Play Smarter with Game Theory
Level 4: Mixed Strategies - How to Keep Your Opponent Guessing
In a penalty kick, the kicker has to choose where to shoot, and the goalkeeper has to decide where to dive. If either side becomes predictable, the other can take advantage. That’s where mixed strategies come in.
To keep the math clear, let’s imagine a simpler version of the game. The kicker can shoot left or right, and the goalkeeper can either dive left or right. The numbers in the table are the kicker’s chance of scoring (higher is better for the kicker).
If the keeper dives the correct way, the kicker’s scoring chance is low (30–40%).
If the keeper dives the wrong way, the kicker’s chance is higher (80–90%).
If the kicker always shoots left, the keeper will just dive left and save most of the shots. If the keeper always dives right, the kicker will just shoot left and score easily. The goal for each side is to stay unpredictable, so neither can be exploited.
To find the kicker’s mix, we calculate the expected values (in this case the average scoring chance).
Suppose the kicker shoots left with probability p and right with probability 1 - p.
If the keeper dives left, kicker’s expected value is: 0.3p + 0.8(1 - p)
If the keeper dives right, kicker’s expected value is: 0.9p + 0.4(1 - p)
At equilibrium, the two values must be equal so that neither side can gain an advantage. Therefore, 0.3p + 0.8(1-p) = 0.9p + 0.4(1-p)
Simplify:
0.3p + 0.8 - 0.8p = 0.9p + 0.4 - 0.4p
0.8 - 0.5p = 0.5p + 0.4
p = 0.4
So the kicker should shoot left 40% of the time and right 60% of the time.
Now for the keeper’s mix, suppose the keeper dives left with probability q and right with probability 1 - q.
If the kicker shoots left: 0.3q + 0.9(1 - q)
If the kicker shoots right: 0.8q + 0.4(1 - q)
Set them equal:
0.3q + 0.9(1 - q) = 0.8q + 0.4(1 - q)
0.3q + 0.9 - 0.9q = 0.8q + 0.4 - 0.4q
0.9 - 0.6q = 0.4q + 0.4
q = 0.5
So the keeper should dive left 50% of the time and right 50% of the time.
The mixed-strategy Nash equilibrium here is:
Kicker: shoot left 40% of the time, right 60% of the time.
Keeper: dive left 50% of the time, right 50% of the time.
The point of mixed strategies isn’t to be random for randomness’s sake, but to keep your opponent from finding a pattern they can exploit.
Next time we’ll step into nature with Evolutionary Game Theory and see how strategies like fighting or cooperating can spread through populations, even when no one is consciously choosing them.
