A Python program to solve the game of darts. More precisely, if the accuracy (2x2 covariance matrix) of a player is known, then the optimal aiming position can be calculated. This is achieved through a brute force method.
This work provides 3 main features :
- Optimal aiming spot to maximize the score per throw
- Deduce a player's accuracy (2x2 covariance matrix) from a list of his/her scores
- Optimal aiming spot for each game situation (e.g. where to aim when your score is 269?)
Should you aim for triple 20? For the bull's eye? Something else? The answer actually depends on your skill level and your current score.
TL&DR:
- Unless you're an excellent player, aiming at the triple 20 is a trap.
- Unless you're a bad player, aiming for the center is also a trap.
- If you're in between, the optimal aiming spot changes depending on your skill level and the current score. You can use this library to output a tailored aim map.
Optimal aiming spots in the early game for (from left to right) a bad player, an average player, a good player and an excellent player
Detailed list of aiming spots for a "good player" for each of the 301 possible current scores: optishots
Update 27/12/2022 Never mind, a paper from february 2022 solved the game. Their work is more complete than ours as they take into account the adversarial nature of the game: https://arxiv.org/pdf/2011.11031.pdf
This work has consistent findings with the paper "A Statistician Plays Darts by Standford's Ryan J. Tibshirani, Andrew Price and Jonathan Taylor" https://www.stat.cmu.edu/~ryantibs/papers/darts.pdf We find a very similar pattern of aiming spots VS player skill. The models used in both studies are very similar (2D gaussian model for the throws), but the approachs used to solve the problem are different. Reaching the same results is reassuring :) However in their work, they only solved half of the problem: where to aim to maximize the score per throw.
Our work also solves the end game as it provides the optimal aiming spot for each possible score
Other posts/articles with great graphs and coherent results: https://www.datagenetics.com/blog/january12012/index.html https://www.codeproject.com/Articles/461044/Throwing-Darts-in-Monte-Carlo
With sigma_x=0.02 and sigma_y=0.02, a player that aims at the middle would get something like this after throwing 2601 darts:
Knowing this, where should the player aim at to maximize the score per throw?
How much better is the optimal aiming spot compared to other common aiming spots?
Ideal shot 37.5 points per shot:
With sigma_x=0.07 and sigma_y=0.07, a player that aims at the middle would get something like this after throwing 2601 darts:
Knowing this, where should the player aim at to maximize the score per throw?
How much better is the optimal aiming spot compared to other common aiming spots?
Ideal shot 16.4 points per shot:
With sigma_x=0.15 and sigma_y=0.09, a player that aims at the middle would get something like this after throwing 2601 darts:
Knowing this, where should the player aim at to maximize the score per throw?
Below, 2601 aiming positions were tried, for each position 10 000 darts were "thrown" in simulation. The expected value for each of the 2061 aiming positions is then calculated and showcased below:
How much better is the optimal aiming spot compared to other common aiming spots?
Ideal shot 13.5 points per shot:
With sigma_x=0.2 and sigma_y=0.2, a player that aims at the middle would get something like this after throwing 2601 darts:
Knowing this, where should the player aim at to maximize the score per throw?
How much better is the optimal aiming spot compared to other common aiming spots?
Ideal shot 12.0 points per shot:
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