The Prisoner's Dilemma

A dominant strategy for each can leave both worse off.

The idea

The Prisoner's Dilemma is the canonical game in which each player's individually best choice leads to an outcome worse for both than one they could have had.

A strategy is strictly dominant for a player when it pays that player strictly more than every alternative, whatever the opponents do. A player who has one needs no beliefs about anyone else: comparing their own payoffs against each opponent choice in turn settles the decision. When every player has a strictly dominant strategy, the profile of those strategies is the game's unique Nash equilibrium.

In the Prisoner's Dilemma, two players each choose to cooperate or to defect. Write $R$ for the reward each earns from mutual cooperation, $P$ for the punishment each earns from mutual defection, $T$ for the temptation earned by defecting against a cooperator, and $S$ for what a cooperator earns against a defector. The dilemma is the ordering

$T > R > P > S.$

Since $T > R$ and $P > S$, defecting pays more than cooperating against either opponent choice, so defection strictly dominates and both players receive $P$. Since $R > P$, both prefer mutual cooperation — but mutual cooperation is not an equilibrium, because from it either player can defect and collect $T$.

Ways to work on it

• Walkthrough. Dominant strategies, the equilibrium, and why it is inefficient.
• Practice. Find the equilibrium payoff in a randomized dilemma.
• Hardest. Verify dominance and measure the efficiency loss.

Not sure where to start? Take the ten-question placement test.