We study a recently introduced extension of normal form games with a phase before the actual play of the game, where each player can make binding offers for payments of utility to the 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 and allow for some degree of cooperation between rational players while preserving the non-cooperative nature of the game. We focus on 2-player negotiations games arising in the preplay phase when offers for payments are made conditional on a suggested matching offer of the same kind being made in return by the receiver. We study and analyze such bargaining games, obtain results describing their possible solutions and discuss the degrees of efficiency and fairness that can be achieved in such negotiation process depending on whether time is valuable or not.
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