Decision Trees and Influence: an Inductive Proof of the OSSS Inequality

We give a simple proof of the OSSS inequality (O’Donnell, Saks, Schramm, Servedio, FOCS 2005). The inequality states that for any decision tree T calculating a Boolean function f : {0,1} → {−1,1}, we have Var[ f ]≤ ∑i δi(T ) Infi( f ), where δi(T ) is the probability that the input variable xi is read by T and Infi( f ) is the influence of the ith variable on f . ACM Classification: G.2, G.3, F.1.3 AMS Classification: 05A20, 60C05, 68R05, 68Q87