Two-player preplay negotiation games with conditional offers

We consider an extension of strategic normal form games with a phase before the actual play of the game, where players can make binding offers for transfer of utilities to other players after the play of the game, contingent on the recipient playing the strategy indicated in the offer. Such offers transform the payoff matrix of the original game but preserve its non-cooperative nature. The type of offers we focus on here are conditional on a suggested matching offer of the same kind made in return by the receiver. Players can exchange a series of such offers, thus engaging in a bargaining process before a strategic normal form game is played. In this paper we study and analyse solution concepts for two-player normal form games with such preplay negotiation phase, under several assumptions for the bargaining power of the players, as well as the value of time for the players in such negotiations. We obtain results describing the possible solutions of such bargaining games and analyse the degrees of efficiency and fairness that can be achieved in such negotiation process. We show the similarities and the differences with a variety of frameworks in the literature of bargaining games and games with a preplay phase.