Skip to main content

Learning Convex Partitions and Computing Game-Theoretic Equilibria from Best Response Queries

  • 776 Accesses

Part of the Lecture Notes in Computer Science book series (LNISA,volume 11316)


Suppose that an m-simplex is partitioned into n convex regions having disjoint interiors and distinct labels, and we may learn the label of any point by querying it. The learning objective is to know, for any point in the simplex, a label that occurs within some distance \(\varepsilon \) from that point. We present two algorithms for this task: Constant-Dimension Generalised Binary Search (CD-GBS), which for constant m uses \(poly(n, \log \left( \frac{1}{\varepsilon } \right) )\) queries, and Constant-Region Generalised Binary Search (CR-GBS), which uses CD-GBS as a subroutine and for constant n uses \(poly(m, \log \left( \frac{1}{\varepsilon } \right) )\) queries. We show via Kakutani’s fixed-point theorem that these algorithms provide bounds on the best-response query complexity of computing approximate well-supported equilibria of bimatrix games in which one of the players has a constant number of pure strategies.


  • Query protocol
  • Equilibrium computation
  • Revealed preferences

Full Online Version of Paper:

F. J. Marmolejo Cossío—Supported by the Mexican National Council of Science and Technology (CONACyT).

This is a preview of subscription content, access via your institution.

Buying options

USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-030-04612-5_12
  • Chapter length: 20 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
USD   69.99
Price excludes VAT (USA)
  • ISBN: 978-3-030-04612-5
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   89.99
Price excludes VAT (USA)
Fig. 1.
Fig. 2.


  1. 1.

    CD-GBS runs in polynomial time for constant m. The time-intensive operation consists of identifying uncovered intervals, but since the dimension of the ambient simplex is constant, each empirical polytope \({\widehat{P}}_i\) has at most a constant number of bounding hyperplanes. These hyperplanes can each be extruded by \(\varepsilon \), and checking whether there exists a point outside all these extrusions can be done in time polynomial in n via brute force. In fact, all other algorithms in this paper have efficient runtimes (in their relevant parameters) due to similar reasoning.


  1. Babichenko, Y.: Query complexity of approximate Nash equilibria. J. ACM 63(4), 36:1–36:24 (2016)

    MathSciNet  CrossRef  Google Scholar 

  2. Babichenko, Y., Rubinstein, A.: Communication complexity of approximate Nash equilibria. In: Proceedings of the 49th STOC, pp. 878–889. ACM (2017)

    Google Scholar 

  3. Babichenko, Y., Barman, S., Peretz, R.: Empirical distribution of equilibrium play and its testing application. Math. Oper. Res. 42(1), 15–29 (2017)

    MathSciNet  CrossRef  Google Scholar 

  4. Baldwin, E., Klemperer, P.: Understanding preferences: “demand types”, and the existence of equilibrium with indivisibilities. Technical report, LSE, October 2016

    Google Scholar 

  5. Brown, G.W.: Some notes on computation of game solutions. RAND corporation report, p. 78, April 1949

    Google Scholar 

  6. Bshouty, N.H., Goldberg, P.W., Goldman, S.A., Mathias, H.D.: Exact learning of discretized geometric concepts. SIAM J. Comput. 28(2), 674–699 (1998)

    MathSciNet  CrossRef  Google Scholar 

  7. Chen, X., Deng, X., Teng, S.-H.: Settling the complexity of computing two-player Nash equilibria. J. ACM 56(3), 14:1–14:57 (2009)

    MathSciNet  CrossRef  Google Scholar 

  8. Daskalakis, C., Pan, Q.: A counter-example to Karlin’s strong conjecture for fictitious play. In: Proceedings of 55th FOCS, pp. 11–20 (2014)

    Google Scholar 

  9. Daskalakis, C., Goldberg, P.W., Papadimitriou, C.H.: The complexity of computing a Nash equilibrium. SIAM J. Comput. 39(1), 195–259 (2009)

    MathSciNet  CrossRef  Google Scholar 

  10. Fearnley, J., Gairing, M., Goldberg, P.W., Savani, R.: Learning equilibria of games via payoff queries. J. Mach. Learn. Res. 16, 1305–1344 (2015)

    MathSciNet  MATH  Google Scholar 

  11. Fudenberg, D., Levine, D.K.: The Theory of Learning in Games. MIT Press, Cambridge (1998)

    MATH  Google Scholar 

  12. Goldberg, P.W., Kwek, S.: The precision of query points as a resource for learning convex polytopes with membership queries. In: Proceedings of the 13th COLT, pp. 225–235 (2000)

    Google Scholar 

  13. Goldberg, P.W., Roth, A.: Bounds for the query complexity of approximate equilibria. ACM Trans. Econ. Comput. 4(4), 24:1–24:25 (2016)

    MathSciNet  CrossRef  Google Scholar 

  14. Goldberg, P.W., Savani, R., Sørensen, T.B., Ventre, C.: On the approximation performance of fictitious play in finite games. Int. J. Game Theory 42(4), 1059–1083 (2013)

    MathSciNet  CrossRef  Google Scholar 

  15. Goldberg, P.W., Turchetta, S.: Query complexity of approximate equilibria in anonymous games. J. Comput. Syst. Sci. 90, 80–98 (2017)

    MathSciNet  CrossRef  Google Scholar 

  16. Hart, S., Mansour, Y.: How long to equilibrium? The communication complexity of uncoupled equilibrium procedures. Games Econ. Behav. 69(1), 107–126 (2010)

    MathSciNet  CrossRef  Google Scholar 

  17. Hart, S., Nisan, N.: The query complexity of correlated equilibria. Games and Economic Behavior (2016). ISSN 0899–8256

    Google Scholar 

  18. Kakutani, S.: A generalization of Brouwer’s fixed point theorem. Duke Math. J. 8(3), 457–459 (1941)

    MathSciNet  CrossRef  Google Scholar 

  19. Klemperer, P.: The product-mix auction: a new auction design for differentiated goods. J. Eur. Econ. Assoc. 8(2/3), 526–536 (2010)

    CrossRef  Google Scholar 

  20. Lipton, R.J., Markakis, E., Mehta, A.: Playing large games using simple strategies. In: Proceedings of the 4th ACM-EC, EC 2003, pp. 36–41. ACM, New York (2003)

    Google Scholar 

  21. Nash, J.: Non-cooperative games. Ann. Math. 54(2), 286–295 (1951)

    MathSciNet  CrossRef  Google Scholar 

  22. Robinson, J.: An iterative method of solving a game. Ann. Math. 54(2), 296–301 (1951)

    MathSciNet  CrossRef  Google Scholar 

  23. Von Stengel, B.: Leadership with commitment to mixed strategies. CDAM Research report (2004)

    Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Paul W. Goldberg .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this paper

Verify currency and authenticity via CrossMark

Cite this paper

Goldberg, P.W., Marmolejo-Cossío, F.J. (2018). Learning Convex Partitions and Computing Game-Theoretic Equilibria from Best Response Queries. In: Christodoulou, G., Harks, T. (eds) Web and Internet Economics. WINE 2018. Lecture Notes in Computer Science(), vol 11316. Springer, Cham.

Download citation

  • DOI:

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-04611-8

  • Online ISBN: 978-3-030-04612-5

  • eBook Packages: Computer ScienceComputer Science (R0)