Nested information does not ensure approximate equilibria in stopping games

We construct a two-player stopping game with two dates and bounded payoffs on a standard Borel common-prior space, with information nested at both dates, that has no ex-ante (Harsanyi) 2⁻⁴²-equilibrium in measurable randomized stopping times. This gives a negative answer to the finite-horizon stopping-game question of Jacobovic and Solan and to its published formulation by Jacobovic, Levy and Solan, which asks for an exact Bayesian equilibrium. The stopping game reduces to a two-player Bayesian game with binary actions and a row on which both payoffs vanish. A quantitative form of the non-endogeny of a noisy majority recursion on the ternary tree rules out approximate equilibria of this Bayesian game.