Nash equilibrium — the prisoner's dilemma

What just happened?

Two suspects questioned separately. Each can cooperate (stay silent) or defect (testify against the other). The payoffs are years in prison — the fewer the better: both stay silent → 1 year each; both testify → 2 years each; if one testifies and the other stays silent → the traitor walks free (0 years), while the silent one gets 3 years.

Payoff matrix (years in prison, A | B):

Check player A’s perspective. If B cooperates: A gets 1 year for cooperating, but 0 years for defecting — defection is better. If B defects: A gets 3 years for cooperating, but 2 years for defecting — again defection is better. So defection is a dominant strategy: it pays off regardless of the opponent’s move. Symmetrically for B.

Nash equilibrium (the invariant):

Since both players rationally choose to defect, they end up in the state (defect, defect) with an outcome of 2 years each — this is a Nash equilibrium. Formally: it is a set of strategies in which no player can improve their outcome by changing only their own decision, with the other’s decision unchanged. When B defects, A switching to cooperation would get 3 instead of 2 years — so it does not pay off. The same for B. The paradox: mutual cooperation (1 year each) would be better for both — this is the $\text{social optimum}$ — but it is not stable, because each has a temptation to deviate and testify.

Individual vs. collective rationality

This is the heart of the dilemma: the individually rational choice (defection) leads to an outcome worse for both than cooperation. The tension between what pays off for the individual and what would be best for the group appears everywhere — from arms races, through price-fixing, to environmental protection.

John Nash and game theory

Nash equilibrium was introduced by the American mathematician John Nash in 1950 (the paper “Equilibrium Points in n-Person Games”). It is one of the fundamental concepts of game theory, with vast applications in economics, evolutionary biology, politics and computer science (auctions, negotiations, market competition). For his contribution to the theory of non-cooperative games, Nash received the 1994 Nobel Prize in Economics (jointly with John Harsanyi and Reinhard Selten). His turbulent life — struggles with schizophrenia — was portrayed in the film “A Beautiful Mind” (2001).

Why this matters:

This is a topic outside the core curriculum — a curiosity showing how mathematics describes human decisions. Worth remembering is one habit of thought: to find an equilibrium, for each player check whether changing only their decision would improve their outcome. If it would for no one — it is a Nash equilibrium.

Remember: in a Nash equilibrium no player gains by changing their strategy unilaterally. In the prisoner’s dilemma that is (defect, defect) — even though joint cooperation would be better for both.