Prisoner's Dilemma: Puzzle and Step-by-Step Solution
Game Theory
Easy
Reported at Jane Street, Citadel, Optiver, IMC
The puzzle
Two suspects are held in separate rooms. If both stay silent, each serves one year. If one testifies and the other stays silent, the one who testifies walks free and the silent one serves ten years. If both testify, each serves eight years. Neither can communicate with the other. What do they do?
Short answer: Both testify and serve eight years, even though staying silent together would cost each of them only one year.
Step 1: Write the payoff matrix
Put the first suspect's choices on the rows and the second's on the columns. Silent and silent gives one year each. Testify against silent gives zero years to the one who testified and ten to the silent one. Testify and testify gives eight each. Years in prison are a cost, so each player wants the smaller number.
Writing the matrix before reasoning is the habit interviewers want to see. Most mistakes in two-player games come from comparing the wrong cells, and the matrix makes the right comparison mechanical.
Step 2: Check each choice against each choice of the other player
Take the first suspect's view. If the other stays silent, she compares one year (silent) with zero (testify), so testifying is better. If the other testifies, she compares ten years (silent) with eight (testify), so testifying is better again.
Testifying is better whatever the other player does. That is a strictly dominant strategy, and since the game is symmetric it is dominant for the second suspect too. Both testify, and both serve eight years.
Step 3: Why the bad outcome is stable
Testify-testify is the unique Nash equilibrium: neither player can do better by changing only their own choice. A suspect who switches to silence while the other keeps testifying moves from eight years to ten.
The frustrating part is that silent-silent is better for both, one year each. It is simply not stable. From there, each player gains by switching to testify, so it collapses. The gap between what is jointly best and what is individually stable is the whole lesson of the puzzle.
Step 4: What actually breaks the trap
Repetition. If the same two players face the game again and again with no known last round, strategies like tit-for-tat make cooperation self-enforcing, because today's defection is punished tomorrow. Enforceable contracts do the same job by changing the payoffs for defecting.
What never works is cheap talk. A promise that costs nothing to break does not change any number in the matrix, so it does not change the equilibrium.
Why interviewers ask it
Use the words dominant strategy and Nash equilibrium about the right cells, then volunteer the repeated-game follow-up before you are asked. Trading desks face repeated games with the same counterparties every day.
Knowing the proof is half of it. Work Prisoner's Dilemma interactively, then take it under a clock with the rest of the Outcry simulators.