Thought Toys · Strategy & computation · Exhibit 27
A Nash equilibrium is a quiet kind of trap: a set of choices where no single player can do better by changing their mind alone. It needn't be the best outcome — only the stable one. Drag the temptation to betray and watch cooperation stop holding together.
Where does the game settle?
Drag the temptation and watch the stable cells change.
Two players, each with the same choice: cooperate or defect. The grid lists what each walks away with — your payoff first, in amber; theirs in cyan. Mutual cooperation pays 3 each; mutual defection pays a meagre 1; and if one defects on a cooperator, the cooperator gets the sucker's 0 while the defector pockets the temptation — the number on the dial.
The little arrows are the whole story. A solid amber arrow says "you'd jump to that row to earn more"; a dashed cyan arrow says the same for the other player and the columns — colour and line style both mark whose move it is, so the grid still reads without colour. A cell that no arrow leaves — that neither of you can improve on by moving alone — is a Nash equilibrium. Drop the token anywhere and hit let them react: each player keeps switching to their best reply, and the token slides downhill along the arrows until it lands in such a cell and stops. The game locks.
Now the dial. While betrayal pays less than honest cooperation (T < 3), cooperating is your best reply to a cooperator, so both mutual cooperation and mutual defection are equilibria — a stag hunt, two stable worlds, one of them good. Drag T up past 3 and the arrows around the cooperative corner flip outward: now you always gain by defecting, so cooperation is no longer self-enforcing and its equilibrium vanishes. What's left is the lone, miserable cell where you both defect for 1 — a prisoner's dilemma. Nobody chose it together; it's just the only place no one can leave. That gap between what's stable and what's best is the whole uneasy point.
improve/verify/27-nash.js): the equilibrium count flips from two
(stag hunt, T=2) to one (dilemma, T=5) at the threshold T=R, and best-response
dynamics from all four starts lands in an equilibrium — collapsing to mutual defection whenever T > R.
Counter-examples: the outcome that's best for both, mutual cooperation (3,3), is
not the equilibrium in the dilemma — "the best outcome is what happens" is exactly the false intuition this
breaks. And some games have no pure equilibrium at all: Matching Pennies, where one player wants to match
and the other to mismatch, sends the arrows chasing in a circle forever — its only equilibrium is a mixed
(randomised) one.
Also in Strategy & computation: The learning-rate cliff →
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