In the Colonel Blotto game, which was initially introduced by Borel in 1921, two colonels simultaneously distribute their troops across different battlefields. The winner of each battlefield is determined independently by a winner-takes-all rule. The ultimate payoff for each colonel is the number of battlefields won. The Colonel Blotto game is commonly used for analyzing a wide range of applications from the U.S. Presidential election to innovative technology competitions to advertising, sports, and politics. There are persistent efforts to find the optimal strategies for the Colonel Blotto game. However, the first polynomial-time algorithm for that has very recently been provided by Ahmadinejad, Dehghani, Hajiaghayi, Lucier, Mahini, and Seddighin. Their algorithm consists of an exponential size linear program (LP), which they solve using the ellipsoid method. Because of the use of the ellipsoid method, despite its significant theoretical importance, this algorithm is highly impractical. In general, even the simplex method (despite its exponential running time in practice) performs better than the ellipsoid method in practice. In this paper, we provide the first polynomial-size LP formulation of the optimal strategies for the Colonel Blotto game using linear extension techniques. Roughly speaking, we consider the natural representation of the strategy space polytope and transform it to a higher dimensional strategy space, which interestingly has exponentially fewer facets. In other words, we add a few variables to the LP such that, surprisingly, the number of constraints drops down to a polynomial. We use this polynomial-size LP to provide a simpler and significantly faster algorithm for finding optimal strategies of the Colonel Blotto game. We further show this representation is asymptotically tight, which means there exists no other linear representation of the strategy space with fewer constraints. We also extend our approach to multidimensional Colonel Blotto games, in which players may have different sorts of budgets, such as money, time, human resources, etc. By implementing this algorithm, we are able to run tests that were previously impossible to solve in a reasonable time. This information allows us to observe some interesting properties of Colonel Blotto; for example, we find out the behavior of players in the discrete model is very similar to the continuous model Roberson solved.
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Finding Minimum Volume Circumscribing Ellipsoids Using Generalized Copositive Programming
We study the problem of finding the Löwner–John ellipsoid (i.e., an ellipsoid with minimum volume that contains a given convex set). We reformulate the problem as a generalized copositive program and use that reformulation to derive tractable semidefinite programming approximations for instances where the set is defined by affine and quadratic inequalities. We prove that, when the underlying set is a polytope, our method never provides an ellipsoid of higher volume than the one obtained by scaling the maximum volume-inscribed ellipsoid. We empirically demonstrate that our proposed method generates high-quality solutions and can be solved much faster than solving the problem to optimality. Furthermore, we outperform the existing approximation schemes in terms of solution time and quality. We present applications of our method to obtain piecewise linear decision rule approximations for dynamic distributionally robust problems with random recourse and to generate ellipsoidal approximations for the set of reachable states in a linear dynamical system when the set of allowed controls is a polytope.
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- Award ID(s):
- 1752125
- PAR ID:
- 10388536
- Date Published:
- Journal Name:
- Operations Research
- Volume:
- 70
- Issue:
- 5
- ISSN:
- 0030-364X
- Page Range / eLocation ID:
- 2867 to 2882
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1287/opre.2021.2156
