Nash Equilibrium

How does one generally solve a game? This is far from self evident, and in many games, there are several different reasonable solutions. The most popular concept for solving games is the Nash equilibrium. There are, however, several other ways in which to solve games, but most often, they are variations of a Nash equilibrium. Note also that, there can be more than one Nash equilibrium in a game.

A Nash equilibrium is:

• A set of strategies, one for each player.
• The strategies should be such that no player can improve her utility by unilaterally changing her own strategy.

Finding the Nash Equilibrium in a Game in Matrix Form

It is often easy to find the Nash equilibrium for a game in matrix form. Look at the game in Figure 13.1 again. We have four squares in the matrix. We can then find the Nash equilibrium by checking each square separately:

• {Do not confess, Do not confess}, i.e. the lower right square. Can any of the players improve her situation by unilaterally changing her own strategy? If, for instance, A changes to "Confess" she will get +1 instead of -1. (Similarly for B.) Consequently, she can improve her situation and this cannot be a Nash equilibrium.
• {Do not confess, Confess}, i.e. the lower left square. If A changes to "Confess”, she will get -2 instead of -10. Consequently, this cannot be a Nash equilibrium either.
• {Confess, Do not confess}, i.e. the upper right square. If B changes to "Confess”, she will get -2 instead of -10. Consequently, this cannot be a Nash equilibrium.
• {Confess, Confess}, i.e. the upper left square. If A would change to "Do not confess”, she would reduce her utility from -2 to -10, and if B would change she would also reduce her utility from -2 to -10. None of the players can therefore improve on her situation by unilaterally changing her strategy, and this must be a Nash equilibrium. The only Nash equilibrium in the Prisoner’s Dilemma is that both players choose “Confess.”

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