On Computing Nash Equilibria of Borel’s Colonel Blotto Game for Multiple Players including in Arbitrary Measure Spaces by Siddhartha Visveswara Jayanti ర శర జయం साथ वेर जयित B.S.E, Princeton University (2017) Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of Master of Science in Computer Science and Engineering at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY May 2020 © Massachusetts Institute of Technology 2020. All rights reserved. Author................................................................... Department of Electrical Engineering and Computer Science May 15, 2020 Certified by . Constantinos Daskalakis Professor of Electrical Engineering and Computer Science Thesis Supervisor Accepted by.............................................................. Leslie A. Kolodziejski Professor of Electrical Engineering and Computer Science Chair, Department Committee on Graduate Students 2 On Computing Nash Equilibria of Borel’s Colonel Blotto Game for Multiple Players including in Arbitrary Measure Spaces by Siddhartha Visveswara Jayanti ర శర జయం साथ वेर जयित Submitted to the Department of Electrical Engineering and Computer Science on May 15, 2020, in partial fulfillment of the requirements for the degree of Master of Science in Computer Science and Engineering Abstract The Colonel Blotto Problem proposed by Borel in 1921 has served as a widely applicable model of budget-constrained simultaneous winner-take-all competitions in the social sciences. Applications include elections, advertising, research and development, ecology and more. However, the classic Blotto problem and variants that have been studied hereto are only about two-player games on a finite set of discrete battlefields. This thesis extends the classic theory to multiplayer games over arbitrary measureable battlegrounds. Furthermore, it characterizes the symmetric mixed Nash equilibria of a wide array of these generalized Blotto games and provides efficient algorithms to sample from these equilibria; thereby unleashing the potential for quantitative analyses of a multitude of new and significant applications, such as multiparty elections and ideological battles across the surface of the earth. Thesis Supervisor: Constantinos Daskalakis Title: Professor of Electrical Engineering and Computer Science 3 4 Acknowledgments A significant portion of this thesis, namely Chapters 1, 2 and 4, are based on a collaboration with Enric Boix and Ben Edelman [10]. I am proud to have had such nice friends through our undergraduate years of study at Princeton and graduate years of study here in Cambridge, MA. None of the results in this thesis would have been possible if it were not for the long sleepless nights of contemplation that the three of us spent together over the course of the last year. I would like to thank my advisor, Costis Daskalakis, for his guidance over the past few years. Costis is an insightful thinker and an amazingly clear teacher. His engaging lectures in the course on Economics and Computation were principally responsible for luring my interest towards game theory. In fact, what started as the final project in his course lasted much longer, and eventually turned into this thesis. I am of course, ever thankful to my undergraduate advisor, Robert Tarjan, who intro- duced me to the joy and fulfillment of research. I would also like to recognize the contributions of the Hindu Students Council (HSC) to keeping me grounded in the greater meaning of life while I tread my academic path. I would also like to recall the efforts of my family in getting me to where I amtoday. If it were not for the endless hours that my mother and father spent teaching me since my childhood, I would not have gotten to MIT and this place in my education and research. But most importantly, I am supported by the unyielding love of my family—from my mother and father, to my sister and brother-in-law, to my grandparents, uncles, aunts, cousins and beyond. The sacrifices and encouragement of my family, including those that are nolonger with us, were pivotal to my happiness and successes, and to the overall well-being of the family; this, I will always remember in my heart. Finally, I would like to thank the United States Department of Defense that has financially supported my graduate studies through the NDSEG Fellowship program. सयमेव जयते नानृतं सयेन पथा वततो देवयानः । येनामयृषयो ातकामा य तत् सयय परमं नधानम् ॥ ∼ मुडकोपनषत् ३.१.६ 5 6 Chapter 1 Introduction The Colonel Blotto game, proposed by Borel in 1921 [12], has been a staple of the game theory literature for the past century. In this game, two warring commanders, Captain Alice and Colonel Blotto, fight on n simultaneous and independent battlefields of different values v1; : : : ; vn. Each commander has a finite budget of power—BAlice; BBlotto—to split over the battlefields, and for each battlefield i, the value vi, will be awarded to the commander that allocates more power there. Each commander aims to maximize the total expected value of his winnings in this game. The game is called symmetric if all players have the same budget, homogenous if all battlefields have the same value, continuous if allocations can be arbitrary positive real numbers, and discrete if allocations are restricted to be whole numbers. In this thesis, we are primarily interested in computing Nash Equilibria1 of the Colonel Blotto game. The Blotto game has proved to be a treasure trove for mathematicians, computer sci- entists, and social scientists. While the Blotto game is easy to describe, its equilibrium structure is surprisingly complex even in the mathematically pleasing continuous setting, due to the hard budget constraint.2 Thus, an analysis of the equilibria of the game has only 1A collection of prescribed strategies, one for each player, is in Nash equilibrium if no player can in- crease his expected winnings by unilaterally deviating from the prescription. If the prescribed strategies are deterministic, the equilibrium is pure; if they are stochasic, the equilibrium is mixed. 2It is well known that even the simplest Blotto games do not admit pure Nash equilibria. Consider the two-player continuous symmetric homogeneous Blotto game with n > 2 battlefields. If Alice fixes any bid vector ~a = (a1; : : : ; an) (where a1 =6 0 without loss of generality), then Bob can maximize his winnings by ~ a1 picking the action b = (0; a2 + ϵ, a3 + ϵ, : : : ; an + ϵ), where ϵ = n−1 , to win all but the first battlefield. This pair of actions is not in equilibrium because Alice can switch her action to a~0 = ~b in order to win half of the total value rather than 1/nth of it. Therefore, in general we are looking for mixed Nash equilibria. 7 emerged in the past few decades due to a long line of research [30, 43, 36, 27, 38, 41] which has culminated in the work of Kovenock and Roberson [28]. When the allocations must be discrete, closed form analyses have given way to fast algorithms that can output equilibria given a problem description; thus, computer scientists have taken the fore, producing sev- eral pseudo-polynomial time algorithms for various settings of the discrete Blotto problem [1, 7, 5, 6]. The Blotto game has also seen innumerable applications in the social sciences, ranging from the modeling of political elections [35, 31, 30, 33] to modeling competitions between species in ecological niches [23]. Equilibria for the (hard budget constraint) Blotto game have been developed based on solutions for the simpler soft budget constraint version of the game, called General Lotto. In the Lotto game, each player plays a distribution over allocations instead of a single allocation, and the budget constraint is relaxed so that the allocations need to sum to the budget only in expectation, rather than with probability one. Strategies in the Lotto game are a superset of mixed strategies in the Blotto game, since Lotto players can play strategies that go over- budget sometimes as long as they compensate by playing strategies that go under-budget some other times.3 It turns out that this freedom usually makes Lotto easier to solve in practice than Blotto. In particular, the distribution over strategies for each battlefield can be computed independently for Lotto games, as long as the expected values of the resultant distributions played on each battlefield add up to the budget constraint. In contrast, a mixed-strategy for Blotto with the same marginal allocation distributions per battlefield would additionally need to couple these marginals into a joint distribution whose support only consists of allocation vectors summing to the budget constraint. Demonstration of such a coupling is often the hardest part of an equilibrium computation for a Blotto game. Modeling two-party elections is a famous application of the Blotto game [30, 36, 33]. Hoping to understand multiparty electoral systems, Myerson alluded to a Blotto game with more than two players in [35], which compares different types of multiparty election systems by studying the equilibrium strategies those systems induce. In this context, the classic plurality vote elections conducted in many parliamentary democracies such as India and the 3 1 0 For example, if Alice has a budget of BAlice = 1 she can play ~a = (1; 1) with probability /2 and a~ = (0; 0) 1 with probability /2 in the Lotto game; but cannot in the Blotto game because, jj~ajj1 = 1+1 = 2 > 1 = BAlice. 8 United Kingdom are naturally modeled by a multiplayer generalization of Colonel Blotto with, e.g., the multiple parties corresponding to players, voting districts corresponding to battlefields, and district advertising expenditures corresponding to the resource allocations. However, stating that “the hardest part of [the Blotto] problem was to construct joint dis- tributions for allocations that always sum to the given total,” Myerson weakened the true budget constraint to the soft one and further assumed that all battlefields must be treated symmetrically by all players, and thus only analyzed what would nowadays be called multi- player homogeneous symmetric General Lotto.