# Allocation rules on networks

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## Abstract

When allocating a resource, geographical and infrastructural constraints have to be taken into account. We study the problem of distributing a resource through a network from sources endowed with the resource to citizens with claims. A link between a source and a citizen depicts the possibility of a transfer from the source to the citizen. Given the endowments at each source, the claims of citizens, and the network, the question is how to allocate the available resources among the citizens. We consider a simple allocation problem that is free of network constraints, where the total amount can be freely distributed. The simple allocation problem is a *claims problem* where the total amount of claims is greater than what is available. We focus on *resource monotonic* and *anonymous* bilateral principles satisfying a regularity condition and extend these principles to allocation rules on networks. We require the extension to preserve the essence of the bilateral principle for each pair of citizens in the network. We call this condition *pairwise robustness* with respect to the bilateral principle. We provide an algorithm and show that each bilateral principle has a unique extension which is *pairwise robust* (Theorem 1). Next, we consider a Rawlsian criteria of distributive justice and show that there is a unique *“Rawls fair”* rule that equals the extension given by the algorithm (Theorem 2). Pairwise robustness and Rawlsian fairness are two sides of the same coin, the former being a pairwise and the latter a global requirement on the allocation given by a rule. We also show as a corollary that any parametric principle can be extended to an allocation rule (Corollary 1). Finally, we give applications of the algorithm for the egalitarian, the proportional, and the contested garment bilateral principles (Example 1).

## Keywords

Allocation Problem Simple Problem Allocation Rule Network Constraint Bipartite Network## Notes

### Acknowledgments

We would like to thank Paula Jaramillo, Herve Moulin, William Thomson, an associate editor, and two anonymous referees for helpful detailed comments on an earlier draft of the paper. We also thank the seminar participants at Pontificia Universidad Javeriana, GAMES 2012, Institute for Economic Analysis (CSIC), First Caribbean Game Theory Conference, Katholieke Universiteit Leuven, University of Tsukuba, Maastricht University, Universidad del Rosario, and Hausdorff Research Institute for Mathematics for valuable discussions. Part of the research was completed when R. İlkılıç and Ç. Kayı were affiliated with Maastricht University. R. İlkılıç acknowledges the support of the European Community via Marie Curie Grant PIEF-GA-2008-220181. Ç. Kayı thanks the Netherlands Organization for Scientific Research (NWO) for its support under grant VIDI-452-06-013 and gratefully acknowledges the hospitality of the Hausdorff Research Institute for Mathematics for inviting as a visiting fellow to Trimester Program on Mechanism Design and Related Topics in 2009.

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