How to find a mixed strategy Nash equilibrium

One of the principles of mixed-strategy equilibrium solving is that mixed-strategy equilibrium gives equal expected returns to all pure strategies with positive probabilities.

Let's say it's a two-player game.

Player A has 2 pure strategies A and B, which cannot dominate each other.

Player B has 2 pure strategies C and D and cannot dominate each other.

Let the probability of A choosing A be p, then the probability of choosing B is 1-p; Let the probability of b choosing c be q, then the probability of choosing d is 1-q

When a takes a certain p=p0, the total utility obtained by b does not change the value of its q; b takes a certain q=q0, and the total utility obtained by a does not change the value of its own p, so we say that (p0,1-p0) and (q0,1-q0) are Nash equilibrium under a mixed strategy.

Let's say it's a two-player game.

Player A has 2 pure strategies A and B, which cannot dominate each other.

Player B has 2 pure strategies C and D and cannot dominate each other.

Let the probability of A choosing A be p, then the probability of choosing B is 1-p; Let the probability of b choosing c be q, then the probability of choosing d is 1-q

When a takes a certain p=p0, the total utility obtained by b does not change the value of its q; b takes a certain q=q0, and the total utility obtained by a does not change the value of its own p, so we say that (p0,1-p0) and (q0,1-q0) are Nash equilibrium under a mixed strategy.

Game Theory and Nash Equilibrium] Wang Zeke: What is a mixed strategy Nash equilibrium?

The Nash equilibrium is solved as follows:Nash equilibrium, also known as non-cooperative game equilibrium, is an important term in game theory, named after John Nash.

In the course of a game, regardless of the opponent's strategy choice, one of the parties will choose a certain strategy, then the strategy is called the dominant strategy. If the combination of strategies of the parties to the two games constitutes their respective dominant strategies, then this combination is defined as a Nash equilibrium.

A combination of strategies is called a Nash balance, and when each player balances the strategy to achieve the maximum expected return, all other players follow this strategy.

There are at least three Nash equilibrium of pure strategy in c.

The stag hunt game (SHG) is also known as the stag hunt model and the Pareto efficiency of the hunter.

Assurance game and coordination game correspond to trust dilemma.

It originated from the French Enlightenment thinker Rousseau.

Jean-Jacques Rousseau, On the Origin and Foundations of Human Inequality

in a story.

Two people go out hunting, their prey is a deer and a rabbit, and they don't know each other's prey. If you choose a deer, you need another person to choose the deer as well, resulting in cooperation for a successful hunt. Choosing a rabbit as a prey does not require cooperation to succeed, but the benefits of hunting rabbits are less than that of hunting deer.

Introduce. The Stag Hunt Game (SHG) is different from the prisoner's dilemma.

prisoner'S Dilemma (PD), SHG has two pure strategies Nash equilibrium: full cooperation or full betrayal, while in PD, although full cooperation is Pareto optimal.

But only total betrayal can reach Nash equilibrium.