Dividing profits three ways: Exactness vs. consensuality

A rule for 3-way division of profits based on peer evaluation reports is impartial if the calculation of each partner’s share ignores her report, exact if it never allocates more or less than the profit to be shared, and consensual if it respects evaluations when the partners’ reports are in agreement. We use a calculus of variations technique to solve exactly a simple version of the problem of finding a 3-way exact, impartial division function of minimal average deviation from consensuality. We also give approximate solutions to three versions of the problem using a numerical analysis technique that is widely applicable. ► A 3-partner division rule cannot be exact, impartial and consensual. ► We solve a simple version of finding a 3-way exact, impartial division function of minimal average deviation from consensuality. ► We give approximate solutions to three versions of that problem using a numerical analysis technique.