What happens when your best move depends on theirs?
Topic 23 keeps one decision-maker but gives them several objectives that genuinely conflict, so “optimal” stops being a point and becomes a curve. Game theory (topics 11 to 15) drops the other assumption: when another party is also choosing in response to you, your best plan depends on theirs and there is no single answer to compute, only an equilibrium to find. It underwrites markets that clear through auctions, security proofs written as games between an adversary and a defender, and populations that settle into stable mixes of strategy.
A Nash equilibrium is a set of choices, one per player, where nobody can do better by unilaterally switching: given what everyone else is doing. Not "the best outcome." Not "fair." Just stable: no individual has a reason to move.
The figure is the classic Prisoner's Dilemma, and its whole point is that stable and good are different properties. (Silent, silent) is better for both players than (confess, confess) (3 beats 1) but it is not an equilibrium, because each player, reasoning selfishly about their own payoff alone, always prefers to confess regardless of what the other does. That preference holds player by player even though it makes the group worse off. Nothing in the structure of the game pulls it back toward the good outcome; only external enforcement (a binding agreement neither can break alone) would.
This is the pattern that recurs across the rest of this module: individually rational choices, made independently, landing somewhere nobody would have chosen on purpose.
Formally, a strategy profile s* = (s₁*, …, sₙ*) is a Nash equilibrium if, for every player i and every alternative strategy sᵢ available to them:
uᵢ(sᵢ*, s₋ᵢ*) ≥ uᵢ(sᵢ, s₋ᵢ*)
. Your actual payoff is at least as good as any unilateral deviation, holding everyone else's strategy fixed. Existence (Nash, 1950): every finite game (finitely many players, each with finitely many strategies) has at least one equilibrium, though it may require mixed strategies (randomising over pure choices) rather than a single deterministic one. The proof applies Kakutani's fixed-point theorem to the best-response correspondence: the map from "what everyone else is doing" to "my best replies" is continuous enough that it must have a fixed point, and a fixed point of best-response is an equilibrium by definition.
The same idea shows up under a different name the moment a game has many small, interchangeable players instead of two named ones: a Wardrop equilibrium, in a network of drivers choosing routes, is reached when no driver can reduce their own travel time by switching: exactly the Nash condition, applied to routing. Worked on the standard textbook network (Roughgarden): two routes from S to T, each combining one road with travel time x/100 (x = drivers on it) and one fixed 45-minute road. With 4,000 drivers split evenly, each route carries 2,000 and costs 2000/100 + 45 = 65 minutes: the equilibrium, because neither route is faster to switch to. Now add a free connector between the two roads' midpoints. Every driver can strictly improve by routing through it, so the only equilibrium left has all 4,000 drivers on the combined path, each paying 4000/100 + 0 + 4000/100 = 80 minutes: about 23% worse for everyone, from a road that could only ever help. That is Braess's paradox: adding capacity to a selfishly-routed network can make every single user worse off, because the new equilibrium is stable, not good.
Seoul, 2003 In July 2003, Seoul closed a six-lane elevated expressway carrying roughly 168,000 vehicles a day, as part of restoring the Cheonggyecheon stream buried underneath it. City officials expected worse congestion. Traffic around the city instead sped up. Now one of the most widely cited real demonstrations that removing road capacity from a selfishly-routed network can improve the equilibrium everyone actually experiences, the mirror image of Braess's paradox above. The general phenomenon is studied quantitatively as the "price of anarchy" (how much worse the selfish equilibrium is than the coordinated optimum) across real transportation networks (Youn, Gastner & Jeong, Physical Review Letters 101, 128701 (2008)).
Check yourself
In the Prisoner's Dilemma grid, why is (silent, silent) not a Nash equilibrium although both players prefer it to (confess, confess)?
Nash equilibrium means stable, not good: nobody gains by switching alone. From (silent, silent) each player does gain by confessing, even though it leaves both worse off.