John Nash developed a criterion for the mutual consistency of players’ strategies which proved mathematically that every finite game has an equilibrium known as a “Nash Equilibrium”, provided that one “generalizes the concept of actions or strategies to allow mixing of moves.” – John Nash. Since 1950, “The Nash Equilibrium” has become “the analytical structure for studying all situations of conflict and cooperation.” – Roger Myerson, 1999
What is this Equilibrium?
A “Nash Equilibrium”, also called strategic equilibrium, is a list of strategies, one for each player, which has the property that no player can unilaterally (without the agreement of others) change his strategy and get a better payoff. Since players are also rational, it is reasonable that each player expects his opponent to follow the recommendation as well.
Nash formally defined equilibrium of a non-cooperative game to be “a configuration of strategies, such that no player acting on his own can change his strategy to achieve a better outcome for himself.” The outcome of such a game must be a Nash equilibrium if “it is to conform to the assumption of rational individual behavior. That is, if the predicted behavior doesn’t satisfy the condition for Nash equilibrium, then there must be at least one individual who could achieve a better outcome if he were simply made aware of his own best interests.” – Sylvia Nasar, The Essential John Nash
Dixit and Nalebuff give a great description of “The Nash Equilibrium” in The Art of Strategy: “John Nash’s equilibrium was designed as a theoretical way to square circles of thinking about thinking about other people’s choices in games of strategy. The idea is to look for an outcome where each player in the game chooses the strategy that best serves his or her own interest, in response to the other’s strategy. If such a configuration of strategies arises, neither player has any reason to change his choice unilaterally. Such an outcome in a game, where the action of each player is best for him given his beliefs about the other’s action, and the action of each is consistent with the other’s belief about it, neatly squares the circle of thinking about thinking. Therefore, this is a potentially stable outcome of a game where the players make individual and simultaneous choices of strategies and has a good claim to be called a resting point of the players’ thought processes, or an equilibrium of the game. Indeed, this is just a definition of Nash equilibrium and we find reasons for cautious optimism for making Nash equilibrium a starting point of analysis of almost all games.”
Thomas C. Schelling, in The Strategy of Conflict, gives us a definition of “The Nash Equilibrium” when he details the solution to a non-cooperative game: “A non-cooperative [game] is said to have a solution in the strict sense if: (1) There exists an equilibrium pair among the jointly admissible strategy pairs and (2) All jointly admissible equilibrium pairs are both interchangeable and equivalent. An equilibrium pair is a pair of strategies for the two players such that each is the player’s best strategy (or as good as any other) that can be coupled with the other’s. Equilibrium pairs are equivalent if, for each player separately, they yield equal payoffs; equilibrium pairs are interchangeable if all pairs formed from the corresponding strategies are also equilibrium points.”
Holt and Roth state that, “When the goal is prediction rather than prescription, a Nash Equilibrium can be interpreted as a potential stable point of a dynamic adjustment process in which individuals adjust their behavior to that of the other players in the game, searching for strategy choices that will give them better results.” – The Nash Equilibrium: A Perspective.
And, according to Wikipedia, “The simple insight underlying Nash’s idea is that one cannot predict the choices of multiple decision makers if one analyzes those decisions in isolation. Instead, one must ask what each player would do taking into account what the player expects the others to do. Nash equilibrium requires that one’s choices be consistent: no players wish to undo their decision given what the others are deciding.”
But, we must read Nash’s own words for a definition of his Equilibrium that is both clear and concise: “It turns out that the set of equilibrium points of a two-person zero-sum game is simply the set of all pairs of opposing ‘good strategies’.” Finally, the central concept of Nash Equilibrium, according to Nash himself, can be defined mathematically as:
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