Mixed Strategies: Why You Should Be Unpredictable
You’ve learned about dominant strategies and Nash equilibria in pure strategies. But what happens when there’s no pure strategy Nash equilibrium? What if being predictable is your worst enemy?
Welcome to the world of mixed strategies — where randomness becomes your most powerful weapon.
The Problem with Being Predictable
Imagine you’re a goalkeeper facing a penalty kick. You can dive left or right. The striker can shoot left or right. If you both go the same way, the striker scores. If you guess correctly, you save.
Here’s the payoff matrix (from the striker’s perspective, 1 = goal, 0 = save):
| Goalkeeper Left | Goalkeeper Right | |
|---|---|---|
| Shoot Left | 0 | 1 |
| Shoot Right | 1 | 0 |
Look closely at this game. There’s no pure strategy Nash equilibrium. If you always dive left, the striker should shoot right. But if the striker always shoots right, you should dive right. But then the striker should shoot left…
It’s an endless cycle. Being predictable means losing.
Enter Mixed Strategies
A mixed strategy is when you randomly choose between your available actions according to some probability distribution.
Instead of always diving left, you dive left with probability p and right with probability (1-p).
The key insight: by mixing your strategies, you can make your opponent indifferent between their choices, which prevents them from exploiting you.
Finding the Mixed Strategy Nash Equilibrium
Let’s solve the penalty kick game. Suppose:
- Goalkeeper dives left with probability p
- Striker shoots left with probability q
For the striker’s expected payoffs:
If striker shoots left:
- Expected payoff = 0·p + 1·(1-p) = 1-p
If striker shoots right:
- Expected payoff = 1·p + 0·(1-p) = p
The striker is indifferent when: 1-p = p
- Therefore: p = 0.5
By the same logic, the striker should also mix 50-50: q = 0.5
Here’s the beautiful result visualized:
The Mathematics of Mixing
When you mix strategies optimally, you’re solving this equation:
Make your opponent indifferent between all their pure strategies.
Let’s formalize this. If player 1 mixes between actions A and B with probabilities (p, 1-p), then:
Player 2’s expected utility from action X = Player 2’s expected utility from action Y
This equation allows us to solve for p.
Example: Rock-Paper-Scissors
In Rock-Paper-Scissors, each option beats one and loses to another. There’s no pure strategy Nash equilibrium.
Payoff matrix (your perspective: 1 = win, -1 = lose, 0 = tie):
| Rock | Paper | Scissors | |
|---|---|---|---|
| Rock | 0 | -1 | 1 |
| Paper | 1 | 0 | -1 |
| Scissors | -1 | 1 | 0 |
By symmetry, the mixed strategy Nash equilibrium is:
- Each player plays each option with probability 1/3
This makes your opponent completely indifferent between all choices, and you expect to break even in the long run.
Why Mixing Works: The Expected Value Calculation
When both players mix optimally, the expected payoff is determined by the probabilities.
For the penalty kick example with both playing 50-50:
Striker’s expected payoff:
- P(score) = P(shoot left, keeper right) + P(shoot right, keeper left)
- P(score) = (0.5 × 0.5) + (0.5 × 0.5) = 0.5
The striker scores 50% of the time — exactly what we’d expect from optimal play on both sides.
Real-World Applications
1. Sports
Professional tennis players mix up their serves. Data shows they serve to different sides with frequencies that match game theory predictions remarkably well.
2. Military Strategy
Bluffing and unpredictable troop movements follow mixed strategy logic. Never let the enemy know your next move.
3. Cybersecurity
Security audits should be randomized. If hackers can predict when you check, they’ll attack in between.
4. Business
Pricing strategies, product launches, and marketing campaigns benefit from unpredictability to prevent competitors from exploiting patterns.
The Counter-Intuitive Truth
Here’s what makes mixed strategies fascinating:
-
You win by being indifferent: The goal isn’t to maximize your own payoff directly, but to make your opponent unable to exploit you.
-
Randomness is strategic: Flipping a coin isn’t giving up — it’s optimal play.
-
Your mixing probabilities depend on opponent payoffs: You choose your probabilities to make them indifferent, not yourself.
Let’s visualize this counter-intuitive relationship:
When Should You Mix?
Not every game requires mixed strategies. Use them when:
- There’s no pure strategy Nash equilibrium
- Being predictable can be exploited
- Your opponent is sophisticated enough to exploit patterns
Against a weak opponent who always chooses the same action, don’t mix — just exploit their predictability!
Common Mistakes
Mistake 1: Mixing When You Have a Dominant Strategy
If you have a dominant strategy, play it. Mixing is for games where predictability hurts you.
Mistake 2: Using the Wrong Probabilities
Remember: your mixing probabilities are determined by making your opponent indifferent, not by what seems fair or balanced.
Mistake 3: Thinking Mixing is Random Guessing
It’s not. The probabilities are calculated precisely to achieve equilibrium. True randomness would be suboptimal.
Practice Problem
Consider this game:
| Column L | Column R | |
|---|---|---|
| Row U | 2, 1 | 0, 0 |
| Row D | 0, 0 | 1, 2 |
Question: Find the mixed strategy Nash equilibrium.
Hint: Let Row play U with probability p. Make Column indifferent between L and R.
Solution
Step 1: Column’s expected payoff from L = p·1 + (1-p)·0 = p
Step 2: Column’s expected payoff from R = p·0 + (1-p)·2 = 2(1-p)
Step 3: Set them equal: p = 2(1-p) → p = 2 - 2p → 3p = 2 → p = 2/3
By symmetry, Column plays L with probability q = 2/3.
Nash Equilibrium: (U: 2/3, D: 1/3) vs (L: 2/3, R: 1/3)
Key Takeaways
- Mixed strategies involve randomizing between pure strategies according to specific probabilities
- They exist when no pure strategy Nash equilibrium exists
- Your mixing probabilities make your opponent indifferent between their options
- Unpredictability is often optimal in competitive, strategic situations
- Professional players intuitively use mixed strategies in sports, business, and warfare
What’s Next?
Now you understand why being unpredictable is mathematically optimal. But what happens when the same game is played repeatedly? Can cooperation emerge from self-interest?
Next in the Game Theory Series, we’ll explore Repeated Games and discover how the shadow of the future changes everything.
This post is part of the Game Theory Series, where we explore the mathematics of strategic decision-making.