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When Dividing Mixed Manna Is Easier Than Dividing Goods: Competitive Equilibria with a Constant Number of Chores

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Algorithmic Game Theory (SAGT 2021)

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 12885))

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Abstract

We study markets with mixed manna, where m divisible goods and chores shall be divided among n agents to obtain a competitive equilibrium. Equilibrium allocations are known to satisfy many fairness and efficiency conditions. While a lot of recent work in fair division is restricted to linear utilities, we focus on a substantial generalization to separable piecewise-linear and concave (SPLC) utilities. We first derive polynomial-time algorithms for markets with a constant number of items or a constant number of agents. Our main result is a polynomial-time algorithm for instances with a constant number of chores (as well as any number of goods and agents) under the condition that chores dominate the utility of the agents. Interestingly, this stands in contrast to the case when the goods dominate the agents utility in equilibrium, where the problem is known to be PPAD-hard even without chores.

J. Garg and P. McGlaughlin—Supported by NSF grant CCF-1942321 (CAREER). M. Hoefer and M. Schmalhofer—Supported by DFG grant Ho 3831/5-1, 6-1 and 7-1.

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Notes

  1. 1.

    More precisely, the allocation satisfies an adaptation of proportionality up to one good (PROP1) to mixed manna.

  2. 2.

    While we conjecture that conceptually all our ideas can be applied also when \(u_{ijk} = 0\) is allowed, the analysis of such segments generates a lot of technicalities, which we leave for future work.

  3. 3.

    Any feasible allocation that gives all agents non-negative utility can be seen as CE. We can compute such an allocation when solving the LP (1) to determine the instance type.

  4. 4.

    Alternatively, if the goal is to compute CEEIs for a fair division instance, we can determine the instance type in polynomial time by solving the LP (1) and then assign appropriate budgets \(e_i = 1\) or \(e_i = -1\) for all \(i \in N\).

  5. 5.

    For consistency with previous sections, we assume that \(u_{ij} \ne 0\) throughout. Our arguments can be adapted easily by assuming that when \(u_{ij} = 0\), j is a good for i.

References

  1. Aziz, H., Caragiannis, I., Igarashi, A., Walsh, T.: Fair allocation of indivisible goods and chores. In: Proceedings of the 28th International Joint Conference on Artificial Intelligence (IJCAI) (2019)

    Google Scholar 

  2. Aziz, H., Moulin, H., Sandomirskiy, F.: A polynomial-time algorithm for computing a Pareto optimal and almost proportional allocation. Oper. Res. Lett. 48(5), 573–578 (2020)

    Article  MathSciNet  Google Scholar 

  3. Aziz, H., Rey, S.: Almost group envy-free allocation of indivisible goods and chores. IJCAI (2020)

    Google Scholar 

  4. Azrieli, Y., Shmaya, E.: Rental harmony with roommates. J. Econ. Theory 153, 128–137 (2014)

    Article  MathSciNet  Google Scholar 

  5. Barman, S., Krishnamurthy, S.K.: On the proximity of markets with integral equilibria. In: Proceedings of the 33rd Conference on Artificial Intelligence (AAAI) (2019)

    Google Scholar 

  6. Bhaskar, U., Sricharan, A., Vaish, R.: On approximate envy-freeness for indivisible chores and mixed resources (2020). arxiv:2012.06788

  7. Bogomolnaia, A., Moulin, H., Sandomirskiy, F., Yanovskaia, E.: Competitive division of a mixed manna. Econometrica 85(6), 1847–1871 (2017)

    Article  MathSciNet  Google Scholar 

  8. Bogomolnaia, A., Moulin, H., Sandomirskiy, F., Yanovskaia, E.: Dividing bads under additive utilities. Soc. Choice Welf. 52(3), 395–417 (2018). https://doi.org/10.1007/s00355-018-1157-x

    Article  MathSciNet  MATH  Google Scholar 

  9. Brams, S.J., Taylor, A.D.: Fair Division - From Cake-Cutting to Dispute Resolution. Cambridge University Press, Cambridge (1996)

    Google Scholar 

  10. Brandt, F., Conitzer, V., Endriss, U., Lang, J., Procaccia, A. (eds.): Handbook of Computational Social Choice. Cambridge University Press, Cambridge (2016)

    Google Scholar 

  11. Branzei, S., Sandomirskiy, F.: Algorithms for competitive division of chores (2019). arXiv:1907.01766

  12. Chaudhury, B.R., Garg, J., McGlaughlin, P., Mehta, R.: Dividing bads is harder than dividing goods: on the complexity of fair and efficient division of chores (2020). arxiv:2008.00285

  13. Chaudhury, B.R., Garg, J., McGlaughlin, P., Mehta, R.: Competitive allocation of a mixed manna. In: Proceedings of the 31st Symposium on Discrete Algorithms (SODA) (2021)

    Google Scholar 

  14. Chen, X., Teng, S.: Spending is not easier than trading: on the computational equivalence of fisher and Arrow-Debreu equilibria. In: Proceedings of the 20th International Symposium on Algorithms and Computation (ISAAC), pp. 647–656 (2009)

    Google Scholar 

  15. Devanur, N., Kannan, R.: Market equilibria in polynomial time for fixed number of goods or agents. In: Proceedings of the 49th Symposium on Foundations of Computer Science (FOCS), pp. 45–53 (2008)

    Google Scholar 

  16. Devanur, N., Papadimitriou, C., Saberi, A., Vazirani, V.: Market equilibrium via a primal-dual algorithm for a convex program. J. ACM 55(5), 1–18 (2008)

    Google Scholar 

  17. Garg, J., McGlaughlin, P.: Computing competitive equilibria with mixed manna. In: Proceedings of the 19th Conference on Autonomous Agents and Multi-Agent Systems (AAMAS), pp. 420–428 (2020)

    Google Scholar 

  18. Garg, J., Mehta, R., Sohoni, M., Vazirani, V.V.: A complementary pivot algorithm for market equilibrium under separable, piecewise-linear concave utilities. SIAM J. Comput. 44(6), 1820–1847 (2015)

    Article  MathSciNet  Google Scholar 

  19. Garg, J., Végh, L.A.: A strongly polynomial algorithm for linear exchange markets. In: Proceedings of the 51st Symposium on Theory of Computing (STOC) (2019)

    Google Scholar 

  20. Huang, X., Lu, P.: An algorithmic framework for approximating maximin share allocation of chores (2019). arXiv:1907.04505

  21. McGlaughlin, P., Garg, J.: Improving Nash social welfare approximations. J. Artif. Intell. Res. 68, 225–245 (2020)

    Article  MathSciNet  Google Scholar 

  22. Moulin, H.: Fair Division and Collective Welfare. MIT Press, Cambridge (2003)

    Google Scholar 

  23. Nisan, N., Tardos, É., Roughgarden, T., Vazirani, V. (eds.): Algorithmic Game Theory. Cambridge University Press, Cambridge (2007)

    Google Scholar 

  24. Orlin, J.: Improved algorithms for computing Fisher’s market clearing prices. In: Proceedings of the 42nd Symposium on Theory of Computing (STOC), pp. 291–300 (2010)

    Google Scholar 

  25. Robertson, J., Webb, W.: Cake-Cutting Algorithms: Be Fair If You Can. AK Peters, MA (1998)

    Google Scholar 

  26. Su, F.E.: Rental harmony: sperner’s lemma in fair division. Am. Math. Mon. 106(10), 930–942 (1999)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. University of Illinois at Urbana-Champaign, Urbana-Champaign, USA

    Jugal Garg & Peter McGlaughlin

  2. Goethe University Frankfurt, Frankfurt am Main, Germany

    Martin Hoefer & Marco Schmalhofer

Authors
  1. Jugal Garg
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  2. Martin Hoefer
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  3. Peter McGlaughlin
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  4. Marco Schmalhofer
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Correspondence to Marco Schmalhofer .

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Editors and Affiliations

  1. Aarhus University, Aarhus, Denmark

    Ioannis Caragiannis

  2. Aarhus University, Aarhus, Denmark

    Kristoffer Arnsfelt Hansen

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Garg, J., Hoefer, M., McGlaughlin, P., Schmalhofer, M. (2021). When Dividing Mixed Manna Is Easier Than Dividing Goods: Competitive Equilibria with a Constant Number of Chores. In: Caragiannis, I., Hansen, K.A. (eds) Algorithmic Game Theory. SAGT 2021. Lecture Notes in Computer Science(), vol 12885. Springer, Cham. https://doi.org/10.1007/978-3-030-85947-3_22

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  • DOI: https://doi.org/10.1007/978-3-030-85947-3_22

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