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Dynamic Games and Applications

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01.03.2015

Strategic Exploitation of a Common-Property Resource Under Rational Learning About its Reproduction

verfasst von: Christos Koulovatianos

Erschienen in: Dynamic Games and Applications | Ausgabe 1/2015

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Abstract

We build a workable game of common-property resource extraction under rational Bayesian learning about the reproduction prospects of a resource. We focus on Markov-perfect strategies under truthful revelation of beliefs. For reasonable initial conditions, exogenously shifting the prior beliefs of one player toward more pessimism about the potential of natural resources to reproduce can create anti-conservation incentives. The single player whose beliefs have been shifted toward more pessimism exhibits higher exploitation rate than before. In response, all other players reduce their exploitation rates in order to conserve the resource. However, the overall conservation incentive is weak, making the aggregate exploitation rate higher than before the pessimistic shift in beliefs of that single player. Due to this weakness in strategic conservation responses, if the number of players is relatively small, then in cases with common priors, jointly shifting all players’ beliefs toward more pessimism exacerbates the commons problem.

Vorheriger Artikel The N-Player War of Attrition in the Limit of Infinitely Many Players

Nächster Artikel Correlated Equilibria in Stochastic Games with Borel Measurable Payoffs

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Debates about acid rain (see, for example, [16, pp. 178–181]), or about a large-scale biodiversity deterioration ([16, pp. 249–257]) are related to shifts in fundamentals regarding the natural ability of resources to reproduce. While environmental changes are not disputed among experts as facts, the magnitude of such changes is a vivid topic of disagreement among experts. For example, the article by Rörsch et al. [20] documents the opposition against [16] book by a part of the scientific community. What we keep from this debate among experts for the purposes of this study is that investor uncertainty about environmental fundamentals is a plausible working hypothesis for analyzing investments in natural resources.

Despite that we analyze a specific example, the stochastic structure of our Bayesian-learning setup is general, i.e., it is not restricted by conjugate-priors assumptions, as it will be clearer in the model’s outline below. This generality allows for a wide range of applications, extending even to the analysis of jump processes. This versatility of the example’s stochastic structure may stimulate further theoretical and empirical investigation, not only because of the theoretical ideas which are demonstrated here, but also because the example may be fitted to actual data itself. Here, we restrict our analysis to demonstrating the example’s mechanics theoretically.

For example, collecting and processing data on each winter’s temperature inform scientists and investors in natural-resource markets about the validity of their prior beliefs regarding a structural break due to “global warming.”

This heterogeneity in prior beliefs is justified by differences in non-informative priors, which are priors formed before any data are available. Ad-hoc differences in non-informative priors can persist and can play a crucial role in the transitional dynamics of a game. For an introduction to non-informative priors and references on alternative criteria for choosing non-informative priors see, for example, [7, pp. 61–66].

Our setup of Bayesian learners who anticipate learning in the future is similar to the case of rational learning examined by [5, 8, 13].

Focusing on Markov-perfect-Nash-equilibrium strategies may seem restrictive. Yet, even this simplified equilibrium concept involves considerable technical complexities. It was several years after the seminal example by Levhari and Mirman [15] that [21] proved equilibrium existence for the more general deterministic version. Dutta and Sundaram [6] were the first to extend analysis to the stochastic version of the game under rational expectations, while Amir [2] contributed a remarkably general characterization of such stochastic games offering existence and uniqueness results. For a general survey of natural-resource games see [17].

Kalai and Lehrer [11] show a key result related to our rational-learning formulation. When Bayesian updating is envisaged by each player, then Bayesian updating of collected information will lead in the long run to accurate prediction of the future play of the game and rational expectations as a limit of behavior with probability one, as time goes to infinity. This consideration on the side of players, which learning will be completed in the long run, and its impact on strategies, is a key distinctive feature of rational learning from other forms of learning. (The way to make players envisage Bayesian updating in our setting that focuses on Markov-perfect Nash strategies, is to incorporate Bayes’ rule in the Bellman equation of each player.) In their survey paper, [4] explain this distinction as well.

A recent paper that also combines strategic interaction extending the [15] model with rational Bayesian learning in a similar fashion to [13] is [19]. A key difference of the present paper from [19] is that here heterogeneous beliefs are allowed, and the impact of belief changes of one player can be characterized using a general stochastic structure for learning.

The proof regarding the solution of the linear system of multiple Nash-equilibrium necessary conditions, which appears as Lemma 3 in the Appendix of this paper, and which relies on the “ matrix determinant lemma,” first appeared in Koulovatianos [12, pp. 27–29] and enabled the identification of the solution in both [1, 12].

We use a player’s index $i\in \left\{ 1,\ldots ,N\right\} $ as subscript for variables and as superscripts for functions, and we drop it whenever it is redundant.

While Proposition 1 holds for cases in which $\theta $ is a multidimensional vector, the comparisons performed in this section rely upon the concept of stochastic dominance. Avoiding the technicalities of multidimensional stochastic dominance is the main reason for assuming that $\theta $ is a single parameter in this section.

For the definition of strict FOSD, see, for example, [10, p. 6].

Propositions 1 and 2 can facilitate such an analysis under heterogeneous beliefs. Yet, belief heterogeneity makes it difficult to obtain intuitive generalizations about how beliefs affect the intensity of the commons problem. Under heterogeneous beliefs, one would need to compare equilibrium extraction rates with a social-planner’s solution under heterogeneous beliefs. This cumbersome task is beyond the scope of this paper.

The commons problem is straightforward to verify from Eq. (19). Increasing $N$, i.e., the number of players with the same priors, always leads to higher aggregate exploitation rate, $G\left( N,c^{o}\left( \xi \right) \right) $.

Recall from our analysis above that such a shift in priors toward pessimism is captured by $\underline{\xi }\prec _{\mathrm{{FOSD}}}\xi $, in which $\xi $ is the initial priors, and $\underline{\xi }$ is beliefs after the shift. The result is $c^{o}\left( \underline{\xi }\right) >c^{o}\left( \xi \right) $.

Notice that, according to Eq. (19), each player’s individual exploitation rate is,

$$\begin{aligned} \frac{G\left( N,c^{o}\left( \xi \right) \right) }{N}=\frac{c^{o}\left( \xi \right) }{1+\left( N-1\right) c^{o}\left( \xi \right) }, \end{aligned}$$

which is increasing in $c^{o}\left( \xi \right) $.

This is verifiable from Eq. (19), which implies that $\partial ^{2}\left[ G\left( N,c^{o}\left( \xi \right) \right) /N\right] /\left[ \partial N\partial c^{o}\left( \xi \right) \right] <0$.

This proof first appeared in [12, pp. 27–29], which is an early version of this paper.

I am indebted to an anonymous referee and the Editor for motivating me to provide the argument that leads to the condition given by (11).

Agbo M (2013) Strategic exploitation with learning and heterogeneous beliefs, SSRN working paper, No 1973793

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Blume L, Easley D (1993) Rational expectations and rational learning, economics working paper archive (EconWPA), series: game theory and information, no. 9307003. http://129.3.20.41/eps/game/papers/9307/9307003.pdf

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Titel: Strategic Exploitation of a Common-Property Resource Under Rational Learning About its Reproduction
verfasst von: Christos Koulovatianos
Publikationsdatum: 01.03.2015
Verlag: Springer US
Erschienen in: Dynamic Games and Applications / Ausgabe 1/2015
Print ISSN: 2153-0785
Elektronische ISSN: 2153-0793
DOI: https://doi.org/10.1007/s13235-014-0113-3

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