For a NTU-bargaining game with two privately informed and patient players and an exogenously fixed disagreement outcome, consider any ex-ante efficient social choice function which is truthfully implementable in Bayesian Nash equilibrium with a fixed number of messages. Conditions are derived under which this social choice function can be implemented in Perfect Bayesian equilibrium of an escalation game. In this game, the players choose simultaneously their demands and then decide in turn whether to accept the opponent's proposal or to escalate until a deadline is reached. Escalation stops the game with an appropriate exogenously imposed probability of the disagreement outcome.