Maximin share guarantee for goods with positive externalities

Abstract

One of the important yet insufficiently studied subjects in fair allocation is the externality effect among agents. For a resource allocation problem, externalities imply that the share allocated to an agent may affect the utilities of other agents. In this paper, we conduct a study of fair allocation of indivisible goods with positive externalities. Inspired by the models in the context of network diffusion, we present a simple and natural model, namely network externalities, to capture the externalities. To evaluate fairness in the network externalities model, we generalize the idea behind the notion of maximin-share (\(\mathsf {MMS}\)) to achieve a new criterion, namely, extended-maximin-share (\(\mathsf {EMMS}\)). Next, we consider two problems concerning our model. First, we discuss the computational aspects of finding the value of \(\mathsf {EMMS}\) for every agent. For this, we introduce a generalized form of partitioning problem that includes many famous partitioning problems such as maximin, minimax, and leximin. We further show that a 1/2-approximation algorithm exists for this partitioning problem. Next, we investigate approximate \(\mathsf {EMMS}\) allocations, i.e., allocations that guarantee each agent a utility of at least a fraction of his extended-maximin-share. We show that under a natural assumption that the agents are \(\alpha\)-self-reliant, an \(\alpha /2\)-\(\mathsf {EMMS}\) allocation always exists. This, combined with the former result yields a polynomial-time \(\alpha /4\)-\(\mathsf {EMMS}\) allocation algorithm.

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Notes

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    Note that these exchanges are only to prove the lemma, and not in the algorithm.

References

  1. Amanatidis G, Birmpas G, Christodoulou G, Markakis E (2017) Truthful allocation mechanisms without payments: Characterization and implications on fairness. In: Proceedings of the 2017 ACM conference on economics and computation. ACM, pp 545–562

  2. Amanatidis G, Birmpas G, Markakis E (2016) On truthful mechanisms for maximin share allocations. In: Proceedings of the twenty-fifth international joint conference on artificial intelligence. AAAI Press, pp 31–37

  3. Amanatidis G, Markakis E, Nikzad A, Saberi A (2015) Approximation algorithms for computing maximin share allocations. In: International colloquium on automata, languages, and programming. Springer, pp 39–51

  4. Anari N, Ehsani S, Ghodsi M, Haghpanah N, Immorlica N, Mahini H, Mirrokni VS (2010) Equilibrium pricing with positive externalities. In: International workshop on internet and network economics. Springer, pp 424–431

  5. Aziz H, Rauchecker G, Schryen G, Walsh T (2017) Algorithms for max-min share fair allocation of indivisible chores. In: AAAI conference on artificial intelligence, vol 17. AAAI Press, pp 335–341

  6. Barman S, Biswas A, Krishnamurthy SK, Narahari Y (2018) Groupwise maximin fair allocation of indivisible goods. In: Thirty-second AAAI conference on artificial intelligence. AAAI Press, pp 917–924

  7. Barman S, Krishna Murthy SK (2017) Approximation algorithms for maximin fair division. In: Proceedings of the 2017 ACM conference on economics and computation. ACM, pp 647–664

  8. Brânzei S, Michalak T, Rahwan T, Larson K, Jennings NR (2013) Matchings with externalities and attitudes. In: Proceedings of the 2013 international conference on autonomous agents and multi-agent systems. International Foundation for Autonomous Agents and Multiagent Systems, pp 295–302

  9. Brânzei S, Procaccia AD, Zhang J (2013) Externalities in cake cutting. In: International joint conference on artificial intelligence, pp 55–61

  10. Budish E (2011) The combinatorial assignment problem: approximate competitive equilibrium from equal incomes. J Polit Econ 119(6):1061–1103

    Article  Google Scholar 

  11. Cormen TH, Leiserson CE, Rivest RL, Stein C (2009) Introduction to algorithms. MIT Press, Cambridge

    Google Scholar 

  12. Dalton H (1920) The measurement of the inequality of incomes. Econ J 30(119):348–361

    Article  Google Scholar 

  13. Deuermeyer BL, Friesen DK, Langston MA (1982) Scheduling to maximize the minimum processor finish time in a multiprocessor system. SIAM J Algebraic Discrete Methods 3(2):190–196

    Article  Google Scholar 

  14. Farhadi A, Hajiaghayi MT, Ghodsi M, Lahaie S, Pennock D, Seddighin M, Seddighin S, Yami H (2017) Fair allocation of indivisible goods to asymmetric agents. In: Proceedings of the 16th conference on autonomous agents and multiagent systems. International Foundation for Autonomous Agents and Multiagent Systems, pp 1535–1537

  15. Ghodsi M, Hajiaghayi MT, Seddighin M, Seddighin S, Yami H (2018) Fair allocation of indivisible goods: improvements and generalizations. In: Proceedings of the 2018 ACM conference on economics and computation. ACM, pp 539–556

  16. Gourvès L, Monnot J (2017) Approximate maximin share allocations in matroids. In: International conference on algorithms and complexity. Springer, pp 310–321

  17. Graham RL (1969) Bounds on multiprocessing timing anomalies. SIAM J Appl Math 17(2):416–429

    Article  Google Scholar 

  18. Haghpanah N, Immorlica N, Mirrokni V, Munagala K (2011) Optimal auctions with positive network externalities. In: Proceedings of the 12th ACM conference on electronic commerce. ACM, pp 11–20

  19. Hardy GH, Littlewood JE, Pólya G, Littlewood DE, Pólya G et al (1952) Inequalities. Cambridge University Press, Cambridge

    Google Scholar 

  20. Heinen T, Nguyen N-T, Nguyen TT, Rothe J (2018) Approximation and complexity of the optimization and existence problems for maximin share, proportional share, and minimax share allocation of indivisible goods. In: Autonomous agents and multi-agent systems, pp 1–38

  21. Hong M, Park J (2018) Core and top trading cycles in a market with indivisible goods and externalities

  22. Kempe D, Mahdian M (2008) A cascade model for externalities in sponsored search. In: International workshop on internet and network economics. Springer, pp 585–596

  23. Kurokawa D, Procaccia AD, Shah N (2015) Leximin allocations in the real world. In: Proceedings of the sixteenth ACM conference on economics and computation. ACM, pp 345–362

  24. Kurokawa D, Procaccia AD, Wang J (2016) When can the maximin share guarantee be guaranteed? In: AAAI conference on artificial intelligence, vol 16. AAAI Press, pp 523–529

  25. Leme RP, Syrgkanis V, Tardos É (2012) Sequential auctions and externalities. In: Proceedings of the twenty-third annual ACM-SIAM symposium on discrete algorithms. SIAM, pp 869–886

  26. Li M, Zhang J, Zhang Q (2015) Truthful cake cutting mechanisms with externalities: do not make them care for others too much! In: International joint conference on artificial intelligence, pp 589–595

  27. Mirrokni VS, Roch S, Sundararajan M (2012) On fixed-price marketing for goods with positive network externalities. In: WINE. Springer, pp 532–538

  28. Procaccia AD, Wang J (2014) Fair enough: guaranteeing approximate maximin shares. In: Proceedings of the fifteenth ACM conference on economics and computation. ACM, pp 675–692

  29. Steinhaus H (1948) The problem of fair division. Econometrica 16(1):101–104

    Google Scholar 

  30. Suksompong W (2018) Approximate maximin shares for groups of agents. Math Soc Sci 92:40–47

    Article  Google Scholar 

  31. Velez RA (2016) Fairness and externalities. Theor Econ 11(1):381–410

    Article  Google Scholar 

  32. Weymark JA (1981) Generalized Gini inequality indices. Math Soc Sci 1(4):409–430

    Article  Google Scholar 

  33. Woeginger GJ (1997) A polynomial-time approximation scheme for maximizing the minimum machine completion time. Oper Res Lett 20(4):149–154

    Article  Google Scholar 

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Correspondence to Masoud Seddighin.

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Seddighin, M., Saleh, H. & Ghodsi, M. Maximin share guarantee for goods with positive externalities. Soc Choice Welf 56, 291–324 (2021). https://doi.org/10.1007/s00355-020-01278-8

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