A single private signal can destroy approximate equilibria in stopping games

Simon, Solan and Tomkowicz proved that a discrete-time multiplayer stopping game whose players share one conditionally atomless filtration, with an integrable payoff process adapted to it, admits an ε-equilibrium for every ε>0. We show that this conclusion can fail once one player receives a single private signal at the start. We construct a two-player stopping game in their format and add one private signal. The public filtration is conditionally atomless and contains the fair-coin filtration. The payoffs are bounded and adapted to that filtration, and the payoff is zero if nobody stops. Player 1 additionally observes a private signal at stage 0, and the game has no 2⁻⁵⁵-equilibrium. Only three stages are payoff relevant, so without the private signal backward induction would already give approximate equilibria. Additional public signals independent of the state, such as uniform sunspots at every stage, do not restore existence. Information is nested, and the less informed player learns everything two stages later. Jacobovic and Solan note that a one-stage delay yields existence by backward induction, so their delay condition cannot be relaxed to two stages. We also record that the theorem of Simon, Solan and Tomkowicz yields sunspot ε-equilibria in every multiplayer stopping game with common information.