Abstract
We present a polynomial-time algorithm that computes an ex-ante envy-free lottery over envy-free up to one item (EF1) deterministic allocations. It has the following advantages over a recently proposed algorithm: it does not rely on the linear programming machinery including separation oracles; it is SD-efficient (both ex-ante and ex-post); and the ex-ante outcome is equivalent to the outcome returned by the well-known probabilistic serial rule. As a result, we answer a question raised by Freeman, Shah, and Vaish (2020) whether the outcome of the probabilistic serial rule can be implemented by ex-post EF1 allocations. In the light of a couple of impossibility results that we prove, our algorithm can be viewed as satisfying a maximal set of properties. Under binary utilities, our algorithm is also ex-ante group-strategyproof and ex-ante Pareto optimal. Finally, we also show that checking whether a given random allocation can be implemented by a lottery over EF1 and Pareto optimal allocations is NP-hard.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
Freeman et al. [15] also presented several other results charting the landscape of possibility and impossibility results when considering fairness and efficiency properties ex post and ex-ante. In particular, they study in detail the rule that maximizes ex-ante Nash welfare. However, they show that the rule cannot be implemented by EF1 allocations.
- 2.
SD-efficiency is also referred to as ordinal efficiency in the literature [8].
- 3.
The statement follows from the well-known Carathéodory’s Theorem.
- 4.
In fact an RB allocation satisfies a stronger properly called strong EF1. Stronger EF1 requires that upon removing the same item from agent i’s bundle, no other agent j envies i, for all i and j. The property was proposed by Conitzer et al. [13].
- 5.
The original EPS algorithm [20] is presented for the case of single-unit demands. However, it can easily be extended to the case of multiple items (see e.g., the Controlled Cake Eating Algorithm (CCEA) algorithm [6]). CCEA is described in the context of cake cutting with piecewise constant valuations. It also applies to allocation of items: each cake segment can be treated as a separate item.
References
Aziz, H.: A probabilistic approach to voting, allocation, matching, and coalition formation. In: Laslier, J.-F., Moulin, H., Sanver, M.R., Zwicker, W.S. (eds.) The Future of Economic Design. SED, pp. 45–50. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-18050-8_8
Aziz, H., Gaspers, S., Mackenzie, S., Walsh, T.: Fair assignment of indivisible objects under ordinal preferences. Artif. Intell. 227, 71–92 (2015)
Aziz, H., Huang, X., Mattei, N., Segal-Halevi, E.: The constrained round robin algorithm for fair and efficient allocation. CoRR abs/1908.00161 (2019), http://arxiv.org/abs/1908.00161
Aziz, H., Mackenzie, S., Xia, L., Ye, C.: Ex post efficiency of random assignments. In: Proceedings of 14th AAMAS Conference, pp. 1639–1640 (2015)
Aziz, H., Walsh, T., Xia, L.: Possible and necessary allocations via sequential mechanisms. In: Proceedings of 24th IJCAI, pp. 468–474 (2015)
Aziz, H., Ye, C.: Cake cutting algorithms for piecewise constant and piecewise uniform valuations. In: Liu, T.-Y., Qi, Q., Ye, Y. (eds.) WINE 2014. LNCS, vol. 8877, pp. 1–14. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-13129-0_1
Birkhoff, G.: Three observations on linear algebra. Univ. Nac. Tacuman Rev. Ser. A 5, 147–151 (1946)
Bogomolnaia, A., Moulin, H.: A new solution to the random assignment problem. J. Econ. Theory 100(2), 295–328 (2001)
Bogomolnaia, A., Moulin, H.: Random matching under dichotomous preferences. Econometrica 72(1), 257–279 (2004)
Budish, E.: The combinatorial assignment problem: approximate competitive equilibrium from equal incomes. J. Polit. Econ. 119(6), 1061–1103 (2011)
Budish, E., Che, Y.K., Kojima, F., Milgrom, P.: Designing random allocation mechanisms: theory and applications. Am. Econ. Rev. 103(2), 585–623 (2013)
Caragiannis, I., Kurokawa, D., Moulin, H., Procaccia, A.D., Shah, N., Wang, J.: The unreasonable fairness of maximum Nash welfare. In: Proceedings of 17th ACM-EC Conference, pp. 305–322 (2016)
Conitzer, V., Freeman, R., Shah, N., Vaughan, J.W.: Group fairness for the allocation of indivisible goods. In: Proceedings of 33rd AAAI Conference (2019)
Devanur, N., Papadimitriou, C.H., Saberi, A., Vazirani, V.: Market equilibrium via a primal-dual algorithm for a convex program. J. ACM 55(5), 1–8 (2008)
Freeman, R., Shah, N., Vaish, R.: Best of both worlds: ex-ante and ex-post fairness in resource allocation. Technical report (2020)
Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1(2), 169–197 (1981). https://doi.org/10.1007/BF02579273
Halpern, D., Procaccia, A.D., Psomas, A., Shah, N.: Fair division with binary valuations: one rule to rule them all. Technical report (2020)
Hopcroft, J.E., Karp, R.M.: An n\({}^{\text{5/2 }}\) algorithm for maximum matchings in bipartite graphs. SIAM J. Comput. 2(4), 225–231 (1973)
Karzanov, A.V.: An exact estimate of an algorithm for finding a maximum flow, applied to the problem on representatives. Probl. Cybern. 5, 66–70 (1973)
Katta, A.K., Sethuraman, J.: A solution to the random assignment problem on the full preference domain. J. Econ. Theory 131(1), 231–250 (2006)
Kojima, F.: Random assignment of multiple indivisible objects. Math. Soc. Sci. 57(1), 134–142 (2009)
Lovász, L., Plummer, M.D.: Matching Theory. AMS Chelsea Publishing, Providence (2009)
Varian, H.R.: Equity, envy, and efficiency. J. Econ. Theory 9, 63–91 (1974)
Vazirani, V.V.: Combinatorial algorithms for market equilibria. In: Nisan, N., Roughgarden, T., Tardos, É., Vazirani, V. (eds.) Algorithmic Game Theory, chap. 5, pp. 103–134. Cambridge University Press, Cambridge (2007)
Acknowledgement
Aziz is supported the Defence Science and Technology (DST) under the project “Auctioning for distributed multi vehicle planning” (DST 9190). He thanks Ethan Brown and Dominik Peters for helpful comments.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Aziz, H. (2020). Simultaneously Achieving Ex-ante and Ex-post Fairness. In: Chen, X., Gravin, N., Hoefer, M., Mehta, R. (eds) Web and Internet Economics. WINE 2020. Lecture Notes in Computer Science(), vol 12495. Springer, Cham. https://doi.org/10.1007/978-3-030-64946-3_24
Download citation
DOI: https://doi.org/10.1007/978-3-030-64946-3_24
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-64945-6
Online ISBN: 978-3-030-64946-3
eBook Packages: Computer ScienceComputer Science (R0)