# Relationship between dominant strategy and nash equilibrium theory

### Nash Equilibrium and Dominant Strategies- Game Theory - Fundamental Finance

In game theory, there are two kinds of strategic dominance: a strictly dominant strategy is that strategy that always provides greater utility to a. Game theory is the study of strategic interactions between players. The key to Equilibrium in Dominant Strategies = An outcome of a game in which each firm is . The relationships between the game theory strategies can be summarized: 1. Does trust explain the difference between Nash equilibrium and observed equilibrium? “Dominant strategy equilibrium” is not a concept used in game theory.

The players know the planned equilibrium strategy of all of the other players. The players believe that a deviation in their own strategy will not cause deviations by any other players. 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.

• Nash equilibrium

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.

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.

### game theory - Dominant-Strategy Equilibrium vs Nash Equilibrium - Mathematics Stack Exchange

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.

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. This conclusion is drawn from the " stability " theory above. In these situations the assumption that the strategy observed is actually a NE has often been borne out by research.

### Nash equilibrium - Wikipedia

The blue equilibrium is not subgame perfect because player two makes a non-credible threat at 2 2 to be unkind U. The Nash equilibrium is a superset of the subgame perfect Nash equilibrium. The subgame perfect equilibrium in addition to the Nash equilibrium requires that the strategy also is a Nash equilibrium in every subgame of that game.

This eliminates all non-credible threatsthat is, strategies that contain non-rational moves in order to make the counter-player change their strategy. The image to the right shows a simple sequential game that illustrates the issue with subgame imperfect Nash equilibria. In this game player one chooses left L or right Rwhich is followed by player two being called upon to be kind K or unkind U to player one, However, player two only stands to gain from being unkind if player one goes left.

If player one goes right the rational player two would de facto be kind to him in that subgame. However, The non-credible threat of being unkind at 2 2 is still part of the blue L, U,U Nash equilibrium.

Some strategies—that were not dominated before—may be dominated in the smaller game. The first step is repeated, creating a new even smaller game, and so on.

## Strategic dominance

The process stops when no dominated strategy is found for any player. This process is valid since it is assumed that rationality among players is common knowledgethat is, each player knows that the rest of the players are rational, and each player knows that the rest of the players know that he knows that the rest of the players are rational, and so on ad infinitum see Aumann, There are two versions of this process.

One version involves only eliminating strictly dominated strategies. If, after completing this process, there is only one strategy for each player remaining, that strategy set is the unique Nash equilibrium [2]. 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 Y. This is the single Nash Equilibrium for this game.