# Game Theory Nash Equilibrium

by paulali1989 (student)

Issues in Microeconomics

Game Theory Coursework 1

By Paul Allen

B00561027

What is the Nash Equilibrium? This is the idea that equilibrium in a physical system, was that players would adjust their strategies until no player could benefit from changing. All players are then choosing strategies that are best (utility maximising) responses to all other players strategies.(i)

Nash Equilibrium

This concept is a set of strategies where any given player can’t get a better payoff by changing their strategy if all players keep their strategic choices constant. Thus once our hypothetical market reaches a Nash Equilibrium, it will be stable, since no individual strategic response would change anything.

One reason to choose the most desirable multiple Nash equilibria on the basis of perfect rationality is the Pareto optimality concept. A Pareto optimum is an equilibrium for which no player can get better off without making the other player worse off.

Rational

In game theory we assume rational behaviour by all players; players are perfectly informed and follow their best strategy. A consistent ranking will be given to all the possible payoffs from chosen strategies. The rationality can be said in two different counts.

• Complete knowledge of self-interests exists.
• Meticulously calculated actions of what interests serve best are considered.

This of course is not the case there are systematic errors. An idiot behaving in hast, or an insane person repeating the same process expecting a different result.

• A rational player will move more conservatively.
• A maximum strategy to expose lose may be selected.

Nash equilibrium is rarely Pareto efficient why?  Dominant strategy equilibria are when there is one optimal choice of strategy for both players no matter what the other player does this rarely happens. Often there is no not one Nash Equilibrium but numerous equilibriums.

Dominant Strategy

This occurs when one player’s best response to any strategy the other player may pick results in the highest playoff for the players. The players best strategy is not dependant on another player this makes it a dominant strategy. Nash equilibrium exists when one player has a dominant strategy, letting all other players respond with their best alternative.

• Strongly dominant strategies yields a greater payoff from all other strategies, no matter what the other opponents chooses.
• Weekly dominant strategy is at least as good as all other strategies no matter what other opponents choose.

Mixed Strategy

The alternatives of strategies are randomly mixed. The players mix various strategies to maximise payoff. It is the probability distribution over or some of all the strategies available to players.

If we consider the game show split or steal we have the perfect example of game theory.  The ...