An Experiment on Cooperation in a CPR Game with a Disapproval Option

Abstract

This paper studies the standard version of the approval mechanism with two players in a common pool resource (CPR) extraction game. In the case of disapproval, the Nash extraction level is implemented. The paper investigates, experimentally, the extent to which the Nash threat leads to Pareto-improving extraction levels. Through our experiment, we confirm the effectiveness of the Nash threat in reducing CPR over-extraction. Although participants’ behavior is mainly explained by rational thinking, inequity in payoff can also motivate their behavior. Moreover, we show that there is neither an order effect nor a framing effect. Finally, the reduction persists when the Nash threat is no longer in place.

1. Introduction

People interact constantly with their peers in social life. Within these social interactions, many situations arise in which selfish, rational agents are unwilling to sacrifice their benefit to promote cooperation and achieve greater social benefits [1,2]. Thus, in the absence of explicit incentives, self-interest tends to dominate social interest and lead to inefficient outcomes, a situation called a social dilemma [3,4,5,6]. In the context of a common pool resource (CPR) game, the social dilemma leads to an over-extraction of the resource [7,8,9,10,11,12,13,14,15]. We experimentally investigate the extent to which the approval mechanism (AM hereafter) can mitigate over-extraction when the disapproval benchmark is the Nash equilibrium. Ref. [16] showed that the AM solves the social dilemma in the case of public good provision. Moreover, ref. [17] documented that the AM is also Pareto-improving in the case of a CPR game. The AM can be thought of as a way to promote cooperation between players by introducing an agreement on a collective sanction in case of disapproval. The underlying principle of the AM is therefore closely related to cooperative games. Indeed, ref. [18] has already shown that it is possible to transform a cooperative game into a non-cooperative game if the negotiation process in the cooperative game is treated as a stage in the non-cooperative game in the way that each player uses all the essential strengths of their position. The AM operates in the same way.

The AM is a simple mechanism that can be used in simultaneous moves games, such as a voluntary contribution game or a CPR extraction game, in order to increase cooperation between players. In the CPR extraction game, each player chooses his/her level of extraction. When the AM is activated, the underlying game is transformed into a two-stage game. In stage 1, participants no longer make extraction decisions, but instead make extraction proposals. In stage 2, all proposals and their associated payoffs are made public, and each player must decide whether to approve or disapprove the proposals. If the group approves, the proposals are implemented. Otherwise, a uniform extraction (called disapproval benchmark or disapproval threat) is imposed on each one. In this paper, we consider the Nash equilibrium as the disapproval benchmark (hereafter called the Nash threat). The main reason for this is that the Nash equilibrium is the prediction of the CPR game. Therefore, it can be the easiest consensus disapproval benchmark and is the easiest to implement. Second, according to the evidence from laboratory experiments, the outcome of the CPR dilemma can be worse than the Nash equilibrium (see [11,19,20]). Since the Nash threat is fixed as an exogenous disapproval benchmark, the negotiation in the AM framework is based on the mutual (dis)approval of the proposals. Mutual (dis)approval defines whether the proposals or the Nash threat is implemented. This feature makes the AM a hybrid game, i.e., it is both cooperative in the sense of [18] and non-cooperative.

The introduction of the Nash threat in the CPR game changes the structure of the game by promoting individual and group extraction levels below the Nash extraction level [17]. Ref. [16] showed that the Nash threat increases contributions in the public good game, and [21] found that it induces cooperation in the prisoner’s dilemma game. In this paper, we implement the Nash threat in a CPR extraction game. The CPR dilemma differs from the public good dilemma in two important ways: first, the payoff function is concave, and second, there is rivalry between players. We aim to investigate whether the Nash threat can lead to lower CPR extraction, i.e., higher cooperation, in this context.

This paper also addresses the persistent effect of the Nash threat. In real life, mechanisms can sometimes lead to a withdrawal [22], outright suppression, or reform. In fact, the change of administrator or regime may lead to a change in economic and social policies that may result in the withdrawal of the mechanism. This was the case when the United States left the Paris Agreement after the election of President Trump. Thus, to test the robustness of the Nash threat, we examine its persistent effect. We therefore explore several questions: Does the mechanism change players’ behaviors? Does it continue to affect them even after it is removed? Are players familiar with the mechanism, or have they learned to cooperate, and can they continue to cooperate even if the mechanism is removed?

Unlike the existing literature, we test the robustness of the Nash threat in a two-player CPR game. First, we extend the dataset to confirm the effectiveness of the Nash threat observed in [17]. We examine in detail the reason that the Nash threat as the disapproval benchmark reduces CPR over-extraction to understand how subjects behave under the threat of Nash extraction. Third, we test the order effect, i.e., participants start with the CPR with the AM in sequence 1 and receive the AM in sequence 2. Fourth, we test the persistent effect of the Nash threat. Finally, we examine the framing effect under the AM. Thus, through our experiment, we confirm the effectiveness of the Nash threat in decreasing the over-extraction of the CPR even with a larger number of observations with respect to [17]. Moreover, our results show that there is neither an ordering effect nor a framing effect. Finally, we show that the Nash threat is persistent.

The rest of the paper is organized as follows. Section 2 describes the experimental design, i.e., the CPR model and treatments. Section 3 deals with the results: a descriptive analysis and detailed results. Section 4 provides a discussion and concludes.

2. Experimental Design

2.1. The Common Pool Resource (CPR) Game

We consider the 2-player CPR game. The CPR model is very similar to those of [17,20]. Let us consider the CPR model in [20]. Ref. [20] assumes a player (or appropriator) who has an endowment noted w. Ref. [20] interprets the endowment (w) as the total time available to player i, and he/she divides this time between leisure and work. Work ($x_i$) is the time that the player i spends fishing in a joint fishery, i.e., in the common pool resource extraction.

In this work, we consider that each player has the same endowment, and $x_i$ is the part of w that player i allocates in the CPR (e.g., in the fishery). Thus, $x_i$ is the effort or investment of player i in the CPR extraction. The CPR payoff function is given by the concave function $aX-b{X}^2$, where X is the total effort of the group members in the CPR. The payoff of player i from the CPR is proportional to his/her effort ($x_i/X$). Therefore, the more player i invests in the CPR, the higher his/her payoff level is. For this reason, his/her effort level can be interpreted as his/her extraction level. In the rest of the paper, the investment in the CPR will be referred to as extraction accordingly.

$w-x_i$ is the private investment of player i (i.e., the real leisure time according to [20]). The CPR game consists of deciding which part of the endowment should be invested in the CPR (i.e., the level of extraction) and which part in a private activity. Obviously, when the player i decides on the amount of his/her extraction from the CPR ($x_i$), he/she also decides on his/her investment in his/her private activity. ($w-x_i$).

The total payoff is given by:

$${\pi }_i(x_i,x_j)=\frac{x_i}{X}(aX-b{X}^2)+p(w-x_i)$$

(1)

The symmetric Nash equilibrium is achieved by maximizing the profit of player i and assuming that all players extract the same amount of CPR. Moreover, the optimal extraction is achieved by maximizing the sum of individual payoffs (see [17,20]). Using the same parameters ($a=23$, $b=0.25$, and $p=5$) as [11] and assuming that 1 token is equal to 3 units of endowment, we obtain Equation (2). Finally, since we are only dealing with groups of two players, the Nash equilibrium at the group level is 16 tokens (48 units) and the optimum is 12 tokens (36 units). At the individual level, the Nash equilibrium is 8 tokens (24 units) and the optimum is 6 tokens (18 units). Note that tokens are used for the experiment.

$${\pi }_1(x_1,x_2)=69x_1-\frac{9}{4}{x}_1(x_1+x_2)+15(w-x_1)$$

(2)

2.2. Treatments

We test the properties of the Nash threat (the Nash equilibrium as a disapproval benchmark) in the approval mechanism (AM) under the CPR. To do so, we considered three treatments (see

Table 1. Summary of treatments.

Treatments	Control	Introduction	Withdrawal
Sequence 1 (S1)	CPR	CPR	CPR + Nash threat
Sequence 2 (S2)	CPR	CPR + Nash threat	CPR
Number of group	17	30	19

Note: CPR refers to the CPR game without any incentive. CPR + Nash threat refers to the CPR game with the AM, i.e., the threat of Nash.

). Each treatment consists of two sequences. In each sequence, the CPR game or CPR game with the AM is performed over ten periods. In Introduction treatment, participants play the CPR game in sequence 1 and the CPR game with the AM (hereafter the Nash threat) in sequence 2. In Withdrawal treatment, the reverse is true, i.e., participants play the CPR game with the Nash threat in sequence 1 and the Nash threat is withdrawn in sequence 2 as the CPR game is applied in sequence 2. Thus, the withdrawal treatment tests the ordering effect and the persistence effect of the Nash threat. Finally, we performed a control treatment. This allowed us to isolate or capture the effectiveness (i.e., the actual effect) of the Nash threat with respect to control.

In each period (see Appendix A), each player has an endowment of 10 tokens and decides on his/her CPR extraction level. The rest of his endowment is automatically invested in his/her private activity. The results for each participant in the same group (each extraction and the corresponding payoff) are displayed and become common knowledge. Thus, the next period can start. When the AM (i.e., CPR+Nash threat) is activated, the underlying game is transformed into a two-stage game. In stage 1, participants no longer make extraction decisions but instead make extraction proposals. In stage 2, all proposals and their corresponding payoffs are made public, and each participant must decide whether to approve or disapprove the proposals. If the group mutually approves, the proposals are implemented. Otherwise, the predicted Nash extraction under unregulated CPR (CPR game) is imposed on everyone. The Nash extraction at the individual level is equal to 8 tokens and the corresponding payoff is 294. Sixty-six groups of two participants were involved in the experiment. Each participant took part in only one treatment. The experiment was programmed in zTree [23] and recruitment was conducted via ORSEE. All sessions were conducted in the Laboratory of Experimental Economics of Montpellier.

3. Results

This section deals with the summary statistics, the empirical strategy and detailed results.

3.1. Summary Statistics

Table 2. Average group extraction by sequence.

Treatments	Control	Introduction	Withdrawal
	(1)	(2)	(3)
sequence 1 (S1)	15.74	15.35	14.34
sequence 2 (S2)	16.23	14.07	14.81
S2–S1	0.49	−1.28	0.47
p-value of signed-rank test	0.168	0.012 **	0.153
p-value of K-S test	0.994	0.002 ***	0.418

Note: ** denotes significance at the 5-percent level and *** at the 1-percent level.

presents the average group extraction by sequence in each treatment. We observed that the group extractions under the CPR game (in control and in sequence 1 of Introduction treatments) are larger than under the CPR with the threat of Nash equilibrium (AM) or in sequence 2 (under CPR game) of the withdrawal treatment. Thus, the threat of Nash equilibrium decreases the over-extraction (see S2–S1 for Introduction treatments and the corresponding p-values of signed rank and K-S tests in Table 2). This reduction is persistent because there is no difference between sequence 1 and sequence 2 under the withdrawal treatments (see the p-values of signed rank and K-S tests in Table 2 for withdrawal treatments).

The detailed results supported by the outcomes of the regressions will be used to validate the properties of the Nash threat.

Basically, the two treatments with the Nash threat (introduction and withdrawal) were performed in two different frames (framing effect: see Figure 1). These frames are Aggregated payoff and Separated payoff (split payoff). In fact, each participant in the CPR game has two activities: the CPR extraction and the private activity. The total payoff is the sum of the payoffs of the two activities. In the experiment, the total payoff amount is communicated in the set of treatments labeled Aggregated payoff, while the payoffs from the two activities are reported separately in the set of treatments labeled Separated payoff. Therefore, participants in the treatments labeled Separated payoff make an extra effort by calculating the total payoff before making their decisions. Thus, the framing effect allowed us to examine the effect of participants’ computational ability on the effectiveness of the Nash threat. The preliminary experiments showed that there is no framing effect (see Figure 1 and the p-values of the rank-sum tests are higher than $10\%$). Therefore, we pooled the data to obtain two AM treatments (Introduction and Withdrawal).

3.2. Detailed Results

We ran Equation (3) for the support of Results 1 and 3.

$$Y_{it}={\alpha }_0+{\alpha }_1{[sequence\times AM]}_{it}+{\alpha }_2{\left[sequence\right]}_t+{\alpha }_3{\left[period\right]}_t+{\beta }_i+{\sigma }_{it}$$

(3)

where Y is group extraction. AM equals 1 for the groups that received the AM with the Nash treat (treatment named Introduction) and 0 for the control treatment. sequence equals 1 for sequence 2 of the introduction treatment and sequence 2 of the control treatment, and 0 for sequence 1 of the introduction and the control treatments. $sequence\times AM$ is the interaction variable, i.e., our variable of interest that captures the effect of the Nash threat in the DiD approach. ${\beta }_i$ is the group fixed effect. ${\sigma }_{it}$ is the term of the error. The regression is reported in column 1 of

Table 3. Main regressions.

	Introduction		Withdrawal
	(1)	(2)	(3)
$sequence\times AM$	$-1.77$ ***		$-0.02$
	$\left(0.307\right)$		$\left(0.307\right)$
$Mutual$$approval$		$-3.06$ ***
		$\left(0.281\right)$
$sequence$	$-0.14$		0.17
	(0.355)		(0.347)
$period$	$0.06$	$-0.01$	$0.03$
	$\left(0.025\right)$	$\left(0.019\right)$	$0.026$
constant	$15.14$ ***	$16.08$ ***	$14.82$ ***
	$\left(0.175\right)$	$\left(0.346\right)$	$\left(0.182\right)$
$GroupFE$	yes	yes	yes
$Prob>F$	$0.000$	$0.000$	$0.011$
$Observation$	940	300	720

Note: Robust standard errors are in parentheses. *** denotes significance at the 10-percent level. The regressions contain period, sequence and group fixed effect. The variable $sequence$ equals 1 for sequence 2 and 0 for sequence 1. In column (1), the variable $AM$ equals 1 for the $Introduction$ groups and 0 for the control groups. In the same way, in column (3), the variable $AM$ equals 1 for the $Withdrawal$ groups, and 0 for the control groups. The variable $sequence\times AM$ captures the effect of the AM (the threat of Nash). In column (2), the variables $AM$, $sequence$ and $sequence\times AM$ are the same and equal 1 since approval stage appears in sequence 2 of $introduction$ treatment. The variable $Mutual$ $approval$ equals 1 if the two players mutually approve and 0 if at least one player disapproves. $Group$ $FE$ stands for “group fixed effect”.

.

3.2.1. Cooperation under the Nash Threat

Our first finding is reported in Result 1.

Result 1: The Nash threat reduces the over-extraction of the CPR and, therefore, increases cooperation among participants.

Support for Result 1: In order to support Result 1, we use the difference-in-differences (DiD) estimation method. The DiD procedure is used to measure the effectiveness of the treatment by comparing the difference between the control groups and the groups subjected to the Nash threat (the AM) before and after the introduction of the Nash threat. It allows us to control for group and temporal characteristics by isolating the real impact of the Nash threat in reducing the group over-extraction observed in the control groups. The first differences concern the differences in group extraction between sequence 2 and sequence 1 (S2–S1) of the $Control$ and $Introduction$ treatments (see columns (1) and (2) of Table 2). These values are $0.49$ and $-1.28$ for the $Control$ and $Introduction$, respectively. The second difference refers to the difference between S2–S1 of the treatments named Introduction and Control, i.e., $-1.28-0.49=-1.77$. The coefficient of interest in the DiD is highly significant, indicating that the Nash threat significantly reduces the over-extraction of the CPR, which is materialized by increasing the cooperation between subjects.

The support for Result 1 is reached by running Equation (3).

3.2.2. The Driver of the Approval Decision

We examine in detail the reason that the Nash extraction as the disapproval benchmark reduces CPR over-extraction. In fact, the approval stage is added in stage 2 under the AM. If the subjects mutually approve, the proposed group extraction is implemented. However, if only one subject approves or in case of mutual disapproval, the Nash threat is implemented. Before deciding to approve or disapprove, each group member has three pieces of information: his extraction and payoff, his/her partner’s extraction and payoff, and the extraction and payoff of the disapproval benchmark, i.e., the Nash threat. The rational player compares his/her payoff to the payoff of the Nash threat, and the other consideration is to compare his/her payoff to that of his/her partner. According to our CPR model and the parameters used, the symmetric Nash equilibrium is eight tokens and the resulting payoff is 294 for each player in a group with two members. Thus, under the AM with the Nash threat, the selfish rational player approves only if his/her payoff from the proposals is greater than or equal to 294, and disapproves only if his/her payoff is lower than 294. We define this type of approval and disapproval decision as rational approval and rational disapproval, respectively. The other cases of approval/disapproval correspond to the other consideration. Consequently, we distinguish between rational (dis)approval and (dis)approval based on other considerations (see Figure 2).

We observed $60\%$ of mutual approval and $40\%$ (nobody approved and only one approved) of disapproval. For the $40\%$ of disapproval, only $5\%$ are mutual disapproval, and the rest, i.e., $35\%$, corresponds to one subject’s approval (see Figure 3). Moreover, overall, the average proposed group extraction with mutual approval is lower than the average proposed group extraction with disapproval. Therefore, mutual approval seems to drive the reduction in over-extraction due to the Nash threat. To confirm this effect, we add the variable $Mutual$ $approval$ to Equation (3) to account for group approval. This variable is coded as 1 if both participants in the group mutually approve of the proposed extractions and 0 if otherwise. Note that the variable "Mutual approval" is exactly the same as the variable $sequence\times AM\times [Mutual$ $approval]$. The regression is reported in column 2 of Table 3. This regression confirms that the reduction in group over-extraction is driven by the mutual approval of the proposals, since the coefficient of the variable $Mutual$ $approval$ ($-3.06$) is significant. In fact, our analysis shows that subjects who approve are those who extract more from the resource, cooperate less, and have higher earnings. Thus, mutual approval implies that both subjects make a cooperative effort, which increases the payoff of both and minimizes the difference in payoff between the two subjects.

At the individual level, we observed $82.67\%$ of approval decisions and $17.33\%$ of disapproval decisions. Moreover, $88.31\%$ of approval decisions are rational, while $76.92\%$ of disapproval decisions are rational. Thus, $11.69\%$ of approval decisions and $23.08\%$ of disapproval decisions are made with other considerations (see Figure 2). Therefore, we examine whether rationality and other considerations influence the decision to approve or disapprove. To do so, we determine the marginal effect of the following probit model:

$$prob\left\{approval\right\}=prob\{{\gamma }_0+{\gamma }_{1}rational+{\gamma }_{2}inequity+ϵ\}$$

(4)

where the variable $approval$ is equal to 1 if the subject approves and 0 if he/she disapproves. The variable $rational$ is coded as 1 if the participant’s payoff is greater than or equal to 294, and 0 if his/her payoff is less than 294. The variable $inequity$, which represents the other consideration, is coded as 1 if the participant receives a lower payoff than his/her partner, and 0 if he/she receives a greater or equal payoff compared to his/her partner. We ran the probit model regression on the approval/disapproval decisions in sequence 2 of treatment $introduction$. The marginal effects are $46.22\%$ and $-13\%$ for the $rational$ and $inequity$ variables, respectively. The coefficients of the two variables are statistically significant at $1\%$. Thus, the probability that a participant approves the proposed extractions increases by $46.22\%$ when his/her payoff is greater than the Nash payoff and decreases by $13\%$ when his/her payoff is lower than his/her partner’s payoff. We can conclude that although (dis)approval is mainly explained by rational thinking, there are also some situations in which inequity motivated participant’s behavior.

3.2.3. The Order Effect of the Nash Threat

Our second finding is reported in Result 2.

Result 2: Whatever the Nash threat is implemented in sequence 1 or in sequence 2, the group extraction is reduced, and therefore the cooperation increases among group members.

Support for Result 2: We performed the rank-sum test for the average group extractions by period in sequence 1 of the control treatment (CPR game) compared to those in sequence 1 of the withdrawal treatment (CPR game with AM). This rank-sum test (p-value = 0.000) shows that the average group extraction under the withdrawal treatment is lower than the average group extraction under the control treatment. Since there is no difference between sequences 1 and 2 under the control treatment (see the p-value of the signed-rank test in column 1 of Table 2), we also performed the rank-sum test for the average group extractions by period in sequence 2 of the control treatment (CPR game) versus those in sequence 2 of the introduction treatment (CPR game with AM). This second rank-sum test (p-value = 0.000) also shows that the average group extraction under the introduction treatment is lower than the average group extraction under the control treatment. According to the results of the above rank-sum tests and with respect to Result 1, we can conclude that regardless of whether the Nash threat is implemented in sequence 1 or in sequence 2, group extraction is reduced, and therefore the cooperation increases.

3.2.4. Persistent Effect of the Nash Threat

Our third finding is reported in Result 3.

Result 3: The Nash threat has a persistent effect on the reduction in the over-extraction of the CPR.

Support for Result 3: First, we conducted the rank-sum test with sequence 2 of the control treatment (CPR game) and sequence 2 of the withdrawal treatment (CPR game), because there is no difference between sequences 1 and 2 under the control treatment and under the withdrawal treatment, respectively (see p-values of the signed-rank test in columns 1 and 3 of Table 2). The rank-sum shows that the average group extraction is lower under sequence 2 of the withdrawal treatment than under sequence 2 of the control treatment. Second, we performed Equation (3), where $sequence$ is coded as 1 for sequence 2 of the withdrawal treatment and sequence 2 of the control treatment, and 0 for sequence 1 of the withdrawal treatment and sequence 1 of the control treatment. $period$ represents the periods under withdrawal and control treatments. $AM$ is coded as 1 for the withdrawal treatment and 0 for the control treatment. The variable $sequence\times AM$ is our variable of interest, capturing the effect of withdrawal of the Nash threat. The regression is reported in column 3 of Table 3. This coefficient is not significant (see column 3 of Table 3). Thus, withdrawing the Nash threat in sequence 2 does not significantly alter the average group extraction and cooperation level. For both reasons, we can conclude that the Nash threat has a persistent effect with regard to reducing CPR over-extraction.

4. Discussion and Conclusions

In this paper, we study the two main properties of the Nash threat in a two-player common-pool resource dilemma through an experiment with three treatments (see Table 1). The two main properties are based on the increase in cooperation between participants and the persistent effect when the Nash threat is removed. We found that the Nash threat significantly reduces the over-extraction of the resource. Moreover, the reduction is persistent when the Nash threat is no longer present. Our results also show that there is neither an ordering effect nor a framing effect.

The introduction of the Nash threat changes the structure of the basic CPR game. This structural change could therefore explain the reduction in CPR over-extraction and hence the increase in cooperation level. Indeed, under the AM, each participant compares his/her payoff from the stage 1 proposals to the payoff from the Nash equilibrium. If his payoff is greater than or equal to the payoff from the Nash equilibrium, he approves; otherwise, he disapproves. In theory, ref. [17] show that the set of extractions that survive to the elimination of the weakly dominated strategies are lower than the extraction level of the Nash equilibrium. This prediction is mostly observed in our experiment when the Nash threat is applied. However, other consideration (inequity), i.e., the difference between the payoff of the considered subject and his/her partner, explain part of the participants’ behavior.

Our results suggest that the Nash threat is credible because it increases cooperation, even if it is no longer present. On the other hand, it does not allow us to implement the optimum. Note that [21] compared the Nash threat in the Prisoner’s Dilemma with the mechanism of [24] and showed that the Nash threat performs better than the mechanism of [24]. It will also be interesting to do so in the case of CPR. We can compare it with the mechanism of [25]. It will be worth considering an alternative withdrawal test by adding a third sequence to the experiment, e.g., CPR game in sequence 1, CPR game with AM (the Nash threat) in sequence 2, and CPR game in sequence 3. We can increase the group size and apply the AM in “give and take” [26], which might help us to compare the AM in CPR and the AM in public goods provision.