2/29/12

How to find a Nash Equilibrium in a 2X2 matrix



Getting to the Nash equilibrium can be tricky, so this post goes over two quick methods to find the Nash equilibrium of any size matrix, but uses a 2X2 matrix as an example.

Summary (dominant strategy method):
  1. Check each column for Row player’s highest payoff, this is their best choice given Column player’s choice.  (if there are two high choices, then the result will be a mixed strategy outcome).
  2. Now check to see if Row’s choice for 1) would also be their choice given any choice by Column player.
  3. If Row always sticks with their choice regardless of Column’s choice, this is their dominant strategy.
  4. Repeat for Column player, and the Nash equilibrium is where the dominant strategies intersect.


Summary (rule of thumb method):
  1. Choose one opponent’s choice and see if the player has an incentive to change their choice. 
  2. If no, circle that payoff, if yes; check another cell within the same choice by the opponent.
  3. Repeat for all choices for both players.
  4. The Nash equilibrium (could be more than 1) occur where both payoffs are circled.

Constructing the payoff matrix, rules:
Total market share equals 10,
Cost of advertising is 4 for high, 2 for low.
If firms both choose the same advertising level they split the market, if one firm chooses high and the other low, than the firm that chose high advertising gets the entire market.



Column Player



High
Low
Row Player
High
(1,1)
(6,-2)

Low
(-2,6)
(3,3)

Example of finding Nash equilibrium using the dominant strategy method:
We can first look at Row player’s payoffs to see that if column chooses high, it is in row’s best interest to choose high because 1>-2, and if column choose low, row will also choose high because 6>3.  So choosing high is row’s dominant strategy.  We can do the same analysis for column player to get the same result.  Since both players have a dominant strategy of choosing high, this will be a Nash equilibrium.

Example of finding Nash equilibrium using rule of thumb method:
Let’s start with the first cell, and see if row player wants to switch choices.  Since 1>-2, row player doesn’t want to switch, so we can circle that payoff (in blue).  The same method for column player shows that they would not want to switch as well so we can circle their payoff (in red).  We can do the same analysis with each choice, to see where all of the circles should go.  The cell with both payoffs circled is a Nash equilibrium.  Remember that it is possible to have a payoff matrix with no Nash equilibrium.


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