The Bargaining Problem PDF - Nash 1950

# The Bargaining Problem - **Author:** John F. Nash, Jr. - **Publication:** Econometrica, Volume 18, Issue 2 (Apr., 1950), 155-162. - **Stable URL:** http://links.jstor.org/sici?sici=0012-9682%28195004%2918%3A2%3C155%3ATBP%3E2.0.CO%3B2-H ## Introduction This paper presents a new treatment of the...

# The Bargaining Problem - **Author:** John F. Nash, Jr. - **Publication:** Econometrica, Volume 18, Issue 2 (Apr., 1950), 155-162. - **Stable URL:** http://links.jstor.org/sici?sici=0012-9682%28195004%2918%3A2%3C155%3ATBP%3E2.0.CO%3B2-H ## Introduction This paper presents a new treatment of the classical economic problem of bargaining, bilateral monopoly, etc., which can be regarded as a nonzero-sum two-person game. The problem arises from situations where two individuals can collaborate for mutual benefit in more than one way. However, each individual's well-being is unaffected by the other's actions, unless consent is obtained. This paper explores different ways "solutions" can be defined, specifically for situations like: - Monopoly versus Monopsony - State trading between nations - Negotiations between employers and labor unions ## The "Solution" And Its Applications The goal is to determine the amount of satisfaction each individual should expect to gain from the situation, or how much each person would benefit if they have an opportunity to bargain. The paper suggests that a "solution" involves the identification of the situation with a nonzero sum two-person game. ### Theoretical Approach The paper applies the theory of games developed by John von Neumann and Oskar Morgenstern in *Theory of Games and Economic Behavior,* which uses numerical utility to represent preferences and tastes. This model is based on the idea that each individual seeks to maximize their gains. In order to apply this utility model, the concept of "anticipation" for each individual is introduced. Anticipation refers to the expectation of future contingencies, which can range from certainties to probabilities. The following assumptions are made about this anticipation: * Each individual can determine the preference order of any two anticipations. * This order is transitive. * Probability combinations of equally desirable states are just as desirable as either. * Any probability combination of equally desirable states is just as desirable. * There is a probability combination of A and C which is just as desirable as C, assuming A, B, and C is ordered as in assumption (2). * If A and B have equal desirability, then, for any p between 0 and 1, pA + (1-p)C and pB + (1-p)C are equally desirable. By proving the existence of a satisfactory utility function, the paper develops a model for two-person games: * **Two Person Theory:** The situation involves two individuals, each with a certain expectation of their future environment. The utility functions are applied to the two-person anticipation, each giving the result for the corresponding individual. * **The Definition of a Two-Person Anticipation:** It's a combination of two one-person anticipations. ## Application of the Theory The paper outlines the conditions for the "solution" to be determined: * **The "Solution" Point:** This point is a set where the utility functions (u1 and u2) for both individuals are maximized and corresponds to a fair bargain. * **Assumptions and Properties:** * The set of points (S), containing the "solution" point, must be compact and convex. * If a point (a) exists in S such that another point (β) exists in S with u1(β) > u1(a) and u2(β) > u2(a), then a ≠ c(S). * If T contains S and c(T) is in S, then c(T) = c(S). * If S is symmetric and u1 and u2 display this symmetry, then the solution point (c(S)) is on the line of equality, where u1 = u2. The main idea here is that the solution point (c(S)) should be the point where the product of the two utilities is maximized. It corresponds to an agreement where both individuals can reach a fair bargain, and the agreement is the most desirable outcome. ## Examples The paper provides an example of applying the theory to a situation where two individuals, Bill and Jack, can trade goods without money. The theory also accounts for situations with money involved, where the utility of money serves as a satisfactory approximation to a good's utility. **Conclusion:** This theory provides a new approach to understanding bargaining situations where the individual seeks to maximize their gains. The theoretical framework and practical applications demonstrate a method for determining a "fair bargain" within a defined set of possibilities.