The Nash Equilibrium Theory

The Game

Let's think of a game in which two players have two cards (an Ace and a Jack) and the decision of each of them is which card to play given: (i) if they both play an ace they each receive USD 1, (ii ) if one plays a Jack and the other an Ace the first receives USD 0 and the second USD 10 and (iii) if both play a Jack receives USD 5 each.

If we graph the possible results according to the decision made by each player we have:

Assuming that:

• Each player knows and has adopted his best strategy, and
• Everyone knows each other's strategies.

We will have the conditions to apply the Nash equilibrium theory and in it any game with a finite number of strategies has at least one Nash equilibrium in mixed strategies.

Thus, each player is executing the best possible "move" taking into account the movements of the other players.

?In other words, a Nash equilibrium is a situation in which all players have implemented, and know they have, a strategy that maximizes their payoffs given the strategies of the others. Consequently, no player has any incentive to individually modify his strategy.