Economic Theory

, Volume 58, Issue 1, pp 183–216 | Cite as

Waging simple wars: a complete characterization of two-battlefield Blotto equilibria

  • Scott T. Macdonell
  • Nick Mastronardi
Research Article


We analyze the strategic allocation of resources across two contests as in the canonical Colonel Blotto game. In the games we study, two players simultaneously allocate their forces across two fields of battle. The larger force on each battlefield wins that battle, and the payoff to a player is the sum of the values of battlefields won. We completely characterize the set of Nash equilibria of all two-battlefield Blotto games and provide the unique equilibrium payoffs. We also show how to extend our characterization to cover previously unstudied games with nonlinear resource constraints.


Colonel Blotto game Zero-sum game Warfare All-pay auction Multi-unit auction 

JEL Classification

C72 H56 D7 


  1. Adamo, T., Matros, A.: A blotto game with incomplete information. Econ. Lett. 105(1), 100–102 (2009)CrossRefGoogle Scholar
  2. Arad, A., Rubinstein, A.: Multi-dimensional iterative reasoning in action: the case of the colonel blotto game. J. Econ. Behav. Organ. 84(2), 571–585 (2012)CrossRefGoogle Scholar
  3. Blackett, D.W.: Some blotto games. Nav. Res. Logist. Q. 1(1), 55–60 (1954)CrossRefGoogle Scholar
  4. Blackett, D.W.: Pure strategy solutions of blotto games. Nav. Res. Logist. Q. 5(2), 107–109 (1958)CrossRefGoogle Scholar
  5. Borel, E.: La théorie du jeu les équations intégralesa noyau symétrique. Comptes Rendus de l’Académie des Sciences 173, 1304–1308 (1921); English translation by Savage, L.: The theory of play and integral equations with skew symmetric kernels. Econometrica 21(1), 97–100 (1953)Google Scholar
  6. Chowdhury, S.M., Kovenock, D., Sheremeta, R.M.: An experimental investigation of colonel blotto games. Econ. Theory 52(3), 1–29 (2013)CrossRefGoogle Scholar
  7. Colantoni, C.S., Levesque, T.J., Ordeshook, P.C.: Campaign resource allocations under the electoral college. Am. Polit. Sci. Rev. 69(1), 141–154 (1975)CrossRefGoogle Scholar
  8. Coughlin, P.J.: Pure strategy equilibria in a class of systems defense games. Int. J. Game Theory 20(3), 195–210 (1992)CrossRefGoogle Scholar
  9. Dziubiński, M.: Non-symmetric discrete general lotto games. Int. J. Game Theory 42(4), 801–833 (2012)CrossRefGoogle Scholar
  10. Golman, R., Page, S.E.: General blotto: games of allocative strategic mismatch. Public Choice 138(3–4), 279–299 (2009)CrossRefGoogle Scholar
  11. Gross, O.: The symmetric blotto game. RAND Corporation RM-424 (1950)Google Scholar
  12. Gross, O., Wagner, R.: A continuous colonel blotto game. RAND Corporation RM-408 (1950)Google Scholar
  13. Hart, S.: Discrete colonel blotto and general lotto games. Int. J. Game Theory 36(3), 441–460 (2008)CrossRefGoogle Scholar
  14. Hortala-Vallve, R., Llorente-Saguer, A.: A simple mechanism for resolving conflict. Games Econ. Behav. 70(2), 375–391 (2010)CrossRefGoogle Scholar
  15. Kovenock, D., Roberson, B.: Coalitional colonel blotto games with application to the economics of alliances. J. Public Econ. Theory 14(4), 653–676 (2012a)CrossRefGoogle Scholar
  16. Kovenock, D., Roberson, B.: The Oxford Handbook of the Economics of Peace and Conflict, Oxford University Press, Chap Conflicts with multiple battlefields (2012b)Google Scholar
  17. Kovenock, D., Mauboussin, M.J., Roberson, B.: Asymmetric conflicts with endogenous dimensionality. Korean Econ. Rev. 26(2), 287–305 (2010)Google Scholar
  18. Laslier, J.: How two-party competition treats minorities. Rev. Econ. Des. 7(3), 297–307 (2002)Google Scholar
  19. Laslier, J., Picard, N.: Distributive politics and electoral competition. J. Econ. Theory 103(1), 106–130 (2002)CrossRefGoogle Scholar
  20. Le Breton, M., Zaporozhets, V.: Sequential legislative lobbying under political certainty. Econ. J. 120(543), 281–312 (2010)CrossRefGoogle Scholar
  21. Merolla, J., Munger, M., Tofias, M.: In play: a commentary on strategies in the 2004 U.S. presidential election. Public Choice 123(1–2), 19–37 (2005)CrossRefGoogle Scholar
  22. Powell, R.: Sequential, nonzero-sum “Blotto”: allocating defensive resources prior to attack. Games Econ. Behav. 67(2), 611–615 (2009)CrossRefGoogle Scholar
  23. Powers, M., Shen, Z.: Colonel blotto in the war on terror: implications for event frequency. J. Homel. Secur. Emerg. Manag. 6(1), (2009)Google Scholar
  24. Roberson, B.: The colonel blotto game. Econ. Theory 29(1), 1–24 (2006)CrossRefGoogle Scholar
  25. Roberson, B., Kvasov, D.: The non-constant-sum colonel blotto game. Econ. Theory 51(2), 397–433 (2012)CrossRefGoogle Scholar
  26. Sahuguet, N., Persico, N.: Campaign spending regulation in a model of redistributive politics. Econ. Theory 28(1), 95–124 (2006)CrossRefGoogle Scholar
  27. Szentes, B., Rosenthal, R.W.: Three-object two-bidder simultaneous auctions: chopsticks and tetrahedra. Games Econ. Behav. 44(1), 114–133 (2003)CrossRefGoogle Scholar
  28. Thomas, C.: N-dimensional colonel blotto game with asymmetric battlefield values. Working paper, The University of Texas at Austin (2012)Google Scholar
  29. Vorob’ev, N.N.: Game theory: lectures for economists and systems scientists. Springer, New York (1977)CrossRefGoogle Scholar
  30. Wu, Y., Wang, B., Liu, K.: Optimal power allocation strategy against jamming attacks using the colonel blotto game. IEEE Global Telecommunications Conference, 2009 (GLOBECOM 2009), pp 1–5 (2009)Google Scholar
  31. Young, H.P.: The allocation of funds in lobbying and campaigning. Behav. Sci. 23(1), 21–31 (1978)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.CNAAlexandriaUSA
  2. 2.US Air Force Academy (USAFA)US Air Force AcademyUSA

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