Nash solution

The cooperative Nash solution is the decisive contribution John Nash for the solution of problems of bargaining theory ( Bargaining problem ). In his essay The bargaining problem , published in 1950 , he succeeded for the first time in mathematically deriving a clear solution for this type of negotiation situation.

Problem and solution

Nash modeled a bargaining problem in the form of a utility space U and a result vector k, which assigns each actor its conflict settlement, i.e. the benefit that the respective actor receives in the event that the negotiation is broken off.

His goal was to base this solution on as few generally accepted axioms as possible (see properties of negotiated solutions ). These should ensure that rational actors had to agree. These axioms in detail are:

weak Pareto optimality ,
Symmetry ,
Independence from linear affine transformations and
Independence from irrelevant alternatives.

On the basis of these conditions, Nash was able to show that a clear result (in the form of a result vector that assigns a specific benefit to each individual actor) can be calculated.

example

The Nash solution is intended to maximize the so-called Nash product , the product of the increase in utility of the negotiating parties or players. In a negotiating situation for two players, the function takes shape . This could involve an amount of € 100 that the players should split among themselves (the threat point is (0.0) for simplification). The players have different utility functions, such as and . The resulting optimization problem looks like this: ${\ displaystyle (B, d)}$ ${\ displaystyle f \ colon \, B \ rightarrow {\ mathbb {R}}, \, (x_ {1}, x_ {2}) \ mapsto (x_ {1} -d_ {1}) \ cdot (x_ { 2} -d_ {2})}$ ${\ displaystyle U_ {1} (x_ {1}) = 20 \ cdot x_ {1}}$ ${\ displaystyle U_ {2} (x_ {2}) = 5 \ cdot x_ {2}}$

{\ displaystyle \ operatorname {max} \; (U_ {1} (x_ {1}) - U_ {1} (0)) (U_ {2} (x_ {2}) - U_ {2} (0)) }

under the condition

{\ displaystyle x_ {1} + x_ {2} = 100}

and can be solved with the help of Lagrange multipliers .

{\ displaystyle {\ begin {aligned} {\ mathcal {L}} & = (20x_ {1}) (5x_ {2}) + \ lambda (x_ {1} + x_ {2} -100) \\ {\ mathcal {L}} _ {\ lambda} & = (x_ {1} + x_ {2} -100) = 0 \\ {\ mathcal {L}} _ {x_ {1}} & = 100x_ {2} + \ lambda = 0 \\ {\ mathcal {L}} _ {x_ {2}} & = 100x_ {1} + \ lambda = 0 \\\ end {aligned}}}

Which leads to an equally divided amount of , whereby the resulting utility values of the players differ: and . ${\ displaystyle x_ {1} = x_ {2} = 50}$ ${\ displaystyle U_ {1} (50) = 1000}$ ${\ displaystyle U_ {2} (50) = 250}$

Individual evidence

^ Berninghaus, SK, KM Erhart, and W. Güth. "Strategic Games: An Introduction to Game Theory, 2nd, revised and expanded" Ed., Berlin et al. O (2010). S.

literature

John Forbes Nash Jr .: The bargaining problem (PDF; 248 kB), Econometrica 18, 1950, pp. 155–162.

[1] Berninghaus, SK, KM Erhart, and W. Güth. "Strategic Games: An Introduction to Game Theory, 2nd, revised and expanded" Ed., Berlin et al. O (2010). S.