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Nash Analysis

John Nash assets Bargaining theories which might be regarded as undertaking a rigorous and mathematical analysis of social contract negotiation. They show that under certain rationality assumptions, bargaining among rational individuals results in agreements that have interesting structural characteristics that may also result from a certain kind of bargaining procedure. When the characteristics have an ideological flavor, one might say that there is a reason to be ideological: an ideological arrangement is the result of rational bargaining.

The best-known bargaining theory is due to Nash (23) and yields the Nash bargaining solution (which should not be confused with the Nash equilibrium of non-cooperative game theory). Figure 5 illustrates the Nash bargaining solution for two persons. The point represents the utility vector of a default position . This is the state of affairs before bargaining starts, and the state that remains if bargaining fails. The region under the curve represents the feasible set, which is the set of possible agreements consistent with available resources. The point a represents the utility vector after the players arrive at a deal . The Nash bargaining solution selects a deal y that maximizes the area of the rectangle shown.

It means that maximizes the finding . If there are players, the solution selects to maximize .

Figure 5. Nash bargaining solution for two persons.

The finding formula tends to result in near-maximum utility without overly depriving any one player and can therefore be seen as enforcing some kind of fairness. It has enjoyed a degree of acceptance in practical application. The motivation is to obtain near maximum throughput while not excessively discriminating against packets from any one source. The Nash bargaining solution has been defended on both axiomatic and procedural grounds.

One axiomatic argument assumes "cardinal non-comparability", which requires that the ranking of utility vectors be invariant under the transformation , where . Note that the scaling factor can be different for each individual, whereas in unit comparability it is the same. The argument also assume s anonymity and a Pareto condition, as well as " independence of irrelevant alternatives", which is necessary if the product criterion is to make sense. It requires that if a is the Nash bargaining solution for a given feasible set, it remains the solution if the set is reduced without excluding .

It is impressive that a finding criterion could be derived from these axioms, and the proof is quite interesting. However, the premises are again strong. While much attention is focused on the independence axiom, it is rather innocuous in this context.

The strongest premise, and the one that does most of the work in the proof, is the assumption of cardinal non comparability. It leaves room for very little interpersonal comparability, because the ranking must be invariant...

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