Every game you’ve analysed so far had a resting place — at least one cell where both players were best-responding and nobody itched to move. But some games refuse to settle. Look hard for a stable cell and you find a merry-go-round instead: whoever is currently losing always wants to switch, which makes the other player want to switch, forever. This lesson is about those games, and about the beautiful trick that tames them — you stop trying to pick the right move and start rolling dice on purpose.
The game that never settles: matching pennies
Here is the cleanest engine of pure conflict in all of game theory. You and a rival each secretly pick Heads or Tails. If the two faces match, you win (+1) and they lose (−1); if they differ, they win and you lose. What’s good for you is exactly bad for them — a zero-sum standoff with no shared interest to coordinate around.
Try to find a stable cell in the matrix below. The analysis panel does the best-response bookkeeping for you.
Payoff matrix
Matching pennies: pure conflict, no resting place
A ringed number is that player's best response to the rival's choice. A cell where both are ringed would be a Nash equilibrium — but look what the panel reports.
| Rival | ||
|---|---|---|
| Heads | Tails | |
| YouHeads | 1-1 | -11 |
| Tails | -11 | 1-1 |
A ringed payoff is that player’s best response to the rival’s choice. A cell where both are ringed is a Nash equilibrium.
What the matrix says
You — dominant strategy: none
Rival — dominant strategy: none
Nash equilibrium (pure): none in pure strategies
Chase it around. Suppose you’re both on Heads — a match, so you win. That instantly makes the rival want Tails to break the match. Once they’re on Tails (a mismatch), you want to switch to Tails as well, to match again. Now they want Heads… and the loop closes with nobody ever content. Every one of the four cells has exactly one happy player and one player halfway out the door. There is no pure-strategy Nash equilibrium — no combination of fixed moves where both are best-responding at once.
So does the whole equilibrium concept just break here? It would, if “a strategy” only ever meant “always play the same move.” The fix is to widen what counts as a strategy.
The fix: a mixed strategy
A pure strategy is what you’ve used until now — commit to one move and always play it (“always Heads”). A mixed strategy is a probability distribution over your pure moves: a rule like “Heads 50% of the time, Tails 50%, decided at random on each play.” You’re not choosing Heads or Tails; you’re choosing the dice — the frequencies — and then letting genuine randomness pick the actual move.
Randomising is a real, legitimate strategy
It’s tempting to feel that “flip a coin to decide” is a cop-out — a refusal to strategise. In games of pure conflict it’s the opposite: it is the most sophisticated thing you can do. A mixed strategy is a fully specified plan; “Heads 50%” is as concrete a commitment as “always Heads.” The only difference is that its output isn’t a move, it’s a lottery — and against a clever opponent, an unpredictable lottery is exactly what you want to be holding.
The moment mixed strategies are on the table, the merry-go-round stops. There is a mix that no rival can beat — and finding it is the whole art.
Why randomising works: unexploitability
The point of mixing isn’t to confuse yourself; it’s to make yourself unexploitable. Think of it from your rival’s side of the table. If you play any lopsided mix — say Heads 80% of the time — a sharp rival simply notices the lean and leans back: they play Tails almost always, cashing in on your predictability. Any bias you have is a handle they can grab.
Drag the slider below through the full range and watch your guaranteed payoff — the bold line, which is the worst a smart rival can do to you at each mix. It’s an inverted V. The two faint lines are what you’d earn if the rival committed to one response; a rational rival always picks whichever is worse for you, so you can only count on the lower of the two.
Unexploitability
Matching pennies: only 50/50 can't be punished
Drag the slider to change how often you play Heads. The faint lines are your payoff if the rival commits to Heads or to Tails; a smart rival always picks the one that's worse for you, so the bold line is what you can actually guarantee. Find its peak.
Playing Heads 80% of the time: if the opponent answers {s0} you get 0.6, if they answer Tails you get -0.6. A smart opponent picks Tails, so you can only count on -0.6. Shift toward the unexploitable mix (50% Heads) to lift your guaranteed payoff to 0.
At every lopsided setting the guaranteed line dips into negative territory: the rival has a punishing response and takes it. The line peaks at exactly Heads 50%, where your guaranteed payoff is 0 — the best you can force in a fair, symmetric fight. At that one spot the two faint response lines cross: the rival earns the same whether they answer Heads or Tails, so they have no exploiting move to reach for. You’ve made yourself a moving target with no lean to grab.
The indifference principle
Notice why the peak sits where the two response lines cross. At that mix, your rival is exactly indifferent between their options — every response yields them the same expected payoff. And that is not a coincidence; it is the mechanism.
The indifference principle: in a mixed-strategy equilibrium, each player chooses their mix precisely so as to make the other player indifferent between their options.
Read that twice, because it’s the counterintuitive “aha” of the whole topic: you tune your own randomisation to control your opponent’s incentives, not your own. If your mix left the rival even slightly preferring one of their responses, they’d pile onto it and drag you down — that preference is an exploit. The only mix with no exploit is the one that erases the rival’s preference entirely, leaving them indifferent. Kill their reason to lean, and you’ve killed their ability to punish you.
Where the equilibrium mix comes from
This gives you a recipe, not just a picture. To find your equilibrium mix, don’t ask “what maximises my payoff?” — ask “what mix of mine makes my opponent indifferent between their two moves?” Set their expected payoff from one response equal to their expected payoff from the other, and solve for your frequencies. In matching pennies the symmetry makes that 50/50; in the next example, it won’t be.
When the sides aren’t equal: penalty kicks
Matching pennies is symmetric, so the answer was a tidy 50/50. Real strategic conflicts are usually lopsided — and then the equilibrium mix leans, but never all the way.
Picture a penalty kick. The kicker aims left or right; the goalie dives left or right; they commit at the same instant. But the kicker isn’t equally good on both sides. The grid below is the kicker’s scoring chance out of 100 in each of the four combinations — they’re much better shooting to their strong side than their weak side, and (as always) a save is likeliest when the goalie guesses right.
Asymmetric conflict
Penalty kicks: lean to your strong side, but don't camp there
Drag to change how often the kicker aims left. The faint lines are the scoring chance if the goalie always dives one way; the goalie will dive whichever way hurts the kicker most, so the bold line is the guaranteed scoring chance. Find its peak.
Playing Dive left 50% of the time: if the opponent answers {s0} you get 60%, if they answer Dive right you get 70%. A smart opponent picks Dive left, so you can only count on 60%. Shift toward the unexploitable mix (38% Dive left) to lift your guaranteed payoff to 65%.
Slide it and the peak lands at Aim-left 37.5%, Aim-right 62.5% — decidedly not 50/50. The kicker leans toward the stronger side, exactly as intuition says. But watch the two extremes: shove the slider to 100% and the goalie simply always dives that way, and the guaranteed chance craters. Predictability is punished even when you’re predictably playing your best shot. The equilibrium leans without ever committing.
And here’s the payoff of playing it: at 37.5% the kicker guarantees a 65% scoring chance regardless of what the goalie does. The two response lines cross there, so the goalie is indifferent between diving left and right — no exploit remains.
| Kicker’s mix (Aim left) | Goalie’s best reply | Guaranteed scoring chance |
|---|---|---|
| 0% (always right) | Dive right | 50% |
| 25% | Dive right | 60% |
| 37.5% (equilibrium) | indifferent | 65% (peak) |
| 50% | Dive left | 60% |
| 100% (always left) | Dive left | 40% |
Read the middle column: away from 37.5% the goalie always has a favourite dive that drags the kicker below 65%. Only at the equilibrium mix does that favourite vanish. The lesson generalises past football: when your options aren’t equally strong, weight your mix toward the strong one — but keep enough weight on the weak one to keep your opponent honest.
This actually shows up in the data
Economists have checked real play against these predictions. Studies of thousands of professional penalty kicks (Palacios-Huerta, 2003) and of tennis first-serve direction find that top players randomise at frequencies strikingly close to their game-theoretic mix — and, crucially, that their sequences are close to serially independent: they don’t fall into exploitable patterns. Elite competitors are, in a real sense, running this exact calculation in their bones.
Nash’s theorem: there’s always an equilibrium
You might worry we got lucky — that mixing rescued these games but some nastier game could still have no equilibrium at all. It can’t. In 1950, a 21-year-old Princeton graduate student named John Nash proved the result that anchors this entire course (and later won him a Nobel Prize and a film, A Beautiful Mind):
Nash’s existence theorem: every finite game — any finite number of players, each with a finite number of pure strategies — has at least one Nash equilibrium, once mixed strategies are allowed.
So “this game has no equilibrium” is never the final word. It might have no pure equilibrium — matching pennies doesn’t — but allow randomisation and one always appears, possibly several. This is exactly why the Nash equilibrium is the universal solution concept of game theory: Nash didn’t just define the resting point, he proved the resting point always exists. There is always somewhere for rational play to land.
Pitfalls
Three ways learners misread mixed strategies, each worth inoculating against.
- Mixing is not being wishy-washy. A mixed strategy is not indecision, hedging, or “I couldn’t make up my mind.” It is a precise, deliberate commitment to specific frequencies, executed with genuine randomness. “Aim left 37.5% of the time, at random” is a sharp instruction, not a shrug. The randomness is the point, not a symptom of confusion.
- You must be genuinely unpredictable. The equilibrium demands real randomness, not a private ritual that only feels random. Alternating Heads, Tails, Heads, Tails is technically 50/50 by frequency — but it’s a fixed, deterministic pattern, so it’s a pure strategy in disguise and a sharp opponent cracks it in seconds and cleans you out. Humans are famously bad at hand-generating randomness (we avoid repeats and drift into rhythms); if it matters, use an actual random source.
- Against a weak opponent, don’t mix — exploit. The equilibrium mix is the safe, unexploitable choice, not always the payoff-maximising one. It’s what you play against a rival who is also playing optimally. If your opponent is not — if they have a tell, a bias, a lean — the best reply is to abandon your mix and pounce on their predictability with a pure best response. Equilibrium is your floor against a genius, not your ceiling against a fool.
You play your exact matching-pennies equilibrium mix: Heads 50%, at random. What does this specifically achieve?
Fill in the core vocabulary of this lesson.
Pick the right option for each blank, then check.
A strategy always plays the same move, while a strategy is a probability distribution over your moves, chosen at random. In a mixed equilibrium you set your own frequencies to make the exactly indifferent between their options, which is what makes your play . And Nash proved that once mixing is allowed, every finite game has equilibrium.
Match each term to its precise meaning.
Pick a term, then click its definition.
When to use it
Reach for mixed strategies in any strictly-competitive, repeated situation where being predictable gets you punished — where your opponent actively studies you and profits from any pattern. The canonical cases:
- Poker bluffing frequencies — bluff too rarely and folds pile up; bluff too often and you get called; the unexploitable frequency sits in between and keeps opponents guessing.
- Tax-audit and inspection selection — a fixed, predictable audit rule tells cheaters exactly when they’re safe; randomised selection keeps everyone honest.
- Security patrol and defence — a guard on a fixed route is a gift to an attacker; randomised timing and paths remove the reliable gap.
- Sports — serve direction in tennis, pitch selection in baseball, the penalty side above: lean toward your strength, but stay a moving target.
- Pricing and promotion games — randomised sales and unpredictable price moves stop rivals and bargain-hunters from perfectly anticipating you.
The through-line: if a smart adversary is watching and predictability is a vulnerability, the safe play isn’t a clever fixed move — it’s a well-tuned dice roll.
So far we’ve taken every game’s payoffs as fixed and hunted for where play lands inside them. But the sharpest players don’t just play the game well — they change the game itself, editing the payoffs, the options, even who’s at the table, so the equilibrium moves in their favour. That’s next, in Change the Game, Not the Players.