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

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## Abstract

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.

## Keywords

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

C72 H56 D7## References

- Adamo, T., Matros, A.: A blotto game with incomplete information. Econ. Lett.
**105**(1), 100–102 (2009)CrossRefGoogle Scholar - 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 - Blackett, D.W.: Some blotto games. Nav. Res. Logist. Q.
**1**(1), 55–60 (1954)CrossRefGoogle Scholar - Blackett, D.W.: Pure strategy solutions of blotto games. Nav. Res. Logist. Q.
**5**(2), 107–109 (1958)CrossRefGoogle Scholar - 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 - Chowdhury, S.M., Kovenock, D., Sheremeta, R.M.: An experimental investigation of colonel blotto games. Econ. Theory
**52**(3), 1–29 (2013)CrossRefGoogle Scholar - 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 - Coughlin, P.J.: Pure strategy equilibria in a class of systems defense games. Int. J. Game Theory
**20**(3), 195–210 (1992)CrossRefGoogle Scholar - Dziubiński, M.: Non-symmetric discrete general lotto games. Int. J. Game Theory
**42**(4), 801–833 (2012)CrossRefGoogle Scholar - Golman, R., Page, S.E.: General blotto: games of allocative strategic mismatch. Public Choice
**138**(3–4), 279–299 (2009)CrossRefGoogle Scholar - Gross, O.: The symmetric blotto game. RAND Corporation RM-424 (1950)Google Scholar
- Gross, O., Wagner, R.: A continuous colonel blotto game. RAND Corporation RM-408 (1950)Google Scholar
- Hart, S.: Discrete colonel blotto and general lotto games. Int. J. Game Theory
**36**(3), 441–460 (2008)CrossRefGoogle Scholar - Hortala-Vallve, R., Llorente-Saguer, A.: A simple mechanism for resolving conflict. Games Econ. Behav.
**70**(2), 375–391 (2010)CrossRefGoogle Scholar - 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 - 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
- Kovenock, D., Mauboussin, M.J., Roberson, B.: Asymmetric conflicts with endogenous dimensionality. Korean Econ. Rev.
**26**(2), 287–305 (2010)Google Scholar - Laslier, J.: How two-party competition treats minorities. Rev. Econ. Des.
**7**(3), 297–307 (2002)Google Scholar - Laslier, J., Picard, N.: Distributive politics and electoral competition. J. Econ. Theory
**103**(1), 106–130 (2002)CrossRefGoogle Scholar - Le Breton, M., Zaporozhets, V.: Sequential legislative lobbying under political certainty. Econ. J.
**120**(543), 281–312 (2010)CrossRefGoogle Scholar - 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 - Powell, R.: Sequential, nonzero-sum “Blotto”: allocating defensive resources prior to attack. Games Econ. Behav.
**67**(2), 611–615 (2009)CrossRefGoogle Scholar - 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 - Roberson, B.: The colonel blotto game. Econ. Theory
**29**(1), 1–24 (2006)CrossRefGoogle Scholar - Roberson, B., Kvasov, D.: The non-constant-sum colonel blotto game. Econ. Theory
**51**(2), 397–433 (2012)CrossRefGoogle Scholar - Sahuguet, N., Persico, N.: Campaign spending regulation in a model of redistributive politics. Econ. Theory
**28**(1), 95–124 (2006)CrossRefGoogle Scholar - Szentes, B., Rosenthal, R.W.: Three-object two-bidder simultaneous auctions: chopsticks and tetrahedra. Games Econ. Behav.
**44**(1), 114–133 (2003)CrossRefGoogle Scholar - Thomas, C.: N-dimensional colonel blotto game with asymmetric battlefield values. Working paper, The University of Texas at Austin (2012)Google Scholar
- Vorob’ev, N.N.: Game theory: lectures for economists and systems scientists. Springer, New York (1977)CrossRefGoogle Scholar
- 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
- Young, H.P.: The allocation of funds in lobbying and campaigning. Behav. Sci.
**23**(1), 21–31 (1978)CrossRefGoogle Scholar

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