بهره برداری استراتژیک از یک منبع مشترک تحت عدم قطعیت

We construct a game of noncooperative common-resource exploitation which delivers analytical solutions for its symmetric Markov-perfect Nash equilibrium. We examine how introducing uncertainty to the natural law of resource reproduction affects strategic exploitation. We show that the commons problem is always present in our example and we identify cases in which increases in risk amplify or mitigate the commons problem. For a specific class of games which imply Markov-perfect strategies that are linear in the resource stock (our example belongs to this class), we provide general results on how payoff-function features affect the responsiveness of exploitation strategies to changes in riskiness. These broader characterizations of games which imply linear strategies (appearing in an Online Appendix) can be useful in future work, given the technical difficulties that may arise from the possible nonlinearity of Markov-perfect strategies in more general settings.

In games of common-property renewable resource exploitation each player partly controls the future evolution of the resource, given the strategies of other players. Models in which there is a dynamic element and in which resources are shared play an important role in economics, e.g., industrial organization models or models with natural resources. The fundamental, infinite-horizon setup, in which all players have full information about the economic environment, has been studied in the economics literature almost exclusively within the deterministic framework. The main finding of this literature is that the equilibrium is characterized by a “commons problem”. Namely, the higher the number of non-cooperating players, the higher the aggregate exploitation rate, so the lower the level of the resource in the long run.1 Our goal in this paper is to examine how noncooperative strategic interaction is affected by uncertainty in the natural law of resource reproduction. Our focus on randomness in resource reproduction is a natural starting point for the study of uncertainty in resource games. In the real world, resources evolve according to stochastic laws of motion. Especially in the context of natural resources, as is the case with biological populations such as forests and fish species, these evolve subject to the existence of predators or climate, that are affected by random disturbances.2 Stochastic dynamic games can be particularly complex and difficult to characterize when the law of resource reproduction, the payoffs and the distributions of random disturbances are all given by general functions.3 At the same time, the task of characterizing decisions in the presence of uncertainty in a general framework can be demanding even in the case of a single decision maker.4 We discuss why technical problems arise in multiple-player dynamic games which use general functional forms. Specifically, in resource games problems arise because each player's objective function directly contains the strategies of other players. When the Markov-perfect Nash strategies of other players are strictly concave, a player's objective function may lose key properties, such as concavity, differentiability, and continuity. These technical difficulties are discussed in Mirman [16]. A special class of dynamic games avoids such technical difficulties related to the concavity of Markov-perfect Nash strategies. It is the class of games which possess primitives such that symmetric Markov-perfect Nash strategies are linear decision rules with respect to the common resource.5 A game that falls in this class is the parametric example of Levhari and Mirman [13]. Yet, strategies in the Levhari and Mirman [13] example are unaffected by introducing uncertainty. Here we provide a new example that nests and extends the Levhari and Mirman [13] example, in which introducing uncertainty and risk changes (in terms of first- or second-order stochastic dominance) affect exploitation strategies. Our analysis is extended to the case of N players, where N can be more than two players. A study that extended the Levhari and Mirman [13] model to N>2N>2 players is Okuguchi [18], who has also emphasized the effects of entry (or exit) in fish war in comparison with cooperative solutions (joint resource management by all players). Understanding how noncooperative strategic behavior changes as we add players to a game is key to understanding whether, under particular forms of regulation, cooperation is sustainable as a subgame-perfect equilibrium. 6 Here, apart from presenting our example and performing comparative analysis of strategies as we increase risk, we do not provide extensions to resource regulations. We do, however, provide an Online Appendix which proposes theoretical tools in order to analyze the comprehensive class of games with primitives that allow for Markov-perfect Nash exploitation strategies which are linear in the resource stock. 7 These tools and theoretical results which are based on a stochastic-dominance analysis of how risk changes affect strategies are applicable to other examples of linear-Markov-perfect–Nash strategy games that one may discover along the way.

The impact of uncertainty on strategic behavior in games of common-resource exploitation is not adequately understood. One reason for this lack of progress is that technical anomalies may arise if Markov-perfect–Nash strategies are nonlinear. One specific class of games which overcomes such anomalies is games that produce Markov-perfect Nash exploitation strategies which are linear in the resource stock. While the famous Levhari and Mirman [13] example falls within this particular class, it nevertheless implies no impact of uncertainty on the strategic behavior of players. In this paper we have contributed another analytical example with linear Markov-perfect Nash exploitation strategies which nests the Levhari and Mirman [13] example as a special case and which offers insights on how changes in uncertainty affects players' strategic behavior. Our example involves additively separable utility with constant relative risk aversion. There are two key findings within our example. First, if the coefficient of relative risk aversion is higher than unity, then, for a given number of symmetric players, each player will conserve the resource once uncertainty (or an increase in risk) is introduced. This conservation response of players to increases in risk resembles investment literatures in which a high coefficient of relative risk aversion can be one of the ingredients leading to precautionary savings. Second, our example suggests that strategic interaction is more complex: we find that the addition of players leads to exacerbated increases in aggregate exploitation rates as we simultaneously increase risk, if the coefficient of relative risk aversion is higher than unity. It seems that for highly risk-averse players the commons problem dominates any conservation incentives that our first finding has suggested arise after increasing risk. So, in our example, increasing risk exacerbates the commons problem if players are highly risk-averse. In an Online Appendix we also develop technical tools for studying the comprehensive class of games that produce Markov-perfect Nash exploitation strategies which are linear in the resource stock. This more general analysis shows that the conservation response of players to increases in risk indeed hinges upon properties of the momentary utility function of players, always within the linear-strategy class of games. Our results regarding the impact of increasing risk on the intensity of the commons problem may be more specific to our example which links the elasticity of intertemporal substitution to parameters of the natural law of resource reproduction. Nevertheless, both our example and our more general analysis in the Online Appendix give shape to specific questions regarding the impact of uncertainty on strategic interaction in common resource games for further investigation in more general setups. Our contribution is related to the understanding of how rules for regulating the commons (see, for example, Ostrom et al. [19]) may depend on the magnitude of risk borne by players in a common resource exploitation environment. Although we do not examine any forms of regulation, our example and the tools we develop in the Online Appendix may help in formally understanding how uncertainty affects strategic exploitation in more general settings in future work. For example, Polasky et al. [20] study whether cooperation can become a subgame-perfect-equilibrium outcome, and Tarui et al. [27] extend this analysis to including imperfect monitoring of each player's harvest. These two papers are closer to tackling questions of regulation than ours. Yet, introducing informational uncertainty, e.g., through Bayesian learning, to such studies, may be natural extensions in order to understand regulation in more realistic environments. Our work here may be a starting point in order to pursue such extensions.