Efficient allocations in common-pool resources cannot be accomplished when appropriators are selfish. In addition, we find that a system of a common-pool resource is locally unstable if there are four or more appropriators. Such instability most likely makes efficiency worse than that in the Nash equilibrium. These results indicate that equilibrium analyses might not capture the essence of the common-pool resource problem. They may also provide an answer to the unexplained pulsing behavior among appropriators and inefficiency observed in experiments.

An open access resource such as a fishing ground, an irrigation system, or a forest is called a common-pool resource (CPR). According to

However, the static analysis based on the Nash equilibrium entails a number of restrictive assumptions, namely that players (i) are rational, (ii) have complete information about others, and (iii) respond simultaneously. The problem is that these assumptions are often violated in realistic situations, including laboratory experiments. For example, actual players may not be rational enough to determine the Nash equilibrium immediately, and may require a series of trials to achieve it. Thus, it is important to consider how the dynamics of players’ allocation strategies occur and whether it converges to a Nash equilibrium. Now, let us say that a Nash equilibrium of a game is dynamically stable (or simply, stable) if it gives an asymptotically stable fixed point of a dynamic version of the game; a fixed point is asymptotically stable if all nearby solutions not only stay nearby, but also tend to the fixed point (

In this study, we investigate the conditions for the dynamic stability of the Nash equilibrium of a CPR dilemma game. However, to define the dynamics, we need to specify the decision-making rule (or the rule for behavioral adjustment) of the players. This is an important issue, given that stability is largely dependent on the rule. Experiments provide a hint as to the solution to this problem. In their experiments,

We first provide a detailed theoretical basis for the instability of the Nash equilibrium of a CPR dilemma game, though our analysis is confined to the deterministic best-response dynamics with regard to the underlying dynamics. Our theory applies to a broad range of production functions, including those adopted by Ostrom and others as special cases (

The dynamic instability of the CPR dilemma game also has an implication for the efficiency of the resource use. Here, we say that a CPR is efficiently (inefficiently) used if the social payoff is equal to (lower than) its possible maximum, where the social payoff is defined as the sum of the payoffs of all appropriators, less the sum of the initial endowments of all appropriators (see Section 4). By definition, efficiency in our terminology implies Pareto efficiency, but not vice versa. A well-known common property of CPR dilemma games is that the Nash equilibrium is inefficient (

The rest of the paper is organized as follows. Sections 2 and 3 provide specific examples of CPR dilemma games with an unstable Nash equilibrium. Section 4 provides a general mathematical result as to the instability of the Nash equilibrium. Section 5 provides statistical analyses relating our results to experimental data. Lastly, Section 6 summarizes our findings and discusses their implications.

The experiment of

In the experiments of

Now, we describe the WGO model, and show that its Nash equilibrium is unstable and a cycle emerges under the parameter values they used in their experiments. Imagine a society with _{i}_{i}_{i}_{i}_{i}^{2} be the production function for the fish. The number of fish that appropriator

where _{–i}=S_{j≠i} _{j}_{i}_{i}_{–i}), subject to 0≤ _{i}_{i}_{–i}. Partial differentiation of eq. (1) with respect to _{i}

which is equal to zero when

Thus, eq. (2b) gives the best response for appropriator _{i}_{i}_{i}_{−i}), is given by

if

if

if

The best-response function (3a–c) does not include appropriator-specific parameters, except _{i}_{i}_{i}

In WGO’s basic setup, the parameter values are _{i}_{−i}=(1/7)_{−i}; on this line, all appropriators invest the same amount of labor. The intersection of lines

The best-response curve in the WGO model and the stability of the Nash equilibrium. The purple line represents the best response of each appropriator to given values of x_{-i}. Gray lines represent the set of points where all appropriators invest equal amounts of labor (0e_{i}

Let us consider the best-response dynamics in the WGO model to demonstrate the instability of the equilibrium and the emergence of a cycle. In the best-response dynamics, by definition, appropriator^{1}=(1, 2, 3, 4, 5, 6, 7, 8) at the first time step, and then consider appropriator 1’s choice in the second time step. Given that ^{2}=(10, 10, 10, 10, 10, 10, 10, 10) (point ^{3}=(1, 1, 1, 1, 1, 1, 1, 1) (point ^{4}=^{2} (point ^{5}=(1, 1, 1, 1, 1, 1, 1, 1) (point

WGO (1990) and _{i}_{i}_{j}_{j}_{i}_{−i}=63. Thus, in terms of our model, the rule of thumb adopted by their subjects reads as “Choose 10 if _{−i} is at most 63,” which roughly corresponds to the best-response behavior.

WGO and OGW conducted two treatments with

To see the potential effect of the number of appropriators, let us change the number of appropriators from eight to six. Then the best-response function is ^{1}=(1, 2, 3, 4, 5, 6), and then consider how the best-response dynamics occur. For example, for appropriator 1, we have ^{2}=(10, 10, 10, 10, 10, 10), which is the Nash equilibrium (

The stability property for

It is easy to see from

In WGO’s model, considered in Section 2, the best-response curve had a “plateau” (i.e. the flat part on the upper boundary), in which the best response is constrained by the initial endowment. The best-response curve was also special in the sense that its non-flat part had a constant negative slope. The latter property results from the quadratic shape of the production function _{i}_{i}_{i}^{2}. While the quadratic production function is mathematically simple, it is somewhat unrealistic, given that the amount of production becomes negative for total labor inputs higher than a threshold

Stability properties when _{–i}. Gray lines represent the set of points where all appropriators invest equal amounts of labor (0c

To see the effect of a reduction in the number of appropriators, consider the case of three appropriators (i.e. the rectangle spanned by diagonal 0-

As already argued, an interior equilibrium is unstable under

As _{-i}_{i}

In this section, we explore the mathematical conditions for an interior equilibrium to be destabilized using a general production function satisfying a set of mild conditions. We assume that the initial endowment of each appropriator is large enough to ensure that any Nash equilibrium is an interior point. We consider a production function

and

where the last condition ensures the concavity of the production function.

Suppose that appropriator

is asymptotically stable at Nash equilibrium

is asymptotically stable, where

Note that the system in eq. (6) is the linear approximation of the system in eq. (5) at the Nash equilibrium

Since the per-capita production

When

Instability tends to reduce efficiency, even compared to that in a Nash equilibrium, although we could not derive precise mathematical conditions on which efficiency reduction occurs. Let us define the social payoff function ^{2}, as shown in

Instability reduces efficiency. The red curve gives the social payoff function of the WGO model (i.e. U(x)^{2}). U is maximized at point P, and e is the Nash equilibrium. For more details, see the text.

In the previous sections, we provided theoretical arguments to explain how the Nash equilibrium in a CPR dilemma game is destabilized. Here, we examine whether our hypothesis is consistent with empirical data. In this section, we conduct statistical analyses to show that the individuals’ data in the WGO experiments can be interpreted as a result of myopic decision rules and deterministic cyclic dynamics around an unstable Nash equilibrium. Note that

_{i}_{,t+1} be subject _{−i,t} be the observed total labor inputs of the other subjects at period _{i}_{,t+1} for period

If γ_{i}=0, the belief is exactly the observation at the previous period. In contrast, if γ_{i}=1, the belief is the average of all previous observations. In this sense, γ_{i} determines the lag length of the information used by subject _{i} might also take a value outside the range [0, 1] (_{i}>1 indicates that subject _{i}<0 indicates that the effect of past information changes its sign in each period.

To capture subjects’ concerns about other group members’ payoffs, following the design of

where _{i}_{i} is a coefficient measuring subject _{i}=0, subject _{i}>0, player _{i}<0, player

With the utility given by equation (9), we assume that the decisions by subjects follow stochastic best-response dynamics (see

where _{i}_{,t+1} is the observed choice of subject _{i}_{,t+1} is the utility of subject _{i}_{i}>0 is a factor that captures the decision errors of subject _{i}_{,t+1} at period _{−i,t}, _{−i,t−1}, …, _{−i,1}. When λ_{i}→0, all choices for subject _{i} becomes large, subject _{i}→∞, player

Based on the meanings of the parameters, it is clear that subjects are exactly following the best-response dynamics when γ_{i}→0, β_{i}→0, and λ_{i}→∞. To obtain parameter estimates, we maximize the following log-likelihood function:

where _{i}, β_{i}, and γ_{i}, individually.

The results of parameter estimation.

Experiments |
||||
---|---|---|---|---|

Periods 2–16 | Periods 16–30 | Periods 2–11 | Periods 11–20 | |

γ_{i} |
0.410 | 0.653 | 0.833 | 0.641 |

(0.007) | (0.003) | (0.009) | (0.003) | |

β_{i} |
–1.581 | –0.491 | –0.998 | –0.290 |

(0.055) | (0.011) | (0.016) | (0.010) | |

λ_{i} |
0.095 | 0.340 | 0.045 | 0.082 |

(0.002) | (0.005) | (0.001) | (0.001) | |

Obs. | 360 | 360 | 240 | 240 |

lnL | –677.80 | –631.11 | –718.50 | –681.30 |

Jackknifed standard errors are shown in parentheses.

In both experiments (_{i} becomes larger over time, which indicates that the decision errors become smaller (χ^{2}(1)=24.98 for the experiment with ^{2}(1)=9.52 for the experiment with

All estimates of β_{i} are negative. This makes sense because the CPR environment is competitive. Interestingly, in both experiments, the estimates of β_{i} become closer to zero over time (χ^{2}(1)=13.14 for the experiment with ^{2}(1)=8.44 for the experiment with

The interpretation for estimates of γ_{i} is not obvious. It seems that players become more myopic in the latter half of the experiment with

Overall, the above results show that decision errors become smaller and subjects become more self-interested over time. This implies that subjects’ behaviors become closer to the best response based on some previous observations (not only the last observation) with repeated trials in both the experiments. Given this statistical result, and the theoretical result that the best-response dynamics induce pulsing behavior, we hypothesize that the group sum pulses more in the latter half of the experiment than it does in the beginning half of the experiment.

To test this hypothesis, we compute the sample autocorrelation for the group sum in each group.

The sample autocorrelations.

Group | Experiments |
|||
---|---|---|---|---|

Periods 2–16 | Periods 16–30 | Periods 2–11 | Periods 11–20 | |

1 | –0.133 | –0.137 | –0.131 | –0.366 |

2 | –0.005 | –0.344 | 0.025 | –0.210 |

3 | 0.086 | –0.279 | 0.140 | –0.221 |

These results support our hypothesis. All groups generate a negative sample autocorrelation in the latter half of the experiment, which indicates pulsing. However, three of the six groups have a positive sample autocorrelation in the beginning half of the experiment.

Now, we consider whether pulsing causes inefficiency in the experiments. As table 5.2 (page 117) and figure 5.4 (page 119) in OGW show, the low efficiency in the experiment with

Statistics to test the hypothesis that pulsing causes inefficiency.

Experiments |
||
---|---|---|

Centile (25%) | 62 | 60 |

Centile (75%) | 68 | 71 |

Interquartile range | 6 | 11 |

Average payoff (standard deviation) | 63.81 (8.16) | 133.56 (22.87) |

Predicted payoff, Nash | 66 | 141 |

t-Statistic* | –2.12 | –3.59 |

*For the t-test, we first compute the average across periods for each individual, and then conduct the t-test over the sample of those averages.

We use the interquartile range to measure the amplitude of the pulsing in the group sum. This shows that the amplitude of the pulsing is larger in the experiment with

We have shown that a system of simultaneous difference equations describing the best-response dynamics of a CPR dilemma game is locally unstable when the number of appropriators is at least four. Works such as

The statistical analyses in Section 5 revealed that in WGO’s experiments, subjects were more self-interested and their choices were closer to the best responses in the latter half of the experiments than they were in the beginning half. On the other hand, using post-experiment questionnaires, OGW found that many subjects used a rule of thumb that is very close to the best-response behavior. It may be that in WGO’s case, subjects were initially exploring the system, but later found the rule of thumb. This could explain why their behaviors were different to the best responses in the first half of the experiments, but were close to best responses in the latter half. It may be interesting to conduct questionnaires not only at the end of the experiments, but also in the middle, in order to reveal the process in which subjects acquire best-response behavior.

We found that interior equilibria are destabilized for a broad range of parameter values, while boundary equilibria are always stable. Importantly, we can create a boundary equilibrium by decreasing the initial endowment sufficiently for virtually any best-response function. Of course, it is difficult to control the initial endowment of a player in a real system if the “endowment” is determined by his/her internal capacities or capabilities (e.g. physical, biological, or monetary). However, we may be able to control the upper limit of the labor input by each player externally, for example, through a political institution. Thus, an external restriction on the use of a CPR may be important, not just for the efficient use of the resource, but also for stable production.

Another possible way to control the instability of a real system may be to manipulate the timing of appropriators’ responses. The equilibrium is unstable in a system with many appropriators because they all respond simultaneously in every round. The system may be stabilized if, in every round, only one (or a few) appropriators are allowed to change their behavior or, similarly, if appropriators choose their behavior in a sequential manner, and each of them responds to the preceding choices of other players. However, this mechanism may require a more complicated political rule to ensure fairness than simply controlling the initial endowment. We may also need further theoretical investigations to reveal the stability conditions for such systems with asynchronous decision-making.

Note that the instability issue in CPR games is closely related to that in oligopoly models (_{i}_{i}

which corresponds to eq. (1). Then, the Nash equilibrium of this game gives a stable fixed point of the best-response dynamics when

(see

Condition (7) (and even (13)) of stability is mathematically clear, but its economic meaning is less so. Unfortunately, we could find neither a simple economic interpretation for each term in the condition, nor an economic reason why four must be the critical number for instability. Each of the terms comes from complicated calculations, and represents joint outcomes of various effects, which makes an intuitive understanding extremely difficult. It might be worth exploring other representations of the stability condition to gain further economic insights.

Thus far, researchers have been arguing that CPRs cause a dilemma between the Pareto efficient and Nash allocations. However, in addition to this dilemma, we have shown that appropriators may suffer from additional inefficiency owing to dynamic instability. Thus, the problem is more complicated than first thought, and we should pay attention to the stability problem and the static inefficiency of the Nash equilibrium. Cyclic solutions caused by the instability may be a possible answer to the pulsing behavior of each appropriator’s labor input in the CPR experiments summarized by

In addition, there is evidence that players might use behavioral rules affected by institutional devices, such as peer punishment. For instance,

Our theory reveals that CPR systems with myopic players are stable only under very special conditions, namely, with linear payoff functions, or with very small numbers of appropriators. Given that real CPR systems are unlikely to satisfy such special conditions, we suspect that the results of many laboratory experiments in previous studies, in which linear payoff functions and a very small number of appropriators are used, have only limited applicability. In future experiments, researchers should focus on the number of appropriators and the payoff structure in order to capture the essential behavior of real systems. We should also analyze data from dynamic viewpoints, rather than from static viewpoints, as we did in Section 5. That is, we should focus on how, or to what extent players’ strategies fluctuate.

We thank Professor James Walker for providing us with the individual data of the experiments in

In order to simplify the stability argument of the system of simultaneous difference equations, the following well-known property is useful (see WGO;

_{i}_{j} for all i and j.

Since _{i}_{j}_{i}_{j}_{i}

_{i}_{i}_{−i})=0, we have

where

and

Since

Since

The following local stability condition is well known in oligopoly theory; for example, see Bischi et al. (2009, 61).

The local stability property does not depend on the initial value ^{1}. The following proposition shows the necessary and sufficient condition for local stability.

That is, 1>|

■

(i)

(ii)

(iii)

■

Since the average productivity

Bifurcation diagram for the WGO model, obtained by numerical simulations. Parameter values are the same as in

Bifurcation diagram for the model with