[complete information static game characteristics of Nash equilibrium]

Solution of game

Given a game , On the solution of the game problem , There are at least three different possible explanations ：

1. Experienced 、 A descriptive explanation ： In a given game , How the players play the game ;
2. Normative interpretation ： In a given game , Participants “ should ” How to play the game ;
3. The explanation of the theory ： Assume that the participant's behavior is “ reasonable ” or “ rational ”, So what can be inferred .

The first explanation is based on the observation of participants' behavior ; The second interpretation is based on agreed principles , It is suitable for studying cooperative game ; The third explanation is non cooperative game .

Nash The meaning of equilibrium

Nash The characteristic of equilibrium ： For any participant $i$, Given the choice of other participants , Equilibrium strategy is one's own optimal strategy . It is the self affirmation of all people .

Given a game $G$,$R$ It is a collection of mixed strategic combinations of participants ,$\phi (G)\in R$ Represents the solution of the game , It satisfies two conditions ：

1. $\forall G,\pi \in \phi (G)$, There is an environment where $\pi$ Become a participant in this game $G$ An accurate prediction of how the will act .（ Anything in the solution set is OK , It doesn't mean it's not possible outside ）
2. $\forall G,\pi \in R\backslash \phi (G)$, There is no environment where $\pi$ Become a participant in this game $G$ An accurate prediction of how the will act .（ No one outside the solution set , It doesn't mean it's OK ）

The solution satisfying the first property is called the lower solution of the game , The solution satisfying the second property is called the upper solution of the game , obviously Nash Equilibrium is the upper solution .

Nash The existence of equilibrium

Theorem 3.1（Nash Existence theorem of equilibrium 1） every last Co., LTD. There is at least one strategic game Nash equilibrium （ Including pure strategy and mixed strategy Nash equilibrium ）.

Theorem 3.2（Nash Existence theorem of equilibrium 2） For strategic games , if $S_i$ Is a nonempty compact convex subset of Euclidean space , Payment function $u_i$ About the strategic mix $s$ continuity 、 About $s_i$ Quasiconcave , Then the game has pure strategic Nash equilibrium .

Theorem 3.3（Nash Existence theorem of equilibrium 3） For strategic games , if $S_i$ Is a nonempty compact subset of a metric space , Payment function $u_i$ About the strategic mix $s$ continuity , Then the game has a mixed strategy Nash equilibrium .

Theorem 3.4（Nash Existence theorem of equilibrium 4） For strategic games , if $S_i$ Is a nonempty compact convex subset of a finite dimensional Euclidean space , Payment function $u_i$ About $s$ Upper semicontinuous 、 About $s_i$ Quasiconcave and ${\max }_{s_i}{u}_i(s_i,s_{-i})$ About $s_{-i}$ continuity , Then the game has a pure strategic Nash equilibrium .

Nash The multiplicity of equilibrium

In the study of traditional game theory , The problem may not be how to find the game Nash equilibrium （ The problem of existence ）, But in the game of multiple Nash Choose a reasonable equilibrium （ Multiplicity problem ）.

Focal effect

The phenomenon ： In a game with multiple equilibria , Tends to focus participants' attention on anything in a balanced way , Can make the participants all expect and then implement this balance . This phenomenon is called “ Focal effect ”.

An equilibrium in the focus effect that has a property that distinguishes it significantly from all other equilibria , go by the name of “ Focus balance ”.

The factors that cause the focus balance are ：

1. Social and cultural customs 、 habit 、 The history of the game …;
2. The nature of equilibrium strategy itself （ In several equilibria , The equilibrium with the greatest utility is often called the final answer ）;
3. Bargain negotiation .

Bargain negotiation means ： Participants before the game starts , Achieved at no cost 、 Negotiations that are not binding on participants .

Focus effect can not lead rational participants to implement a non - Nash Balanced strategic mix .

cheap talk It doesn't work under any circumstances .

Correlation equilibrium

Let participants choose actions based on a commonly observed signal .

If the rule that the participants choose to act according to the signal itself can constitute a Nash equilibrium , Then participants may choose actions based on a commonly observed signal .

The rules of action with participants constitute Nash Equilibrium is “ Correlation equilibrium ”

Definition 3.1 A given finite $n$ Correlative equilibrium of man strategic game , Include ：

1. Finite probability space $(\Omega ,\pi )$;
2. $\forall i\in \Gamma$, State set $\Omega$ A division of $P_i$;
3. if $\delta =({\delta }_1^{\ast },...,{\delta }_n^{\ast })$ Is a correlation equilibrium if and only if it is true for $\forall i\in \Gamma$ And arbitrary ${\delta }_i$, Yes $$\sum _{\omega \in \Omega }\pi (\omega )\cdot u_i({\delta }_i^{\ast },{\delta }_{-1}^{\ast })\ge \pi (\omega )\cdot u_i({\delta }_i,{\delta }_{-1}^{\ast })$$