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Saturday, November 17, 2007

Curling Myths: How do we calculate the Truth?

I was recently doing some research on available documents related to strategy for Curling. Needless to say, there is very little available. I discovered Russ Howard is writing a book and, given the lack of current information, I’m looking forward to seeing if it includes much strategic material. If anyone can point me to other papers, articles or other strategy books, please send a Private Message to milobloom at Curlingzone, or post a comment here.

I did come across a PDF document posted at a CCA website entitled 4rockstrat. In it, I read many of the common “traditional” ideas regarding basic curling strategy. The purpose of this article is to challenge some of its logic. I certainly hope the CCA plans on expanding their available material in future. It is clear the authors were trying, but the lack of detailed information is apparent. I will attempt to dispute a couple of the Myths this document supports and are commonly accepted by many curlers and fans.

Myth #1: Keep it simple in the early ends
Excerpted from 4rockstrat.pdf:


Early ends (1 to 3) Without Last Rock

Most teams will attempt to implement a defensive game plan during this segment of the game especially as it pertains to avoiding high risk finesse shots. Remember, you do not have to score (steal) in the early ends without last rock to ensure victory. It is more important to keep the score close as you build your team’s confidence while learning the ice and assessing the abilities of the opposition. A general objective is to limit the opposition to scoring a single point when you do not have last rock. Even a two-ender is acceptable.

Early Ends (1 to 3) - With Last Rock

Teams may be a little more aggressive in early ends when they have the advantage of last rock but generally speaking, still try to avoid risky situations that require the making of finesse shots. Last rock skips will also tend to play a defensive style of play as they build the confidence of their teammates while assessing the ability of the opposition and learning the ice. They will attempt to score their 2+ points to the side of the sheet but will not be overly concerned about scoring a single point, blanking the end or giving up a steal of one.

The author comments in both cases about building confidence of your team and evaluating ice conditions. In today’s game this is less significant than years past. Consistent ice surfaces throughout an event, practice ice and the skill level of top teams lead me to believe this is minimal. If you’re a skip and worried about your team adjusting, then you won’t be competing at the highest levels.

Let’s address “with hammer” situation first. Assume a steal of a single is the most likely outcome of a poor end. Why then, would a team play defensive, risking at worse being down one starting the second end? The correct strategy for a team with hammer in the first end should be the most aggressive play possible, perhaps more than in later ends. Reasons:

A deuce in the first end will result in a win 73% of the time (74% for an 8 end game). A three results in 85% chance and a four is 91%.

A steal leaves a 43% chance to win, being 1 down with hammer after first end.

A steal of two, though unlikely, still leaves a 27% chance to win and is no different (mathematically, perhaps not psychologically) than surrendering a deuce without last rock.

Combining the significant advantage a large score gives you, with the greater odds, because of the number of ends remaining, to come from behind if a steal occurs, leads me to suspect aggression is the correct approach. Let’s try to use math to prove our theory.

Let’s estimate some outcomes based on our strategy and calculate or win percentage:

Expected Results (ER) with 9(7) ends remaining:
Odds of winning if tied with hammer (x) = 60.3% (60.7%)
Odds of winning if one down with hammer (y) = 43% (42.7%)
Odds of winning if two down with hammer (z) = 27.1% (26%)
Odds of winning if three down with hammer (m) = 14.7% (15%)

Notice that an 8 end game (7 ends remaining) does not change the outcomes much at all, so we will disregard for now.

Option 1: Aggressive starting the game
Blank (b) = 0%
Take 1 (t) = 30%
Take 2 (u) = 30%
Take 3 (v) = 10%
Steal = (s) 30%
Win (W) = bx + t(1-y)+u(1-z)+v(1-m)+sy
W = 60%

Option 2: Conservative starting the game
Blank = 20%
Take 1 = 50%
Take 2 = 20%
Take 3 = 0%
Steal = 10%

W = 59%

A nearly identical outcome. If we expect our chance of allowing a steal in the first end less than 30%, when playing Aggressive, we increase our chances further. For example, in Option 1 (Aggressive) if s=.2 and t=.4 we now win 62%.

Given the early stages of a game, I suspect the differences in actual results between teams would be significant. For example, Kevin Martin will win higher than 60% if tied with hammer after 1 end and, I suspect, the lower teams on the WCT rankings would be the opposite. The more ends remaining, we would expect a greater discrepancy between teams. But as a comparison, the analysis shows us being conservative is not the correct strategy as believed, it is actually the same or slightly worse than being aggressive.

The outcome for 8 end games is, oddly enough, nearly identical.

Now, what about when we do not have last rock? This decision appears to be obvious by answering this simple question:

When tied without hammer, starting the 3rd end of a ten end game, what strategy would we use?

I expect nearly every team would choose to place a centre guard, and attempt to play for a steal. Mathematically, the 3rd end of a 10 end game is identical to the 1st end of an 8 end game. I’ll leave any further estimation to be done by the reader, as I am already convinced based on this fact alone. There is no doubt, risk of a deuce or three is high in the first end and puts us into a poor position. However, it doesn’t get any better as the game progresses and the more ends remaining to come back, the better our chances. A team without hammer should play the first end to steal or force opposition to one. In my opinion, the best way to achieve this is by aggressive play early (centre guard, draws to middle of house).

It is perhaps simplistic to say Aggressive vs. Conservative play. Commonly a team may move from one strategy to the other, and back, during a single end. However, the common misconception that supports conservative play in early ends does not appear correct.

Myth #2: A deuce early is not significant
In the CCA document, it is suggested that, when starting without hammer:

A general objective is to limit the opposition to scoring a single point when you do not have last rock. Even a two-ender is acceptable.

The author then goes on to suggest placing the first rock of the game in the four foot! Per Myth 1, this is not the approach I would recommend. So, is surrendering a deuce acceptable? From Myth 1, we now know down two after the first end puts us at a 26% or 27% chance of winning (based on 8 or 10 end game). This doesn’t sound acceptable to me. We may need to surrender a deuce because that is how the end develops, but we certainly should not play the end to make a deuce acceptable.

Let’s take a scenario where two rocks remain in the first end and we do not have last rock. Our team has missed a few shots and we are in trouble. Our opposition, holding hammer, lays two. One rock is open in the twelve foot; the other is partially buried in the top four foot. We can hit the open rock and surrender a deuce or try a corner freeze on shot rock to force the opposition to 1. What percentage do we need to succeed to make the draw the correct call? We’ll use the statistics from 8 end games for this analysis.

W (if down two) = z = 26%

The team with hammer either scores three or is held to one. The chance for a steal or deuce is considered negligible.

Successful draw (hold team to one) = d
W = dy + (1-d)m
Setting W to 26%
.26 = d(.43) + (1-d)(.26)
Solving for d = .395

We need to make the draw greater than 40% of the time for it to be correct call. Given skills of top skips, I would suggest the correct play is the draw and “accepting a deuce” is not the correct decision. Better to risk a three at the possibility of forcing opposition to a single.

Myth #3: Play conservative against a stronger team
It is a commonly held belief that when playing a team that is stronger than yours; play conservative, keep the game close and hope to win a close battle at the end. The U.S. team in the 2007 World Championship appeared to use this strategy to perfection against Team Canada during the round robin. I don't entirely agree with this approach and will attempt to explain why….in another article.