Some Economics (and Finance and Statistics) of Curling

My guess is that I am the only person in the world who combines an economics Ph.D., experience in curling, and a blog. Tyler Cowen at Marginal Revolution initially posted about "The Economics of Curling", but here's my two cents (see the previous post if you need pointers about how the game works).

The goal in curling is to win by scoring more points than your opponents. The game is composed of rounds (called ends), and the final score is simply the sum of the score across those rounds. There is no big deal with those two features. But, within an round, only one team scores, so the optimal behavior in curling is to make sure that you are the team that scores in each round, and then to make the most points. This is where it gets interesting.

Play in curling is sequential, and one team always goes last. Since only one team scores, going last is valuable; in fact, valuable enough to have a name - the hammer. For fairness, the hammer is awarded to the team that did not score in the previous round. If neither team scores, the team with the hammer keeps it for another round.

Scoring within a round accentuates the value of the hammer. The team that scores is the one with the stone closest to the center of the bullseye (called the house). That team gets a point for each successively further out stone until a stone of the other team is reached.

Clearly, a round of curling is a sequential game, for which backward induction is useful.

The optimal outcome of a hammer throw is easy to see: get as close to the center of the bullseye as you need to be to score (and perhaps nudge your opponents stones out of the way). This can either mean displacing an opponents stone that is closest to the center, or adding your hammer stone to a set of your stones that are already closest to the center.

Going backward another step, what position does the team without the hammer want to be in after their last throw? This is not quite as clear. Fairly obviously, they want to have the stone closest to the center of the bullseye - the only way to score if you don't have the hammer is to be in position and hope for a bad hammer throw. If you can have more than one stone in scoring position after then penultimate throw, so much the better. Both of those require an aggressive stance towards your opponents stones in scoring position - get them out if you can. After that it becomes murkier.

Should you leave your opponents stones in play or knock them out if they are not in scoring position? Where should you position your stones that are not in scoring position: in front where they may be knocked into scoring position, or behind the bullseye where they can be sacrificed to stop a stone being knocked out of scoring position? Should they be in front of the house, blocking access for the hammer throw (other than through a ricochet)? I don't think economics offer clear answers on these. Continuing backwards, it is easy to see that all 15 throws of the round (excluding the hammer) are characterized by an optimization problem too difficult to solve directly. But that's why we play games: we don't know what is optimal so playing amounts to a Monte Carlo technique to test the different possibilities.

There is a finance aspect to curling as well. When investing, it is optimal (within reason) to accept increased risk in exchange for increased returns (and vice-versa). Yet, within curling, the only throw on which it is clear that this relationship exists is with the hammer. The low risk-return strategy is just to hit the bullseye and claim a point - this is fairly common in tournament play. A somewhat more risky proposition is to try to score multiple points. However, when we talk about other throws, it is not at all clear what the risk and return actually are. For the penultimate throw, the return is probably very low (because the hammer may eliminate your chance to score), but the risk is quite high (since there is the possibility that you will be the beneficiary of a bad hammer throw when the most stones are still in play). As we progress backwards through the throws, it is clear that both risk and return will decline, with the latter declining more slowly (since there isn't much chance of return on any individual stone anyway). The nice thing about this is that finance then tells us how a team should be structured: since the risk-return tradeoff gets increasingly unfavorable as the round progresses, it pays to have your most accurate and steady throwers going later in the round. This is precisely what we see in curling - experience counts for a lot.

For statisticians, scoring in each round has a minimum of zero, a maximum of eight (the number of stones for each team). The modal score is probably one - although I never measured that. The distribution of scores is then a count distribution: something like a Poisson with censoring. My guess is that scoring is underdispersed (the variance is less than the mean), because of the preponderance of scores of one due to the ability of one team to control its risk-return tradeoff better than the other team. So a seminonparametric version of a censored Poisson probably has the best chance of fitting the data.

In the financial game personal loans could be seen as a time out. Use a personal loan to take a step back and look at your finances, then see what the money from your personal loan can do to help you. Just make sure you can repay the personal loan so you don't end up in a downward spiral.

Comments

[William Polley]

John Palmer (who blogs at EclectEcon) also likes a good bonspiel. I have never curled, but consider it a goal to do so at least once in my life. I have taken to recording the Olympic matches that are televised overnight--I like it that much.

People have described curling as "chess on ice". I agree, and the backward induction analogy fits that description.

[Dave Tufte]

And I though I was so special ... harumph

I hope you guys enjoy the viewing. I spoke to my parents in Vero Beach last weekend, and they had all the curling slots ready to be recorded.