We consider a model of renewable resource extraction described by a differential game with infinite horizon. The environmental problems are often considered from cooperative prospective as selfish behavior of the players may negatively affects not only on other players’ profits, but also on the environment. The reason is the joint stock of resource which is influenced by all players. We characterize the Berge and altruistic equilibrium in a differential game of renewable resource extraction and compare them with the Nash equilibrium. According to the concept of altruistic equilibrium players can choose the part of the other players’ payoffs they support and summarize with the part of their own profit. This equilibrium can be considered as an intermediate between Berge and Nash equilibria. We make numerical simulations and demonstrate theoretical results for the case of n symmetric players.

Altruistic-Like Equilibrium in a Differential Game of Renewable Resource Extraction

Biotechnology Among Computer Science and Data Science: A Review of Scientific Development

Various concepts of solutions can be employed in the non-cooperative game theory. The Berge equilibrium is one of such solutions. The Berge equilibrium is an altruistic concept of equilibrium. In this concept, the players act on the principle “One for all and all for one!” The Berge equilibrium solves such well known paradoxes in the game theory as the “Prisoner’s Dilemma”, “Battle of the sexes” and many others. At the same time, the Berge equilibrium rarely exist in pure strategies. Moreover, in finite games, the Berge equilibrium may not exist in the class of mixed strategies. The paper proposes the concept of a weak Berge equilibrium. Unlike the Berge equilibrium, the moral basis of this equilibrium is the Hippocratic Oath “First do no harm”. On the other hand, all Berge equilibria are some weak Berge equilibria. The properties of the weak Berge equilibrium have been investigated. The existence of the weak Berge equilibrium in mixed strategies has been established for finite games. A numerical weak Berge equilibrium approximate search method, based on 3LP-algorithm, is proposed. The weak Berge equilibria for finite 3-person non-cooperative games are computed.

Weak Berge Equilibrium

Various concepts of solutions can be employed in non-cooperative game theory A Berge equilibrium is one such solutions The Berge equilibrium is altruistic concept of equilibrium In this concept, the players act on the principle of “One for all and all for one!” The Berge equilibrium solves such well known paradoxes in game theory as the “Prisoner’s Dilemma”, “Battle of the sexes” and many others At the same time, Berge equilibrium rarely exists in pure strategies Moreover, in finite games, the Berge equilibrium may not exist in the class of mixed strategies The paper proposes the concept of a weak Berge equilibrium Unlike the Berge equilibrium, the moral basis of this equilibrium is the Hippocratic Oath “First do no harm” On the other hand, all Berge equilibria are some weak Berge equilibria The existence of weak Berge equilibrium in mixed strategies has been established for finite games A numerical weak Berge equilibrium approximate search method, based on 3LP-algorithm, was proposed The weak Berge equilibrium for finite three-person non-cooperative games were computed

Computation the Weak Berge Equilibrium in Finite 3-Person Games

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Weak Berge Equilibrium in Finite Three-person Games: Conception and Computation

The effectiveness of using the local search algorithm (mountain climbing) and the Lemke–Howson method for searching for Nash equilibrium in 4-person games in general and multi-matrix settings using the Matlab, Python, and FORTRAN software environments are studied. The local search procedure implemented in the Python environment, involving the use of multiplication of a multi-dimensional matrix by a vector, turned out to be an effective tool. The modification of the Lemke–Howson method for multi-matrix formulation showed very good results.

Ustav Malkov

Papers

Weak Berge Equilibrium

Computation the Weak Berge Equilibrium in Finite 3-Person Games

Weak Berge Equilibrium in Finite Three-person Games: Conception and Computation

Effectiveness of Nash Equilibrium Search Algorithms in Four-Person Games in General and Multi-matrix Settings