Assuming that there are n players participating in the game, if no one player can act alone to increase the income (that is, in order to maximize his own interests, no one is willing to change his strategy [1]), then this strategy combination is called Nash equilibrium. All players' strategies form a strategy profile.
Nash equilibrium is essentially a non-cooperative game state.
When Nash equilibrium is reached, it does not mean that both sides of the game are in a state of immobility. In sequential games, this equilibrium is achieved in the continuous actions and reactions of players. Nash equilibrium does not mean that both sides of the game have reached the overall optimal state. It should be noted that only the optimal strategy can achieve Nash equilibrium, and the strictly inferior strategy cannot be the best response, while the weakly superior and weakly inferior strategies can achieve Nash equilibrium. A game may have multiple Nash equilibria, but the prisoner's dilemma has only one Nash equilibrium.
Nash's main academic contributions are embodied in his two papers 1950 and 195 1. It was only in 1950 that he published his research results in the Monthly Bulletin of the American Academy of Sciences, which immediately caused a sensation. Speaking of it, it all depends on the work of Brother David Gale. Just a few days after being belittled and ridiculed by von Neumann, he met Gail and told him that he had found the promotion method and balance point of von Neumann's "minimum-maximum principle", just like talking in a dream. The fledgling Nash has no idea about the dangers of competition and has never thought about the consequences of academic cheating. So, David Gale acted as his "agent" and drafted a short message to the Academy of Sciences, while Lev Shetz, the head of the department, used the convenient relationship to personally submit the manuscript to the Academy of Sciences. Nash didn't write many articles. He advocated that less is the best. Morris, winner of the Nobel Prize in Economics from 65438 to 0996, did not publish any articles when he was a professor of economics in edgeworth at Oxford University. Special talents should have special selection methods.
Nash equilibrium refers to the situation in the game. For each participant, as long as others don't change their strategies, they can't improve their situation. Nash proved that Nash equilibrium must exist on the premise that each player has limited strategy choices and allows mixed strategies. Taking the price war between the two companies as an example, Nash equilibrium means that both parties may lose: under the condition that the other party does not change the price, it can neither raise the price, otherwise it will further lose the market; You can't reduce the price because you will lose money. So the two companies can change the original interest pattern and seek a new interest evaluation and distribution scheme through consultation, that is, Nash equilibrium. Similar reasoning can of course be applied to elections, conflicts of interest between groups, deadlock before the outbreak of potential wars, debates on bills in parliament and so on.
Nash equilibria can be divided into two categories: pure strategic Nash equilibria and mixed strategic Nash equilibria.
To explain pure strategy Nash equilibrium and mixed strategy Nash equilibrium, we must first explain pure strategy and mixed strategy.
The so-called pure strategy is to provide players with a complete definition of how to play the game. In particular, pure strategy determines the exercise to be done in any situation. A strategy set is a set of pure strategies that players can execute. Mixed strategy is a strategy formed by assigning a probability to each pure strategy. Mixed strategy allows players to randomly choose a pure strategy. The game equilibrium of mixed strategies should be calculated by probability, because each strategy is random, and when it reaches a certain probability, the optimal payment can be realized. Because the probability is continuous, even if the strategy set is limited, there will be infinite mixed strategies.
Of course, strictly speaking, every pure strategy is a "degenerate" mixed strategy. The probability of a specific pure strategy is 1, and others are 0.
So "pure strategy Nash equilibrium", that is, everyone involved plays pure strategy; And the corresponding "mixed strategy Nash equilibrium", in which at least one player plays mixed strategy. Not every game will have a pure strategic Nash equilibrium. For example, the "coin problem" only has a mixed strategic Nash equilibrium, and there is no pure strategic Nash equilibrium. But there are still many games with pure strategy Nash equilibria (such as coordination game, Prisoner's Dilemma, Deer Hunting Game). Even, some games can have both pure strategy and mixed strategy equilibrium.
Finally, tell me an example similar to the prisoner's dilemma.