Approximate Equilibria in Non-constant-sum Colonel Blotto and Lottery Blotto Games with Large Numbers of Battlefields

In the Colonel Blotto game, two players with a fixed budget simultaneously allocate their resources across n battlefields to maximize the aggregate value gained from the battlefields where they have the higher allocation. Despite its long-standing history and important applicability, the Colonel Blotto game still lacks a complete Nash equilibrium characterization in its most general form-the non-constant-sum version with asymmetric players and heterogeneous battlefields. In this work, we propose a simply-constructed class of strategies-the independently uniform strategies-and we prove them to be approximate equilibria of the non-constant-sum Colonel Blotto game; moreover, we also characterize the approximation error according to the game's parameters. We also introduce an extension called the Lottery Blotto game, with stochastic winner-determination rules allowing more flexibility in modeling practical contexts. We prove that the proposed strategies are also approximate equilibria of the Lottery Blotto game

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

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

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

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

Path Planning Problems with Side Observations-When Colonels Play Hide-and-Seek

Resource allocation games such as the famous Colonel Blotto (CB) and Hide-and-Seek (HS) games are often used to model a large variety of practical problems, but only in their one-shot versions. Indeed, due to their extremely large strategy space, it remains an open question how one can efficiently learn in these games. In this work, we show that the online CB and HS games can be cast as path planning problems with side-observations (SOPPP): at each stage, a learner chooses a path on a directed acyclic graph and suffers the sum of losses that are adversarially assigned to the corresponding edges; and she then receives semi-bandit feedback with side-observations (i.e., she observes the losses on the chosen edges plus some others). We propose a novel algorithm, EXP3-OE, the first-of-its-kind with guaranteed efficient running time for SOPPP without requiring any auxiliary oracle. We provide an expected-regret bound of EXP3-OE in SOPPP matching the order of the best benchmark in the literature. Moreover, we introduce additional assumptions on the observability model under which we can further improve the regret bounds of EXP3-OE. We illustrate the benefit of using EXP3-OE in SOPPP by applying it to the online CB and HS games.Comment: Previously, this work appeared as arXiv:1911.09023 which was mistakenly submitted as a new article (has been submitted to be withdrawn). This is a preprint of the work published in Proceedings of the 34th AAAI Conference on Artificial Intelligence (AAAI