# Introduction to Game Theory/Nash equilibrium

## Basic Definition:

A Nash Equilibrium is a set of mixed strategies for finite, non-cooperative games between two or more players whereby no player can improve his or her payoff by unilaterally changing their strategy. Each player's strategy is an 'optimal' response based on the anticipated rational strategy of the other player(s) in the game.

Nash outlined a new paradigm for mathematical and economic thinkers with his pioneering use of Equilibrium Theory. He had been accepted to study in New Jersey from his native West Virginia on a Scholarship for (Pure) Mathematics, and worked briefly under the advice of Albert Einstein. Many of Nash's contemporaries refer to him as a (post)modern day 'genius' for his reformations to some of Adam Smith's views on Economics and when considering his more personal characteristics, including his unorthodox teaching and research procedures, and his past experiences with schizophrenia.

## Equilibrium Theory:

The Nash Theorem maintains its focus on rivalries with mutual gain; a perceptual focus of Nash's mathematical vision found in the light of Leon Walras' General Equilibrium Theory (published 1874) and John von Neumann's and Oskar Morgenstern's Theory of Games (1944), now simply called Game Theory. Nash later established his own idea of dominant strategy equilibria through maximization solutions for zero-sum games. He did this with original mathematical techniques to demonstrate the existence of methods for finding a measurable equilibrium in a general class of non-cooperative games.

Other academic theorists used the concept of 'equilibrium' in the 19th century (Maxwell, Walras, Gibbs), for chemical and economic equilibrium in the early stages of the 20th century (van der Waals, Onnes, Keynes) before Nash used it in the middle of the 20th century. Others improved upon Nash's original formulation in the 1950's and 60's (Selten, Harsanyi) and explored different possible aspects of a General Equilibrium Theory (GET) from the 70's to the 90's (Arrow, Hicks, Debreu). Many students of economics still study GET today. Recent academic applications of equilibrium theory from non-economists (Kolmogorov, S. Nagel), and in directions other than those laid down by the neo-Classicalists (cf. neo-Keynesians below), persist as well. The proposed universality of equilibrium theory (i.e. that all situations or economic conditions can/should be considered in the paradigm of general or specific equilibrium thinking) by some theorists makes it difficult to analyze subjects or objects with(-in) a single interpretive framework, and often demands multiple perspectives or even 'relativized' equilibria to explain and interpret diverse economies across the disciplines. Ordeshook above shows us the reach of Nash's equilibrium theory over disciplinary boundaries, and its relevance for practical mathematical-economic applications; now (post)modern theorists must again find ways to deal with its implications in the 21st century academy.

The possible theoretical limitations of equilibrium theory have led to disequilibrium neo-Keynesian theories during the last fifty years (Hahn, Fisher). And stubborn questions remain, even with uncertainty: what if there is more than one Nash Equilibrium in a given game; or if the players in a game have incomplete information; or if the rationality thesis fails to clearly convince readers who live beyond the so-called Age of Reason...? How does a unified Nash Equilibrium maintain itself in a theoretically plural academic culture and with increasingly complex indicators and instruments used in the scientific pursuit? Such may be the cases, for example, when turn-of-the-century relativity theories or even practical evolutionism (i.e. applied morphology) in Economics disallow unification on any shared equilibrium goal(s) or value(s). But then again, this message is itself being written in a time of (post)war.

John Nash can be credited against astonishing odds with making a normative distinction between cooperative and non-cooperative games, and for using mathematical models to support and exemplify his research. But his contribution to theory has become (neo-Nash) a web of explanation(s) and justification(s) far beyond the original conceptions of the author, and has also progressed steadily into the social world—somewhere Nash himself may not have wanted it to go.