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.