Generalized Colonel Blotto Game

Competitive resource allocation between adversarial decision makers arises in a wide spectrum of real-world applications such as in communication systems, cyber-physical systems security, as well as financial, political, and electoral competition. As such, developing analytical tools to model and analyze competitive resource allocation is crucial for devising optimal allocation strategies and anticipating the potential outcomes of the competition. To this end, the Colonel Blotto game is one of the most popular game-theoretic frameworks for modeling and analyzing such competitive resource allocation problems. However, in many real-world competitive situations, the Colonel Blotto game does not admit solutions in deterministic strategies and, hence, one must rely on analytically complex mixed-strategies with their associated tractability, applicability, and practicality challenges. In this paper, a generalization of the Colonel Blotto game which enables the derivation of deterministic, practical, and implementable equilibrium strategies is proposed while accounting for the heterogeneity of the battlefields. In addition, the proposed generalized game enables accounting for the consumed resources in each battlefield, a feature that is not considered in the classical Blotto game. For the generalized game, the existence of a Nash equilibrium in pure-strategies is shown. Then, closed-form analytical expressions of the equilibrium strategies, are derived and the outcome of the game is characterized; based on the number of resources of each player as well as the valuation of each battlefield. The generated results provide invaluable insights on the outcome of the competition. For example, the results show that, when both players are fully rational, the more resourceful player can achieve a better total payoff at the Nash equilibrium, a result that is not mimicked in the classical Blotto game.Comment: 8 pages, 5 figure

Operations Research Methods for Multi-Domain Campaign Phase Planning

In Antiaccess Warfare as Strategy, Tangredi posits the question and need to consider multiple domains and governmental and warfighting functions in various phases of campaign execution. Multi-domain integration within and across various phases of the joint campaign presents a host of non-linear factors that are compounded and amplified by uncertainties. Colonel Blotto is a simple game that is suited to compare traditional force-on-force military engagements where mass wins the day, but have had limited application to more complex military planning. This thesis explores the formulation schema, data-driven parameters, methods of calculation, and scenarios applicable to a generalized Colonel Blotto (General Blotto) game. It explores this generalized game theory framework, its applicability to multi-domain operations, and recommends future research areas that could help to extend its applicability, enabling planners and commanders to gain similar insight as to those straight-forward applications

Characterizing the interplay between information and strength in Blotto games

In this paper, we investigate informational asymmetries in the Colonel Blotto game, a game-theoretic model of competitive resource allocation between two players over a set of battlefields. The battlefield valuations are subject to randomness. One of the two players knows the valuations with certainty. The other knows only a distribution on the battlefield realizations. However, the informed player has fewer resources to allocate. We characterize unique equilibrium payoffs in a two battlefield setup of the Colonel Blotto game. We then focus on a three battlefield setup in the General Lotto game, a popular variant of the Colonel Blotto game. We characterize the unique equilibrium payoffs and mixed equilibrium strategies. We quantify the value of information - the difference in equilibrium payoff between the asymmetric information game and complete information game. We find information strictly improves the informed player's performance guarantee. However, the magnitude of improvement varies with the informed player's strength as well as the game parameters. Our analysis highlights the interplay between strength and information in adversarial environments.Comment: 8 pages, 2 figures. Accepted for presentation at 58th Conference on Decision and Control (CDC), 201

Conflicts with Multiple Battlefields

This paper examines conflicts in which performance is measured by the players' success or failure in multiple component conflicts, commonly termed “battlefields”. In multi-battlefield conflicts, behavioral linkages across battlefields depend both on the technologies of conflict within each battlefield and the nature of economies or diseconomies in how battlefield out-comes and costs aggregate in determining payoffs in the overall conflict.conflict, contest, battlefield, Colonel Blotto Game, auction, lottery

Conflicts with Multiple Battlefields

This paper examines conflicts in which performance is measured by the players' success or failure in multiple component conflicts, commonly termed "battlefields." In multi-battlefield conflicts, behavioral linkages across battlefields depend both on the technologies of conflict within each battlefield and the nature of economies or diseconomies in how battlefield out- comes and costs aggregate in determining payoffs in the overall conflict.Con ict, Contest, Battleeld, Colonel Blotto Game, Auction, Lottery

A Class of N-Player Colonel Blotto Games with Multidimensional Private Information

We consider a class of incomplete-information Colonel Blotto games in which N ≥ 2 agents are engaged in (N + 1) battlefields. An agent’s vector of battlefield valuations is drawn from a generalized sphere in Lp-space. We identify a Bayes-Nash equilibrium in which any agent’s resource allocation to a given battlefield is strictly monotone in the agent’s valuation of that battlefield. In contrast to the single-unit case, however, agents never enjoy any information rent. We also outline an extension to networks of Blotto games

A Class of N-Player Colonel Blotto Games with Multidimensional Private Information

In this paper, we study N-player Colonel Blotto games with incomplete information about battlefield valuations. Such games arise in job markets, research and development, electoral competition, security analysis, and conflict resolution. For M ≥ N + 1 battlefields, we identify a Bayes-Nash equilibrium in which the resource allocation to a given battlefield is strictly monotone in the valuation of that battlefield. We also explore extensions such as heterogeneous budgets, the case M ≤ N, full-support type distributions, and network games

Finding mixed-strategy equilibria of continuous-action games without gradients using randomized policy networks

We study the problem of computing an approximate Nash equilibrium of continuous-action game without access to gradients. Such game access is common in reinforcement learning settings, where the environment is typically treated as a black box. To tackle this problem, we apply zeroth-order optimization techniques that combine smoothed gradient estimators with equilibrium-finding dynamics. We model players' strategies using artificial neural networks. In particular, we use randomized policy networks to model mixed strategies. These take noise in addition to an observation as input and can flexibly represent arbitrary observation-dependent, continuous-action distributions. Being able to model such mixed strategies is crucial for tackling continuous-action games that lack pure-strategy equilibria. We evaluate the performance of our method using an approximation of the Nash convergence metric from game theory, which measures how much players can benefit from unilaterally changing their strategy. We apply our method to continuous Colonel Blotto games, single-item and multi-item auctions, and a visibility game. The experiments show that our method can quickly find high-quality approximate equilibria. Furthermore, they show that the dimensionality of the input noise is crucial for performance. To our knowledge, this paper is the first to solve general continuous-action games with unrestricted mixed strategies and without any gradient information