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

Open-ended Learning in Symmetric Zero-sum Games

Zero-sum games such as chess and poker are, abstractly, functions that evaluate pairs of agents, for example labeling them `winner' and `loser'. If the game is approximately transitive, then self-play generates sequences of agents of increasing strength. However, nontransitive games, such as rock-paper-scissors, can exhibit strategic cycles, and there is no longer a clear objective -- we want agents to increase in strength, but against whom is unclear. In this paper, we introduce a geometric framework for formulating agent objectives in zero-sum games, in order to construct adaptive sequences of objectives that yield open-ended learning. The framework allows us to reason about population performance in nontransitive games, and enables the development of a new algorithm (rectified Nash response, PSRO_rN) that uses game-theoretic niching to construct diverse populations of effective agents, producing a stronger set of agents than existing algorithms. We apply PSRO_rN to two highly nontransitive resource allocation games and find that PSRO_rN consistently outperforms the existing alternatives.Comment: ICML 2019, final versio

The Economics of Lotteries: An Annotated Bibliography

This paper presents an annotated bibliography of all papers relating to the economics of lotteries as of early to mid 2011. All published scholarly papers that could be identified by the authors are included along with the published abstract where available.lotto, lottery, public finance, gambling

Colonel Blotto games with a head start

This paper studies Colonel Blotto games with two battlefields where one player has a head start in the form of additional troops on one of the battlefields. Such games arise naturally in marketing, electoral competition, and military conflict. Sion and Wolfe (1957) have shown that, if the strategy space is continuous, a mixed-strategy Nash equilibrium need not exist. Therefore, we consider a finite approximation. Using the iterated elimination of (weakly) dominated strategies, we identify an equilibrium for all parameter constellations and discuss its uniqueness properties. In equilibrium, resource decisions are typically not uniform but tend to concern units that roughly correspond in size to multiples of the head start. Moreover, competition takes the form of a hide-and-seek game, where the favorite tries to outguess the number of units that the underdog commits to the balanced battlefield. Somewhat unexpectedly, equilibrium payoffs of finite approximations of the Sion-Wolfe game accumulate around precisely three values. We also discuss the relation to the model with heterogeneous budgets but no head start

Strategically Revealing Intentions in General Lotto Games

Strategic decision-making in uncertain and adversarial environments is crucial for the security of modern systems and infrastructures. A salient feature of many optimal decision-making policies is a level of unpredictability, or randomness, which helps to keep an adversary uncertain about the system’s behavior. This paper seeks to explore decision-making policies on the other end of the spectrum – namely, whether there are benefits in revealing one’s strategic intentions to an opponent before engaging in competition.We study these scenarios in a well-studied model of competitive resource allocation problem known as General Lotto games. In the classic formulation, two competing players simultaneously allocate their assets to a set of battlefields, and the resulting payoffs are derived in a zero-sum fashion. Here, we consider a multi-step extension where one of the players has the option to publicly pre-commit assets in a binding fashion to battlefields before play begins. In response, the opponent decides which of these battlefields to secure (or abandon) by matching the pre-commitment with its own assets. They then engage in a General Lotto game over the remaining set of battlefields. Interestingly, this paper highlights many scenarios where strategically revealing intentions can actually significantly improve one’s payoff. This runs contrary to the conventional wisdom that randomness should be a central component of decision-making in adversarial environments

Strategically Revealing Intentions in General Lotto Games

Strategic decision-making in uncertain and adversarial environments is crucial for the security of modern systems and infrastructures. A salient feature of many optimal decision-making policies is a level of unpredictability, or randomness, which helps to keep an adversary uncertain about the system’s behavior. This paper seeks to explore decision-making policies on the other end of the spectrum – namely, whether there are benefits in revealing one’s strategic intentions to an opponent before engaging in competition.We study these scenarios in a well-studied model of competitive resource allocation problem known as General Lotto games. In the classic formulation, two competing players simultaneously allocate their assets to a set of battlefields, and the resulting payoffs are derived in a zero-sum fashion. Here, we consider a multi-step extension where one of the players has the option to publicly pre-commit assets in a binding fashion to battlefields before play begins. In response, the opponent decides which of these battlefields to secure (or abandon) by matching the pre-commitment with its own assets. They then engage in a General Lotto game over the remaining set of battlefields. Interestingly, this paper highlights many scenarios where strategically revealing intentions can actually significantly improve one’s payoff. This runs contrary to the conventional wisdom that randomness should be a central component of decision-making in adversarial environments