Skip to main content
Humanities LibreTexts

26.1: Prisoner’s Dilemmas

  • Page ID
    95269
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

    ( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\id}{\mathrm{id}}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\kernel}{\mathrm{null}\,}\)

    \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\)

    \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\)

    \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    \( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

    \( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

    \( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vectorC}[1]{\textbf{#1}} \)

    \( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

    \( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

    \( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)

    The notorious felons A and B rob a bank. A few days later they are apprehended and booked into jail. The District Attorney feels certain that they committed the robbery but doesn’t have enough evidence to convict either prisoner unless one of them confesses. The DA puts the two prisoners in separate interrogation rooms and makes the following offer to each:

    1. If you confess but the other prisoner does not, you can go free (in return for turning state’s evidence). The other prisoner will be sentenced to 10 years.
    2. If you both confess, you each get 5 years.
    3. If neither of you confess, you will both get 1 year (I can’t get a conviction on robbery, but both of you were carrying unlicensed concealed weapons when you were arrested).

    The DA informs each prisoner that they both received the same offer, and leaves them both to ponder their options.

    Each prisoner may feel some loyalty to the other prisoner, or may consider the possibility that if they squeal the other will seek revenge. But suppose each is so terrified of prison that such considerations play a negligible role in their deliberations.

    What would you do?

    A two-by-two matrix (Figure 26.1.1) allows us to visualize the possible outcomes. The rows of the matrix represent A’s options (Confess, Don’t Confess). The columns represent B’s options (the same as A’s). Each of the four cells of the matrix specifies one of the four possible outcomes. The first number in each cell represents the number of years A gets, and the second number in the cell represents the number of years B gets.

    Thus, the top left cell of the matrix represents the outcome where A and B both confess. The numbers in this cell are 5, 5, which indicate that in this situation both prisoners get a 5-year sentence. The top right-hand cell represents the outcome where A confesses but B does not. The numbers in this cell are 0, 10, which indicates that in this situation A gets 0 years and B gets 10. The two bottom cells work the same way.

    Now put yourself in A’s position. How would you reason? A doesn’t know whether B will confess, but B only has two options: confess or not. So, A considers each possibility in turn.

    1. B confesses (this means we are considering the first column). In this condition, I get 5 years if I confess and 10 years if I do not. So, if B does confess, I’m better off (by five years) if I confess too.
    2. B does not confess (we are now in the second column). In this condition, I get 0 years if I confess and 1 year if I do not. So, if B doesn’t confess, I’m better off (by one year) if I confess.
    Screenshot (106).png
    Figure \(\PageIndex{1}\): Prisoner’s Dilemma

    Either way, A is better off confessing. So, confessing looks like the rational thing to do. B will go through the same pattern of reasoning, so it’s also rational for him to confess.

    The unsettling result is that when each prisoner pursues their own selfinterest, the outcome is collectively self-defeating. They both end up with a condition (5 years in prison) that is four year’s worse than the fate they would have had (1 year) if they’d cooperated. Each prisoner acts to maximize his own self-interest, so from the point of view of each prisoner it makes good sense to confess. But the result for each is much worse.

    Would it make a difference if the prisoners could communicate before making their decisions? It depends. In many cases, people reach an agreement to do something in the future. Suppose that is the case here. A and B discuss the DA’s offer and agree to cooperate.

    But when alone, both A and B’s thoughts return to their earlier line of reasoning. They ask themselves: Is it rational for me to keep my agreement? And the same line of reasoning they went through before convinces them that they are better off breaking their promise and confessing. Since B’s reasoning will parallel A’s, both confess. The ability to communicate isn’t enough to solve the problem; if A and B do not trust each other, then their most fervent promises of cooperation won’t help.


    This page titled 26.1: Prisoner’s Dilemmas is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Jason Southworth & Chris Swoyer via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.