Nash Equilibrium

A Nash equilibrium is a combination of choices — one for each player — where no single player can make their own outcome better by changing their choice alone. Everyone is doing the best they can, given what everyone else is doing. It is a definition of stability, not a guarantee of happiness.

Finding equilibria

You don't find an equilibrium by staring. You find it by checking every cell against one question: would anyone rather move?

Best responses

The check runs in two passes. First, the row player's: fix the column player's choice and ask where the row player's best payoff in that column sits. Mark it red. Then switch chairs — fix the row, find the column player's favorite cell in it, and mark it green.

Figure 1 runs both passes on a coordination game: two players picking between System A and System B, both happier on the same one. If Player 2 commits to A, Player 1's best reply is A — 2 beats 0 — so the top-left cell earns its red corner. If Player 2 commits to B, the best reply is B, and the bottom-right cell earns one too. The green corners answer the same question from the other chair and land on the same two cells.

A cell carrying both marks is a cell where both players are best-responding at once. Nobody wants to move — the definition, checked by hand. Two cells pass the test — this game has two equilibria. Both are stable; neither is guaranteed.

Dominance & iterated elimination

Sometimes you can skip the cell-by-cell check, because one strategy simply never wins. The walkthrough below is the whole argument on the Prisoner's Dilemma grid: Defect pays exactly one more than Cooperate in every column, so Cooperate is strictly dominated for the row player — and the same check from the other chair dominates Cooperate for the column player too.

Delete the dominated row, then the dominated column; what survives is the equilibrium. Figure 2 shows the grid that argument runs on.

The order of deletion doesn't matter here — take the column out first and the row's dominance is still standing, and the same cell survives either way. Some games need several rounds of this, each deletion making the next check easier. And in bigger games, iterated elimination can prune half the board and still leave a crowd: it narrows the field; it doesn't promise a winner.

Mixed strategies

Every equilibrium so far has players picking one thing and holding it. Some games refuse that. When any fixed choice can be read and punished, the only plan that survives is a random one — and finding the mix that makes it work is three lines of algebra. Here it is on Matching Pennies, told as a penalty kick.

You can feel this in The Trap lab — load the pennies game and watch the analysis find ½ for both players.

When there are several equilibria

Two equilibria in one grid asks an awkward question: which one should we expect? Equilibrium analysis tells you where a game can settle — not where it will. Selection is its own problem, and the Stag Hunt is where you feel it.

Two hunters can take a stag together and eat well — (4, 4) — or each can quietly hunt rabbit alone and eat modestly — (2, 2). Both outcomes are equilibria; check the marks and both cells are double-marked. But they are not equal.

Payoff-dominance is the optimistic tiebreaker: if one equilibrium pays every player more than another, expect that one. (Stag, stag) beats (rabbit, rabbit) for both hunters, so it payoff-dominates — it is where the game wants to go.

Risk-dominance is the nervous tiebreaker, and it is about the cost of guessing wrong. Hunt stag while your partner quietly switches to rabbit and you come home with 1 — the worst meal on the board. Hunt rabbit and you get 2 or 3 no matter what anyone else does. When you are unsure of your partner, rabbit is the safer play — and enough hunters reasoning that way make rabbit the self-fulfilling outcome. The safe equilibrium can beat the rich one.

Which instinct wins depends on the people, not the matrix. For the full solved set →, including this game's third, mixed equilibrium.

The trap, in grid form

The fastest way to feel this is the Prisoner's Dilemma inside The Trap lab. Two players each choose between Cooperate and Defect. Defect pays better than Cooperate no matter what the other player does, so the best response to anything is to defect. When both players follow that logic, they land on Defect–Defect — and once there, neither wants to switch. That is the equilibrium.

The uncomfortable part: both would be better off at Cooperate–Cooperate. But that outcome is not stable, because either player could quietly improve by defecting while the other stays put. Nash equilibrium does not promise you a good outcome. It promises an outcome with no incentive to move. The gap between those two promises is where most real-world frustration lives.

Why the idea matters

The value of a Nash equilibrium is often negative space. If a plan, a price, or a treaty is not an equilibrium, then at least one person has a reason to break it — and that reason does not go away just because you wish it would. Spotting the equilibrium tells you what is likely to stick, and what is likely to unravel the moment nobody is watching.

This is why economists, diplomats, and platform designers all reach for the same question: is this outcome self-enforcing, or does it need constant enforcement to survive? The equilibrium answer marks the boundary between the two.

The fine print

The idea leans on heavy assumptions: that players understand the game, that they know each other's incentives, and that each one is optimizing their own payoff. Real humans are messier — we overreact to losses, care about fairness, and copy the crowd. None of that invalidates the concept; it just means equilibria are landmarks on a map that people keep wandering off.

Try it

Open The Trap lab and move the numbers. Watch the equilibrium markers follow your edits, and notice which cells stay stuck and which ones dissolve. Flip a single payoff and see the whole board reorganize — sometimes the equilibrium barely moves, and sometimes one changed number sends it somewhere else entirely. Then ask yourself: if I were really sitting in this grid, would I play the equilibrium, or would I try to talk the other player into something better?