We study the probabilistic distribution of identical successive units. We represent the allocation process as the filling of an urn with balls of different colors (one color per agent). Applications include the scheduling of homogeneous tasks among workers and allocating new workers between divisions. The fixed chances methods allocate each unit independently of the current distribution of shares. The Polya-Eggenberger methods place in an urn a fixed number of balls and draw from the urn with replacement of two balls of the color drawn. These two families of urn-filling methods emerge uniquely from our axiomatic discussion involving: a version of the familiar Consistency property; Share Monotonicity (my probability of receiving the next ball is non-decreasing in my current share); Independence of Transfers (transferring balls across agents is not profitable), and Order Independence (a sequence of successive allocations is as likely as any permuted sequence). We also explore the impact of Share Monotinicity (my probability of receiving the next ball is non-increasing in my current share), leading to an equalization of individual shares along a fixed standard of comparison.
