Some games hand you a move that’s just plain best no matter what the other player does — and those are lovely, because you don’t have to think about them. Pick it and walk away. But most of the interesting ones aren’t like that. When nobody holds a dominant strategy, your best move depends on theirs, and theirs depends on yours, and yours on theirs again — a hall of mirrors where every reflection points at another reflection and nothing sits still long enough to grab.
You could try to think your way straight through that regress and go slightly mad. Or you could reach for the one tool that tames it: the best response. It’s the atom of the whole equilibrium — the smallest, hardest-to-argue-with unit of strategic reasoning. Get this one idea solid and the Nash equilibrium (next lesson) falls out almost for free, because an equilibrium turns out to be nothing more than a pile of best responses that happen to agree.
Before you read — take a guess
You and a rival startup are each about to pick a video format, A or B, for a new device. You have no idea yet what they'll choose. A teammate asks: 'So which format is best for us?' Before we've defined anything — what's the honest problem with that question as stated?
Define the best response
Your best response to a particular choice by the other player is the move that gives you the highest payoff, given that they make that choice. That’s it. Highest-payoff move, conditional on a specific thing the opponent does.
The word doing all the work is given. A best response is not “my best move” in some free-floating, absolute sense — it’s “my best move if they play Format A.” Change the if and the answer can change. So a best response always comes as a pair: a condition (what they do) and a reply (what you should do about it). “If they play A, my best response is A. If they play B, my best response is B.” Two conditions, two replies, one for each thing they might do.
Here’s why that reframing is a genuine rescue and not just a semantic dodge. “What’s my best move, period?” is often unanswerable in an interdependent game — it depends on a thing you don’t control and may not know. But “what’s my best move if they do this specific thing?” is always answerable: you’ve frozen the one variable that was making the question slippery, and now you’re just reading numbers off a fixed column. You have traded one impossible question for two easy ones.
Conditional beats unconditional
You can always name a best response, even in a game with no dominant strategy, because you’re answering the easier question. “Best, period” asks you to be right against an unknown. “Best if they do X” asks you to be right against a known — you just pretend, for a moment, that their choice is settled, and pick your top payoff. Do that for each of their possible moves and you’ve mapped your entire strategic response, one condition at a time.
How to compute it
The mechanics are almost insultingly simple, which is the point — the atom should be easy to split.
Fix the other player’s column, then pick your highest-payoff row in it. That’s the entire procedure. The column is the condition (“suppose they play this”); scanning down it for your biggest number is finding your best reply. Do it once per column and you’ve found your best response to everything they could do.
Below is the standards game: you and a rival startup each pick Format A or Format B. The payoffs reward matching — if you both land on the same format, the device wins the market and you split the spoils; mismatch and the product flops. The interactive matrix rings each player’s best-response payoff automatically. Read the numbers first, work out the best responses in your head, then check yourself against the rings.
Payoff matrix
Best responses in the standards game
A ringed number is that player's best response to the rival's choice. Read down each column (fixing the rival's move) and find the row with your highest payoff.
| Rival startup | ||
|---|---|---|
| Format A | Format B | |
| Your startupFormat A | NE33 | 11 |
| Format B | 00 | NE22 |
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 startup — dominant strategy: none
Rival startup — dominant strategy: none
Nash equilibrium (pure): (Format A, Format A) · (Format B, Format B)
Let’s walk it cell by cell, from your point of view (the row player), one column at a time.
- Suppose the rival plays Format A (the left column). Your two options are the top-left cell, where you score 3, and the bottom-left cell, where you score 0. Three beats zero, so your best response to A is A. Match them.
- Suppose the rival plays Format B (the right column). Now your options are the top-right cell, worth 1, and the bottom-right cell, worth 2. Two beats one, so your best response to B is B. Match them again.
Notice your best response flipped between the two columns — A when they play A, B when they play B. That flip is the fingerprint of a game with no dominant strategy: there’s no single row that’s best down both columns. Your reply genuinely depends on the condition.
The rival reasons the same way, but across rows (they pick a column). If you play Format A (top row), they compare their 3 (A) against their 1 (B) and pick A. If you play Format B (bottom row), they compare their 0 (A) against their 2 (B) and pick B. Their best responses mirror yours — match whatever you did.
Here’s the whole thing as a lookup table. This is what “computing the best response” actually produces: a little dictionary from their move to your reply.
| If the rival plays… | Your payoff from A | Your payoff from B | Your best response |
|---|---|---|---|
| Format A | 3 | 0 | Format A (3 > 0) |
| Format B | 1 | 2 | Format B (2 > 1) |
Build the mirror-image table for the rival and you’d find: if you play A they answer A, if you play B they answer B. Stack the two dictionaries together and something jumps out — there are two cells where both players are simultaneously giving their best response: both-A at (3, 3) and both-B at (2, 2). Those doubly-ringed cells are exactly where the reasoning stops chasing itself. Hold that thought; it is the equilibrium, and it’s the whole reason the next lesson exists.
In the standards game above, suppose you've somehow learned the rival is committed to Format B. A colleague argues: 'Format A gave us the biggest single number in the whole grid — a 3 — so we should play A.' Where's the mistake?
Best response vs dominant strategy
Now the payoff for building the atom carefully: it lets us define a dominant strategy in one clean line.
A dominant strategy is a move that is a best response to everything the other player could do.
That’s the whole relationship. A dominant strategy isn’t a different kind of thing from a best response — it’s a best response that happens to win in every column at once. Where the standards game gave you a reply that flipped (A against A, B against B), a dominant strategy would be the same row ringed all the way down. When that happens, the “given what they do” caveat evaporates: it doesn’t matter what they do, your answer is the same, so you can stop thinking about them entirely.
The cleanest example is the prisoner’s dilemma. Two suspects are questioned separately; each can Cooperate (stay silent, protect the other) or Defect (rat the other out for a lighter sentence). Higher numbers are better outcomes. Watch what Defect does across both of the rival’s choices.
Payoff matrix
Defect is a best response to everything
Check each column separately. If the same row is ringed down both columns, that row is a dominant strategy — a best response no matter what the rival does.
| Prisoner Col | ||
|---|---|---|
| Cooperate | Defect | |
| Prisoner RowCooperate | 33 | 05 |
| Defect | 50 | NE11 |
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
Prisoner Row — dominant strategy: Defect
Prisoner Col — dominant strategy: Defect
Nash equilibrium (pure): (Defect, Defect)
Run the column-by-column check on Prisoner Row:
- If the rival Cooperates (left column): Cooperate scores 3, Defect scores 5. Five wins — best response is Defect.
- If the rival Defects (right column): Cooperate scores 0, Defect scores 1. One wins — best response is Defect again.
Same answer in both columns. Defect is a best response to everything, so it’s a dominant strategy — and by the symmetry of the payoffs, it’s dominant for the other prisoner too. Because the caveat has vanished, both can reason it out with zero mind-reading: whatever you do, I defect. The tragedy that both would rather have landed on mutual Cooperation (3, 3) than mutual Defection (1, 1) is a story for a later lesson — for now, just notice how we proved dominance: nothing more than “best response, checked in every column, same answer.”
Dominant strategies are the exception, not the rule
Most games are not this generous. In the standards game your best response flipped between columns, so no move dominated — and that’s the common case. When there’s no dominant strategy, you can’t collapse the problem to a single answer; you’re stuck with a best response that genuinely depends on the column. That dependence is precisely the difficulty the Nash equilibrium was invented to resolve, which is why the next lesson can’t start until this one is in your bones.
Here’s the infinite regress in full, using the standards game with no dominant strategy. You start reasoning: “I should match them — but that depends on what they pick. They’ll want to match me — but that depends on what I pick. So to know my move I need to know theirs, and to know theirs I need to know mine, and to know mine I need to know theirs…” The reasoning spins forever because every step passes the question back unanswered. This is the hall of mirrors: two mirrors facing each other, each reflection defined only by the next.
The best response cuts it with a single move: stop asking the unconditional question. Instead of “what’s my best move, period?” (the question that bounces back and forth forever), you ask “what’s my best move given they play A?” and “what’s my best move given they play B?” Each of those is a dead-end in the good sense — it terminates. You freeze the rival’s move as an assumption, read your best reply straight off the frozen column, and the mirror stops reflecting because you’ve painted one of them opaque.
You don’t yet know which assumption is the right one to make — that’s the equilibrium’s job, next lesson. But you’ve converted an infinite chase into a small, finite table of “if this, then that.” The regress dies the instant you make the question conditional. That’s the entire trick, and it’s why the best response is the atom: every larger structure in game theory is built by snapping these conditional answers together.
Practice
Match each term to its precise meaning.
Pick a term, then click its definition.
State the definition in your own words.
Pick the right option for each blank, then check.
A best response is your move given a choice by the other player. A move that is a best response to the other player could do is called a dominant strategy.
When to use it
Reach for the best response the moment payoffs are interdependent — the instant your best move depends on someone else’s move and theirs on yours. Before you try to predict anything about how a strategic situation will resolve, do the small, patient work first: for each thing the other side might do, ask “what’s my best response to that?” and write it down. You’ll produce a little dictionary from their moves to your replies, and you’ll do the same from their side.
This is the move that stops you from freezing in the hall of mirrors or, worse, grabbing the biggest number on the board regardless of whether it’s reachable. Negotiations, pricing wars, standards battles, standoffs — all of them yield to the same first step: map the best responses, one condition at a time. Only once that map exists can you ask the bigger question of where the two sides’ best responses agree.
Where this goes next
You now own the atom. A best response is your highest-payoff move given a specific opponent choice; you compute it by fixing their column and picking your top row; and a dominant strategy is simply the lucky case where the same response wins in every column, so the “given” stops mattering. You saw both — a game where your reply flips (standards) and a game where one move dominates outright (prisoner’s dilemma) — and you watched the best response cut the infinite regress by trading one unanswerable question for a finite table of conditional ones.
But we left a thread dangling. In the standards game, two cells lit up with both players’ rings at once — (3, 3) and (2, 2) — places where your best response and the rival’s best response pointed at the same cell simultaneously. That mutual agreement is not a coincidence; it’s the destination. Next up, Mutual Best Response: Finding the Equilibrium — where we show that a Nash equilibrium is nothing more exotic than a cell where every player’s best response meets at the same time, and that the whole apparatus of predicting strategic outcomes is just this atom, snapped together.