# Nash equilibrium existence

The Nash equilibrium [ Nash, ] is applied. The "market" is represented by a collection of independent servers. Each server tries to maximize its profit by setting optimal service prices and optimal server rates. Indeed, for cell B,A 40 is the maximum of the first column and 25 is the maximum of the second Nash equilibrium existence. For A,B 25 is the maximum of the second column and 40 is the maximum of the first row.

Same for cell C,C. For other cells, either one or both of the duplet members are not the maximum of the corresponding rows and columns. This said, the actual mechanics of finding equilibrium cells is obvious: If these conditions are met, the cell represents a Nash equilibrium. Check all columns this way to find all NE cells.

Stability[ edit ] The concept of stabilityuseful in the analysis of many kinds of equilibria, can also be applied to Nash equilibria. A Nash equilibrium for a mixed-strategy game is stable if a small change specifically, an infinitesimal change in probabilities for one player leads to a situation where two conditions hold: If these cases are both met, then a player with the small change in their mixed strategy will return immediately to the Nash equilibrium.

The equilibrium is said to be stable. If condition one does not hold then the equilibrium is unstable. If only condition one holds then there are likely to be an infinite number of optimal strategies for the player who changed.

In the "driving game" example above there are both stable and unstable equilibria. If either player changes their probabilities slightly, they will be both at a disadvantage, and their opponent will have no reason to change their strategy in turn. Stability is crucial in practical applications of Nash equilibria, since the mixed strategy of each player is not perfectly known, but has to be inferred from statistical distribution of their actions in the game.

In this case unstable equilibria are very unlikely to arise in practice, since any minute change in the proportions of each strategy seen will lead to a change in strategy and the breakdown of the equilibrium. The Nash equilibrium defines stability only in terms of unilateral deviations.

In cooperative games such a concept is not convincing enough. Strong Nash equilibrium allows for deviations by every conceivable coalition.

In fact, strong Nash equilibrium has to be Pareto efficient. As a result of these requirements, strong Nash is too rare to be useful in many branches of game theory. However, in games such as elections with many more players than possible outcomes, it can be more common than a stable equilibrium.

A refined Nash equilibrium known as coalition-proof Nash equilibrium CPNE  occurs when players cannot do better even if they are allowed to communicate and make "self-enforcing" agreement to deviate.

Every correlated strategy supported by iterated strict dominance and on the Pareto frontier is a CPNE. CPNE is related to the theory of the core. Finally in the eighties, building with great depth on such ideas Mertens-stable equilibria were introduced as a solution concept.

Mertens stable equilibria satisfy both forward induction and backward induction.Theorem 1 (Nash, ) There exists a mixed Nash equilibrium. Here is a short self-contained proof. We will deﬁne a function Φ over the space of mixed strategy proﬁles. (Nash) Every ﬁnite game has a mixed strategy Nash equilibrium. Implication: matching pennies game necessarily has a mixed strategy equilibrium. Why is this important? Without knowing the existence of an equilibrium, it is diﬃcult (perhaps meaningless) to try to understand its properties. pi(s,α) ≤ 0 for every i and every s ∈ Si, hence must be a Nash equilibrium.

This concludes the proof of the This concludes the proof of the existence of a Nash equilibrium. Reny™s () equilibrium existence theorem states that every compact, quasiconcave, better-reply secure game has a Nash equilibriumin pure strate-gies.

Theorem 1 (Reny,) If G = (X i;u i) i2I is compact, quasiconcave, and better-reply secure, then it possesses a pure strategy Nash equilibrium. @John Baez existence of a Nash equilibrium for two person finite zero sum games is a linear programming problem.

The existence of symmetric equilibrium for a two person finite game with symmetric payoff matrices that are symmetric is a quadratic programming problem. Game Theory: Lecture 5 Existence Results Existence Results We start by analyzing existence of a Nash equilibrium in ﬁnite (strategic form) games, i.e., games with ﬁnite strategy sets.

Theorem (Nash) Every ﬁnite game has a mixed strategy Nash equilibrium. Implication: matching pennies game necessarily has a mixed strategy equilibrium.

Simple proof of the existence of Nash equilibria for 2-person games? - MathOverflow