: Determinants of mutual cooperation
Random-effects probit and marginal-effects estimates in SPNE region
Game 1
Game 2
Game 3
All Games
All Games:
Marginal
NE Border §
-2.224***
-1 294***
-0.080
-0.943***
-0.321***
(0.527)
(0.350)
(0.353)
(0.210)
(0.060)
Sum of Transfers
0.053
-0.001
0.102***
0.067***
0.026***
(0.053)
(0.041)
(0.030)
(0.021)
(0.008)
Equal Transfers 8
0.895
0.170
0.682*
0.543**
0.213**
(0.637)
(0.390)
(0.356)
(0.236)
(0.092)
Final Payments are Closer 8
-0.296
0.401
0.609**
0.350**
0.132**
(0.465)
(0.316)
(0.297)
(0.188)
(0.070)
Game 1 §
0.666***
0.259***
(0.237)
(0.090)
Game 2 §
1.268***
474***
(0.247)
(0.081)
Constant
-0.782
0.350
-3.808***
-2.526***
(1.598)
(0.995)
(0.928)
(0.634)
Observations
169
197
266
632
632
Number of Group
69
66
85
220
220
Standard errors in parentheses * significant at 10%; ** significant at 5%; *** significant at 1%
§ Marginal effects report the change in probability when the regressor goes from to 1 .
Being on the NE border statistically hurts mutual cooperation, decreasing its probability
by 32 percentage points and making it one of the most important effects in magnitude. The sum
of the transfers also has a statistically-significant effect on the probability of mutual cooperation.
On average, the transfers sum to 24, with a range of 16 to 42. Hence going from lowest to
largest would affect cooperation by 68 percentage points and a movement of one standard
deviation (which is 5) would affect the probability by 13 percentage points. Having equal
transfers make cooperation more likely than transfers that make final payoffs further apart by 21
percentage points while making them closer only increase by 13 percentage points. Game 2 has
a higher probability of mutual cooperation, other things equal than all other games and Game 1
has a higher probability of mutual cooperation than Game 3. The significant coefficients for the
24
Game 1 and Game 2 dummies in the rightmost columns are evidence against Hypothesis 4, as
cooperation rates do vary across our games when transfers are permitted.
5. Discussion
We find a much higher rate of cooperation when players can choose contingent rewards
for cooperation than when they cannot, in three different asymmetric variants of the prisoner's
dilemma. The increased cooperation rate occurs not simply because transfers are feasible, but
also depends substantially on the values of the reward pairs. In all of our transfer games,
cooperation rates are at least double for reward pairs in the interior of the NE transfer region than
for reward pairs that lead to mutual cooperation not being a Nash equilibrium. Furthermore,
reward pairs on the border of the NE-region lead to intermediate levels of cooperation that are
significantly different from both the low levels with no transfers permitted or the higher levels in
the interior of the transfer region.
An innovative feature of our design is that the endogenous SPE reward pairs induce
coordination games in which mutual cooperation and mutual defection are both Nash equilibria.
While mutual defection can be ruled out as a SPE action pair for all SPE reward pairs in Games
1 and 2 (and nearly half of the feasible SPE reward pairs in Game 3), doing so requires
somewhat sophisticated arguments. Nevertheless, we see fairly high cooperation rates, with a
range of between 42.9% and 68.1% in the three games; this compares to the 50.5% cooperation
rate for the game in Andreoni and Varian (1999), which features cooperation being a dominant
strategy for the unique (in integers) SPE-consistent reward pair.
Our cooperation rates when transfers were not permitted ranged from 11% to 18%; this is
in line with rates of cooperation observed in previous studies - Roth and Murnigham (1978),
25
Cooper, DeJong, Forsythe, and Ross (1996) and Andreoni and Miller (1993) observed 10%,
25%, and 18% cooperation rates, respectively. As the individual cooperation rate in the no-
transfer game in Andreoni and Varian (1999) was 26% and so higher than our base rates, it
appears that endogenous transfers were at least as effective, relative to the no-transfer case, in
enhancing cooperation via induced coordination games as when mutual cooperation is the unique
Nash equilibrium in the subgame.
We find that cooperation is substantially more likely when it happens that the sufficient
rewards chosen are equal. In a sense, this effect of the equality of payoffs on cooperation seems
intuitive and focal, but it is nevertheless completely outside the current economic models of
social preferences. This effect could possibly be seen as a mutual recognition of the problem, or
as a simplification in the process that makes it seem more likely that the other player will
cooperate. Perhaps there is something attractive about reaching the original targets of the
payoffs from mutual cooperation. This could be similar to firms that collude or have mutually
beneficial arrangements in which they try to control market shares.
A more standard form of social preference appears to also influence behavior, as we see a
player is more likely to cooperate when the reward pair decreases the difference between the
players' payoffs with mutual cooperation. However, this only has a significant effect when
equality favors the chooser, so that one can perhaps view this as a form of self-serving bias.
Reward pairs that reduce the difference in the payoffs from mutual cooperation are more than
twice as common as reward pairs that cause this difference to increase.
Looking across our three experimental games, we see some substantial differences in
rates when contingent rewards are feasible. The size of the gains from mutual cooperation
20 "In the case of the JEC [Joint Executive Committee], the cartel took the form of market share allotments rather
than absolute amounts of quantities shipped" (Porter 1983).
26
relative to the size of the mutual defection payoffs does seem to correspond to the cooperation
rates in our games. These gains are 40 in both Games 1 and 2, compared to 20 in Game 3, and
the ratio of these gains to the sum of the payoffs from mutual defection is 0.77, 0.91, and 0.33,
respectively; the corresponding cooperation rates without contingent rewards are 15.8%, 17.5%,
and 10.8%. There is also considerable differentiation among these games when transfers are
allowed, with overall cooperation rates of 53.9%, 68.1%, and 42.9% in Games 1, 2, and 3,
respectively. Perhaps one explanation for this stems from the symmetry of the SPE-region in
Game 2, as seen in Appendix B. In our experiments, players are frequently changing roles;
21
perhaps the symmetry makes it easier to use the same transfer choice or to identify it.
There is an additional dimension that differentiates Game 3 from the other two games.
Recall that mutual cooperation is the only action pair consistent with SPE transfers in Games 1
and 2, but that mutual defection is also consistent with SPE transfers in Game 3 whenever
neither Hi nor H2 is larger than 16 (as is true for more than 75% of the chosen transfer pairs in
the SPE-region). In any event, it seems plausible that the increased uncertainty with lower
transfers acts as a damper on the attraction toward mutual cooperation in Game 3, helping to
explain the lower rate of mutual cooperation seen in Table 1.
The success of the reward mechanism in our induced coordination games is achieved
with contingent rewards that are much smaller than in games where there is a unique SPE reward
pair that leads to mutual cooperation being the only Nash equilibrium in the induced subgame.
For example, in Andreoni and Varian (1999), the transfer payments required for SPE in integers
is nine, compared to total payoffs of seven with mutual defection (78%). In contrast, in Game 1
21 In fact, there were 27 instances where the reward pair in Game 2 was (9,9), the minimum point inside the SPE
region (of which 23 resulted in mutual cooperation); the corresponding minimum point in Game 1, (9,13), is less
'focal' and is never observed.
27
the equivalent total transfer required is 22, compared to the mutual-defection total payoffs of 52
(42%). In Game 2, the comparison is 18 to 44 (41%); in Game 3, the comparison is 18 to 60
(30%). In this sense, perhaps the players could be said to achieving success with less exposure.
Nevertheless, we hesitate to draw any strong conclusion in this regard, as the populations are
different and there may well be other uncontrolled sources of variation as well.
6. Conclusion
Achieving cooperation in the Prisoner's Dilemma has challenged theorists for decades.
Most previous studies have focused on repetitions for solutions: either finite (see Bereby-Meyer
and Roth 2003) or infinite (see Aoyagi and Frechette 2004, Dal Bo 2002, and Duffy and Ochs
2003). We test experimentally an endogenous reward mechanism described in Varian (1994)
and further elaborated in Qin (2002), where 'gifts' are contingent upon cooperation in a one-shot
environment. Typically, the contracting parties bind themselves in a way that usually leads to
the efficient outcome in Games 1 and 2, with cooperation rates roughly quadrupled in all three
experimental games. This increase in efficiency occurs despite the fact that the reward pairs
consistent with mutual cooperation being a subgame -perfect action pair induce a coordination
game rather than a subgame in which mutual cooperation is the unique Nash equilibrium, as with
the game in Andreoni and Varian (1999). Our results provide support for the Coase theorem in
a very difficult environment with a two-sided externality.
Our games have a substantial range of integer transfer pairs consistent with mutual
cooperation being part of a SPE, and so we can examine patterns within this region to see which
ones tend to be beneficial in fostering mutual cooperation and efficiency. Our analysis suggests
at least two main prescriptions: First, sufficient transfers that make cooperation a strict best
28
response to expected cooperation are considerably more effective than transfers that lead to
indifference with expected cooperation. Second, sufficient transfers that are identical are
particularly effective, followed by transfers that narrow the gap between the players' payoffs in
the event of mutual cooperation; avoid reward pairs that make these payoffs diverge.
Despite the difficulties inherent with asymmetric payoffs and sophisticated inferences,
we observe a reasonably high degree of cooperation in our games, achieved with rewards for
cooperation that are modest in size relative to the payoffs in the game. This seems a hopeful sign
for efficiency in contracting, as the choice of play in a coordination game is not as obvious as
when one has a dominant strategy. There is substantial scope for this compensation mechanism
to achieve beneficial social outcomes in commerce and in international affairs, and reason to be
concerned about the ability of firms to design collusive agreements.
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Fehr, E., and K. Schmidt (1999). "A Theory of Fairness, Competition, and Cooperation."
Quarterly Journal of Economics, 114, 817-868.
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Harrison, G. and M. McKee (1985), "Experimental evaluation of the Coase theorem," Journal of
Law and Economics, 28, 653-670.
Hoffman, E. and M. Spitzer (1982), "The Coase theorem: Some experimental tests," Journal of
Law and Economics, 25, 73-98.
Jackson, M. and S. Wilkie (2003), "Endogenous games and mechanisms: Side payments among
Players," mimeo.
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and the Coase theorem," Journal of Political Economy, 98, 1325-1348.
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1220.
Porter, M. (1983), "A Study of Cartel Stability: the Joint Executive Committee, 1880-1886," The
Bell Journal of Economics, 14, 301-314.
Qin, C.-Z. (2002), "Penalties and rewards as inducements to cooperate," UCSB Working Paper
in Economics #13-02 (available on line at www.econ.ucsb.edu/~qin).
Rapoport, A and A. Chammah (1965), Prisoner's Dilemma, Ann Arbor: University of Michigan
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Roth, A. (1988), "Laboratory experimentation in economics: A methodological overview,
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30
APPENDIX A
INSTRUCTIONS
Thank you for participating in our experiment. You will receive $5 for showing up on
time, plus you will receive your earnings from the choices made in the session.
There will be 25 periods. In each period, each person will be matched with one other
person. The person with whom you are matched will be randomly re-drawn after every period.
You are paired anonymously, which means that you will never learn the identity of the other
person in any of the periods.
One person will have the role of ROW and the other person will have the role of
COLUMN. Your role will also be randomly re-drawn in each period, so that sometimes you will
have the role ROW and sometimes you will have the role of COLUMN.
Here is the basic game:
COLUMN
Left Right
ROW Up
Down
40, 52
8, 60
52,8
28,24
The ROW and COLUMN players make choices simultaneously . The ROW player chooses Up
or Down; the COLUMN player chooses Left or Right.
The 1st number in each cell refers to the payoff (in cents) for the ROW player, while the 2 nd
number in each cell refers to the payoff (in cents) for the COLUMN player. Thus, for
example, if ROW chooses Up and COLUMN chooses Left, the ROW player would receive 40
and the COLUMN player would receive 52.
However, before these game choices are made, ROW may choose a binding amount to be paid
{transferred) by him or her to COLUMN if and only if COLUMN chooses Left. COLUMN (at
the same time) may offer a binding amount to be transferred to ROW if and only if ROW
chooses Up. These amounts must be non-negative integers.
The amounts that you each choose will be communicated to each of you prior to your choices in
the game above. You will then make your game choice (Up or Down if you are ROW, or Left or
Right if you are COLUMN). You will then learn your payoff for the period, from which you can
infer the game choice made by the person with whom you are paired.
This completes one period of play. We'll do 25 periods and pay people individually and
privately.
31
FURTHER EXPLANATION
Offers to pay money contingent on the other person choosing Up (or Left, if the other person is a
COLUMN player) have the effect of changing the payoff matrix. Note that whatever amount
you state will be transferred to the other person if he or she plays Up as a ROW player or Left as
a COLUMN player; this money will be transferred regardless of your game choice.
Suppose, for example, that ROW offers to pay $x to COLUMN if COLUMN plays Left and
COLUMN offers (independently and simultaneously) to pay $y to ROW if ROW plays Up.
Then the payoff matrix becomes:
COLUMN
Left
ROW
Up
Down
Right
40 + y - x, 52 + x - y
8 + y, 60 - y
52-x,8+x
28,24
We explain the 4 possible outcomes below. Remember, the values of x and y are always
determined by the ROW and COLUMN players, respectively, before making game choices.
1) If ROW chooses Up and COLUMN chooses Left, then ROW must pay x units to
COLUMN and COLUMN must payj units to ROW. Thus, ROW would receive 40 + y -
x and COLUMN would receive 52 + x - y.
2) If ROW chooses Up and COLUMN chooses Right, then COLUMN must pay y units to
ROW, but ROW pays nothing to COLUMN (because COLUMN did not choose Left).
Thus, ROW would receive 8 + y and COLUMN would receive 60 -y.
3) If ROW chooses Down and COLUMN chooses Left, then ROW must pay x units to
ROW, but COLUMN pays nothing to ROW (because ROW did not choose Up). Thus,
ROW would receive 52 - x and COLUMN would receive 8 + x.
4) If ROW chooses Down and COLUMN chooses Right, then neither player pays the other
anything. Thus, ROW would receive 28 and COLUMN would receive 24.
We don't wish to illustrate this with an example with realistic numbers, as this could bias your
behavior. However, we can use an example where x = 999 and y = 1000. (We don't expect
anyone to choose these values forx and y.) In this case, the payoff matrix becomes:
COLUMN
Left Right
ROW Up
Down
41, 51
1008, -940
-947, 1007
28, 24
We encourage people to work out scenarios on paper, drawing a game matrix for each
possibility.
Are there any questions? Please feel free to ask, by raising your hand.
32
APPENDIX B
Transfer-pair regions consistent with (C,C) being a subgame-perfect action pair
Game 1
(8,20) (16,20)
H 2
(8,12)
(16,12)
Hi
Game 2
(8,16) (16,16)
(8,8)
(16,8)
Hi
Game 3
(16,24)
(28,24)
H,
(8,16)
(28,16)
(8,8)
(20,8)
Hi
33
APPENDIX C
Determinants of cooperation
Random-effects probit estimates in NE region
Game 1:
Row
Game 1:
Column
Game 2:
Row
Game 2:
Column
Game 3:
Row
Game 3:
Column
Would Pay
-0.03
-0.013
0.056
-0.011
-0.013
0.029
(0.019)
(0.049)
(0.056)
(0.036)
(0.029)
(0.026)
Would Receive
0.102*
0.151***
0.113**
0.012
0.172***
0.076***
(0.054)
(0.055)
(0.050)
(0.051)
(0.033)
(0.022)
NE Border
-0.562
-1 359***
-0.990***
-1.087***
-0.067
-0.45
(0.371)
(0.386)
(0.344)
(0.326)
(0.280)
(0.288)
Equal Transfers
0.935*
0.054
0.432
-0.435
0.711**
0.522*
(0.535)
(0.556)
(0.362)
(0.486)
(0.315)
(0.301)
Final Payments are Closer
-0.262
-0.368
1.129***
-1.060**
0.425*
0.725***
(0.353)
(0.372)
(0.393)
(0.438)
(0.249)
(0.247)
Constant
-0.185
-0.349
-1.702*
2.055**
-2.085***
-1.671***
(0.939)
(1.304)
(0.946)
(0.986)
(0.636)
(0.578)
Observations
228
228
298
298
294
294
Number of Subjects
31
32
31
32
32
32
Standard errors in parentheses * significant at 10%; ** significant at 5%; *** significant at 1%
34
APPENDIX D
Determinants of mutual cooperation
Random-effects probit with one way subject error terms
and marginal-effects estimates in SPE region
Game 1
Game 2
Game 3
All Games
All Games:
Marginal
NE Border
-2.138***
-1.275***
-0.113
-0.936***
-0.321***
(0.460)
(0.328)
(0.286)
(0.183)
(0.053)
Sum of Transfers
0.035
0.035
0.076***
0.057***
0.022***
(0.045)
(0.050)
(0.024)
(0.018)
(0.007)
Equal Transfers
0.875
0.183
0.512*
0.487**
0.191**
(0.561)
f (0.363)
(0.281)
(0.199)
(0.078)
Final Payments are Closer
-0.179
0.357
0.468**
0.340**
0.129**
(0.411)
(0.297)
(0.219)
(0.157)
(0.058)
Game 1
0.608***
0.237***
(0.214)
(0.082)
Game 2
1 .008***
0.385***
(0.212)
(0.075)
Constant
-0.401
-0.535
-2.858***
-2.146***
(1.387)
(1.198)
(0.721)
(0.543)
Observations
169
197
266
632
632
Number of Groups
28
28
32
88
88
Standard errors in parentheses * significant at 10%; ** significant at 5%; *** significant at 1%
35