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Register a new userPlay Like the Pros? Solving the Game of Darts as a Dynamic Zero-Sum Game
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Chun Wang
Publication date
2020
fields
Mathematical Statistics
and research's language is
English
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The game of darts has enjoyed great growth over the past decade with the perception of darts moving from that of a pub game to a game that is regularly scheduled on prime-time television in many countries including the U.K., Germany, the Netherlands and Australia among others. In this paper we analyze a novel data-set on sixteen of the top professional darts players in the world during the 2019 season. We use this data-set to fit skill-models to the players and use the fitted models to understand the variation in skills across these players. We then formulate and solve the dynamic zero-sum-games (ZSGs) that darts players face and to the best of our knowledge we are the first to do so. Using the fitted skill models and our ZSG problem formulation we quantify the importance of playing strategically in darts. We are also able to analyze interesting specific game situations including some real-world situations that have been the subject of some debate among darts fans and experts.
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In this paper the extended model of Minority game (MG), incorporating variable number of agents and therefore called Grand Canonical, is used for prediction. We proved that the best MG-based predictor is constituted by a tremendously degenerated system, when only one agent is involved. The prediction is the most efficient if the agent is equipped with all strategies from the Full Strategy Space. Each of these filters is evaluated and, in each step, the best one is chosen. Despite the casual simplicity of the method its usefulness is invaluable in many cases including real problems. The significant power of the method lies in its ability to fast adaptation if lambda-GCMG modification is used. The success rate of prediction is sensitive to the properly set memory length. We considered the feasibility of prediction for the Minority and Majority games. These two games are driven by different dynamics when self-generated time series are considered. Both dynamics tend to be the same when a feedback effect is removed and an exogenous signal is applied.
Robots deployed in real-world environments should operate safely in a robust manner. In scenarios where an ego agent navigates in an environment with multiple other non-ego agents, two modes of safety are commonly proposed -- adversarial robustness and probabilistic constraint satisfaction. However, while the former is generally computationally intractable and leads to overconservative solutions, the latter typically relies on strong distributional assumptions and ignores strategic coupling between agents. To avoid these drawbacks, we present a novel formulation of robustness within the framework of general-sum dynamic game theory, modeled on defensive driving. More precisely, we prepend an adversarial phase to the ego agents cost function. That is, we prepend a time interval during which other agents are assumed to be temporarily distracted, in order to render the ego agents equilibrium trajectory robust against other agents potentially dangerous behavior during this time. We demonstrate the effectiveness of our new formulation in encoding safety via multiple traffic scenarios.
Computational advertising has been studied to design efficient marketing strategies that maximize the number of acquired customers. In an increased competitive market, however, a market leader (a leader) requires the acquisition of new customers as well as the retention of her loyal customers because there often exists a competitor (a follower) who tries to attract customers away from the market leader. In this paper, we formalize a new model called the Stackelberg budget allocation game with a bipartite influence model by extending a budget allocation problem over a bipartite graph to a Stackelberg game. To find a strong Stackelberg equilibrium, a standard solution concept of the Stackelberg game, we propose two algorithms: an approximation algorithm with provable guarantees and an efficient heuristic algorithm. In addition, for a special case where customers are disjoint, we propose an exact algorithm based on linear programming. Our experiments using real-world datasets demonstrate that our algorithms outperform a baseline algorithm even when the follower is a powerful competitor.
Machine learning processes, e.g. learning in games, can be viewed as non-linear dynamical systems. In general, such systems exhibit a wide spectrum of behaviors, ranging from stability/recurrence to the undesirable phenomena of chaos (or butterfly effect). Chaos captures sensitivity of round-off errors and can severely affect predictability and reproducibility of ML systems, but AI/ML communitys understanding of it remains rudimentary. It has a lot out there that await exploration. Recently, Cheung and Piliouras employed volume-expansion argument to show that Lyapunov chaos occurs in the cumulative payoff space, when some popular learning algorithms, including Multiplicative Weights Update (MWU), Follow-the-Regularized-Leader (FTRL) and Optimistic MWU (OMWU), are used in several subspaces of games, e.g. zero-sum, coordination or graphical constant-sum games. It is natural to ask: can these results generalize to much broader families of games? We take on a game decomposition approach and answer the question affirmatively. Among other results, we propose a notion of matrix domination and design a linear program, and use them to characterize bimatrix games where MWU is Lyapunov chaotic almost everywhere. Such family of games has positive Lebesgue measure in the bimatrix game space, indicating that chaos is a substantial issue of learning in games. For multi-player games, we present a local equivalence of volume change between general games and graphical games, which is used to perform volume and chaos analyses of MWU and OMWU in potential games.
We introduce a new non-zero-sum game of optimal stopping with asymmetric information. Given a stochastic process modelling the value of an asset, one player has full access to the information and observes the process completely, while the other player can access it only periodically at independent Poisson arrival times. The first one to stop receives a reward, different for each player, while the other one gets nothing. We study how each player balances the maximisation of gains against the maximisation of the likelihood of stopping before the opponent. In such a setup, driven by a Levy process with positive jumps, we not only prove the existence, but also explicitly construct a Nash equilibrium with values of the game written in terms of the scale function. Numerical illustrations with put-option payoffs are also provided to study the behaviour of the players strategies as well as the quantification of the value of information.
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