Nash Equilibrium Set Function in Dyadic Mixed-Strategy Games

Abstract

Dyadic two-person mixed strategy games form the simplest case for which we can determine straightforwardly Nash equilibrium sets. As in the precedent chapters, the set of Nash equilibria in a particular game is determined as an intersection of graphs of optimal reaction mappings of the first and the second players. In contrast to other games, we obtain not only an algorithm, but a set-valued/multi-valued Nash equilibrium set function (Nash function) ${ NES}(A,B)$ that gives directly as its values the Nash equilibrium sets corresponding to the values of payoff matrix instances. The function is piecewise and it has totally 36 distinct pieces, i.e. the domain of the Nash function is divided into 36 distinct subsets for which corresponding Nash sets have a similar pattern. To give an expedient form to such a function definition and to its formula, we use a code written in the Wolfram language (Wolfram, An elementary introduction to the Wolfram language, Wolfram Media, Inc., Champaign, XV+324 pp, 2016, [1]; Hastings et al., Hands-on start to Wolfram mathematica and programming with Wolfram language, Wolfram Media, Inc., Champaign, X+470 pp, 2015, [2]) that constitutes a specific feature of this chapter in comparison with other chapters. To prove the main theoretic result of this chapter, we apply the Wolfram language code too. The Nash function ${ NES}(A,B)$ is a multi-valued/set-valued function that has in the quality of its domain the Cartesian product $\mathbb {R}^{2\times 2}\times \mathbb {R}^{2\times 2}$ of two real spaces of two $2\times 2$ matrices and in the quality of a Nash function image all possible sets of Nash equilibria in dyadic bimatrix mixed-strategy games. These types of games where considered earlier in a series of works, e.g. Vorob’ev (Foundations of game theory: noncooperative games, Nauka, Moscow (in Russian), 1984, [3]; Game theory: lectures for economists and systems scientists, Nauka, Moscow (in Russian), 1985, [4]), Gonzalez-Diaz et al. (An introductory course on mathematical game theory, American Mathematical Society, XIV+324 pp, 2010, [5]), Sagaidac and Ungureanu (Operational research, CEP USM, Chişinău, 296 pp (in Romanian), 2004, [6]), Ungureanu (Set of nash equilibria in $2\times 2$ mixed extended games, from the Wolfram demonstrations project, 2007, [7]), Stahl (A gentle introduction to game theory, American Mathematical Society, XII+176 pp, 1999, [8]), Barron (Game theory: an introduction, 2nd ed, Wiley, Hoboken, XVIII+555 pp, 2013, [9]), Gintis (Game theory evolving: a problem-centered introduction to modeling strategic interaction, 2nd ed, Princeton University Press, Princeton and Oxford, XVIII+390 pp, 2009, [10]). Recently, we found the paper by John Dickhaut and Todd Kaplan that describes a program written in Wolfram Mathematica for finding Nash equilibria (Dickhaut and Kaplan, Economic and financial modeling with mathematica®, TELOS and Springer, New York, pp 148–166, 1993, [11]). The program is based on the game theory works by Rapoport (Two-person game theory: the essential ideas, University of Michigan Press, 229 pp, 1966, [12]; N-person game theory: concepts and applications, Dover Publications, Mineola, 331 pp, 1970, [13]), Friedman (Game theory with applications to economics, 2nd ed, Oxford University Press, Oxford, XIX+322 pp, 1990, [14]), and Kreps (A course in microeconomic theory, Princeton University Press, Princeton, XVIII+839 pp, 1990, [15]). Some examples from Harsanyi and Selten book (Harsanyi and Selten, General theory of equilibrium selection in games, The MIT Press, Cambridge, XVI+378 pp, 1988, [16]) are selected for tests. Unfortunately, the program and package need to be updated to recent versions of the Wolfram Mathematica and the Wolfram Language.