The halflie problem

In Ulamʹs game Paul tries to find one of n possibilities with q Yes–No questions, while responder Carole is allowed to lie a fixed number k of times. We consider an asymmetric variant in which Carole must say yes when that is the correct answer (whence the halflie). We show that the maximal Ak(q) for which Paul wins has the asymptotic formAk(q)=2q+kk!q−k+Θ(2qq−k−12).