Equilibrium analysis for a leader-follower game with noisy observations: A pace forward in Witsenhausen's counterexample conjecture

Abstract

In this paper, we view Witsenhausen's problem as a leader-follower coordination game in which the action of the leader is corrupted by an additive normal noise, before reaching the follower. The leader who observes the realization of the state, chooses an action that minimizes her distance to the state of the world and the ex-ante expected deviation from the follower's action. The follower then makes a noisy observation of the leader's action and chooses an action minimizing her expected deviation from the leader's action. To this end, we analyze the perfect Bayesian equilibria of this game and show that strong complimentarily between the leader and the follower combined with a prior with poor enough precision can give rise to what we call “near piecewise-linear equilibria”. As a major consequence, we prove local optimality of a class of slopey quantization strategies which had been suspected of being the optimal solution in the past, based on numerical evidence for Witsenhausen's counterexample.