Alea Iacta Est: Auctions, Persuasion, Interim Rules, and Dice

Authors Shaddin Dughmi, David Kempe, Ruixin Qiang

Thumbnail PDF


  • Filesize: 0.64 MB
  • 20 pages

Document Identifiers

Author Details

Shaddin Dughmi
  • University of Southern California, Los Angeles, CA, USA
David Kempe
  • University of Southern California, Los Angeles, CA, USA
Ruixin Qiang
  • University of Southern California, Los Angeles, CA, USA

Cite AsGet BibTex

Shaddin Dughmi, David Kempe, and Ruixin Qiang. Alea Iacta Est: Auctions, Persuasion, Interim Rules, and Dice. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 31:1-31:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


To select a subset of samples or "winners" from a population of candidates, order sampling [Rosén, 1997] and the k-unit Myerson auction [Myerson, 1981] share a common scheme: assign a (random) score to each candidate, then select the k candidates with the highest scores. We study a generalization of both order sampling and Myerson's allocation rule, called winner-selecting dice. The setting for winner-selecting dice is similar to auctions with feasibility constraints: candidates have random types drawn from independent prior distributions, and the winner set must be feasible subject to certain constraints. Dice (distributions over scores) are assigned to each type, and winners are selected to maximize the sum of the dice rolls, subject to the feasibility constraints. We examine the existence of winner-selecting dice that implement prescribed probabilities of winning (i.e., an interim rule) for all types. Our first result shows that when the feasibility constraint is a matroid, then for any feasible interim rule, there always exist winner-selecting dice that implement it. Unfortunately, our proof does not yield an efficient algorithm for constructing the dice. In the special case of a 1-uniform matroid, i.e., only one winner can be selected, we give an efficient algorithm that constructs winner-selecting dice for any feasible interim rule. Furthermore, when the types of the candidates are drawn in an i.i.d. manner and the interim rule is symmetric across candidates, unsurprisingly, an algorithm can efficiently construct symmetric dice that only depend on the type but not the identity of the candidate. One may ask whether we can extend our result to "second-order" interim rules, which not only specify the winning probability of a type, but also the winning probability conditioning on each other candidate's type. We show that our result does not extend, by exhibiting an instance of Bayesian persuasion whose optimal scheme is equivalent to a second-order interim rule, but which does not admit any dice-based implementation.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algorithmic game theory
  • Interim rule
  • order sampling
  • virtual value function
  • Border's theorem


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Nibia Aires, Johan Jonasson, and Olle Nerman. Order sampling design with prescribed inclusion probabilities. Scandinavian journal of statistics, 29(1):183-187, 2002. Google Scholar
  2. Saeed Alaei, Hu Fu, Nima Haghpanah, Jason D. Hartline, and Azarakhsh Malekian. Bayesian optimal auctions via multi-to single-agent reduction. In Proc. 13th ACM Conf. on Electronic Commerce, page 17, 2012. Google Scholar
  3. Kim C. Border. Implementation of Reduced Form Auctions: A Geometric Approach. Econometrica, 59(4):1175-1187, 1991. Google Scholar
  4. Kim C. Border. Reduced form auctions revisited. Economic Theory, 31(1):167-181, 2007. Google Scholar
  5. Yang Cai, Constantinos Daskalakis, and S. Matthew Weinberg. An algorithmic characterization of multi-dimensional mechanisms. In Proc. 44th ACM Symp. on Theory of Computing, pages 459-478, 2012. Google Scholar
  6. Yang Cai, Constantinos Daskalakis, and S. Matthew Weinberg. Optimal multi-dimensional mechanism design: Reducing revenue to welfare maximization. In Proc. 53rd IEEE Symp. on Foundations of Computer Science, pages 130-139, 2012. Google Scholar
  7. Shaddin Dughmi. Algorithmic information structure design: a survey. ACM SIGecom Exchanges, 15(2):2-24, 2017. Google Scholar
  8. Shaddin Dughmi and Haifeng Xu. Algorithmic Bayesian Persuasion. In Proc. 48th ACM Symp. on Theory of Computing, pages 412-425, 2016. Google Scholar
  9. Parikshit Gopalan, Noam Nisan, and Tim Roughgarden. Public projects, Boolean functions and the borders of Border’s Theorem. In Proc. 16th ACM Conf. on Economics and Computation, page 395, 2015. Google Scholar
  10. Martin Grötschel, László Lovász, and Alexander Schrijver. Geometric algorithms and combinatorial optimization, volume 2. Springer Science &Business Media, 2012. Google Scholar
  11. Jason D. Hartline. Mechanism design and approximation. Now Publishers, 2013. Google Scholar
  12. Emir Kamenica and Matthew Gentzkow. Bayesian persuasion. American Economic Review, 101(6):2590-2615, 2011. Google Scholar
  13. Jean-Bernard Lasserre. Moments, positive polynomials and their applications, volume 1. World Scientific, 2010. Google Scholar
  14. Eric Maskin and John Riley. Optimal auctions with risk averse buyers. Econometrica, 52(6):1473-1518, 1984. Google Scholar
  15. Steven A. Matthews. On the implementability of reduced form auctions. Econometrica, 52(6):1519-1522, 1984. Google Scholar
  16. Konrad Mierendorff. Asymmetric reduced form auctions. Economics Letters, 110(1):41-44, 2011. Google Scholar
  17. Roger B. Myerson. Optimal Auction Design. Mathematics of Operations Research, 6(1):58-73, 1981. Google Scholar
  18. James G Oxley. Matroid theory, volume 3. Oxford University Press, USA, 2006. Google Scholar
  19. Mallesh M. Pai and Rakesh Vohra. Optimal auctions with financially constrained buyers. Journal of Economic Theory, 150:383-425, 2014. Google Scholar
  20. Bengt Rosén. Asymptotic theory for order sampling. Journal of Statistical Planning and Inference, 62(2):135-158, 1997. Google Scholar
  21. Hassler Whitney. On the abstract properties of linear dependence. American Journal of Mathematics, 57(3):509-533, 1935. Google Scholar