The Multiplayer Colonel Blotto Game

We initiate the study of the natural multiplayer generalization of the classic continuous Colonel Blotto game.The two-player Blotto game, introduced by Borel as a model of resource competition across nsimultaneous fronts, has been studied extensively for a century and seen numerous applications throughout the social sciences. Our work defines the multiplayer Colonel Blotto gameand derives Nash equilibria for various settings of k(number of players) and n(number of battlefields)---in particular, we mostly solve the symmetric three-player case. We also introduce a "Boolean" version of multiplayer Blotto. The main technical difficulty of our work, as in the two-player theoretical literature, is the challenge of coupling various marginal distributions into a joint distribution satisfying a strict sum constraint. In contrast to previous works in the continuous setting, we derive our couplings algorithmically in the form of efficient sampling algorithms. The full paper can be found at https://arxiv.org/abs/2002.05240.

[1]  G. Simons,et al.  On the theory of elliptically contoured distributions , 1981 .

[2]  Pern Hui Chia,et al.  Colonel Blotto in the Phishing War , 2011, GameSec.

[3]  Sergiu Hart,et al.  Discrete Colonel Blotto and General Lotto games , 2008, Int. J. Game Theory.

[4]  É. Borel.  The Theory of Play and Integral Equations with Skew Symmetric Kernels , 1953 .

[5]  Mervin E. Muller,et al.  A note on a method for generating points uniformly on n-dimensional spheres , 1959, CACM.

[6]  Philip B. Stark,et al.  From Battlefields to Elections: Winning Strategies of Blotto and Auditing Games , 2018, SODA.

[7]  Jean-François Laslier,et al.  Distributive Politics and Electoral Competition , 2002, J. Econ. Theory.

[8]  Daniel G. Arce,et al.  Terrorism Experiments , 2011 .

[9]  Mohammad Taghi Hajiaghayi,et al.  From Duels to Battlefields: Computing Equilibria of Blotto and Other Games , 2016, AAAI.

[10]  Marcin Dziubinski,et al.  Non-symmetric discrete General Lotto games , 2013, Int. J. Game Theory.

[11]  R. Myerson.  Incentives to Cultivate Favored Minorities Under Alternative Electoral Systems , 1993, American Political Science Review.

[12]  Oliver Alfred Gross,et al.  A Continuous Colonel Blotto Game , 1950 .

[13]  Jonathan Weinstein,et al.  Two Notes on the Blotto Game , 2012 .

[14]  Aniol Llorente-Saguer,et al.  Pure strategy Nash equilibria in non-zero sum colonel Blotto games , 2012, Int. J. Game Theory.

[15]  Mohammad Taghi Hajiaghayi,et al.  Optimal Strategies of Blotto Games: Beyond Convexity , 2019, EC.

[16]  Jean-François Laslier,et al.  How two-party competition treats minorities , 2002 .

[17]  Michael R. Baye,et al.  The all-pay auction with complete information , 1990 .

[18]  Dan Kovenock,et al.  Generalizations of the General Lotto and Colonel Blotto games , 2015, Economic Theory.

[19]  Michael C. Munger,et al.  In Play: A Commentary on Strategies in the 2004 U.S. Presidential Election , 2005 .

[20]  Dmitriy Kvasov,et al.  Contests with limited resources , 2007, J. Econ. Theory.

[21]  Robert M. Bell,et al.  Competitive Optimality of Logarithmic Investment , 1980, Math. Oper. Res..

[22]  Scott E. Page,et al.  General Blotto: games of allocative strategic mismatch , 2009 .

[23]  Jiayu Lin.  On The Dirichlet Distribution , 2016 .

[24]  B. Roberson.  The Colonel Blotto game , 2006 .

[25]  Caroline D. Thomas,et al.  N-dimensional Blotto game with heterogeneous battlefield values , 2018 .

[26]  Martin Shubik,et al.  Systems Defense Games: Colonel Blotto, Command and Control , 1981 .

[27]  E. Maskin,et al.  The Existence of Equilibrium in Discontinuous Economic Games, I: Theory , 1986 .

[28]  E. Maskin,et al.  The Existence of Equilibrium in Discontinuous Economic Games, II: Applications , 1986 .

[29]  D. Song,et al.  Lp-NORM UNIFORM DISTRIBUTION , 1996 .

[30]  Roman M. Sheremeta,et al.  A survey of experimental research on contests, all-pay auctions and tournaments , 2012, Experimental Economics.

[31]  Joachim H. Ahrens,et al.  Computer methods for sampling from gamma, beta, poisson and bionomial distributions , 1974, Computing.

[32]  L. Friedman.  Game-Theory Models in the Allocation of Advertising Expenditures , 1958 .

[33]  D. W. Blackett.  Pure strategy solutions of blotto games , 1958 .

[34]  D. Song,et al.  _{}-norm uniform distribution , 1997 .

[35]  B. Roberson,et al.  The non-constant-sum Colonel Blotto game , 2008, SSRN Electronic Journal.

[36]  Lawrence Freedman.  The Problem of Strategy , 1980 .

[37]  S. Shankar Sastry,et al.  The heterogeneous Colonel Blotto game , 2014, 2014 7th International Conference on NETwork Games, COntrol and OPtimization (NetGCoop).

[38]  Mohammad Taghi Hajiaghayi,et al.  Faster and Simpler Algorithm for Optimal Strategies of Blotto Game , 2016, AAAI.