# Allocation rules on networks

- 328 Downloads
- 1 Citations

## 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.

## References

- Ambec S, Ehlers L (2008) Sharing a river among satiable agents. Games Econ Behav 64:35–50CrossRefGoogle Scholar
- Ambec S, Sprumont Y (2002) Sharing a river. J Econ Theory 107:453–462CrossRefGoogle Scholar
- Ansink E, Weikard HP (2009) Contested water rights. Eur J Political Econ 25:247–260CrossRefGoogle Scholar
- Ansink E, Weikard HP (2012) Sequential sharing rules for river sharing problems. Soc Choice Welf 38:187–210CrossRefGoogle Scholar
- Aumann RJ, Maschler M (1985) Game theoretic analysis of a bankruptcy problem from the Talmud. J Econ Theory 36:195–213CrossRefGoogle Scholar
- Bjørndal E, Jörnsten K (2010) Flow sharing and bankruptcy games. Int J Game Theory 39:11–28CrossRefGoogle Scholar
- Bochet O, İlkılıç R, Moulin H (2013) Egalitarianism under earmark constraints. J Econ Theory 148:535–562CrossRefGoogle Scholar
- Bochet O, İlkılıç R, Moulin H, Sethuraman J (2012) Balancing supply and demand under bilateral constraints. Theor Econ 7:395–423CrossRefGoogle Scholar
- Branzei R, Ferrari G, Fragnelli V, Tijs S (2008) A flow approach to bankruptcy problems. AUCO Czech Econ Rev 2:146–153Google Scholar
- Brown J (1979) The sharing problem. Oper Res 27:324–340CrossRefGoogle Scholar
- Chun Y (1999) Equivalance of axioms for bankruptcy problems. Int J Game Theory 28:511–520CrossRefGoogle Scholar
- Dagan N, Volij O (1997) Bilateral comparisons and consistent fair division rules in the context of bankruptcy problems. Int J Game Theory 26:11–25CrossRefGoogle Scholar
- Hall NG, Vohra R (1993) Towards equitable distribution via proportional equity constraints. Math Program 58:287–294CrossRefGoogle Scholar
- Hoekstra A (2006) The global dimension of water governance: nine reasons for global arrangements in order to cope with local problems. Value of Water Research Report Series 20. UNESCO-IHE Institute for Water EducationGoogle Scholar
- Hokari T, Thomson W (2008) On the properties of division rules lifted by bilateral consistency. J Math Econ 44:211–231CrossRefGoogle Scholar
- İlkılıç R (2011) Networks of common property resources. Econ Theory 47:105–134CrossRefGoogle Scholar
- Kar A, Kıbrıs O (2008) Allocating multiple estates among agents with single-peaked preferences. Soc Choice Welf 31:641–666CrossRefGoogle Scholar
- Kıbrıs O, Küçükşenel S (2009) Uniform trade rules for uncleared markets. Soc Choice Welf 32:101–121CrossRefGoogle Scholar
- Klaus B, Peters H, Storcken T (1997) Reallocation of an infinitely divisible good. Econ Theory 10:305–333CrossRefGoogle Scholar
- Klaus B, Peters H, Storcken T (1998) Strategy-proof division with single-peaked preferences and individual endowments. Soc Choice Welf 15:297–311CrossRefGoogle Scholar
- Megiddo N (1974) Optimal flows in networks with multiple sources and sinks. Math Program 7:97–107CrossRefGoogle Scholar
- Megiddo N (1977) A good algorithm for lexicographically optimal flows in multi-terminal networks. Bull Am Math Soc 83:407–409CrossRefGoogle Scholar
- Moulin H (1999) Rationing a commodity along fixed paths. J of Econ Theory 84:41–72CrossRefGoogle Scholar
- Moulin H, Sethuraman J (2013) The bipartite rationing problem. Oper Res 61:1087–1100CrossRefGoogle Scholar
- O’Neill B (1982) A problem of rights arbitration from the Talmud. Math Soc Sci 2:345–371CrossRefGoogle Scholar
- Özdamar O, Ekinci E, Küçükyazıcı B (2004) Emergency logistics planning in natural disasters. Ann Oper Res 129:217–245CrossRefGoogle Scholar
- Rawls J (1971) A theory of justice. Harvard University Press, CambridgeGoogle Scholar
- Sprumont Y (1991) The division problem with single-peaked preferences: a characterization of the uniform allocation rule. Econometrica 59:509–519CrossRefGoogle Scholar
- Szwagrzak KF (2011) The replacement principle in networked economies with single-peaked preferences. mimeo. University of Southern Denmark, OdenseGoogle Scholar
- Thomson W (2003) Axiomatic analysis of bankruptcy and taxation problems: a survey. Math Soc Sci 45:249–297CrossRefGoogle Scholar
- Thomson W (2006) How to divide when there isnt enough: from the Talmud to game theory. mimeo. University of Rochester, RochesterGoogle Scholar
- Young HP (1987) On dividing an amount according to individual claims or liabilities. Math Oper Res 12:398–414CrossRefGoogle Scholar