Sequential bargaining (also known as alternate-moves bargaining, alternating-offers protocol, etc.) is a structured form of bargaining between two participants, in which the participants take turns in making offers. Initially, person #1 has the right to make an offer to person #2. If person #2 accepts the offer, then an agreement is reached and the process ends. If person #2 rejects the offer, then the participants switch turns, and now it is the turn of person #2 to make an offer (which is often called a counter-offer). The people keep switching turns until either an agreement is reached, or the process ends with a disagreement due to a certain end condition. Several end conditions are common, for example:

Several settings of sequential bargaining have been studied.

An alternating-offers protocol induces a sequential game. A natural question is what outcomes can be attained in an equilibrium of this game. At first glance, the first player has the power to make a very selfish offer. For example, in the Dividing the Dollar game, player #1 can offer to give only 1% of the money to player #2, and threaten that "if you do not accept, I will refuse all offers from now on, and both of us will get 0". But this is a non-credible threat, since if player #2 refuses and makes a counter-offer (e.g. give 2% of the money to player #1), then it is better for player #1 to accept. Therefore, a natural question is: what outcomes are a subgame perfect equilibrium (SPE) of this game? This question has been studied in various settings.

Ariel Rubinstein studied a setting in which the negotiation is on how to divide $1 between the two players. Each player in turn can offer any partition. The players bear a cost for each round of negotiation. The cost can be presented in two ways:

Rubinstein and Wolinsky studied a market in which there are many players, partitioned into two types (e.g. "buyers" and "sellers"). Pairs of players of different types are brought together randomly, and initiate a sequential-bargaining process over the division of a surplus (as in the Divide the Dollar game). If they reach an agreement, they leave the market; otherwise, they remain in the market and wait for the next match. The steady-state equilibrium in this market it is quite different than competitive equilibrium in standard markets (e.g. Fisher market or Arrow–Debreu market).

Fudenberg and Tirole study sequential bargaining between a buyer and a seller who have incomplete information, i.e., they do not know the valuation of their partner. They focus on a two-turn game (i.e., the seller has exactly two opportunities to sell the item to the buyer). Both players prefer a trade today than the same trade tomorrow. They analyze the Perfect Bayesian equilibrium (PBE) in this game, if the seller's valuation is known, then the PBE is generically unique; but if both valuations are private, then there are multiple PBE. Some surprising findings, that follow from the information transfer and the lack of commitment, are:

Grossman and Perry study sequential bargaining between a buyer and a seller over an item price, where the buyer knows the gains-from-trade but the seller does not. They consider an infinite-turn game with time discounting. They show that, under some weak assumptions, there exists a unique perfect sequential equilibrium, in which:

Nejat Anbarci studied a setting with a finite number of outcomes, where each of the two agents may have a different preference order over the outcomes. The protocol rules disallow repeating the same offer twice. In any such game, there is a unique SPE. It is always Pareto optimal; it is always one of the two Pareto-optimal options of which rankings by the players are the closest. It can be found by finding the smallest integer k for which the sets of k best options of the two players have a non-empty intersection. For example, if the rankings are a>b>c>d and c>b>a>d, then the unique SPE is b (with k=2). If the rankings are a>b>c>d and d>c>b>a, then the SPE is either b or c (with k=3).