  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 analyze solution concepts for two-player normal
form games with such preplay negotiation phase, under several assumptions for
the bargaining power of the players, such as the possibility of withdrawing
previously made offers and opting out from the negotiation process, 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 analyze the
degrees of efficiency and fairness that can be achieved in such negotiation
process.
