Title: Perron–Frobenius eigenfunctions of perturbed stochastic matrices
Speaker: Rajeeva L. Karandikar (CMI, Chennai)
Date: 27 September 2019
Time: 3 – 5 pm (with a 15 minute break at 3:45)
Venue: LH-1, Mathematics Department
Consider a stochastic matrix $P$ for which the Perron–Frobenius eigenvalue
has multiplicity larger than 1, and for $\epsilon > 0$, let
P^\epsilon := (1 - \epsilon) P + \epsilon Q,
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.