Play Like the Pros? Solving the Game of Darts as a Dynamic Zero-Sum Game

Abstract

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 such as the United Kingdom, Germany, the Netherlands, and Australia, among others. It involves strategic interactions between two players, but to date, the literature has ignored these interactions. In this paper, we formulate and solve the game of darts as a dynamic zero-sum game (ZSG), and to the best of our knowledge, we are the first to do so. We also estimate individual skill models using a novel data set based on darts matches that were played by the top 16 professional players in the world during the 2019 season. Using the fitted skill models and our ZSG problem formulation, we quantify the importance of playing strategically—that is, taking into account the score and strategy of one’s opponent—when computing an optimal strategy. For top professionals, we find that playing strategically results in an increase in win probability of just 0.2%–0.6% over a single leg but as much as 2.2% over a best-of-31-legs match. Summary of Contribution: Dynamic zero-sum games (ZSGs) are of considerable interest, as they arise in many applications including sports, the management of communication networks, interdiction games, and heads-up poker—an important topic in modern artificial intelligence. In this study we consider the game of darts, which is growing increasingly popular around the world today. We formulate the game of darts as a ZSG and solve it iteratively by formulating each player’s best-response problem as a stochastic shortest-path (SSP) problem. We then solve these SSPs using standard dynamic programming methods. In solving the ZSG, we are able to accurately quantify the importance of top professionals playing strategically.