Kalai–Smorodinsky bargaining solution

Setting

A two-person bargain problem consists of a pair ( F , d ) :

It is assumed that the problem is nontrivial, i.e, the agreements in F are better for both parties than the disagreement.

A bargaining solution is a function f that takes a bargaining problem ( F , d ) and returns a point in its feasible agreements set, f ( F , d ) ∈ F .

Requirements from bargaining solutions

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 f ( F , d ) must be Pareto-efficient.

Symmetry is also necessary. The names of the players should not matter: if player 1 and player 2 switch their utilities, then the agreement should be switched accordingly.

Invariant to 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 (IIR). This means that, if the set of possible agreements grows (more agreements become possible), but the bargaining solution picks an agreement that was contained in the smaller set, then this agreement must be the same as the agreement reached when only the smaller set was available, since the new agreements are irrelevant. For example, suppose that in Sunday we can agree on option A or option B, and we pick option A. Then, in Monday we can agree on option A or B or C, but we do not pick option C. Then, Nash says that we must pick option A. The new option C is irrelevant since we do not select it anyway.

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. The fact that option C becomes available allows player 2 to say: "if I give up my best option - C, I have a right to demand that at least my second-best option will be chosen".

Therefore, KS remove the IIR requirement. Instead, they add a monotonicity 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.

The formal definition of monotonicity is based on the following definitions.

The monotonicity requirement says that, if ( F , d ) and ( F ′ , d ) 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

The KS solution can be calculated geometrically in the following way.

Let b ( F ) be the point of best utilities ( B e s t 1 ( F ) , B e s t 2 ( F ) ) . Draw a line L from d (the point of disagreement) to b (the point of best utilities).

By the non-triviality assumption, the line L has a positive slope. By the convexity of F , the intersection of L with the set F is an interval. 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. I.e, it is a point μ on the Pareto frontier of F , such that:

Examples

Alice and George are businessmen and they have to choose between three options, that give them the following monetary revenues:

They can also mix these options in arbitrary fractions. E.g, they can choose option a for a fraction x of the time, option b for fraction y, and option c for fraction z, such that: x + y + z = 1 . Hence, the set F 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 x = 0 and y = 7 / 8 and z = 1 / 8 (i.e, option b is used 87.5% of the time and option c is used in the remaining time). The utility-gain of Alice is $17.5 and of George - $35.

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 x = 0 and y = 21 / 26 and z = 5 / 26 . The utility-gain of Alice is $16.1 and of George - $37.7.

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. This solution is equivalent to letting x = 1 / 2 and y = 0 and z = 1 / 2 , which gives a utility of only $15 to Alice and $35 to George.