Mixed Strategy Nash Equilibrium

An introduction to solving for Mixed Strategy Nash Equilibrium

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One of my favorite sporting events to watch is the World Cup. The level of competition, intensity, and drama make it a very fun event that brings out a sense of national pride. However, it gets even more exciting when regulation ends in a tie resulting in penalty kicks. The goalie has a very difficult job of trying to anticipate where the kicker is going to kick the ball.  In addition, the kicker has to make sure the ball will be out of the goalie’s reach once the ball is kicked.

What would this look like in a game theory model? Possibly something like this:

Kicker\Goalie Left Right
Left 0,1 1,0
Right 1,0 0,1

The goalie receives a payoff of 1 when he or she correctly anticipates the direction that the soccer ball is kicked. The kicker receives a payoff of 1 when he or she is able to kick the ball in the opposite direction of the goalie’s choice. However, you can see that there is no pure strategy Nash Equilibrium.

Kicker\Goalie Left Right
Left 0,1 1,0
Right 1,0 0,1

Instead, we need to solve for the mixed strategy Nash Equilibrium. A little bit of probability and algebra is needed to solve this game. Let p represent the probability that the kicker kicks left and 1-p be the probability that the kicker kicks right. Let q represent the probability that the goalie moves to the left and 1-q be the probability the goalie moves to the right.

q 1-q
Kicker\Goalie Left Right
p Left 0,1 1,0
1-p Right 1,0 0,1

To solve for p and 1-p of the kicker we must identify the potential payoffs of the goalie should he or she choose to move left or right. They are highlighted in red and purple above.

We can solve using the following equation based on probability:

1(p) + 0(1 – p) = 0(p) + 1(1-p) =>

p = 1 – p =>

2p = 1 =>

p = 1/2 =>

Solve for 1 – p: 1 – 1/2 = 1/2

p = 1/2, 1-p = 1/2 (your probability totals must equal 1)

This means that the kicker has a mixed strategy of kicking left 1/2 of the time and kicking right 1/2 of the time. This makes sense if you think about it as the kicker does not want to be predictable on kick placement.

Now to solve for the goalie’s mixed strategy:

q 1-q
Kicker\Goalie Left Right
p Left 0,1 1,0
1-p Right 1,0 0,1

Solving for q and 1-q involves the same equation:

0(q) + 1(1 – q) = 1(q) + 0(1 – q) =>

1 – q = q =>

1 = 2q =>

q = 1/2 =>

Solve for 1-q: 1 – 1/2 = 1/2

q = 1/2, 1-q = 1/2 (your probability totals must equal 1)

Similar to the kicker, the goalie should move left 1/2 of the time and move right 1/2 if the time.  The mixed strategy Nash Equilibrium for this game would be Kicker: left(1/2), right(1/2) and Goalie: left(1/2),right(1/2). Payoff for choosing a mixed strategy is also simple to solve.

You take the original equations and plug in the values for (p, 1-p) and (q, 1-q):

Kicker: 1(p) + 0(1 – p) = 0(p) – 1(1-p)  => 1(1/2) + 0(1/2) = 0(1/2) + 1(1/2) =>

1/2 = 1/2 (both sides should equal each other)

Goalie: 0(q) + 1(1 – q) = 1(q) + 0(1 – q) => 0(1/2) + 1(1/2) = 1(1/2) + 0(1/2) =>

1/2 = 1/2

The payoffs of a mixed strategy for both Goalie and Kicker = (1/2,1/2). In the end, it is important for both players to not be predictable.

 

 

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