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Submodular optimization views on the random assignment problem

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Abstract

We present a non-pricing allocation scheme of divisible goods to agents with utility functions and submodular constraints on goods. The main contribution of the present paper is that through our non-pricing allocation scheme we reveal the close relation between (1) the recent results in the allocation schemes of the random assignment problem and its extensions with ordinal or lexicographic preferences on goods and (2) the monotone algorithms of fair (egalitarian) allocations with separable utility functions and submodular constraints investigated a few decades ago. The underlying submodularity structure plays a crucial rôle, so that the probabilistic serial mechanism of Bogomolnaia and Moulin and other related mechanisms can naturally be extended to problems with submodular constraints.

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Notes

  1. It should, however, be noted that the advantage of the random assignment problem with the probabilistic serial mechanism is that we need only agents’ preferences but not their utility functions which are expensive or hard to identify in practice.

References

  1. Bochet, O., İlkılıç, R., Moulin, H.: Egalitarianism under earmark constraints. J. Econ. Theory 148, 535–562 (2013)

    Article  MathSciNet  Google Scholar 

  2. Bochet, O., İlkılıç, R., Moulin, H., Sethuraman, J.: Balancing supply and demand under bilateral constraints. Theor. Econ. 7, 395–423 (2012)

    Article  MathSciNet  Google Scholar 

  3. Bogomolnaia, A.: Random assignment: redefining the serial rule. J. Econ. Theory 158, 308–318 (2015)

    Article  MathSciNet  Google Scholar 

  4. Bogomolnaia, A., Heo, E.J.: Probabilistic assignment of objects: characterizing the serial rule. J. Econ. Theory 147, 2072–2082 (2012)

    Article  MathSciNet  Google Scholar 

  5. Bogomolnaia, A., Moulin, H.: A new solution to the random assignment problem. J. Econ. Theory 100, 295–328 (2001)

    Article  MathSciNet  Google Scholar 

  6. Bogomolnaia, A., Moulin, H.: Random matching under dichotomous preferences. Econometrica 72, 257–279 (2004)

    Article  MathSciNet  Google Scholar 

  7. Budish, E., Che, Y.-K., Kojima, F., Milgrom, P.: Designing random allocation mechanisms: theory and applications. Am. Econ. Rev. 103, 585–623 (2013)

    Article  Google Scholar 

  8. Chandramouli, S., Sethuraman, J.: Groupstrategyproofness of the egalitarian mechanism for constrained rationing problems. Math. Soc. Sci. 90, 111–118 (2017)

    Article  MathSciNet  Google Scholar 

  9. Che, Y.-K., Kim, J., Mierendorff, K.: Generalized reduced-form auctions: a network-flow approach. Econometrica 81, 2487–2520 (2013)

    Article  MathSciNet  Google Scholar 

  10. Dutta, B., Ray, D.: A concept of egalitarianism under participation constraints. Econometrica 57, 615–635 (1989)

    Article  MathSciNet  Google Scholar 

  11. Edmonds, J.: Submodular functions, matroids, and certain polyhedra. In: Proceedings of the Calgary International Conference on Combinatorial Structures and Their Applications (R. Guy, H. Hanani, N. Sauer and J. Schönheim, eds., Gordon and Breach, New York), pp. 69–87 (1970)

  12. Ekici, Ö., Kesten, O.: An equilibrium analysis of the probabilistic serial mechanism. Int. J. Game Theory 45, 655–674 (2016)

    Article  MathSciNet  Google Scholar 

  13. Flores-Szwagrzak, K.: The replacement principle in networked economies with single-peaked preferences. Soc. Choice Welfare 47, 763–789 (2016)

    Article  MathSciNet  Google Scholar 

  14. Flores-Szwagrzak, K.: Efficient, fair, and strategy-proof (re)allocation under network constraints. Soc. Choice Welfare 48, 109–131 (2017)

    Article  MathSciNet  Google Scholar 

  15. Fujishige, S.: Algorithms for solving the independent-flow problems. J. Oper. Res. Soc. Jpn. 21, 189–204 (1978)

    Article  MathSciNet  Google Scholar 

  16. Fujishige, S.: Lexicographically optimal base of a polymatroid with respect to a weight vector. Math. Oper. Res. 2, 186–196 (1980)

    Article  MathSciNet  Google Scholar 

  17. Fujishige, S.: Submodular Functions and Optimization, 2nd edn. Elsevier, Amsterdam (2005)

    MATH  Google Scholar 

  18. Fujishige, S., Sano, Y., and Zhan, P.: A solution to the random assignment problem with a matroidal family of goods. RIMS Preprint RIMS-1852, Kyoto University, May (2016)

  19. Fujishige, S., Sano, Y., Zhan, P.: An extended probabilistic serial mechanism to the random assignment problem with multi-unit demands and polymatroidal supplies. RIMS Preprint RIMS-1866, Kyoto University, November (2016)

  20. Fujishige, S., Sano, Y., Zhan, P.: The random assignment problem with submodular constraints on goods. ACM Trans. Econ. Comput. 6(1), 28 (2018)

    Article  MathSciNet  Google Scholar 

  21. Gallo, G., Grigoriadis, M.D., Tarjan, R.E.: A fast parametric maximum flow algorithm and applications. SIAM J. Comput. 18, 30–55 (1989)

    Article  MathSciNet  Google Scholar 

  22. Groenevelt, H.: Two algorithms for maximizing a separable concave function over a polymatroid feasible region. Eur. J. Oper. Res. 54, 227–236 (1991)

    Article  Google Scholar 

  23. Hashimoto, T., Hirata, D., Kesten, O., Kurino, M., Ünver, M.U.: Two axiomatic approaches to the probabilistic serial mechanism. Theor. Econ. 9, 253–277 (2014)

    Article  MathSciNet  Google Scholar 

  24. Heo, E.J.: Probabilistic assignment problem with multi-unit demands: a generalization of the serial rule and its characterization. J. Math. Econ. 54, 40–47 (2014)

    Article  MathSciNet  Google Scholar 

  25. Hokari, T.: Monotone-path Dutta-Ray solution on convex games. Soc. Choice Welfare 19, 825–844 (2002)

    Article  MathSciNet  Google Scholar 

  26. Hokari, T., van Gellekom, A.: Population monotonicity and consistency in convex games: some logical relations. Int. J. Game Theory 31, 593–607 (2002)

    Article  MathSciNet  Google Scholar 

  27. Katta, A.-K., Sethuraman, J.: A solution to the random assignment problem on the full preference domain. J. Econ. Theory 131, 231–250 (2006)

    Article  MathSciNet  Google Scholar 

  28. Kojima, F.: Random assignment of multiple indivisible objects. Math. Soc. Sci. 57, 134–142 (2009)

    Article  MathSciNet  Google Scholar 

  29. Kojima, F., Manea, M.: Incentives in the probabilistic serial mechanism. J. Econ. Theory 145, 106–123 (2010)

    Article  MathSciNet  Google Scholar 

  30. Megiddo, N.: Optimal flows in networks with multiple sources and sinks. Math. Program. 7, 97–107 (1974)

    Article  MathSciNet  Google Scholar 

  31. Moulin, H.: Entropy, desegregation, and proportional rationing. J. Econ. Theory 162, 1–20 (2016)

    Article  MathSciNet  Google Scholar 

  32. Moulin, H.: Consistent bilateral assignment. Math. Soc. Sci. 90, 43–55 (2017)

    Article  MathSciNet  Google Scholar 

  33. Moulin, H.: One-dimensional mechanism design. Theor. Econ. 12, 587–619 (2017)

    Article  MathSciNet  Google Scholar 

  34. Moulin, H., Sethuraman, J.: The bipartite rationing problem. Oper. Res. 61, 1087–1100 (2013)

    Article  MathSciNet  Google Scholar 

  35. Murota, K.: Discrete Convex Analysis (SIAM Monographs on Discrete Mathematics and Applications 10, SIAM) (2003)

  36. Murota, K.: Discrete convex analysis: a tool for economics and game theory. J. Mech. Inst. Des. 1, 151–273 (2016)

    Google Scholar 

  37. Murota, K., Shioura, A.: M-convex function on generalized polymatroid. Math. Oper. Res. 24, 95–105 (1999)

    Article  MathSciNet  Google Scholar 

  38. Saban, D., Sethuraman, J.: A note on object allocation under lexicographic preferences. J. Math. Econ. 50, 283–289 (2014)

    Article  MathSciNet  Google Scholar 

  39. Shioura, A., Tamura, A.: Gross substitutes condition and discrete concavity for multi-unit valuations: a survey. J. Oper. Res. Soc. Jpn. 58, 61–103 (2015)

    Article  MathSciNet  Google Scholar 

  40. Schulman, L.J., Vazirani, V.V.: Allocation of divisible goods under lexicographic preferences. In: Leibniz International Proceedings in Informatics (35th IARCS Annual Conf. Foundation of Software Technology and Theoretical Computer Science (FSTTCS)) (Eds., P. Harsha and G. Ramalingam), pp. 543–559 (2015)

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Acknowledgements

We are very grateful to Britta Peis for communications and useful discussions about the subject of this paper, which motivated our present work and improved the presentation. We also appreciate useful comments of an anonymous referee.

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Authors and Affiliations

  1. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan

    Satoru Fujishige

  2. Division of Information Engineering, Faculty of Engineering Information and Systems, University of Tsukuba, Tsukuba, Ibaraki, 305-8573, Japan

    Yoshio Sano

  3. Department of Communication and Business, Edogawa University, Nagareyama, Chiba, 270-0198, Japan

    Ping Zhan

Authors
  1. Satoru Fujishige
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  2. Yoshio Sano
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  3. Ping Zhan
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Corresponding author

Correspondence to Satoru Fujishige.

Additional information

S. Fujishige’s work is supported by JSPS KAKENHI Grant Numbers JP25280004 and JP26280001 and Y. Sano’s work by JSPS KAKENHI Grant Numbers JP15K20885 and JP16H03118.

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Fujishige, S., Sano, Y. & Zhan, P. Submodular optimization views on the random assignment problem. Math. Program. 178, 485–501 (2019). https://doi.org/10.1007/s10107-018-1310-4

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