Mixed Strategy Nash Equilibrium

Futuristic control room with blue glowing screens and monitors

Pure vs Mixed Strategy

A Pure Strategy is when a player picks one specific move and sticks to it.

Take a look at the classic Prisoner's Dilemma. In that game, the smartest move is a pure strategy ... both players usually end up choosing to "tell" on each other every single time because it's the safest bet.

But what happens when being predictable completely ruins your chances? That's where things get interesting. In those games our best move is to keep our opponent guessing by randomly choosing a mix of actions: a Mixed Strategy.

Rock-Paper-Scissors

Three hands simultaneously displaying the rock, paper, and scissors symbols

Let's play rock-paper-scissors! There are three choices:

Rock beats scissors
Scissors beats paper
Paper beats rock

If we both pick the same, we tie.

Any single choice (such as paper) can't win all the time.

So, how do you play to win?

If you pick rock too often, I will notice this and start picking paper. Then, you lose!

The solution is a Mixed Strategy

Don't pick the same move every time. Pick rock, paper, or scissors randomly.

This keeps you a mystery. I can't find a pattern to beat you.

Usually, you want to pick each about one-third of the time but people are a little predictable!

New players often start with Rock (35%)
Experienced players often start with Paper (35%)
Scissors is least likely (30%)

Black and white portrait photo of mathematician John Nash
John Nash

Nash Equilibrium

A brilliant mathematician named John Nash figured out a special concept called a Nash Equilibrium. It's a point in a game where no player can get a better score by changing their own strategy. It's like a stable standoff.

Nash Equilibrium in Mixed Strategies

Here's the good part: we can apply this stability to our random mixed strategies!

We want probabilities so each option gives the same expected result.

When this balance is reached, switching strategies won't improve the outcome. Any predictable change could be used by the opponent.

We call this balance a Mixed Strategy Equilibrium.

Worked Example: The Tennis Serve

In tennis, a server can serve Wide or down the T. The receiver must guess which way to lean. If the receiver guesses right, the server is less likely to win the point.

Receiver

Lean Wide

Lean T

Server

Serve Wide

30%, 70%

80%, 20%

Serve T

90%, 10%

20%, 80%

To find the Mixed Strategy Nash Equilibrium, the Server must choose a probability that makes the Receiver "indifferent" (meaning the Receiver gets the same result no matter what they choose).

Why make them indifferent? If you serve Wide too often, the Receiver will notice and always Lean Wide to win more points. By choosing the exact mix that makes both of their options equal, you leave them with no way to exploit your strategy!

Step 1: Assign Probabilities

Let p be the probability the Server serves Wide
Therefore, 1 − p is the probability the Server serves T

Step 2: Set the Receiver's Expected Payoffs to Equal

We look at the Receiver's win percentages (the second numbers in the table):

If Receiver Leans Wide: 70p + 10(1 − p)
If Receiver Leans T: 20p + 80(1 − p)

For equilibrium, we set these as equal:

70p + 10 − 10p = 20p + 80 − 80p

Step 3: Solve for p

Simplify: 60p + 10 = 80 - 60p
Add 60p to both sides: 120p + 10 = 80
Subtract 10: 120p = 70
Divide: p = 70/120 or 0.58

The Result: To be unpredictable and unexploitable, the Server should serve Wide 58% of the time and serve T 42% of the time.

If the Server uses this exact mix, the Receiver will win 45% of points regardless of whether they lean Wide or T. The Server has successfully protected their strategy!

Summary

When no single choice is perfect, randomness can be the best strategy
By using mixed strategies we aren't guessing: we are strategically choosing when to be unpredictable
We want probabilities so each option gives the same expected result