Play Like the Pros? Solving the Game of Darts as a Dynamic Zero-Sum Game Martin B. Haugh Department of Analytics, Marketing & Operations Imperial College Business School, Imperial College [email protected] Chun Wang Department of Management Science and Engineering School of Economics and Management, Tsinghua University [email protected] November 24, 2020 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 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 (ZSG's) 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 arXiv:2011.11031v1 [stat.AP] 22 Nov 2020 darts fans and experts. 1 1 Introduction and Literature Review In recent years the game of darts has experienced a surge in popularity with the game now having a substantial presence in many countries including the U.K., Ireland, Germany, the Netherlands and Australia among others. Indeed a recent headline in The Economist(2020) explains \How darts flew from pastime to prime time" and how it's not uncommon today to have tens of thousands of fans attend darts tournaments. The game is also becoming increasingly attractive to women as evidenced by the exploits of Fallon Sherrock who in 2019 became the first woman to win a match at the PDC World Darts Championship. The aforementioned article from The Economist also notes how in recent years darts has become the second-most-watched sport over the Christmas period on Sky Sports1 in the U.K., coming second only to soccer. Further growth is also anticipated with professional tournaments now taking place in Shanghai and scheduled to take place in Madison Square Garden in New York in 2021. At its core, darts is a game between two players who both start on a fixed score (typically 501) and take it in turns to throw darts at a dartboard approximately 8 feet away. Loosely speaking, each dart scores a number of points depending on where it lands on the board and these points are then subtracted from the player's score to yield an updated score. The first player to have an updated score of exactly zero wins the game or leg but the winning dart must be a so-called double. Darts is therefore a strategic game and the strategic component is two-fold: 1. Where should the player aim in order to minimize the expected number of throws (or \turns" as we shall see) to get to zero? This is a dynamic programming (DP) problem and optimal policies for this problem take no account of the score of his / her (hereafter \his") opponent. 2. How should the player adjust his DP strategy to account for his opponent's current score? This is a game-theoretic question and leads to formulating the game as a dynamic zero-sum game (ZSG). In order to distinguish between the two strategic components we will only use the term \strategic" when referring to the ZSG policies and in fact, will generally use the term \non-strategic" when referring to DP policies. But of course there is a strategic component to the DP problem and in fact it's worth noting at this point that the optimal DP strategy for a player will often coincide with his optimal ZSG strategy. To date there has been little analytic work on the game of darts in the literature. The earliest paper that we are aware of is the work by Kohler(1982) who uses a DP formulation and a branch- and-bound approach to solve a DP version of the game. Stern and Wilcox(1997) and Percy(1999) considered the easier problem of where on the dartboard to aim in order to maximize the score of a dart. The former used the Weibull distribution to model and fit dart throws. More recently Tibshirani et al.(2011) proposed several models based on the Gaussian and skew Gaussian bivariate distributions to model the throwing skills of players. They use the EM algorithm together with importance sampling to fit these distributions to dart-throwing data. While we borrow and extend the skill modeling approach of Tibshirani et al.(2011) (hereafter TPT), we extend the literature in several directions. First, we present a novel data-set based on the throws of 16 top professional players from the 2019 season and we use this data to fit a simple extension of TPT's bivariate normal skill model and to analyze the throwing skills of the top pros. It is particularly interesting to note the variation in fitted skill models across the players, particularly as it pertains to their abilities to hit \doubles". For example, some players have smaller 1Sky Sports is the premium sports television channel in the U.K. and Ireland and is comparable to ESPN in the U.S. They are the more-or-less exclusive providers of live Premier League soccer in the U.K., the globally most popular league of the most popular sport in the world. 2 error variances along the x-coordinate whereas other players have smaller variances along the y- coordinate. We also observe that the magnitude and sign of the covariance between the x and y errors can vary quite significantly across the players. Taken together, these idiosyncrasies explain why different players prefer to target (or should prefer to target) different doubles on the dartboard. Second, we formulate and solve the game of darts as a dynamic ZSG and to the best of our knowledge we are the first to do this. Using the fitted skill models, we construct optimal DP, best-response (BR) and equilibrium strategies for each of the players. We quantify the difference in win-probability between playing a DP strategy instead of a BR strategy against a player playing a DP strategy. We note that over a single leg this difference is quite small and on the order of approx. 0.2% - 0.6% for the top players. However, a darts match is typically played across many legs and we note that this 0.2% - 0.6% difference across a single leg can translate to a difference of as much as 2.3% across a best-of-35 legs match which is typical in many big tournaments. A third contribution is our use of dartboard heat-maps for displaying strategic information in various game scenarios. While TSS also used2 dartboard heat-maps, their focus was on analyzing the expected dart score as a function of the target. Because we compute optimal strategies in this paper we are also able to use heat-maps to display the so-called Q-values3 for each potential target on the dartboard as well as other quantities of interest. Our use of these heat-maps also enables us to analyze visually interesting game situations that have arisen in real-world tournaments. Given the increasing significance of analytics throughout sports, we believe it's quite possible that the darts world will at some point adopt the heat-map approach to displaying information. One problem that we don't consider here is estimating just how close the professional players are to playing optimally. We don't tackle this problem because we don't have specific throw-by-throw match-play data where we can see the sequence of throws and scores in each match. If we had this data then we could easily estimate the extent of sub-optimal play by the top players using the framework we present here. That said, we do look at some interesting specific real-world situations later in the paper and find evidence that even the best players in the world do sometimes play sub-optimally. Nonetheless, a more thorough analysis based on superior data would be required to confirm this and quantify the loss in win-probability from playing sub-optimally. Finally we note that the OR / MS community has taken a great interest in data analytics over the past couple of decades and concurrent with this has been an increased interest in sports analytics. Some examples of work in the field include Kaplan and Garstka(2001) on office pools motivated by the \March madness" NCAA basketball tournament, Kaplan et al.(2014) on ice hockey, Che and Hendershott(2008) on the NFL and more recently Hunter et al.(2019), Bergman and Imbrogno(2017) and Haugh and Singal(2020) on fantasy sports. We see this paper as adding to this growing literature on OR / MS applications in sports. The remainder of this paper is organized as follows. In Section2 we explain the rules of darts while in Section3 we describe our professional darts data-set. Section4 explains the skill-model we use for throwing darts and we discuss the estimation of this model in Section5 where we also use the fitted models to compare the skills of the different professional players.