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Two Notes on the Blotto Game

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  • Weinstein Jonathan

Abstract

We exhibit a new equilibrium of the classic Blotto game in which players allocate one unit of resources among three coordinates and try to defeat their opponent in two out of three. It is well known that a mixed strategy will be an equilibrium strategy if the marginal distribution on each coordinate is U[0,(2/3)]. All classic examples of such distributions have two-dimensional support. Here we exhibit a distribution which has one-dimensional support and is simpler to describe than previous examples. The construction generalizes to give one-dimensional distributions with the same property in higher-dimensional simplices as well.As our second note, we give some results on the equilibrium payoffs when the game is modified so that players have unequal budgets. Our results suggest a criterion for equilibrium selection in the original symmetric game, in terms of robustness with respect to a small asymmetry in resources.

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  • Weinstein Jonathan, 2012. "Two Notes on the Blotto Game," The B.E. Journal of Theoretical Economics, De Gruyter, vol. 12(1), pages 1-13, March.
    • Handle: RePEc:bpj:bejtec:v:12:y:2012:i:1:n:7
    DOI: 10.1515/1935-1704.1893
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    1. Brian Roberson & Dmitriy Kvasov, 2012. "The non-constant-sum Colonel Blotto game," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 51(2), pages 397-433, October.
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    Cited by:

    1. Subhasish M Chowdhury & Dan Kovenock & David Rojo Arjona & Nathaniel T Wilcox, 2021. "Focality and Asymmetry in Multi-Battle Contests," The Economic Journal, Royal Economic Society, vol. 131(636), pages 1593-1619.
    2. Dan Kovenock & Sudipta Sarangi & Matt Wiser, 2015. "All-pay 2 $$\times $$ × 2 Hex: a multibattle contest with complementarities," International Journal of Game Theory, Springer;Game Theory Society, vol. 44(3), pages 571-597, August.
    3. Dan Kovenock & Brian Roberson & Roman M. Sheremeta, 2019. "The attack and defense of weakest-link networks," Public Choice, Springer, vol. 179(3), pages 175-194, June.
    4. Dan Kovenock & Brian Roberson, 2018. "The Optimal Defense Of Networks Of Targets," Economic Inquiry, Western Economic Association International, vol. 56(4), pages 2195-2211, October.
    5. Duffy, John & Matros, Alexander, 2017. "Stochastic asymmetric Blotto games: An experimental study," Journal of Economic Behavior & Organization, Elsevier, vol. 139(C), pages 88-105.
    6. Kaplan, Todd R. & Zamir, Shmuel, 2015. "Advances in Auctions," Handbook of Game Theory with Economic Applications,, Elsevier.
    7. Christian Ewerhart, 2022. "A “fractal” solution to the chopstick auction," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 74(4), pages 1025-1041, November.
    8. Jonathan Lamb & Justin Grana & Nicholas O’Donoughue, 2022. "The Benefits of Fractionation in Competitive Resource Allocation," Computational Economics, Springer;Society for Computational Economics, vol. 59(2), pages 831-852, February.
    9. Yosef Rinott & Marco Scarsini & Yaming Yu, 2012. "A Colonel Blotto Gladiator Game," Mathematics of Operations Research, INFORMS, vol. 37(4), pages 574-590, November.
    10. Boix-Adserà, Enric & Edelman, Benjamin L. & Jayanti, Siddhartha, 2021. "The multiplayer Colonel Blotto game," Games and Economic Behavior, Elsevier, vol. 129(C), pages 15-31.
    11. Cortes-Corrales, Sebastián & Gorny, Paul M., 2018. "Generalising Conflict Networks," MPRA Paper 90001, University Library of Munich, Germany.
    12. Kostyantyn Mazur, 2017. "A Partial Solution to Continuous Blotto," Papers 1706.08479, arXiv.org, revised Sep 2017.
    13. Caroline Thomas, 2018. "N-dimensional Blotto game with heterogeneous battlefield values," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 65(3), pages 509-544, May.
    14. Dan Kovenock & Brian Roberson, 2021. "Generalizations of the General Lotto and Colonel Blotto games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 71(3), pages 997-1032, April.
    15. AmirMahdi Ahmadinejad & Sina Dehghani & MohammadTaghi Hajiaghayi & Brendan Lucier & Hamid Mahini & Saeed Seddighin, 2019. "From Duels to Battlefields: Computing Equilibria of Blotto and Other Games," Management Science, INFORMS, vol. 44(4), pages 1304-1325, November.
    16. Florian Gauer & Christoph Kuzmics, 2020. "Cognitive Empathy In Conflict Situations," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 61(4), pages 1659-1678, November.
    17. Deck, Cary & Sarangi, Sudipta & Wiser, Matt, 2017. "An experimental investigation of simultaneous multi-battle contests with strategic complementarities," Journal of Economic Psychology, Elsevier, vol. 63(C), pages 117-134.
    18. Ghosh, Gagan, 2021. "Simultaneous auctions with budgets: Equilibrium existence and characterization," Games and Economic Behavior, Elsevier, vol. 126(C), pages 75-93.

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    Keywords

    Blotto; zero-sum games;

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