Consider a stochastic matrix $P$ for which the Perron–Frobenius eigenvalue has multiplicity larger than 1, and for $\epsilon > 0$, let

\begin{equation} P^\epsilon := (1 - \epsilon) P + \epsilon Q, \end{equation}

where $Q$ is a stochastic matrix for which the Perron–Frobenius eigenvalue has multiplicity 1. Let $\pi^\epsilon$ be the Perron–Frobenius eigenfunction for $P^\epsilon$. We will discuss the behavior of $\pi^\epsilon$ as $\epsilon \to 0$.

This was an important ingredient in showing that if two players repeatedly play Prisoner’s Dilemma, without knowing that they are playing a game, and if they play rationally, they end up cooperating. We will discuss this as well in the second half.

The talk will include the required background on Markov chains.