The Prisoner's Dilemma

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Christmas 1914: enemy soldiers climbed out of the trenches and shook hands. Why does trust sometimes break out, and sometimes collapse? It starts with one small game.

The Prisoner’s Dilemma

Two gas stations sit across the street from each other. Each morning each owner makes one choice: hold prices, or cut them to pull in the other’s customers. If both hold, both do well. If one cuts while the other holds, the cutter takes the street.

From earlier lessons
Two quick ones before we start.

A move that beats your alternatives no matter what the other player does is called what?

Both players could gain by switching actions together. Does that prove either would gain by switching alone?

Predict, then play

Exhibit 1  Payoffs per round (you, them), in profit
Both owners are sensible and self-interested. It is an ordinary morning. What happens?

Why did that happen?

Take a guess, then reveal the reason

Cutting pays more for you whatever the other owner does. If they hold, cutting wins the street. If they cut, you cannot afford to be the only one holding. So both cut, and both earn less than if they had left prices alone. Neither owner is careless. They are stuck in the logic of the situation.

The math

Write the game in normal form as G=(N,(Si),(ui))G=(N,(S_i),(u_i)): players N={1,2}N=\{1,2\}, a strategy set Si={C,D}S_i=\{C,D\} for each (hold or cut), and a payoff uiu_i for every pair of choices.

Label the four payoffs with the usual names: RR for both holding (reward), TT for cutting against a holder (temptation), PP for both cutting (punishment), SS for holding against a cutter (sucker). The ranking that defines a Prisoner’s Dilemma is

T>R>P>S.T > R > P > S.

Here R=3, T=5, P=1, S=0R=3,\ T=5,\ P=1,\ S=0. Repeated-game versions often add 2R>T+S2R>T+S. That extra condition says two rounds of mutual cooperation beat taking turns exploiting one another. It is useful for studying sustained cooperation, but it is not part of the one-shot definition.

Cutting strictly dominates holding, since the cutter does better against either move (5>35>3 and 1>01>0):

ui(D,a)>ui(C,a)∀ a.u_i(D, a) > u_i(C, a) \quad \forall\, a.
Definition, Nash equilibrium

A profile s∗s^* is a Nash equilibrium if ui(si∗,s−i∗)≥ui(si,s−i∗)u_i(s_i^*, s_{-i}^*) \ge u_i(s_i, s_{-i}^*) for every sis_i and every ii.

A strictly dominant strategy is played in every equilibrium, so (D,D)(D,D) is the only one, paying (1,1)(1,1). But (C,C)(C,C) pays (3,3)(3,3) and beats it for both players. The equilibrium is the outcome nobody wanted.

Find the equilibrium yourself

Show that both-cut is the only equilibrium

Check each player’s best reply. If they hold, you get 33 from holding but 55 from cutting, so you cut. If they cut, you get 00 from holding but 11 from cutting, so you cut again. Cutting is your best reply to everything, and the same is true for them. The only cell where neither wants to move is (cut,cut)(\text{cut}, \text{cut}), paying (1,1)(1,1).

Key idea

Key idea

Each player, acting sensibly alone, drives both to the outcome they were trying to avoid.

Check yourself

Check yourselfapply
Which outcome is the Nash equilibrium, the one neither owner regrets given the other's choice?
Check yourselfbrings back: dominance
Two airlines keep undercutting fares until both barely profit, though both would earn more holding prices. In game terms, cutting fares is each airline's what?

Sources

  1. Axelrod, R. (1984). The Evolution of Cooperation. Basic Books.
  2. Flood, M. M. (1958). Some experimental games. Management Science, 5(1), 5-26.
  3. Nash, J. F. (1950). Equilibrium points in n-person games. PNAS, 36(1), 48-49.
  4. Osborne, M. J., and Rubinstein, A. (1994). A Course in Game Theory. MIT Press.