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The Spectrum of Equilibria for the Colonel Blotto and the Colonel Lotto Games

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Book cover Algorithmic Game Theory (SAGT 2017)

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 10504))

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Abstract

We study Nash equilibria of a symmetric Colonel Blotto and Colonel Lotto games. In these game two players, with \(N \ge 1\) units of resources each, distribute their resources simultaneously across \(K \ge 2\) battlefields. We introduce a characteristic of equilibria in this game called spectrum which stands for the fraction of battlefields receiving given numbers of units. We provide complete characterization of spectra of Nash equilibria in Colonel Lotto game as well as necessary conditions on spectra of Nash equilibria in Colonel Blotto game for the case of \({K}\! \mid \!{N}\).

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

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Notes

  1. 1.

    Colonel Blotto game is symmetric when both players have the same amount of resources and it is asymmetric otherwise.

  2. 2.

    For convenience reasons, throughout the paper we number the rows of matrices starting for 0 and we number the columns starting from 1.

  3. 3.

    Given natural numbers \(c_1,\ldots ,c_r\), \({{\mathrm{lcm}}}(c_1,\ldots ,c_m)\) is their least common multiple.

  4. 4.

    If \({K}\! \mid \!{N}\) and K is even, then N must be even as well, so K being odd is the only remaining case.

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Authors and Affiliations

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

    Marcin Dziubiński

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  1. Marcin Dziubiński
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Correspondence to Marcin Dziubiński .

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Editors and Affiliations

  1. University of Salento, Lecce, Italy

    Vittorio Bilò

  2. University of L'Aquila, L'Aquila, Italy

    Michele Flammini

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Dziubiński, M. (2017). The Spectrum of Equilibria for the Colonel Blotto and the Colonel Lotto Games. In: Bilò, V., Flammini, M. (eds) Algorithmic Game Theory. SAGT 2017. Lecture Notes in Computer Science(), vol 10504. Springer, Cham. https://doi.org/10.1007/978-3-319-66700-3_23

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  • DOI: https://doi.org/10.1007/978-3-319-66700-3_23

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