This is a review of the Three-Dimensional Markov Model research conducted by Paul Brenzel, William Shock, and Harvey Yang.
Curling enjoys great popularity in Canada and is on the rise in the United States. This study models curling as a Markov process to estimate win probabilities of different states during a curling match.
Score information was entered into the model from the matches played from 1998 to 2014 in the Canadian Men’s Curling Championships, including the year of the match, round of the tournament, match location, teams competing, score in each end, final score, time remaining for each team, and which team started with the hammer in the first end.
In order to use the Markov method all possible scenarios that can occur during a game must be defined. All possible state transitions and their associated probabilities must also be known. This data is used to create a three-dimensional Markov model using three states – the end being played, hammer state, and score differential. The purpose of the three-dimensional Markov model is to determine the expected win probability for any team based on the current state of the game and taking into account all possible future transition states and their associated probabilities.
Two models were created: 1) a homogeneous model that assumed that state transitions were strictly a function of hammer possession and independent of any other parameters and, 2) a heterogeneous model, which assumed state transitions were dependent on other parameters. The results from both models were very similar with the exception of increased accuracy of the heterogeneous model in predictions of state transition towards the end of the game, specially the tenth end. Strategy at that point of the game is typically different from that employed earlier in the game. A team that is down by two points will choose very different strategies from a team that is up one point.
Teams can use the information to determine when to score one point and give up the hammer or blank the end to maintain possession of the hammer. The Markov process takes into account not only the team’s probability of scoring a point but also the effects of lost opportunities. The model will also aide teams in deciding when they should concede a game. The analysis actually determines that teams should concede less often than they currently do. Current decisions appear to be at least partially based on psychological conditions rather than statistical analysis.
Analysts can use the Markov model to graph the expected win probabilities for each team over the course of a game. This provides the ability to relay information to viewers in a clearly understandable manner. It also gives them opportunities to more deeply delve into how effective a team’s choices are, or if alternative choices should be made in any given situation.
This Markov model provides the ability to analyze the game of curling at a deeper level than was previously possible.
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