Play Like the Pros? Solving the Game of Darts as a Dynamic Zero-Sum Game
In recent years, the game of darts has experienced a surge in popularity and now has a substantial following in many countries including the U.K., Ireland, Germany, the Netherlands, and Australia among others. Indeed, a recent headline in The Economist (2020) proclaims “How darts flew from pastime to prime time” and how it’s not uncommon to have tens of thousands of fans attend darts tournaments today. 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. Following this surge in popularity, a recent paper by Haugh and Wang (2020) (hereafter HW) analyzes a novel data set on dart throws from sixteen of the top professional darts players in the world during the 2019 season. They use the data set to fit skill-models and to understand the variation in skills across the top darts players. They also formulate and solve the so-called dynamic zero-sum-games (ZSG’s) that darts players play thereby enabling them to quantify the importance of playing strategically in darts. But before describing some of this work, we must first remind ourselves of the scoring-system in darts and the rules of the game.
1 The Rules of Darts
A dartboard is displayed in Figure 1 and depending on where a dart lands on the board, a particular score is realized. The small concentric circles in the middle of the board define the “double bullseye” (DB) and the “single bullseye” (SB) regions. If a dart lands in the DB region (the small red circle) then it scores 50 points while a dart landing in the SB region (the green annulus surrounding the DB region) scores 25. Beyond the SB region, the dartboard is divided into 20 segments of an equal area corresponding to the numbers 1 to 20. If a dart lands in the “20” segment, for example, then it will land in the treble twenty (T20) region, the double twenty region (D20), or the single twenty region (S20) for scores of 60, 40, or 20, respectively. The double region is the region between the two outermost circles on the dartboard whilst the treble region is the region between the two circles beyond the SB region. The single region is then the union of the two disjoint regions between the SB and treble region and between the treble and double regions. If a dart lands beyond the double region then it scores zero.
Figure 1: A standard dartboard
The rules of darts that we describe here are for the most commonly played form of the game, namely “501”. This is the form of the game typically played in pubs and in most professional tournaments. In 501, each player starts on a score of 501 and takes turns in throwing three darts. These three darts constitute a “turn.” After a player’s turn, his score on the three darts are added together and subtracted from his score at the beginning of the turn. The first player to reach a score of exactly zero wins the game. In order to win, however, a player’s final dart must be a double, i.e. it must land on D1, D2, …, D20 or DB. If a player’s turn would result in an updated score of 1 or a negative score, then the turn is invalidated, the player remains on the score he had before the turn and his opponent then takes his turn. Note that it’s possible for a player to win without having to throw all three darts in his turn. For example, suppose a player has a score of 20 just before his turn. If he then scores D10 with the first dart of his turn he wins. Alternatively, if he missed D10 with his first dart and scored S10, then he could still win with the second dart of his turn by throwing a D5.
The game we have just described is known as a “leg” and in practice, darts matches are typically played over many legs with the winner being the first to win some fixed number of legs. Alternatively, some tournaments have a legs and “sets” structure whereby the winner of the match is the first to win a fixed number of sets and the winner of a set is the first to win a fixed number of legs. This latter structure then is similar to other sports such as tennis, for example. Finally, we note that because the player that throws first has an advantage, players alternate in starting legs (and sets). Regardless of the match structure, however, the key component of a darts match is the leg and it should be clear that a player will maximize his chances of winning a match if he optimizes his strategy for playing each leg.
2 A Skill Model for Throwing Darts
Taking the center of the DB region to be the origin, we will use µ ∈ R2 to denote the target of the dart throw and (x, y) ∈ R2 the outcome of the throw. Tibshirani, Price, and Taylor (2011) proposed several models for capturing the skill of a darts player but their main model assumed
where N2(µ, Σ) denotes the bivariate normal distribution with mean µ (the intended target) and covariance matrix Σ. They proposed an EM algorithm to estimate the covariance matrix Σ assuming the intended target µ was known for each dart-throw in their data-set but that only the realized score of the dart was observed rather than the realized location (x, y). This is also true of HW’s data-set.
A feature of professional darts players, however, is that a simple model like (1) is unlikely to be suitable for modeling their skills. This is because professional darts players tend to focus on (and practice throwing at) specific parts of the darts board, e.g. T20, T19, DB etc. This means that their skill levels, as determined by Σ, is likely to be a function of µ. Indeed this is what HW observed in their data. For example, the Dutch player Michael van Gerwen was successful 45.3% of the time when targeting T20 but only 30.2% of the time when targeting T17. (Van Gerwen is widely considered to be the best player in the world!) HW, therefore, partitioned the dart-board into a finite number of regions, say R1, . . . , RM, so that (1) can then be generalized to
Note that using a model such as (2) (or indeed (1)) allows us to consider any potential target µ on the dartboard when solving for the optimal strategies for the players.
2.1 Computing the Expected Dart Score as a Function of µ
Once we have fitted a player’s skill model (2) it is straightforward to compute the expected dart score as a function of µ. In Figure 2, for example, we have displayed heat-maps for the expected score as a function of µ for two of the professionals, namely Michael van Gerwen and Joe Cullen. The target that maximizes the expected score is identified via a green cross. In the case of Joe Cullen, we see his expected score is maximized by aiming at a point in the T19 region. In contrast, the optimal point for Michael van Gerwen (and indeed all of the other professionals in the data-set) is a point in the T20 region. That said, the difference in expected score between Cullen’s best target and his best target in the T20 region is only approx. 0.2.
Figure 2: Heat-maps displaying expected scores as a function of µ for Michael van Gerwen and Joe Cullen. The target with the highest expected score is identified via a green cross.
2.2 Comparing the Skill Profiles of the Players
Let’s briefly consider the skill model fitted to the region R6 which in HW’s work consisted of all the double regions on the board. So let [x y]T > ∼ N2(µ, Σ6) be the fitted skill model for a particular player for µ ∈ R6, i.e. the doubles region, where
By fitting a single skill model to all of the doubles in a zero-sum game it allows us to more easily understand the skill profiles of the players. For example, by comparing σx with σy we can understand which players are relatively more skillful along the x-axis versus the y-axis. This is important because the double regions have different orientations as may be seen from Figure 1 (or indeed Figure 2). For example, D20 should suit players with a relatively smaller σy than σx. This is because D20 runs in an east-west direction and as a target is, therefore, more forgiving of errors along the x-axis. In contrast, D11 runs in a north-south direction and should therefore be relatively better suited for players with a larger σy than σx. Similarly, consideration of both the magnitude and sign of the covariance term σx,y is also informative. For example, a player with a relatively large but negative σx,y would be relatively better off targeting doubles like D4 or D7 which run from northwest (NW) to southeast (SE). In contrast, a player with a relatively large but positive σx,y would be relatively better off targeting doubles like D9 or D15 which run from southwest (SW) to northeast (NE).
To give just one example from HW’s paper, we display the 95% confidence ellipses for Gerwyn Price and Peter Wright in Figures 3(a) and 3(b) below. (A 95% confidence ellipse means that if the target µ was the center of the ellipse then the dart-throw will fall inside the ellipse 95% of the time. The magnitude and orientation of the ellipse are completely determined by the estimated Σ for that player. For example, the smaller the area of the ellipse then the more skillful the player is.) We see their main skill directions run NW-SE for Price and SW-NE for Wright. This then translates into the pattern we see in Figure 3(c) where Price is superior on doubles that largely run SW-NE and Wright is superior on doubles that run NW-SE. This can be very important near the end of a leg when the players need to strategize over what double to try and “exit”, i.e. win, the leg on.
Note: Price is more skilled in NW-SE direction and Wright is more skilled in SW-NE direction. Figure 3: Price vs Wright When Targeting Doubles
3 Solving Dynamic Zero-Sum Games
Before proceeding, we need a little notation to denote the current state of a game between professional players A and B. Then the state of the game can be described by the vector (sA, sB, t, i, u) where sA and sB denote the scores of A and B, respectively, at the beginning of the current turn, t ∈ {A, B} denotes whose turns it is, i ∈ {1, 2, 3} denotes how many throws are left in the current turn, and u denotes the total score of the darts that have already been thrown within the current turn. So for example, if i = 3 then we must have u = 0 as there have been no throws yet within the turn.
After estimating the skill-models for the players, HW solves several interesting problems including solving for the optimal strategies that each player should follow. This involves formulating and solving dynamic zero-sum games (ZSG). (The “zero-sum” terminology refers to the fact that there is exactly one winner and one loser in any leg of darts.) Note also that playing darts is not just a matter of trying to reach zero as quickly as possible and that sometimes you need to take your opponent’s score into account. For example, suppose the current state of the game is (sA = 170, sB = 50, t = A, i = 1, u = 120). This means that player A has one throw remaining in his turn and if he hits DB with this throw then he will win the leg since DB is a double and 120+50 = 170. Indeed in this situation, it is typically optimal for A to aim for DB on his final throw since if he doesn’t, it’s likely B will win on his next turn since he just needs 40 to win. In contrast, if the state of the game is (sA = 170, sB = 150, t = A, i = 1, u = 120), then player B is unlikely to win on his next turn since scoring 150 in a turn (which must include exiting on a double) is challenging even for the pros. This means A doesn’t need to gamble and go for the DB on the final throw of his turn and so in this case it may well be optimal for him to aim instead at S10 and leave himself three attempts at D20 on his next turn (assuming B fails to exit from 150).
HW compute optimal strategies using the fitted skill-models for all the players and also quantify the importance of playing strategically in darts. But rather than going into those details, we’ll just describe one example from their paper where they consider an interesting situation that arose in a recent tournament.
3.1 Analysis of Wade vs White in the 2019 Austrian Open Semi-Final as a Zero-Sum game
This example from HW is based on s semi-final of the 2019 Austrian Open between James Wade and Peter White. This was a best-of 15 legs match and the score was tied at one leg apiece. In the third leg, White (player B) was on a score of sB = 161 and Wade (player A) was on sA = 127 with the latter about to begin his turn. His first two throws hit T20 and S17 leaving him needing 50 to exit and therefore win. Rather than aiming at DB on his third throw (which if successful meant he would win the leg), he instead aimed and hit S10 leaving White an opportunity to exit and therefore win the leg on his next turn. A video1 of this game is available on YouTube at Shanos (2019).
HW’s analysis of this situation may be seen in Figure 4 where each of the three heat-maps analyzes each of the three throws in Wade’s turn. It’s clear from Figures 4(a) and 4(b) that Wade’s first two throws were consistent with the optimal actions indicated by their zero-sum game model. Indeed at the beginning of his turn, we see from Figure 4(a) that Wade has a win-probability of 79.7% if he targets T20 with his first throw. Having succeeded in hitting T20, we see from Figure 4(b) that his win-probability has climbed to 85% if he now targets S17 with his second throw. And as it happens Wade did indeed target and hit S17 with this second throw. This brings us to Figure 4(c) where we see his win-probability is now 85.1% if he targets the DB with his final throw.
However, Wade opted for targeting S10 with his third throw and according to HW’s model, this resulted in his win-probability falling from 85.1% to 81.3%. This decision was quite surprising at the time and indeed the commentary on the YouTube video above suggested that in his recent play Wade had been successful in targeting the DB. The commentary also suggested, however, that some “mind games” might have been involved with Wade’s refusal to go for the DB implicitly telling White that he didn’t think he would succeed in exiting from 161. If such a message was part of Wade’s strategy then it may have been worthwhile going for S10 rather than DB but a pure “by-the-numbers” analysis suggests that he erred in not going for the DB! (As it happens, White failed to exit from 161 on his turn and Wade succeeded in exiting and winning the leg on his next turn.)
References for Darts and Zero-Sum Games
Haugh, Martin B., Chun Wang. 2020. Play like the pros? solving the game of darts as a dynamic zero-sum game
Shanos. 2019. Ian White vs. James Wade Austrian Darts Open 2019 Semi-Final. https://youtu. be/ve2gGuCGmHw?t=562. Accessed: 2020–10–21.
The Economist. 2020. How darts flew from pastime to prime time. URL https://www.economist. com/britain/2020/01/02/how-darts-flew-from-pastime-to-prime-time.
Tibshirani, Ryan J., Andrew Price, Jonathan Taylor. 2011. A statistician plays darts. Journal of the Royal Statistical Society: Series A (Statistics in Society) 174(1) 213–226.
Note: Optimal actions for first and second throws coincide with match decisions by Wade as a zero-sum game. On the third throw, Wade will win the leg with a probability of 85.1% if he targets DB or 81.3% if he targets S10.
Figure 4: Leg 3 of Wade (player A) vs White (player B) in SF of 2019 Austrian Open 6
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