# Relationship between dominant strategy and nash equilibria prisoner dilemma

### Nash equilibrium - Wikipedia

What's the difference between dominant-strategy solution and Nash Equilibrium? I could not tell the difference judging from the definitions. In game theory, strategic dominance (commonly called simply dominance) occurs when one The classic game used to illustrate this is the Prisoner's Dilemma. Strictly dominated strategies cannot be a part of a Nash equilibrium, and as such, . Edit links. This page was last edited on 28 December , at 02 (UTC). Nash Equilibrium is a term used in game theory to describe an equilibrium where Here's the game (remember that in the Prisoners' Dilemma, the numbers.

Indeed, for cell B,A 40 is the maximum of the first column and 25 is the maximum of the second row. For A,B 25 is the maximum of the second column and 40 is the maximum of the first row.

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

Same for cell C,C. For other cells, either one or both of the duplet members are not the maximum of the corresponding rows and columns. This said, the actual mechanics of finding equilibrium cells is obvious: If these conditions are met, the cell represents a Nash equilibrium.

Check all columns this way to find all NE cells. Stability[ edit ] The concept of stabilityuseful in the analysis of many kinds of equilibria, can also be applied to Nash equilibria. A Nash equilibrium for a mixed-strategy game is stable if a small change specifically, an infinitesimal change in probabilities for one player leads to a situation where two conditions hold: If these cases are both met, then a player with the small change in their mixed strategy will return immediately to the Nash equilibrium.

The equilibrium is said to be stable. If condition one does not hold then the equilibrium is unstable. If only condition one holds then there are likely to be an infinite number of optimal strategies for the player who changed. In the "driving game" example above there are both stable and unstable equilibria. If either player changes their probabilities slightly, they will be both at a disadvantage, and their opponent will have no reason to change their strategy in turn.

Stability is crucial in practical applications of Nash equilibria, since the mixed strategy of each player is not perfectly known, but has to be inferred from statistical distribution of their actions in the game.

In this case unstable equilibria are very unlikely to arise in practice, since any minute change in the proportions of each strategy seen will lead to a change in strategy and the breakdown of the equilibrium. The Nash equilibrium defines stability only in terms of unilateral deviations.

In cooperative games such a concept is not convincing enough.

## Nash equilibrium

Strong Nash equilibrium allows for deviations by every conceivable coalition. In fact, strong Nash equilibrium has to be Pareto efficient. As a result of these requirements, strong Nash is too rare to be useful in many branches of game theory. However, in games such as elections with many more players than possible outcomes, it can be more common than a stable equilibrium.

A refined Nash equilibrium known as coalition-proof Nash equilibrium CPNE [13] occurs when players cannot do better even if they are allowed to communicate and make "self-enforcing" agreement to deviate.

**Nash equilibrium**

Every correlated strategy supported by iterated strict dominance and on the Pareto frontier is a CPNE. CPNE is related to the theory of the core. Finally in the eighties, building with great depth on such ideas Mertens-stable equilibria were introduced as a solution concept. Mertens stable equilibria satisfy both forward induction and backward induction. In a game theory context stable equilibria now usually refer to Mertens stable equilibria.

Occurrence[ edit ] If a game has a unique Nash equilibrium and is played among players under certain conditions, then the NE strategy set will be adopted. Sufficient conditions to guarantee that the Nash equilibrium is played are: The players all will do their utmost to maximize their expected payoff as described by the game. The players are flawless in execution. The players have sufficient intelligence to deduce the solution.

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

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.

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. 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. However, unlike the first process, elimination of weakly dominated strategies may eliminate some Nash equilibria.

As a result, the Nash equilibrium found by eliminating weakly dominated strategies may not be the only Nash equilibrium. In some games, if we remove weakly dominated strategies in a different order, we may end up with a different Nash equilibrium. O is strictly dominated by N for Player 1. Therefore Player 1 will never play strategy O.