Mixed Strategy Nash Equilibrium II

Further analysis of Mixed Strategy Nash Equilibrium

Last time I introduced the concept of a mixed strategy Nash Equilibrium. We will continue to focus on solving for this type of equilibrium. The following example will have two players. In addition, no pure strategy Nash Equilibrium exist for either players and each player will have complete information of the available strategies and payoffs. The game matrix is represented as:

1\2 LEFT RIGHT
UP 1,5 5,3
DOWN 3,1 2,2

According to Nash’s Theorem there guarantees the existence of a set of mixed strategies for a finite (limited) number of non cooperative games of two or more players. So let’s see if a mixed strategy Nash Equilibrium exists.

Let p represent the probability that player 1 chooses UP and 1-p represent the probability that player 1 chooses DOWN.

Let represent the probability that player 2 chooses LEFT and 1-q represent the probability that that player 2 chooses RIGHT.

q 1-q
1\2 LEFT RIGHT
p UP 1,5 5,3
1-p DOWN 3,1 2,2

To find the mixed strategy Nash Equilibrium we need to solve for: p, 1-p, q, & 1-q.

First let’s solve for p and 1-p by using the following equation:

5p + 1(1-p) = 3p + 2(1-p)

A few things to note:

  • The coefficients represent Player 2’s payouts highlighted in red
  • The left side of the equation represents Player 2’s payoffs if choosing LEFT
  • The right side of the equation represents Player 2’s payoffs if choosing RIGHT
  • Both sides of the equation must equal 100% (or 1)

5p + 1(1-p) = 3p + 2(1-p)

=> 5p + 1 – p = 3p + 2 – 2p

=> 4p + 1 = p + 2

=> 3p = 1

=> p = 1/3

=> 1-p = 1 – 1/3 = 2/3

=> 1/3 + 2/3 = 1 or 100%

So player 1’s mixed strategy is to choose UP 1/3 of the time and choose DOWN 2/3 of the time.

Now let’s solve for q and 1-q using the same equation but different coefficients:

1q + 5(1-q) = 3q + 2(1-q)

A few things to note:

  • The coefficients represent Player 1’s payouts highlighted in blue
  • The left side of the equation represents Player 1’s payoffs if choosing UP
  • The right side of the equation represents Player 1’s payoffs if choosing DOWN
  • Both sides of the equation must equal 100% (or 1)

1q = 5(1-q) = 3q + 2(1-q)

=> q + 5 – 5q = 3q + 2 – 2q

=> -4q + 5 = q + 2

=> 5 = 5q + 2

=> 3 = 5q

=> q = 3/5

=> 1-q = 1 – 3/5 = 2/5

=> 3/5 + 2/5 = 1 or 100%

Player 2’s mixed strategy is to choose LEFT 3/5 or the time and RIGHT 2/5 of the time. Let’s update the matrix below with the mixed strategies.

q = 3/5 1-q = 2/5
1\2 LEFT RIGHT
p = 1/3 UP 1,5 5,3
1-p = 2/3 DOWN 3,1 2,2

Now we need to determine the payouts of both players playing these mixed strategies.

First we need to determine the probabilities of players 1 & 2 choosing:

(UP, LEFT) = (p)(q) = (1/3)(3/5) = 3/15

(UP, RIGHT) = (p)(1-q) = (1/3)(2/5) = 2/15

(DOWN, LEFT) = (1-p)(q) = (2/3)(3/5) = 6/15

(DOWN, RIGHT) = (1-p)(1-q) = (2/3)(2/5) = 4/15

Second,  we take summation of each individual player’s pure strategy payoff multiplied by the corresponding probability in that quadrant.

Player 1 payoff: (1)(3/15) + (5)(2/15) + (3)(6/15) + (2)(4/15) = 2.6

Player 2 payoff: (5)(3/15) + (3)(2/15) + (1)(6/15) + (2)(4/15) = 2.3

In summary, the mixed strategy Nash Equilibrium is Player 1 choosing (UP, DOWN) = (1/3, 2/3) with a payoff of 2.6  and Player 2 choosing (LEFT, RIGHT) = (3/5, 2/5)  with a payoff of 2.3. Next we will look at games that have both pure strategy and mixed strategy Nash Equilibrium. Your feedback is much appreciated and I look forward to the next post.

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