= Prisoner's dilemma
{title2=$T>R>P>S$}
{wiki}
Two people can each have a clear incentive to choose an outcome that leaves both worse off. The <prisoner's dilemma> isolates this tension between individual <best responses> and <Pareto efficiency>. For example, two partners can each save effort by withholding a promised contribution, yet both would prefer a successful joint project to mutual withholding. The model applies when each person's <payoffs> really have the ordering below; the names “cooperate” and “defect” alone do not establish a <prisoner's dilemma>.
In the symmetric two-player <normal-form game>, each player chooses cooperation $C$ or defection $D$ simultaneously, without observing the other's current choice. The four <payoffs> are the mutual-cooperation reward $R$, mutual-defection punishment $P$, sole-defector temptation $T$, and sole-cooperator <payoff> $S$, with $T>R>P>S$. Defection is a <strictly dominant strategy>: it beats cooperation against either opposing <pure strategy>. This definition and the resulting one-shot <Nash equilibrium> are discussed in https://plato.stanford.edu/entries/prisoner-dilemma/[Steven Kuhn, Prisoner's Dilemma, Stanford Encyclopedia of Philosophy, section 1].
Take the concrete <payoffs> $T=5$, $R=3$, $P=1$, $S=0$. Each entry below gives the row player's <payoff> followed by the column player's <payoff>:
$$
\begin{array}{c|cc}
& C & D\\\hline
C & (3,3) & (0,5)\\
D & (5,0) & (1,1)
\end{array}
$$
\Image[/prisoners-dilemma-payoff-and-best-responses.png]
{title=Prisoner's dilemma: best-response arrows lead to mutual defection despite the gains from mutual cooperation}
For the row player, switching from $C$ to $D$ improves the <payoff> from $3$ to $5$ against $C$, and from $0$ to $1$ against $D$. The column player has the same comparisons, so $D$ is a <strictly dominant strategy> for each. Even against a <mixed strategy> choosing $C$ with <probability> $q$, the <expected payoff> advantage is $2q+(1-q)=1+q>0$. Thus cooperation cannot appear with positive <probability> in a <best response>, and $(D,D)$ is the unique one-shot <Nash equilibrium>, including <mixed strategies>.
At $(D,D)$, a <unilateral deviation> to $C$ lowers the deviator's <payoff> from $1$ to $0$. A joint change to $(C,C)$ instead raises both <payoffs> from $1$ to $3$: mutual cooperation Pareto dominates mutual defection, making the <Nash equilibrium> fail <Pareto efficiency>. A <Nash equilibrium> tests <unilateral deviations>, so it need not maximize shared gains. This is the useful lesson: improving the available outcome may require changing incentives, rather than telling each participant to choose a better <best response>.
The additional inequality $2R\ge T+S$ says that mutual cooperation gives each player at least as much average <payoff> as equal-time alternation between the two asymmetric outcomes. It is a further assumption, not a consequence of $T>R>P>S$. Here it holds strictly: $6>5$, and alternating exploitation gives each player average <payoff> $5/2<3$. This comparison concerns equal-time averages, not every discounted sequence. See https://plato.stanford.edu/entries/prisoner-dilemma/[Kuhn, section 3, on the additional cardinal-payoff condition].
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