Resource Exploitation in a Stochastic Horizon under Two Parametric Interpretations

Abstract

<jats:p>This work presents a two-player extraction game where the random terminal times follow (different) heavy-tailed distributions which are not necessarily compactly supported. Besides, we delve into the implications of working with logarithmic utility/terminal payoff functions. To this end, we use standard actuarial results and notation, and state a connection between the so-called actuarial equivalence principle, and the feedback controllers found by means of the Dynamic Programming technique. Our conclusions include a conjecture on the form of the optimal premia for insuring the extraction tasks; and a comparison for the intensities of the extraction for each player under different phases of the lifetimes of their respective machineries.</jats:p>

A Geologic-Actuarial Approach for Insuring the Extraction Tasks of Non-Renewable Resources by One and Two Agents

“…The expression (9) invites us to relate it to the net level premium referred to in any basic text on actuarial mathematics (such as Chapter 6 in [5]), as well as to establish expressions such as (1). Moreover, Theorem 3 in [4] considers the case of two participants that we study in this work, and proves that if the players' running utility functions are logarithmic (i.e., h i (x, u i ) = ln u i for i = 1, 2), and the terminal payoff function of the i-th player is…”

“…Proposition 1 in [4] proves that, if t 0 h i (x * (s), u * 1 (s), u * 2 (s))ds < ∞ for all t > 0 (where x * (s) represents the trajectory that (3) follows when the strategies referred by Definition 2 are used) and under our hypotheses, the optimal expected payment for each player is…”

“…where f τ j (•) is a density function for τ j . Moreover, Theorem 1 in [4] uses common stochastic dynamic programming techniques to see that if a single agent exploits a well of a nonrenewable resource and its utility function is of logarithmic type, i.e., h(x, u) = ln u, then, the optimal controller for such agent is of a closed-loop form (In fact, what ([4], Theorem 1) proves is the particular case where the random variable τ follows Weibull's or Chen's law. However, it is not difficult to extend that exact same proof to the general case where the distribution meets (4) in Asssumption 2).…”

“…The works [3,4] use the principle of dynamic programming to show that, when the utility function of an agent extracting a non-renewable resource (for example, oil) is logarithmic, then a kind of equivalence principle (see p. 2 and Example 6.1.1 in [5]), namely…”

The commons problem in the presence of negative externalities

“…In more recent articles, Dasgupta et al (2019) focus on the importance of population size, harvesting costs and the ecosystem's productivity on the extraction of the commons and Quaas and Tahvonen (2019) explore the strategic exploitation of an agestructured common resource, where agents target different age classes. In a two-player extraction game, L opez- Barrientos et al (2020) study the case where the random terminal times follow (different) heavy-tailed distributions, which are not necessarily compactly supported.…”