Nash equilibrium - Wikipedia
dominant strategy equilibrium or we say it is the outcome of iterated deletion of strictly dominated The difference between IDSDS and IDWDS. If IDSDS Finding Nash equilibria in strategic form can quickly be done in two ways: ( Elimination). A strategy is dominated if, regardless of what any other players do, the strategy earns a player a smaller payoff than some other strategy. Hence. In game theory, the Nash equilibrium, named after the late mathematician John Forbes Nash Jr. Relationship In terms of game theory, if each player has chosen a strategy, and no player can benefit by changing .. This said, the actual mechanics of finding equilibrium cells is obvious: find the maximum of a column and.
But both of those are true. So this is definitely not a Nash equilibrium. I gave two examples in which a participant can gain by a change of strategy as long as the other participant remains unchanged.
This move was one example, and this was a move by Al, with Bill's denial constant. This was a move by Bill, with Al's denial constant. Not a Nash equilibrium.
Now let's think about-- let's think about state 2. If we are sitting in state 2, assuming Bill is constant, can Al change to improve his outcome? So can Al change to improve his outcome? In state 2, Al was only getting one year. If Al goes from a confession to a denial he's going to get two years. So Al cannot change his strategy and get a gain here. So far it's looking good.
But let's think of it from Bill's point of view. So if we are in state 2, if we are in state 2 right over here, and we assume Al is constant, can Bill do something that changes things?
Well, sure, Bill can go from denying to confessing. If he goes from denying to confessing he goes from 10 years in prison to three years in prison. So I've given an example of a participant who can gain by a change of strategy as long as all of the other participants remain unchanged.
Both of them don't have to be able to do this. You just need to have one of them for it to not be a Nash equilibrium. Because Bill can have a gain by a change of strategy holding Al's strategy constant, so holding Al's strategy in the confession, then this is not a Nash equilibrium. So this is not Nash, because you have this movement can occur to a more favorable state for Bill holding Al constant. Now, let's go to state 3.
Let's think about this. So if we're in state 3, so this is Bill confessing and Al denying-- so let's first think about Al's point of view. If we assume Bill is constant in his confession, can Al improve his scenario? He can go from denying, which is what would have to be in state 3, to confessing.D.5 Dominant strategies and Nash equilibrium - Game Theory - Microeconomics
So he could move in this direction right over here. C is strictly dominated by A for Player 1. Therefore Player 1 will never play strategy C. Player 2 knows this. Therefore, Player 2 will never play strategy Z. Player 1 knows this. Therefore, Player 1 will never play B. Therefore, Player 2 will never play X.
This is the single Nash Equilibrium for this game. Another version involves eliminating both strictly and weakly dominated strategies.
If, at the end of the process, there is a single strategy for each player, this strategy set is also a Nash equilibrium. There is common knowledge that all players meet these conditions, including this one. So, not only must each player know the other players meet the conditions, but also they must know that they all know that they meet them, and know that they know that they know that they meet them, and so on.
Where the conditions are not met[ edit ] Examples of game theory problems in which these conditions are not met: The first condition is not met if the game does not correctly describe the quantities a player wishes to maximize. In this case there is no particular reason for that player to adopt an equilibrium strategy. Intentional or accidental imperfection in execution.
For example, a computer capable of flawless logical play facing a second flawless computer will result in equilibrium.
- Nash equilibrium
- How to find a Nash Equilibrium in a 2X2 matrix
- Strategic dominance
Introduction of imperfection will lead to its disruption either through loss to the player who makes the mistake, or through negation of the common knowledge criterion leading to possible victory for the player. An example would be a player suddenly putting the car into reverse in the game of chickenensuring a no-loss no-win scenario.
In many cases, the third condition is not met because, even though the equilibrium must exist, it is unknown due to the complexity of the game, for instance in Chinese chess. The criterion of common knowledge may not be met even if all players do, in fact, meet all the other criteria. This is a major consideration in " chicken " or an arms racefor example. Where the conditions are met[ edit ] In his Ph. One interpretation is rationalistic: This idea was formalized by Aumann, R.
Brandenburger,Epistemic Conditions for Nash Equilibrium, Econometrica, 63, who interpreted each player's mixed strategy as a conjecture about the behaviour of other players and have shown that if the game and the rationality of players is mutually known and these conjectures are commonly know, then the conjectures must be a Nash equilibrium a common prior assumption is needed for this result in general, but not in the case of two players.
How to find a Nash Equilibrium in a 2X2 matrix - posavski-obzor.info, Learning Economics Solved!
In this case, the conjectures need only be mutually known. A second interpretation, that Nash referred to by the mass action interpretation, is less demanding on players: What is assumed is that there is a population of participants for each position in the game, which will be played throughout time by participants drawn at random from the different populations. If there is a stable average frequency with which each pure strategy is employed by the average member of the appropriate population, then this stable average frequency constitutes a mixed strategy Nash equilibrium.
For a formal result along these lines, see Kuhn, H. Due to the limited conditions in which NE can actually be observed, they are rarely treated as a guide to day-to-day behaviour, or observed in practice in human negotiations. However, as a theoretical concept in economics and evolutionary biologythe NE has explanatory power. The payoff in economics is utility or sometimes moneyand in evolutionary biology is gene transmission; both are the fundamental bottom line of survival.
Researchers who apply games theory in these fields claim that strategies failing to maximize these for whatever reason will be competed out of the market or environment, which are ascribed the ability to test all strategies.