Economists are interested in bargaining not merely because many transactions are negotiated (as opposed to being entirely determined by market forces) but also because, conceptually, bargaining is precisely the opposite of the idealized "perfect competition" among infinitely many traders, in terms of which economists often think about markets. Bargaining situations concern as few as two individuals, who may try to reach agreement on any of a range of transactions which leave them both at least as well off as they could be if they reached no agreement. As early as Edgeworth (1881), it was noted that modeling traders only by their initial endowments and indifference curves, while often adequate to determine a unique competitive equilibrium in a market, would nevertheless leave indeterminate the outcome of bargaining, although it could determine a "contract curve" in which the outcomes of successful bargaining might be found.
With the advent of game theory, attempts were made to develop theories of bargaining which would predict particular outcomes in the contract curve. John Nash (1950, 1953) initiated two related, influential approaches. In his 1950 paper he proposed a model which predicted an outcome of bargaining based only on information about each bargainer's preferences, as modeled by an expected utility function over the set of feasible agreements and the outcome which would result in case of disagreement. In his 1953 paper Nash considered a simple model of the strategic choices facing bargainers, and argued that one of the strategic equilibria of this game, which corresponded to the outcome identified in his 1950 paper, was particularly robust.
Nash's approach of analyzing bargaining with complementary models-- abstract models which focus on outcomes, in the spirit of "cooperative" game theory, and more detailed strategic models, in the spirit of "non-cooperative" game theory--has influenced much of game theory. Modern contributions to this tradition include influential work on bargaining by Ariel Rubinstein and Ken Binmore.
One direction this work has taken has been to connect bargaining theory with the theory of competitive equilibrium in markets, by examining market models in which agents meet and negotiate transactions, with the option of returning to the market in case of disagreement. (See e.g. Binmore and Dasgupta 1987; Osborne and Rubinstein 1990.)
One shortcoming of the classical game theoretic models of bargaining was that they provided little help in understanding disagreements, except to suggest that disagreements resulted primarily from bargainers' mistakes. Incomplete information models help to remedy this, by showing how a positive probablility of disagreement may be inescapable at equilibrium, when agents do not know how other agents value all transactions. The underlying intuition is that if you reach an agreement whenever there are gains from trade, then you are not making as much profit on each agreement as you could (see e.g. the papers in Roth, 1985).
Because many game theoretic models of bargaining depend on information difficult to observe in the field (e.g. bargainers'detailed information and preferences over alternatives), these models were long resistant to all but the most indirect empirical tests. However with the growth of experimental economics, many laboratory experiments were designed to test the predictions of these theories. Although some of their qualitative predictions have received some support, the existing models have performed poorly as point predictors. Most recently, this has led to new (or sometimes renewed) interest in different kinds of theories, having to do with coordination and/or learning and adaptive behavior. (For an account of experimental work see Roth, 1995).
Binmore, Ken and Partha Dasgupta, editors , The Economics of Bargaining, Blackwell, Oxford.
Edgeworth, F.Y. , Mathematical Psychics, London, Kegan Paul.
Nash, John , "The Bargaining Problem," Econometrica, 18, 155- 162.
Nash, John , "Two-Person Cooperative Games," Econometrica, 21, 128-40.
Osborne, Martin J and Ariel Rubinstein , Bargaining and Markets, Academic Press, San Diego.
Roth, Alvin E., editor  Game-Theoretic Models of Bargaining, Cambridge.
Roth, Alvin E.  "Bargaining Experiments," Handbook of Experimental Economics, John Kagel and Alvin E. Roth, editors, Princeton University Press, 253-348.