Bivariate Uniqueness in the Logistic Recursive Distributional Equation [Technical Report # 629, Department of Statistics, UC, Berkeley. (November 2002)]

Abstract : In this work we prove the bivariate uniqueness property of the logistic recursive distributional equation, which arise in the study of the random assignment problem, as described by Aldous (2001). Using this and the genral framework of Aldous and Bandyopadhyay (2005), we then conclude that the associated recursive tree process is endogenous. Thus the logistic variables defined by Aldous (2001) for finding the limiting constant for the optimal cost in random assignment problem turns out to be measurable with respect the edge weights. This then answers the question raised by Aldous (2001). The method involves construction of an explicit recursion to show that the associated integral equation has unique solution.