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A research team from the University of Toronto is rethinking how darts can be played more fairly.

The team, led by Timothy Chan, a professor in the department of mechanical and industrial engineering in the Faculty of Applied Science & Engineering, recently took home first place in the by designing a new framework to level the playing field in dart games. 

“Winning this competition, at the world’s most prominent and competitive sports analytics conference, is a testament to the excellence and ingenuity of our students here at the University of Toronto,” says Chan, who is also U of T’s associate vice-president and vice-provost, strategic initiatives.  

With millions of players around the world, including an estimated 17 million in the United States, according to the National Sporting Goods Association, the game of darts continues to grow in popularity.   

“Darts is a great sport because almost anyone can play and it doubles as a fun mental puzzle,” says Rachael Walker, an alumna who is a co-author of  along with PhD candidate Craig Fernandes and Chan. 

The research, which expands on work Walker did for her undergraduate thesis, focuses on the game of 501 darts, where players begin with a score of 501 and take turns throwing darts at the dartboard. Points are deducted from their total depending on where the darts land. The first player to reach zero wins.   

“We looked at 501 darts played in recreational and professional settings,” says Fernandes. “In a recreational setting, the game is often played amongst players that have different skill sets – and when that happens, the stronger player often wins, which can lead to unexciting matches.”  

As in golf, the imbalance is often addressed by introducing a system that gives the less-skilled players an advantage so that all players have an equal chance at winning.  

“[But] our research first proved that the current approach of giving the weaker player a head start doesn’t actually give all players a fair chance at victory,” says Fernandes. “Instead, we used a Markov decision process to understand the nuances of the game and then come up with a new system that actually leads to mathematical fairness.”  

The new framework first determines a player’s skill level by having them throw several darts at the centre of the board before the start of a game. Players are then assigned a skill level based on where their darts land – players who get most of their darts in the centre are determined to be higher skilled, while those whose darts are spread out across the board are deemed less-skilled players who would benefit from an advantage.   

The new system gives the lesser-skilled player credits that they can cash in at any point in the game. The credits can be used to claim the outcome of a throw – that is, the region of the board they intend the dart to land in – without physically throwing the dart.   

The researchers found that credits create fairness by using a Markov decision process, a mathematical framework that models scenarios where the outcomes are partly in control of the decision-maker and partly random. However, the number of possible decisions and outcomes in darts made the model difficult to implement and solve at scale.    

“To accurately model a dart game that assigns an advantage to a single player, we needed to consider over half a million possible game states and hundreds of possible actions at each state,” says Walker.  

“In a traditional implementation, you optimize across all states simultaneously, which may require considering billions, or even trillions, of possible outcomes.”  

The researchers overcame the challenge of scale by starting simply and slowly adding complexity to the model. The first version did not consider the fact that darts is played in turns of three throws for each player; this helped build intuition and develop implementation tricks that later allowed them to solve the true model.   

The first-place finish at the MIT Sloan Sports Analytics Conference was affirming for the researchers.  

“It was a very strong competition featuring many major North American sports such as football, baseball and basketball, and a lot of research was focused on generative artificial intelligence and machine learning,” says Fernandes, who presented the research at the Sloan conference.   

“Winning with our operations research and optimization approach was exciting for us.”  

The team is now looking to implement the framework with collaborators, including local dart leagues, to see it work in practice.   

“My lab tackles complex decision-making problems in health care and sports using techniques from operations research,” says Chan.  

“The tools we develop are general, so the insights we obtain from solving a problem in darts may then be applied towards solutions in patient scheduling or medical decision-making.”