Abstract
Two-person games are used in many multi-agent mathematical models to describe pair interactions. The type (pure or mixed) and the number of Nash equilibria affect fundamentally the macroscopic behavior of these systems. In this paper, the general features of Nash equilibria are investigated systematically within the framework of matrix decomposition for n strategies. This approach distinguishes four types of elementary interactions that each possess fundamentally different characteristics. The possible Nash equilibria are discussed separately for different types of interactions and also for their combinations. A relation is established between the existence of infinitely many mixed Nash equilibria and the zero-eigenvalue eigenvectors of the payoff matrix.
Highlights
The traditional concepts of game theory [1] have been extensively used for a long time in quantitative investigations of phenomena that occur in biological and social systems [2,3,4,5,6]
The average number of Nash equilibria was studied by Berg [17,18] in n-strategy games with payoff elements determined by random numbers, and the introduction of potential games [19,20,21] has initiated the analysis of the decomposition of games [22,23,24,25,26,27,28]
This example and the flow graph method in general highlight that only one pure Nash equilibrium can exist in each row and column of the payoff matrix, which caps the number of pure Nash equilibria at n in an nstrategy game
Summary
The traditional concepts of game theory [1] have been extensively used for a long time in quantitative investigations of phenomena that occur in biological and social systems [2,3,4,5,6]. In the simplest non-cooperative game there are two equivalent players with two available strategies and their strategy-dependent income is defined by a payoff matrix of four possible payoff elements In these games, the search for the Nash equilibria can be simplified by exploiting two features, namely that the rank of incomes is not affected if the matrix elements are multiplied by a positive number and shifted by a constant. The corresponding two-dimensional map of the possible Nash equilibria distinguishes four characteristic types of behavior illustrated generally by the traditional “prisoner’s dilemma”, “chicken” or “snow-drift”, “staghunt”, and “harmony” or “trivial” games These games are well discussed in the above-mentioned textbooks and reviews. We briefly describe some methods that can be used to determine Nash equilibria
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