Agent-wise–replication invariance, the Walrasian solution and the uniform rule

We consider classical problems of fair division and generalize the standard requirement on solution mappings known as “replication invariance”. Consider an economy and an allocation chosen by a solution for it. Allowing the replication parameter to vary from agent to agent, we correspondingly adjust the social endowment so that it be possible to assign to each agent and their clone(s) what the agent has been assigned at the allocation under consideration. We require that this list of assignments be chosen by the solution mapping for the resulting economy . We show various ways in which this property is related to other properties that have been discussed in the literature on economic design. We characterize the “Walrasian solution operated from equal division” on the basis of efficiency, the equal-division-lower-bound, and this property, thereby obtaining an analog of the Debreu, Scarf (Int Econ Rev 4:235–246, 1963) convergence theorem in which the no-blocking requirements that define the core are replaced by the equal-division-lower-bound. For the problem of allocating a social endowment of a single commodity among a group of agents with single-peaked preferences (Sprumont in Econometrica 59:509–519, 1991), we characterize the so-called “uniform rule” in a similar way.