The Number of Outcomes in the Pareto-Optimal Set of Discrete Bargaining Games

A set of n players is to bargain over which one of m possible outcomes will be chosen. The preference of outcome i to player j is assumed to be a random variable with all preference rankings equally likely and without ties. This would arise in case the payoffs are chosen as independent samplings from a continuous distribution F_j. The mean and distribution of the number of outcomes in the Pareto-optimal set are calculated for finite m. As m → ∞ the mean is asymptotic to (log m + 0.577)ⁿ⁻¹/(n − 1)! and for n = 2, m → ∞, the distribution approaches the normal distribution. The results are also applied to a problem in multiattribute utility theory. Suppose we wish to select an individual with high values on two personal attributes that are independent and continuously measurable. Of a world population of four billion, the efficient set would have an expectation of 22.7 individuals.