بهره برداری استراتژیک با آموزش و باورهای ناهمگون

We study the effect of learning with heterogeneous beliefs on the exploitation of a renewable common-pool resource. To that end, we extend the Great Fish War model of Levhari and Mirman (1980) to a learning environment in which several agents interact strategically and learn about the distribution of the stochastic evolution of the resource. We find that the effect of anticipation of learning with heterogeneous beliefs is twofold. First, the anticipation of learning makes future payoffs more uncertain, which induces the agents to decrease present exploitation due to the precautionary motive. Second, under heterogeneity of beliefs, there is a differential informational externality that induces the agents to increase or decrease present exploitation. We also perform a comparative analysis on the Cournot–Nash equilibrium with learning by studying the effect of optimism and riskiness on resource exploitation.

In the presence of a common-pool resource, strategic interactions play an important role in determining the behavior of agents and thus the evolution of the resource. Strategic exploitation of common-pool resource was first studied by Levhari and Mirman (1980) in a deterministic framework. However, agents exploit a resource under less information. This was initially considered by Brock and Mirman (1972) in the context of optimal stochastic growth, building on earlier studies of positive growth under uncertainty (Mirman, 1972 and Mirman, 1973). Recently, Antoniadou et al. (2013) studied strategic interactions and the tragedy of the commons in the context of an exploitation of a common-pool resource under uncertainty.1 The motivation for studying stochastic rather than deterministic growth was to reduce the information available to the agent in order to provide more realistic results. Indeed, in a deterministic environment, each agent is assumed to have perfect foresight of the effect of his exploitation on the evolution of the resource. Adding uncertainty about the evolution of the stock means that the agent does not need to know the precise effect of his exploitation on the future stock. In addition to facing uncertainty via random variables, there is another important type of uncertainty that has not yet been studied in dynamic games, i.e., uncertainty about the distribution of the random variables. In other words, agents face uncertainty not only about the evolution of the natural resource, but also about the true distribution generating the shocks in the economy. Therefore, because the true distribution is unknown, agents form initial (prior) beliefs about it. Moreover, with the observation of informative signals, rational agents learn by updating their prior beliefs to reduce uncertainty.2 In a recent paper, Koulovatianos et al. (2009) (KMS henceforth)3 analyzed the behavior of a single-agent decision maker, in a growth model of Bayesian learning in which the true distribution of the stochastic variable is not known. However, in many cases, several agents jointly exploit a common stock of a resource and face uncertainty about the structure of the economy (e.g. a river on the border of two countries, a gold field being exploited by many goldsmiths). In other words, there are strategic interactions implying that agents anticipate not only what the other agents do, but also their future learning. Moreover, with several agents, prior beliefs might not be the same across agents.4 The purpose of the paper is to study the effect of Bayesian learning in a dynamic game of resource exploitation in which agents have different beliefs. To that end, we extend the Great Fish War model of Levhari and Mirman (1980) to a learning environment. We consider several agents exploiting a common-pool renewable resource. The renewability function of the resource is affected by a stochastic variable whose distribution is unknown to the agents. Specifically, the distribution depends on an unknown parameter. The exploiters of the resource form prior beliefs about this parameter. We allow these prior beliefs to vary across agents. After observing the past realizations of the renewability variable, and using Bayesian methods, each agent updates his prior belief. Given prior beliefs, an agent exploits the resource while considering the effect of his own exploitation and the exploitation of the others on future stock. In addition, the agents anticipate learning by taking into account belief updating for all agents.5 As mentioned in KMS, the anticipation of learning is a source of risk because it makes the future more stochastic in that future beliefs are unknown. We present two sets of results. We begin by characterizing the unique Cournot–Nash equilibrium in which agents learn using Bayesian methods. We study the effect of anticipating learning on the equilibrium. To that end, we compare the Cournot–Nash equilibrium with learning with the benchmark equilibrium of adaptive learning. In adaptive learning, agents form beliefs but are rationally bounded because they do not anticipate changes in their beliefs. The difference between the learning Cournot–Nash equilibrium and adaptive learning Cournot–Nash equilibrium allows us to capture the effect of learning as a source of risk. In the single-agent case studied by KMS, present exploitation decreases when the agent faces risk from anticipation of learning. In a game with heterogeneous beliefs, this is not necessarily the case. The direction of the effect of learning depends on the degree of optimism of the agents about the future stock of the resource. The effect of learning is twofold. First, as in the single-agent case, anticipating learning induces the agents to decrease present exploitation due to the precautionary motive. Second, with heterogeneous beliefs, agents are a source of an informational externality on one another. Indeed, heterogeneity in beliefs adds a differential informational externality. Specifically, each agent has to take into account the beliefs and the anticipation of other agents whose beliefs are different from his own belief. For a given agent, unlike the precautionary motive, the differential informational externality may increase or decrease his exploitation. For instance, if an agent is too optimistic (about the availability of the future stock) relative to the other agents, he increases his exploitation when anticipating learning. The overall effect of anticipating learning depends on the relative strength of the precautionary motive effect and the differential informational externality effect. 6 We perform a comparative analysis by studying the effect of changes in beliefs. We first study the impact of an increase in an agent's optimism on present exploitation. More optimism on the part of one agent reduces the marginal cost of exploitation of that agent, while it increases the marginal cost of exploitation of the other agents. Therefore, the agent's own present exploitation is increased while the other agents decrease their present exploitations. This result follows from the fact that a more optimistic agent anticipates a higher future stock of the resource, and believes that a future shortage is less likely. In turn, because the more optimistic agent exploits more, the other agents see the need to make a greater effort to contribute to the savings in order to have the stock they desire for the future. Second, we analyze the effect of a riskier belief on exploitation using the concept of second-order stochastic dominance. A riskier belief about the future stock leads the agent to decrease his exploitation while it leads the other agents to increase their exploitation. The reason is that riskier belief about the unknown distribution increases the marginal cost of exploitation of the agent who considers that the renewability becomes riskier. However, because the agent with a riskier belief decreases his exploitation, the other agents get more leeway to exploit the resource, even though their beliefs remain unchanged. The need for them not to harvest is lesser. The issue of learning has also been addressed extensively in the literature on climate change (CO2 emission and International Environment Agreements (IEA)), pioneered by Ulph and Ulph (1997), Na and Shin (1998), Ulph (2004), Baker (2005), Kolstad (2007), Kolstad and Ulph, 2008 and Kolstad and Ulph, 2011, Dellink and Finus (2012), and Karp (2012). In these papers, the uncertainty concerns the damage from emissions, and the effect of learning on the success of an IEA, CO2 emission, and on global welfare is studied. In some respects, this literature on learning has some characteristics that make it different from our framework. In most of the papers cited above, learning is perfect, in the sense that agents know the true value of the stochastic parameter as soon as they learn. 7 In our model, learning is not perfect because our agents receive noisy signals and use Bayesian methods to update beliefs. Consequently, our agents face risk from uncertain future beliefs, which affects behavior when agents interact strategically. In our model, if we assume that learning is perfect, anticipation of learning is no longer a source of risk. The paper is organized as follows. In the section “Model”, we present the model and derive the Cournot–Nash-equilibrium of the game. The section “The effect of anticipation of learning: the role of heterogeneity” studies the role of the heterogeneity of beliefs in the effect of anticipating learning on individual exploitation decisions. In the section “Comparative analysis”, we perform a comparative analysis using the concepts of first-order and second-order stochastic dominance. Concluding remarks are presented in the section “Final remarks”.

In this paper, we analyze the effect of learning on the exploitation of a common-pool renewable resource when the users do not know the distribution for the stochastic evolution of the resource. We show that the degree of heterogeneity in prior beliefs plays an important role in determining the effect of learning on the equilibrium. When all the agents have the same belief, each of them exploits the resource more in adaptive learning than in learning. This is due to the fact that anticipation of learning is a source of risk. Because of that risk, each agent saves more of the resource (precautionary savings). However, in the case of heterogeneous beliefs, some agents could exploit more in learning than in adaptive learning. Indeed, when agents do not have the same belief, an agent cares not only about the anticipation risk, but also about the behavior of the other agents. In addition to the anticipation risk, each agent also pays attention to the others’ degree of optimism. Specifically, in case of beliefs heterogeneity, through strategic interactions the more pessimistic agents bear more of the anticipation risk by saving more. That savings behavior of the more pessimistic agents benefits the more optimistic agent, who can exploit more of the resource without fearing a shortage in the future. In other words, when the degree of heterogeneity is high, the burden of anticipation risk could be completely (or more than needed) supported by the more pessimistic agents. Therefore, the pessimism of the more pessimistic is a sort of positive externality in favor of the more optimistic agents. Conversely, the optimism of the more optimistic agents is a negative externality to the detriment of the more pessimistic ones. As a consequence, the more optimistic agents could exploit the resource more in learning than in adaptive learning if the pessimistic agents are too pessimistic and support the anticipation risk more than necessary. This externality that the agents impose on each other, due to difference in beliefs, is the differential informational externality. Another related result is that, both in learning and adaptive learning, an increase in optimism of one agent leads that agent to increase his exploitation while it leads the others to decrease their exploitation. Moreover, the increase (decrease) in exploitation due to optimism is higher in learning than in adaptive learning. The reason is that the learning environment is riskier than the adaptive learning environment. In learning there are two types of risk: risk due to structural uncertainty and risk due to anticipation of learning. In adaptive learning there is only the risk due to structural uncertainty. For each information structure (adaptive learning and learning), we also study how individual exploitation changes when, through their beliefs, the agents think that the renewability of the resource becomes riskier. We consider an increase in riskiness that preserves the mean of the renewability parameter. The result is that the agents’ decision in adaptive learning is not affected by an increase in riskiness, because of certainty equivalence. However, in learning, the increase in riskiness of one agent decreases the exploitation amount of that agent, while it increases the exploitation of the others. There are several ways to extend our work. We could study Bayesian learning in a situation where the exploitation of the resource not only reduces the stock, but also increases the probability of environmental damage. For example, when companies extract limestone, the consequence is not only a reduction of the limestone stock, but also an increase in the likelihood of an earthquake. When we extract oil or uranium, we also emit CO2 or pollute the environment. Deforestation not only reduces the size of the forest available for economic use by future generations, it also induces environmental problems like the greenhouse effect, loss of biodiversity and soil erosion. In other words, natural resource exploitation and environmental change are closely related (see for example Jacobsen, 2011, Ollivier, 2012, Backlund et al., 2008 and IPCC, 2007). Thus, studying how learning can affect the likelihood of an environmental change is an avenue of future research.