It was suggested by Ehud Kalai and Meir Smorodinsky,[1] as an alternative to Nash's bargaining solution suggested 25 years earlier.
The main difference between the two solutions is that the Nash solution satisfies independence of irrelevant alternatives, while the KS solution instead satisfies resource monotonicity.
A two-person bargain problem consists of a pair
: It is assumed that the problem is nontrivial, i.e., the agreements in
and returns a point in its feasible agreements set,
The Nash and KS solutions both agree on the following three requirements: Pareto optimality is a necessary condition.
For every bargaining problem, the returned agreement
Invariant to positive affine transformations also seems like a necessary condition: if the utility function of one or more players is transformed by a linear function, then the agreement should also be transformed by the same linear function.
This makes sense if we assume that the utility functions are only representations of a preference relation, and do not have a real numeric meaning.
In addition to these requirements, Nash requires Independence of irrelevant alternatives (IIA).
Kalai and Smorodinsky differ from Nash on this issue.
They claim that the entire set of alternatives must affect the agreement reached.
In the above example, suppose the preference relation of player 2 is: C>>B>A (C is much better than B, which is somewhat better than A) while the preference relation of 1 is reversed: A>>B>>C.
Therefore, KS remove the IIA requirement.
This requirement says that, for each player, if the utility attainable by this player for each utility of the other player is weakly larger, then the utility this player gets in the selected agreement should also be weakly larger.
In other words, a player with better options should get a weakly-better agreement.
are two bargaining problems such that: Then, the solution f must satisfy: In the words of KS: By symmetry, the same requirement holds if we switch the roles of players 1 and 2.
The KS solution can be calculated geometrically in the following way.
The KS solution is the top-right point of this interval.
Mathematically, the KS solution is the maximal point which maintains the ratios of gains.
, such that: Alice and George have to choose between three options, that give them the following amounts of money:.
[2]: 88–92 Assume for the purposes of the example that utility is linear in money, and that money cannot be transferred from one party to the other.
They can also mix these options in arbitrary fractions.
of feasible agreements is the convex hull of a(60,80) and b(50,110) and c(30,150).
The disagreement point is defined as the point of minimal utility: this is 30 for Alice and 80 for George, so d=(30,80).
For both Nash and KS solutions, we have to normalize the agents' utilities by subtracting the disagreement values, since we are only interested in the gains that the players can receive above this disagreement point.
Hence, the normalized values are: The Nash bargaining solution maximizes the product of normalized utilities: The maximum is attained when
The KS bargaining solution equalizes the relative gains - the gain of each player relative to its maximum possible gain - and maximizes this equal value: Here, the maximum is attained when
Note that both solutions are Pareto-superior to the "random-dictatorial" solution - the solution that selects a dictator at random and lets him/her selects his/her best option.
, which gives a utility-gain of only $15 to Alice and $35 to George.