Price of fairness

In the theory of fair division, the price of fairness (POF) is the ratio of the largest economic welfare attainable by a division to the economic welfare attained by a fair division. The POF is a quantitative measure of the loss of welfare that society has to take in order to guarantee fairness.

In general, the POF is defined by the following formula:$${\displaystyle POF={\frac {\max _{D\in {\text{Divisions}}}{(\mathrm {welfare} (D))}}{\max _{D\in {\text{Fair Divisions}}}{(\mathrm {welfare} (D))}}}}$$

The exact price varies greatly based on the kind of division, the kind of fairness and the kind of social welfare we are interested in.

The most well-studied type of social welfare is utilitarian social welfare, defined as the sum of the (normalized) utilities of all agents. Another type is egalitarian social welfare, defined as the minimum (normalized) utility per agent.

In this example we focus on the utilitarian price of proportionality, or UPOP.

Consider a heterogeneous land-estate that has to be divided among 100 partners, all of whom value it as 100 (or the value is normalized to 100). First, let's look at some extreme cases.

The extreme cases described above already give us a trivial upper bound: UPOP ≤ 10000/100 = 100. But we can get a tighter upper bound.

Assume that we have an efficient division of a land-estate to 100 partners, with a utilitarian welfare U. We want to convert it to a proportional division. To do this, we group the partners according to their current value: