Nash Equilibrium

Nash equilibrium is a fundamental concept in the theory of games and the most widely used method of predicting the outcome of a strategic interaction in the social sciences.

Any game (in strategic or normal form) consists of the
Following key ingredients:
1. a set of players
2. a set of actions (or pure-strategies) available to each player,
3. and a payoff(also said as utility function) for each player.

A normal form is generally represented in matrix form

The payoff functions represent each player’s preferences over action profiles,where an action profile is simply a list of actions, one for each player. A pure-strategy Nash equilibrium is an action profile with the property that no single player can obtain a higher payoff by deviating unilaterally from this profile.

For example :

Consider first a game involving two players,each of whom has two available actions, which we call A and B. If the players choose different actions, they each get a payoff of 0. If they both choose A, they each get 2,and if they both choose B, they each get 1. This “coordination” game may be represented as follows,(Why this is called a coordination is also explained below) :

If player 1 chooses a row, player 2 chooses a column, and the resulting payoffs are listed in parentheses, with the first component corresponding to player 1’s payoff:

Player1/Player 2	A	B
A	(2,2)	(0,0)
B	(0,0)	(1,1)

                                             Figure1: Coordination Game

The action profile (B,B) is an equilibrium, since a unilateral deviation to A by any one player would result in a lower payoff for the deviating player. Similarly, the action profile (A,A) is also an equilibrium. In simple, it is if Player 1 chooses A that gives him payoff ‘2’ then Player 2 gets payoff of 2 ,if he go with A. If he deviates from the action of player 1 (here it is A) then they payoff here is 0 for both in this case both won’t want this.

So ,this is a coordination game where each player coordinates with the action of others and gets a payoff.

As another example, consider the game “matching pennies,” which again involves two players, each with two actions. Each player can choose either heads (H) or tails (T); player 1 wins a dollar from player 2 if their choices are the same, and loses a dollar to player 2 if they are not.

Player1/Player 2	H	T
H	(1,-1)	(-1,1)
T	(-1,1)	(1,-1)

                                           Figure2: Matching Pennies Game


This game has no pure-strategy Nash equilibria.In some cases, instead of simply choosing an action, players may be able to choose probability distributions over the set of actions available to them. Such randomizations over the set of actions are referred to as mixed strategies. Any profile of mixed strategies induces a probability distribution over action profiles in the game.

Under certain assumptions, a player’s preferences over all such lotteries can be represented by a function (called a von Neumann-Morgenstern utility function) that assigns a real number to each action profile. One lottery is preferred to another if and only if it results in a higher expected value of this utility function, or expected utility. A mixed strategy Nash-equilibrium is then a mixed strategy profile with the property that no single player can obtain a higher value of expected utility by deviating unilaterally from this profile.

The American mathematician John Nash (1950) showed that every game in which the set of actions available to each player is finite has at least one mixed-strategy equilibrium. In the matching pennies game, there is a mixed-strategy equilibrium in which each player chooses heads with probability 1/2.Similarly, in the coordination game of the above example, there is a third equilibrium in which each player chooses action A with probability 1/3 and B with probability 2/3.