Skip to main content
SpringerLink
Log in

Hide and Seek Game with Multiple Resources

  • Conference paper
  • First Online:
Book cover Algorithmic Game Theory (SAGT 2018)

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 11059))

Included in the following conference series:

Abstract

We study a generalization of hide and seek game of von Neumann [14], where each player has one or more resources. We characterize the value and Nash equilibria of such games in terms of their unidimensional marginal distributions. We propose a \(\mathcal {O}(n \log (n))\) time algorithm for computing unidimensional marginal distributions of equilibrium strategies and a quadratic time algorithm for computing mixed strategies given the margins. The characterization allows us to establish a number of interesting qualitative features of equilibria.

This work was supported by Polish National Science Centre through grant nr 2014/13/B/ST6/01807.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Throughout the paper, given a positive integer m we will follow the usual practice of using [m] to denote the set \(\{1,\ldots ,m\}\).

  2. 2.

    The system of equations can be easily used to obtain full characterization in the cases with multiple equilibria. We decided to leave it out of the paper due to presentation considerations.

References

  1. Ahmadinejad, A., Dehghani, S., Hajiaghayi, M., Lucier, B., Mahini, H., Seddighin, S.: From duels to battlefields: computing equilibria of Blotto and other games. In: Schuurmans, D., Wellman, M.P. (eds.) Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence, Phoenix, Arizona, USA, 12–17 February 2016, pp. 376–382. AAAI Press (2016)

    Google Scholar 

  2. Behnezhad, S., et al.: From battlefields to elections: winning strategies of Blotto and auditing games. In: Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, 7–10 January 2018, pp. 2291–2310 (2018)

    Google Scholar 

  3. Behnezhad, S., Dehghani, S., Derakhshan, M., Hajiaghayi, M.T., Seddighin, S.: Faster and simpler algorithm for optimal strategies of Blotto game. In: Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence, San Francisco, California, USA, 4–9 February 2017, pp. 369–375 (2017)

    Google Scholar 

  4. Birkhoff, D.: Tres observaciones sobre el algebra lineal. Universidad Nacional de Tucuman Revista Serie A 5, 147–151 (1946)

    Google Scholar 

  5. Borel, É.: La Théorie du Jeu et les Équations Intégrales à Noyau Symétrique. Comptes Rendus de l’Académie des Sciences 173, 1304–1308 (1921). Translated by Savage L.J.: The theory of play and integral equations with skew symmetric kernels. Econometrica 21, 97–100 (1953)

    Google Scholar 

  6. Crawford, V., Iriberri, N.: Fatal attraction: salience, naïveté, and sophistication in experimental “hide-and-seek” games. Am. Econ. Rev. 97(5), 1731–1750 (2007)

    Article  Google Scholar 

  7. Dulmage, L., Halperin, I.: On a theorem of Frobenius-König and J. von Neumann’s game of hide and seek. Proc. Trans. R. Soc. Can. 49(3), 23–29 (1955)

    MATH  Google Scholar 

  8. Fisher, D.: Two person zero-sum games and fractional graph parameters. Congr. Numerantium 85, 9–14 (1991)

    MathSciNet  MATH  Google Scholar 

  9. Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1(2), 169–197 (1981)

    Article  MathSciNet  Google Scholar 

  10. Kiekintveld, C., Jain, M., Tsai, J., Pita, J., Ordóñez, F., Tambe, M.: Computing optimal randomized resource allocations for massive security games. In: Proceedings of The 8th International Conference on Autonomous Agents and Multiagent Systems-Volume 1, AAMAS 2009, IFAAMAS, Richland, SC, pp. 689–696 (2009)

    Google Scholar 

  11. Korzhyk, D., Conitzer, V., Parr, R.: Complexity of computing optimal Stackelberg strategies in security resource allocation games. In: Proceedings of the 24th AAAI Conference on Artificial Intelligence, AAAI 2010. AAAI Press, Menlo Park (2010)

    Google Scholar 

  12. Korzhyk, D., Conitzer, V., Parr, R.: Security games with multiple attacker resources. In: Proceedings of the 22nd International Joint Conference on Artificial Intelligence, IJCAI 2011, pp. 273–279. AAAI Press, Menlo Park (2011)

    Google Scholar 

  13. Korzhyk, D., Yin, Z., Kiekintveld, C., Conitzer, V., Tambe, M.: Stackelberg vs. Nash in security games: an extended investigation of interchangeability, equivalence, and uniqueness. J. Artif. Intell. Res. 41, 297–327 (2011)

    Article  MathSciNet  Google Scholar 

  14. von Neumann, J.: A certain zero-sum two-person game equivalent to the optimal assignment problem. In: Contributions to the Theory of Games (AM-28), vol. II, pp. 5–12. Princeton University Press (1953)

    Google Scholar 

  15. Scheinerman, E., Ullman, D.: Fractional Graph Theory. Wiley, New York (1997)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Informatics, University of Warsaw, Banacha 2, 02-097, Warsaw, Poland

    Marcin Dziubiński

  2. Department of Economics, School of Social Science, University of Bath, Claverton Down, Bath, BA2 7AY, UK

    Jaideep Roy

Authors
  1. Marcin Dziubiński
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jaideep Roy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marcin Dziubiński .

Editor information

Editors and Affiliations

  1. Peking University, Beijing, China

    Xiaotie Deng

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Dziubiński, M., Roy, J. (2018). Hide and Seek Game with Multiple Resources. In: Deng, X. (eds) Algorithmic Game Theory. SAGT 2018. Lecture Notes in Computer Science(), vol 11059. Springer, Cham. https://doi.org/10.1007/978-3-319-99660-8_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-99660-8_8

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-99659-2

  • Online ISBN: 978-3-319-99660-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation