The axiomatic bargaining approach
The axiomatic approach to bargaining has developed alongside the non-cooperative approach, but it attacks the problem in a very different way. Instead of spelling out the rules of a bargaining situation or negotiation process and looking for equilibrium strategies and outcomes of a well-defined game, the axiomatic approach specifies a set of axioms or rules that a bargaining solution should obey, and then derives a formula or bargaining outcome which is logically equivalent to those rules.
In the axiomatic approach, a two-person bargaining problem consists of the following:
- • A set of players (in our example, the factory and the residents);
- • A set S of possible agreements, which the players can jointly implement should they reach agreement; and
- • A unique disagreement point D which is the outcome if the players fail to reach an agreement. The disagreement point here is simply the utilities the factory and the residents would receive if they could not reach an agreement.
Let us again return to the example presented in Chapter 3. In that example the disagreement point depends on the initial assignment of legal or property rights. Assume that both parties have identical quasilinear preferences. Let Q0 = XQ be the level of Q which the factory can produce without having to gain permission from the residents or pay compensation to them. The parameter X e[0,1] here has a natural interpretation: it is an index of the factory's initial security of property rights. If X = 0, then the factory must obtain the residents' permission to produce any units at all. If X = 1, then the factory has the right to produce the maximum Q if it wishes. If 0 < X < 1, then the factory is entitled to produce XQ without obtaining the residents' permission. The disagreement point or threat point for the factory is:
and for the residents it is:
The benefits that they enjoy at this disagreement point are:
The set S of possible agreements here is simply the utility possibilities set: the combination of utilities or benefits such that the amount of money distributed between the parties is equal to the total amount of money available, and such that the production of Q is feasible:
Figure 11.3.1 The set of possible agreements between the factory and the residents
The disagreement point D and the set of possible agreements S constitutes a bargaining situation.
A bargaining solution is simply an algorithm or set of rules that, for each bargaining situation (S, D), picks out a particular combination of utilities or benefits, denoted by b(S, D). For any given bargaining problem (S, D) the goal of bargaining theory is to write down a set of conditions or axioms which would seem 'natural' or 'reasonable' for a solution b(S, D) to satisfy, and then to discover whether these conditions are consistent with the requirement that the solution is unique in all situations. This is the motivation for Nash's axioms, which are loosely stated as follows: 
For the bargaining problem between the factory and the residents, the function defined by:
is the unique bargaining solution that satisfies Nash's axioms. That is, Nash's axioms are logically equivalent to maximising the product of the parties' net benefits. This provides a mathematically convenient solution to bargaining problems in which Nash's axioms are appropriate.
As an example of this result in action, consider the simplest case where the factory and the residents are bargaining over a surplus of p dollars. Each of them starts out with zero dollars. The payoffs are nF and nR, with nF + nR = n. The Nash solution solves:
subject to nF + nR = n. Substituting this into the expression above yields:
The solution is
In other words, the parties split the surplus equally. More generally, if dF is the disagreement point of the factory and dR is the disagreement point of the residents, then the Nash solution solves:
subject to nF + nR = n. The solution is:
Recall that in section 22.214.171.124 we showed that under the non-cooperative approach, as bargaining becomes frictionless, the equilibrium shares
of the parties each converge to nF = nR = . Thus, we have a remarkable feature of the cooperative Nash bargaining solution: it can also be justified within the framework of non-cooperative game theory. This is an important result, because it tells us when the use of the Nash solution is most appropriate. To wit, if the parties are assumed to have identical discount rates and if bargaining frictions are non-existent, the parties will 'split the surplus' in the non-cooperative approach, and so behave as if they are implementing Nash's axioms. Thus, the Nash solution can be justified as the limit of the non-cooperative subgame perfect outcome when there are good reasons to assume that the absolute magnitudes of frictions in the bargaining situation are very small, or small enough to be safely ignored.
-  Scale invariance: Suppose that a bargaining solution is proposed. Ifthere is a linear scaling up of each party's benefits, then the originalbargaining solution should also be scaled up accordingly. • Symmetry: If the parties are perfectly symmetric and have the samedisagreement point and the same payoff possibilities, then theyshould receive the same final payoffs after bargaining. • Independence of irrelevant alternatives (IIA): If the original bargaining solution proposes some outcome which lies within somesmaller set of possible outcomes, then if we were to start from thatsmaller set of possible outcomes, the outcome proposed under thatbargaining solution should remain unchanged. That is, removing'irrelevant' alternatives should not change the bargaining outcome. • Efficiency: The bargaining solution should not leave any gains fromtrade unexploited.