In the last lesson you learned to write down a game. Now we start solving them, and we begin with the happiest possible case — so happy it almost feels like cheating. Sometimes, when you compare your options, one of them comes out ahead no matter what the other player does. When that happens you don’t need to predict your opponent, read their mind, or worry about their cleverness. You just play your winning move and let them do whatever they like. That move is called a dominant strategy, and recognising one is the fastest win in all of game theory.
What “dominant” precisely means
A strategy is dominant for a player when it gives them a higher payoff than their other option, in every column the opponent might choose. The test is column-by-column: look at what you’d get from Option 1 versus Option 2 if the rival plays their first strategy, then again if they play their second. If the same option of yours wins both comparisons, it’s dominant.
Here’s a clean one. Two competing corner shops each decide whether to accept credit cards or stay cash-only. Taking cards costs a little in fees but wins customers — and it turns out to pay off whatever the rival does. Use the ± steppers to test the claim yourself: compare each shop’s two rows (or columns) one rival-move at a time.
Payoff matrix
Cards or cash? Test each option column by column
Each cell shows (your shop, rival shop) in monthly profit. To check for a dominant strategy, fix the rival's choice and ask which of your two moves pays more — then check the other rival choice too.
| Rival shop | ||
|---|---|---|
| Accept cards | Cash only | |
| Your shopAccept cards | NE44 | 52 |
| Cash only | 25 | 33 |
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
Your shop — dominant strategy: Accept cards
Rival shop — dominant strategy: Accept cards
Nash equilibrium (pure): (Accept cards, Accept cards)
Walk the logic for your shop. If the rival accepts cards (left column), you get 4
for accepting versus 2 for cash-only — accepting wins. If the rival goes
cash-only (right column), you get 5 for accepting versus 3 for cash-only —
accepting wins again. Accepting cards beats cash-only in both worlds, so it’s your
dominant strategy. You can play it without knowing or caring what the rival decides.
By the identical argument down the columns, accepting is dominant for them too — so
the game resolves itself to the top-left, (4, 4).
The shortcut a dominant strategy gives you
A dominant strategy collapses a hard problem into an easy one. Normally your best move depends on predicting the other player. A dominant strategy means you can skip the prediction entirely — your answer is the same against every opponent, genius or fool. When you spot one, take it and stop deliberating.
Its mirror image: the dominated strategy
Flip the idea over. A strategy is dominated when some other option of yours does at least as well in every column and strictly better in some — in other words, there’s a different move that’s never worse and sometimes better. In the shop game, “cash only” is a dominated strategy: accepting cards beats it whatever the rival does, so a rational shop should never play cash-only.
This gives us the field’s most reliable rule of thumb:
A rational player never plays a strictly dominated strategy.
It’s the closest thing game theory has to a law you can lean on. Even when a game has no clean answer, you can almost always start by crossing out the dominated moves — nobody sensible will pick them — which often shrinks a baffling situation to a manageable one.
A strategy D is strictly dominated by strategy U: U pays more than D against every choice the opponent could make. What should a rational player do with D?
Solving a game by elimination
Crossing out dominated strategies is powerful enough to solve some games on its own — even when no single move is outright dominant. The trick is that eliminating one player’s bad option can create a dominant move for the other, which you then eliminate around, and so on. This is called iterated elimination of dominated strategies, and it’s worth seeing once.
Suppose you have moves Top/Bottom and the rival has Left/Middle/Right. You notice the rival’s “Right” is dominated by their “Middle” — Middle always pays them more — so you confidently cross Right off: a rational rival will never go there.
But here’s the move: now that Right is gone, the game is smaller, and in that smaller game it might turn out that your “Top” beats “Bottom” in both remaining columns — even though it didn’t when Right was still in play. So now Top is dominant for you, and you play it. Eliminating their bad option created a clean answer for you.
You assume your opponent is rational (won’t play Right), and that assumption hands you a sharper move. Chains like this can collapse a big, intimidating matrix down to a single predicted outcome — pure logic, no guessing.
The catch: dominant strategies are rare
Now the honest part, because dominant strategies are so satisfying that beginners go hunting for one in every game and force a non-existent answer. Most games have no dominant strategy. Look back at the standards game from the last lesson: there, your best move flipped depending on what the rival did — match Format A if they pick A, match Format B if they pick B. Neither of your options won in both columns, so neither was dominant. That’s the normal situation: your best response depends on theirs, which is exactly the hall-of-mirrors problem we still have to solve.
For each described game, does the player in question have a dominant strategy?
Place each item in the right group.
- Swerve only if you expect the other driver to hold straight
- Drive on the left only if everyone else drives on the left
- Match Format A if the rival picks A, but match Format B if they pick B
- Confessing gets you a lighter sentence whether or not your partner confesses
- Accepting cards pays more than cash-only against every rival move
- Studying raises your grade regardless of what classmates do
Lock in the definitions.
Pick the right option for each blank, then check.
A strategy beats your other options no matter what the opponent does, so you needn't predict them; its opposite, a strategy, is beaten by another option of yours in every case, so a rational player plays it.
When to use it
Whenever you face a strategic choice, run the dominance check first — it’s cheap and occasionally hands you the whole answer. For each of your options, ask: does this beat my alternatives no matter what they do? If yes, you’re done; stop deliberating and play it. If no option dominates, at least cross off any move that’s dominated — never play those — and then bring in the heavier machinery. That machinery is the Nash equilibrium, the idea that finally tells you where to land when your best move and theirs are tangled together. It’s next.