Envy-Free Theory

Envy-free theory addresses the dilemma of allocating scarce resources through the use of complex mathematical formulae so that each individual believes that his or her share is equal to or better than that of anyone else who takes part in this resource sharing. The purpose of devising these mathematical formulae is to reduce or eliminate the envy that one individual may feel against another after the resource in question is allocated.

Over the past 50 years, mathematicians have developed several theories on how to fairly distribute scarce resources and have used the cutting of a cake to represent these devised formulae. Curiously, the application of envy-free theory dates back to more than 2,800 years ago. In his literary work Theogeny, Hesiod cited an example in which Zeus and Prometheus killed an ox, which had to be divided between them. Prometheus cut the ox in half but hid the tender portion under the hide of the ox to make it appear unappealing, with the result that Zeus selected the inferior, bony portion, which enraged him. This unsuccessful “cut and choose” method led to more arduous attempts to create quantitatively based, envyfree theories so that when one party cut the symbolic meat in half and the other party chose his or her half, they both would be satisfied. However, it was not until the 1940s that mathematicians such as Hugo Stenhaus questioned whether an envy-free theory could be developed for application to more than two persons.

Following Stenhaus, several mathematical methods were devised to create an envy-free division of resources for more than two individuals. One of these envy-free theorists, William Webb, combined his theory with several others.

In Webb's method, three pieces of a rectangular cake are distributed. The first person places his knife at the left edge of the cake and is instructed by the second person when to cut it, as the knife is moved to the right. The second person now believes that this cut piece is equal to one third of the entire cake and it is his or her turn to cut the cake. The second person is then told by the first person when to cut the slice. At this point, the first and second persons believe that all cut slices are equal. Now the pieces of cake are to be distributed and are selected in order: first by the third person, next by the second person, and last by the first person.

The distribution of the cake is now envy free because all three persons believe that their choice is equal to or better than the choices of the others. This method is actually mirrored in John Rawls's “difference theory,” which supports any action if the least advantaged party believes that he or she is in a better situation than before. Some of the situations where an envy-free theory may be applied include heirs inheriting an estate, employees splitting a list of duties, parties in a divorce mediation, or students renting a house together.