21st ECMI Modelling Week Final Report

21st ECMI Modelling Week Final Report 2007.08.26 2007.09.02 Team 9 Stategies for darts Borset Kari Norwegian University of Science and Technology, Trondheim Hemmelmeir Claudia Johannes Kepler Universität, A-4040 Linz, Altenberger Str. 69, Austria Klotz Christian TU Eindhoven, The Netherlands Magdziarz Marcin Wroclaw University of Technology, Poland Mas Blesa Albert Universitat Autï 1 noma de Barcelona, Spain 2 Instructor: Martin Frank TU Kaiserslautern, Germany 1

Abstract Playing darts is a very popular game and also a real sport. For a good player itï 1 2 s clear what strategy he should prefer, to aim the triple 20, but whatï 1 2 s the strategy for a not so good or a really bad player? To get a mathematical model we deal with the problem in a probailistic way, seperated into two phases and solve one of these by integration and simulation. Contents 1 Introduction 3 1.1 Rules for playing darts............................. 3 1.2 Assumptions................................... 3 2 Maximization of average scoring 3 2.1 Determination of the expectation value.................... 4 2.2 Results...................................... 4 3 Quantiles play safe or risk? 6 3.1 Determination of the quantiles......................... 6 3.2 Results...................................... 6 4 The doubling out 8 5 A discrete approach 10 5.1 Selection of aiming points........................... 10 5.2 Formulation of the optimization problem................... 11 5.3 Determination of the strategy......................... 11 5.4 Results...................................... 13 2

1 Introduction When amateurs play dart, they often aim for the center of the dart board or at the triple 20. The reason for this is probably because these points represents the aiming point where you are most likely to hit the board and the point with the highest score. In this project we have tried to find what the optimal strategy is for an average Joe - where he should aim for at the different stages of the game to minimize the number of shots. 1.1 Rules for playing darts Evidently, the dartboard measures and the distance between itself and the player are standard for all darts games, so we just have to comment the rules for playing the 501 game. Every player starts with a score of 501 and has to throw 3 darts per turn. The score for each turn is calculated and deducted from the players total score. Bullseye scores 50, the outer ring scores 25 and a dart in the double or triple ring counts double or triple the segment score. The objective is to be the first player (to use a minimal number of turns) to reduce the score to exactly zero with the shortest time. The main difficulty arises in what we call the doubling out. The dart that brings the score down to zero must be a double. Doubles consist of the numbers in the outside narrow scoring band and the center (small) bullseye. If a player reduces the score to 1 or goes below zero, the score is bust, that turn ends immediately and the score is returned to what it was at the start of that turn. For example if a player has 32 to go out and the first dart is a 16, the second is a 15, the player is bust and the score is returned to 32. So on the last turn, it is not necessary to throw all 3 darts (a player can win with the first or second dart of the turn). 1.2 Assumptions Because we had no real data of people playing darts and their playing distributions, we have done all our simulations with the two dimensional normaldistriution, where the variances in each direction are independent of each other, which is as we thought the best to estimate a real distribution.the standard deviation tells us how good the player is. But as we see in practise, no one of us five has a normal distributed hitting. The important thing is, that all of our calculations and simulations can be done with any other distriution. 2 Maximization of average scoring In the first phase of the game, the aim of the player is to gain as much score as possible. In this section this is approached by searching the aiming point with the highest expected score. As this depends highly on the skills of the player, the results are presented for a choice of example players. 3

2.1 Determination of the expectation value For a given aiming point p, the hitting point is normally distributed with expectation value p and a variance that depends on the players skills. Therefor the distribution of the hitting point has the density function of the two dimensional independent normal disribution: f p = 1 2πσ x σ y e 1 2( x px σx )2 1 2 ( y py σy ) 2 (2.1) The expectation value of the score E[ς(p)] is defined as the integral: E[ς(p)] = ς(x, y)f p (x, y) (x, y) (2.2) R 2 The score is 0 outside the dartboard and constant on the different fields of the board. So if you divide the dartboard into the fields (A i ) 62 i=1, equation (2.2) can be rewritten: 62 E[ς(p)] = ς i i=1 A i f p (x, y) (x, y) (2.3) All the fields of the dartboard can easily be described with polar coordinates. The transformation writes as: ( ) ( ) x sin(φ) = r y cos(φ) This yields: 62 E[ς(p)] = i=1 ς i ri max r=r min i φ max i φ=φ min i rf p (r sin(φ), r cos(φ)) φ r (2.4) The integral of equation (2.4) is then evaluated numerically by means of the MATLAB - function dblquad. 2.2 Results The values of E[ς(p)] can now be compared all over the dartboard. Then the point p max, where E[ς(p)] reaches its maximum, is determined with the routine fminsearch from the MATLAB - Optimization toolbox. In this context we consider 4 players with a standard deviation for x and y of 10, 20, 30 and 45 mm. Figure 1 displays the situation for a quite good player who won t often miss his target by more than 1 cm. Therefor he has the best result when he tries to hit the triple 20, which gives him an average score of 29.5. Figure 1: Expectation value of the score for a standard deviation of 10, the star marks the optimum 4

The situation is different for a player with a standard deviation of 20. Picture 2 shows that the expected value is much smaller and reaches its maximum of 17.8 points near triple 19. For him it doesn t matter a lot if he chooses triple 19 or 20 since the expectation values differ only by less than 0.5 points. Figure 2: Expectation value of the score for a standard deviation of 20, the star marks the optimum For players who are worse than that, aiming for triple 20 is not recommendable as it can be seen in figure 3. They should prefer the southwest of the board, e.g the area between triple 16 and 19. This is also obvious in figure 4. Figure 3: Expectation value of the score for a standard deviation of 30, the star marks the optimum Figure 4: Expectation value of the score for a standard deviation of 45, the star marks the optimum The reason is, that the chance to hit the neighbours of the aiming field is high, and the average score of the neighbours of 20 is just 3, while it is 5 for 19. With that method, a player with 3 cm deviation can obtain an average score of 14.7, and a player with 4.5 cm can still reach 13.2. Then, the more the standard deviation increases, the more the optimal point tends to the bullseye, as arrows who miss the board and thus score 0 start to become important. Table 1 gives an overview: σ x = σ y x max y max E[ς(p)] remarks 10-1 103 29.5 aim for triple 20 20-37 -101 17.8 aim for triple 19 30-43 -93 14.7 aim for triple 19 / triple 7 45-52 -56 13.2 aim for southwest 80-16 -5 11.3 19 but very central 130-8 -2 7.3 just try to hit the board at all 170-7 -2 5.0 same Table 1: overview In Figure 5 you can see the development of the expected score for some points. For all but one of them, the average score is monotoneously decreasing with decreasing accuracy, except the score of the southwest point, which becomes better with sinking accuracy. The reason is that this point has a score of only 7, but is surrounded by 19 and 16. So accurate 5

shots get less points than inaccurate, which will hit in an area with high score values. Also it turns ot that from a deviation of 70, the best you can do is go for the bull. Figure 5: Expected score of some aiming points 3 Quantiles play safe or risk? In the previous section we have determined the aiming point with the highest expected score depending on the skills of the player. However, the situation during the game varies. Once we have a big advantage over our opponent, in order to keep the lead, we tend to play safe. On the other hand, when we are far behind the other player, we are more willing to risk, so as to reduce the difference. To set the strategy in these two particular cases, we need to extend our knowledge about the distribution of the score. In what follows, we determine quantiles of the score distribution for a choice of player skills analogous to the one from the previous section. The results will give us some insight into the strategy for certain situations during the game. 3.1 Determination of the quantiles To determine quantiles of the score distribution, we use the Monte Carlo simulations technique. Recall that for a given aiming point p = (p x, p y ), the hitting point h = (h x, h y ) has two-dimensional normal distribution, with independent coordinates h x N(p x, σ x ) and h y N(p y, σ y ). The parameters σ x and σ y depend on the skill of the player. Thus, in order to simulate numerically a single throw, we generate two independent normally distributed random variables N(p x, σ x ) and N(p y, σ y ), which give us coordinates of the hit. Fig. 6 illustrates exemplary results of 500 throw simulations for different parameters (σ x, σ y ). Figure 6: Exemplary results of simulation of 500 throws for different parameters σ x and σ y. In what follows, for each point p from the grid P = {(x, y) : x, y { 170, 160,..., 160, 170}} and for fixed parameters (σ x, σ y ), we simulate 10 4 independent throws. Basing on the obtained random sample, we determine the quantiles of the score distribution. 3.2 Results To set the strategy in the case when the player tends to play safe, we analyze the 40% quantiles. The area, in which the 40% quantiles reach their maximum, is a reasonable 6

target, when the player is winning and wants to control the situation. With such strategy, it is less likely to hit the low values. On the other hand, if the opponent is more experienced or obtained a considerable advantage, it is reasonable to risk and to aim for the higher values. In such a case, 70% quantiles of the score distribution can be very helpful in setting the strategy. Aiming for the area in which the 70% quantiles reach their maximum, makes it more likely to hit the sequence of high values. First, let us consider the case σ x = σ y = 10. The computation and analysis of the quantiles is just the confirmation of the statement, that being such a good player, it is always the best strategy to aim for triple 20. The situation is a bit different for the case σ x = σ y = 20. Then, if we want to play safe, the best strategy is to aim for central 20. However, a more risky player will rather aim for triple 20 and the area around, as the plot of 70% quantiles indicates. Different conclusions are drawn from the analysis of the case σ x = σ y = 30. As the 40% quantile plot indicates, the best safe strategy is to aim for the west side of the board. This is just the confirmation of the well known rule that being an average player it is best to aim for the left side of the dart board. However, if an average player is forced to risk, the best strategy is to aim for triple 20, as the 70% plot indicates. Aiming for the west side of the board is also the best safe strategy for the absolute beginners (σ x = σ y = 45). Nevertheless, an unexpected conclusions can be drawn from the analysis of the 70% quantiles. Apparently, a weak player should aim for 1 or 3, if he wants to score the high number. Such conclusions seem to be senseless, since 1 and 3 are very small numbers. However, the explanation of this phenomenon is the following: if you are a really weak player and you aim for 1, you ll probably miss and hit 20 or 18 instead. Figure 7: 40% and 70% quantiles of the score for the standard deviation σ x = σ y = 10. Figure 8: 40% and 70% quantiles of the score for the standard deviation σ x = σ y = 20. Figure 9: 40% and 70% quantiles of the score for the standard deviation σ x = σ y = 30. Figure 10: 40% and 70% quantiles of the score for the standard deviation σ x = σ y = 45. In Table 3.2 we summarize all the conclusions and strategies drawn from the analysis of the quantiles. 7

σ x = σ y play safe risk 10 aim for triple 20 aim for triple 20 20 aim for central 20 aim for triple 20 and the area around 30 aim for west aim for triple 20 45 aim for middle west aim for 1 or 3 Table 2: Overview of the conclusions drawn from the analysis of quantiles of the score distribution. 4 The doubling out A heuristic argument for the last part of the game is to make a decision tree to make the best aiming point for the next dart. In this decision tree we have not taken into account weather it is the first, second or third dart in that round. This is because we assume that the player is so bad that he will need a lot of shots to make the final doubling out anyway, so he doesn t make a special strategy or make a specific path for the ending, but only optimize the possibility to have the chance to double out. Obviously the player needs to have an even number to double out, and the largest double is the bulls eye. When aiming at the bulls eye, the chance of hitting an odd number is approximately 50% if he or she miss and then an even number has to be obtained again to try to double out. When aiming at the doubles in the outer circle, he/she have approximately 25% chance of hitting an odd number of the board, 25% chance of hitting an even number on the board and 50% to be on the same score, i.e. 75% chance to land at an even number and 25% chance of an odd number. Therefore we exclude that the player should aim at the bulls eye, and the sum has to be below 40 to double out. If we divide the numbers in two groups, the small numbers going from 1 to 10 and the big ones from 11 to 20, we see that on the board they are roughly placed so that we have one small, then one large and then one small and so on. The player has three arrows per round, and if he for example get more points on the first arrow than his score, he loose the rest of the shots. This means that it is better to aim for a doubling out at a large number (below 40) not to loose to many throws. Our idea was then to make a minimization statement that maximized the chance of landing at an even score and to have a fairly high score below 40. Figure 11 shows the different cases we have in this part of the game. To optimize getting a score in the range from 20 to 40 and also have an even number, we made a formula shown in equation 4.1. This formula is based in the arguing made before. p aim = max [α P (odd/even value aim for p) + (1 α)p (S 40 score(p) S 20)] p (4.1) There are two weighted probabilities in the formula. One is the probability to land on an odd/even number and the other one is the probability to land on a score between 20 8

and 40. The weighting constant α depends on the case we re in. To understand better the meaning of α let us suppose that the player has an odd score between 40 and 40 + E[P ], where E[P ] is the biggest mean expected value. If this score is near 40 (41 or 43, for example) it might be better to aim to an odd number than trying to keep near 40, because if he shots and gets an even number probably he will be below 40 with an odd score, so he will have to spend some darts on changing his score to an even number before the double out and his resulting score would be very small. On the other side, if his score is near 40 + E[P ] he might try to keep between 20 and 40 (that is, between double 10 and double 20) after his shoot, in order to have more opportunities to make the doubling out. But obviously it is important to obtain an odd number too. The role of α is to take into account this two situations, so in the first one α must be bigger than in the second one. One can use similar arguments to argue the even case. After this discussion we can state the recursive algorithm based on the formula 4.1 which solves the last part of the darts game 501. We have to remark that we are assuming the player does not know so much on playing darts, so he can not make a plan of where their three darts must go before shooting the first one, he is not accurate. First of all, the player must check if his score S is even or odd and if it is below 40 or not. Then, if: S 40 and even: The player must aim to the double out. S 40 and odd: Use the formula 4.1 with α = 1 and with the odd version in the first probability to find the point to aim in the next shot. S > 40 and odd: Use the formula 4.1 with α = 0, 8 and with the odd version in the first probability to find the expected score for the optimal odd aiming point, which is called E[O]. Then, in order to quantify if the score is near 40 or it is not, check if S (40, 40 + E[O]). If it is, then the player must aim to the p aim found with 4.1. If S > 40 + E[O], use 4.1 with α = 0, 5 and find a new p aim which takes into account that the player s score should be as big as possible below 40. S > 40 and even: Use the same method as the preceding case changing odd by even and with α = 0, 8. If S > 40 + E[E] (where E[E] is the expected score for the optimal even aiming point) use 4.1 with α = 0, 5 to find the new p aim. After the shot, the player has to check in what case his new score is and repeat the method. 9

Figure 11: Decision tree in the last part of the game. S is the current score, E[P] is the expected value of one shot in the first part of the game, E[O] is the expected score for the optimal odd aiming point and E[E] is the expected score of the optimal even aiming point 5 A discrete approach In the previous sections we took the whole continuum of the dartboard as possible aiming points. With this approach it was possible to maximize the expected score or certain quantiles of the score distribution. But this approach makes it difficult to determine optimal strategies especially for the ending phase of the game, where the distribution of whole series of throws start to play an important role. Therefor in this section, we only consider a discrete set of possible aiming points, and then formulate a solvable discrete optimization problem. Furthermore we simplified the problem a little bit, so that this section is more like a conceptual study of what is possible. 5.1 Selection of aiming points For the sake of simplicity we only chose 105 possible aiming points. We took the inner bulls eye, 4 points of the outer bulls eye, and the middle point of each other field on the dartboard. We also included points outside the board, which reduce the danger of getting busted. The choice of points is illustrated in figure??. In order to reference them, they are given short names as follows: 10

name meaning 20is inner section of 20 8os outer section of 8 9x3 triple 9 12x2 double 12 50ib inner bullseye 12mb 12 miss board (outside)...... 5.2 Formulation of the optimization problem First of all, to reduce the complexity of the problem.we dont want to regard risk management here, which means, our strategy will not regard the opponents score. Furthermore we ignore the fact, that each player has 3 darts which he can throw, but assume he has only one, and afterwards the opponents will have their turns. This influences the strategy, because there is the danger of getting busted. If you hit a score which is higher than your actual number of points left, you loose the rest of your turns. Therefor the strategy has to consider this. Definition 1 (strategy) A strategy is a mapping S : N T. With: T = {t j j = 1,..., n} the set of target points The strategy tells you for each score, at which point to aim next. This is a very simple model, as the strategy does not incorporate information about the opponents, or how many darts you have left. Now, in order to find an optimal startegy, a measure of success is defined in order to compare the different options. As a simple option we take the expected number of throws left to finish. For each throw the player has the choice between 105 aiming points, and he chooses the one that gives him the lowest expected value of darts left to finish the game, assumed that all the following throws will also be determined by that strategy. So this definition is somehow recursive. Let s N: S opt (s) := arg min t T E [throws left if for all the following throws you take S optagain] (5.1) This definition is reasonable because at each score the player will like to finish as quick as possible. This startegy can be defined from the bottom to the top. If your score is zero, the expected number of throws left is zero. If you have 2 points left, your only possiblity to double out and finish is to hit double one. So you will aim at the point that maximizes the probability of hitting 2x1. So one can start to determine the startegy for low score, and then proceed to the higher scores with the help of the data yet available. 5.3 Determination of the strategy First we compute the following probability matrix: P = (p ij ) p ij = P robability(hit field i aim at point j) (5.2) 11

Then we initialize the list of expected values: E 0 := 0. Assume now s N is the actual score, the startegy has to be calculated for.e 0,..., E s 1 and S(2),..., (s 1)are already given. The task is now to calculate S(s). There are options where to aim: t 1,..., t 105. We will calculate the expected value: Then we take the optimal target as S(s) E j := E[throws left to finish S(s) = t j ] (5.3) S(s) := arg min E j (5.4) j=1,...,105 So the task is to find a way to calculate E j. Let s, j be fixed. You have s points and your aim is t j. Define the set trans := {i you score decreases if you hit i} and miss as its complement. You might hit each field i at the board with the probability p ij. Therefor the probability of a score transition calculates as follows: p trans = and i trans p ij p miss = 1 p trans. Let E trans be the expected value of rounds to go if you hit a field of trans, and E miss the opposite. It can be calculated by the following equation: Then we find: E trans = 1 p trans i trans E j = (1 + E trans )p trans + (1 + E miss )p miss p ij E s score(i) (5.5) If you hit a field of miss, your score doesn t change. Therefor E miss = E j because you are in state s again. Thus: And because p trans + p miss = 1, we get: E j (1 p miss ) = (1 + E trans )p trans + p miss E j = 1 p trans + E trans (5.6) With the help of this method we can calculate all the E j and therefor choose the optimal strategy S(s). In that way we start at s = 2 and then wind up the score step by step. The complexity presented algorithm is linear in the number of options, and for each s N the effort to calculate S(s) is uniformly bounded by a constant, if all the former startegy and expectation values are already fixed. 12

5.4 Results Exemplarily we computed the optimal startegies for a very good player(σ = 10) and a quite bad player(σ = 45). Its interestig to see that for high scores, the recommendation for the skilled player is 3x20 and for the poor player its 16x2, because this coincides with the results of the former section. But the good player has to change hist strategy at around 100 points to prepare his doubling out, which is surprisingly early. The worse player can score on until he has reached 70 points before preparing to double out. The calculations were really fast. The biggest effort was the computation of the big probability matrix, and then it was not too costy to compute the startegy up to 501 points. All that only took around 10 minutes on a PentiumIV. The concept presented here was very simple. In order to be more realistic, one should consider that one turn consists of 3 throws. That makes the strategy depending on the throws left in the actual round. One could also think of other measures of success which could also encorporate risk management. Finally the number of aims should be raised. Nevertheless we could illustrate a way to gain optimal strategies for darts, and more complex versions could base on the same principles. 13

Points 2 3 4 5 6 7 8 9 10 11 Recommendation 1x2 1os 2x2 1os 3x2 5os 4x2 1os 5x2 3os Points 12 13 14 15 16 17 18 19 20 21 Recommendation 6x2 11os 7x2 13os 8x2 13os 9x2 17os 10x2 19os Points 22 23 24 25 26 27 28 29 30 31 Recommendation 11x2 19os 12x2 17os 13x2 19os 14x2 13os 15x2 15os Points 32 33 34 35 36 37 38 39 40 41 Recommendation 16x2 17os 17x2 19os 18x2 5os 19x2 7os 20x2 9os Points 42 43 44 45 46 47 48 49 50 51 Recommendation 10os 11os 12os 13os 14os 15os 16os 17os 18os 19os Points 52 53 54 55 56 57 58 59 60 61 Recommendation 20os 13os 14os 15os 16os 17os 18os 19os 20os 50ib Points 62 63 64 65 66 67 68 69 70 71 Recommendation 10x3 13x3 16x3 50ib 14x3 17x3 20x3 19x3 18x3 13x3 Points 72 73 74 75 76 77 78 79 80 81 Recommendation 20x3 19x3 14x3 17x3 20x3 19x3 18x3 19x3 20x3 19x3 Points 82 83 84 85 86 87 88 89 90 91 Recommendation 50ib 17x3 50ib 50ib 18x3 17x3 16x3 19x3 50ib 17x3 Points 92 93 94 95 96 97 98 99 100 101 Recommendation 20x3 19x3 18x3 19x3 20x3 19x3 20x3 19x3 20x3 19x3 Points 102 103 104 105 106 107 108 109 110 111 Recommendation 20x3 19x3 20x3 50ib 20x3 19x3 19x3 20x3 20x3 20x3 Points 232 233 234 235 236 237 238 239 240 241 Recommendation 20x3 20x3 20x3 20x3 20x3 20x3 20x3 20x3 20x3 20x3 Table 3: Recommendations for the skilled player Table 4: Recommendations for a poorer player 14