Question

prove by constructing an example that the Kalai-Smorodinsky solution (Kalai and Smorodinsky,1975) violates the strong monotonicity axiom.

Answer 1

Alice and George are businessmen and they have to choose between three options, that give them the following monetary revenues:^[2]:88–92

	a	b	c
Alice	$60	$50	$30
George	$80	$110	$150

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: {\displaystyle x+y+z=1} $x+y+z=1$ . Hence, the set {\displaystyle 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:

	a	b	c
Alice	$30	$20	$0
George	$0	$30	$70

The Nash bargaining solution maximizes the product of normalized utilities:

{\displaystyle \max log(30x+20y)\cdot log(30y+70z)} ${\displaystyle \max log(30x+20y)\cdot log(30y+70z)}$

The maximum is attained when {\displaystyle x=0} $x=0$ and {\displaystyle y=7/8} ${\displaystyle y=7/8}$ and {\displaystyle z=1/8} ${\displaystyle z=1/8}$ (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:

{\displaystyle \max {30x+20y \over 30}={30y+70z \over 70}} ${\displaystyle \max {30x+20y \over 30}={30y+70z \over 70}}$

Here, the maximum is attained when {\displaystyle x=0} $x=0$ and {\displaystyle y=21/26} ${\displaystyle y=21/26}$ and {\displaystyle z=5/26} ${\displaystyle z=5/26}$ . 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 {\displaystyle x=1/2} $x=1/2$ and {\displaystyle y=0} $y=0$ and {\displaystyle z=1/2} ${\displaystyle z=1/2}$ , which gives a utility of only $15 to Alice and $35 to George.