Prisoner’s Dilemma: A Comprehensive Guide

The Prisoner’s Dilemma, formalized in 1950 by Flood and Dresher, illustrates the conflict between individual self-interest and collective benefit. Named by Albert W. Tucker, it shows why rational actors may choose suboptimal outcomes, a key concept in game theory.

This MathMultiverse guide explores its scenarios, payoff structures, visualizations, and applications across economics, psychology, and more.

Scenarios

Classic Scenario

Two prisoners, A and B, choose to confess (C) or stay silent (S):

Both silent: 1 year each.
Both confess: 5 years each.
One confesses: 0 years, other 10 years.

Iterated Version

Repeated rounds, total payoff:

\[ U_i = \sum_{t=1}^n u_i(t) \]

Three-Player Variant

Adjusted payoffs for three prisoners.

Continuous Version

Effort \( e_i \in [0, 1] \):

\[ u_i(e_i, e_{-i}) = -e_i + k (1 - e_{-i}) \]

Payoff Analysis

Payoff Matrix

Prison years (negative utility):

\[ \begin{array}{c|cc} & C & S \\ \hline C & (-5, -5) & (0, -10) \\ S & (-10, 0) & (-1, -1) \end{array} \]

Confess dominates:

\[ u_i(C, C) = -5 > -10 = u_i(S, C) \]
\[ u_i(C, S) = 0 > -1 = u_i(S, S) \]

Nash Equilibrium

(C, C) is stable:

\[ u_i(C, C) \geq u_i(S, C) \]

Pareto Optimality

(S, S) is better but unstable:

\[ u_i(S, S) = -1 > -5 = u_i(C, C) \]

Payoff Heatmap

Payoff matrix visualized as a heatmap (Player A’s years in prison).

Applications

Economics

Price wars:

\[ \pi_i = p_i q_i - c q_i \]

Psychology

Cooperation rate:

\[ C_r = \frac{\text{Cooperative choices}}{\text{Total choices}} \]

Biology

Altruism vs. selfishness:

\[ F_i = b (1 - s_{-i}) - c s_i \]

Politics

Arms races model escalation vs. cooperation.