Theory and Methodology
Modelling curling as a Markov process

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Abstract

Markov processes have been used to model a variety of sports such as football, jai alai and baseball. Curling has a scoring system similar to baseball; both progress on an inning by inning basis. This suggests that curling could also be modeled as a Markov process. We will develop such a model and compare some basic strategies.

Introduction

Curling has been described as a combination of bowling and chess. There are a variety of strategic nuances and subtleties, but as in most sports there are two basic strategies; play cautiously waiting for an opportunity to exploit your opposition's mistakes, or play aggressively where the “best defense is a strong offense”. There are no documented instances of a probabilistic or simulation based curling model. We must develop a probabilistic model to objectively analyze our basic strategies.

Curling is a winter team sport with origins in Scotland. Each team is made up of four players and the game is played on sheets of ice 14 ft 2 in. wide and 144 ft long. At each end of the sheet is a house. The house is a set of four concentric circles of various diameters (12, 8, 4, and 1 ft).

Curling is played with circular disks of polished granite weighing approximately 45 pounds. The teams take alternate turns sliding each of their eight rocks down the sheet of ice. Teams attempt to strategically position their rocks in front of, or within the house. A team may also deliver their rocks to remove their opposition's rock(s) from play. Once all 16 rocks have been played the score is tallied. A team scores a point for each stone closer to the middle of the house than their opposition's closest stone. For example, suppose your team has one rock in the house and your opposition has three. If the opposition's rocks are closer to the middle of the house than yours, your opposition would score three points (Fig. 1). Conversely, if you have a rock closer than any of your opposition's rocks then you will score a point (Fig. 2). If no rocks are in the house at the completion of an end, this is referred to as a blank end.

There is a strategic advantage to having the last rock in an end (also known as having the hammer). Teams will flip a coin to determine who will have the last rock in the first end. In subsequent ends, the team that was scored on in the previous end receives the hammer in the next end. A standard curling game consists of 10 ends or innings. If the game is tied after 10 ends, extra ends are played until one of the teams scores at least one point.

A literature search on curling would generate information that could be classified in one of four categories: rules and instruction (CCA, 1996, Lukowich et al., 1981, Lukowich, 1993), historical (Murray, 1981, Weeks, 1995), examination of the sport's physics (Johnston, 1981, Shegalski et al., 1996, Denny, 1998), and scheduling (Kostuk, 1997). Unlike several other sports, analytical models have not been developed.

Baseball was the first, and is likely the most analyzed of all sports. Various modelling techniques have been applied; statistical analysis (D'Esopo and Lefkowitz, 1977), Markov processes (Howard, 1960, Bukiet et al., 1997, Trueman, 1977) and dynamic programming (Bellman, 1977). Other sports which have been modelled include: Australian rules football (Clarke and Norman, 1998), cricket (Clarke and Norman, 1999), tennis (Hannan, 1976), track and field (Stefani, 1996), American football (Casti, 1971), basketball (Carlin, 1996; Schwertman et al., 1996), jai alai (Byrne and Hesse, 1996), snooker (Percy, 1994) and hockey (Schutz and Liu, 1996).

Curling has a scoring system similar to baseball (but much more similar to bowls); points are tallied at discrete intervals during play. This suggests that one of the models used to analyze baseball could be applied in a similar fashion to curling. We have elected to model curling as a Markov process, and will use this model to compare basic strategies.

The rest of this paper will be divided into four sections. Section 2 will develop our model formulation. The structure of a Markov process will be discussed, and its components will be introduced in the context of our curling model. Section 3 of the paper will illustrate how we developed our transition probability matrix. With the model fully developed, Section 4 will compare basic strategies. This will be followed by a discussion of the results and a conclusion in Section 5.

Section snippets

Curling as a Markov process

For a problem to be modeled as a Markov process we must define a series of states, the probability of moving from state to state, and establish the time interval over which these transitions will occur (Hastings, 1973). The classic example of a Markov process is a frog sitting in a pond full of lily pads (Howard, 1960). Each pad represents a state. At the end of a fixed time interval the frog will jump from pad i to any other pad j with probability p(i,j). The difficulty in defining the state

Transition probability matrix development

Our base data are statistical information recorded from 1985 to 1997 at the Canadian Men's Curling Championships (also known as the Brier). Table 4 is a summary of that data. Each column represents the number of points scored in an end. Probabilities [P(k|j(t))] are generated from the frequency table. The rows represent the ends; the bottom row is the sum of each column. Note that a team may concede defeat at any point in the game. This fact is reflected in the last column of the table. This

Analysis

All games start in a scoreless tie. The only difference from game to game is whether the baseline team wins the toss (and consequently has the hammer). Consequently our initial state will be either {0,0} or {0,1}. The likelihood of reaching any one of the states at the end of the game is determined by the TPM(s) applied in the analysis.

The first question we wished to address was whether a time invariant model would adequately represent the outcome of a match. Fig. 7, Fig. 8 illustrate the

Conclusions

This paper has shown that curling, in a fashion similar to other `discrete-event' sports, can be modeled as a Markov process. We have illustrated the outcomes of respective strategies. It would appear that teams generally enjoy an advantage of just over 1 point when beginning the game with the hammer. This result has been developed through various modeling scenarios such as myopic full-game strategies and teams of unequal abilities.

Additional research into this area appears to be warranted.

Acknowledgements

We acknowledge Brian Cassidy, University of New Brunswick, for his assistance in gathering the data.

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