Curl With Math

Notes from the Pre-Trials (AKA “Road to the Roar”)

I’m struggling with this event called the Road to the Roar or more aptly named Pre-Trials. It reminds me of the NFL deciding to increase playoff teams from 10 to 12 in order to generate more games (and more revenue) on opening “Wild-card” weekend. It’s not that the entertainment isn’t there, but why not just have 16 teams at the Trials in December? Money must have something to do with it. Or perhaps the 400 or so people in the stands in Prince George demanded their opportunity to shine in the spotlight.
Some random thoughts and analysis….

TSN please show me the first stones of the end! Not doing so makes it hard to understand what strategy is being played. In 7^th end Gunnlaugson is up one without hammer (5-4) and plays a centre guard, rather than in rings. Perhaps he read my article from February 2009 where I stated “If we are one up in the 4^th or 7th, we should force our opponent into scoring, even at the risk of a deuce, in order to have hammer in following end.”

Linda Moore, please think about what you say related to hammer in the 10^th end. You are the “stats” announcer, relaying all of the appropriate shooting percentages for the fans, but you fail to recognize the basic premise of statistics related to the final end of a game. During the 9^th end of the C Final between Gunnlaugson and McEwen, you said “that hammer is still the advantage in that last end”. No, the advantage is to be one up without hammer vs one down with. You then state “There’s lots of things that Jason could do but I don’t think it’s wise, I think he’s going to wait till the 10^th as well and there lots of things you can do in the 10^th end…he can do the job without the hammer”. So which is it, advantage with hammer or not? The correct strategy in the 9th when one up without hammer is to play for a steal at the risk of a deuce, and if you force your opponent to one you increase your chances to win from 60% to over 75%. I didn’t see if Jason intended to play their first rock in the rings because, per note 1 above , TSN didn’t show it! I don’t like the 9^th playing out the way it did for Jason. Much better for him to be aggressive in this situation.

Simmons vs Stoughton, A Final. Interesting choice by Simmons to hit and roll in the 8^th end. It appeared he could have played a draw and sit shot stone frozen to the Stoughton rock behind the button. Possible he was concerned by lack of curl or afraid of leaving the angle raise for a big end.

This is an important point in the game. Simmons is two up and holding Stoughton to one is a huge advantage. Simmons chances of winning goes from 67% (tied with hammer and just barely Control) to 86% (and Dominant Control) if he holds Stoughton to a single instead of the two he surrendered. Even if Stoughton makes a circus shot and takes three, Simmons still has 37% chance to win. I prefer the draw for these reasons and if he comes short he still leaves a very difficult shot for two.

Yes Mike McEwen, you should have played the double in the 8^th end against Jason Gunnlaugson in the C Final. Your margin for error was substantially smaller with the hit and stick you attempted. Also, you left your opponent a chance, the other option did not. Kevin Martin would have removed both those stones before you got in the hack. I suspect in years to come you will also.
The B Final, 8^th end. Simmons is up 5 to 4, McEwen has hammer. Simmons plays the run-back again. He could have chosen to draw instead. A successful run back likely forces Mike to one anyway. It appeared an out-turn draw to the face of the stone could provide the same outcome with more margin for error. However, the amount of curl may have weighed in on the decision. Let’s take a look at Pat’s thinking:

Draw: Let’s expect Pat makes a reasonable attempt to draw to the face of Mike’s stone but Mike makes his draw 50% of the time.
W = .5(.67) + .5(.37) = 52%
Run-back: How often does Pat need to make the shot for it to be the correct call?
W = .52 = x(.67) + (1-x)(.37)
Note we assume Mike always scores two – in the actual game his rock caught debris and he did not, but that is assumed to be minimal for our analysis and then can be factored into our decision later.
We can see that if x (Pat’s chance of the run-back) is greater than 50% then it is the correct call. If we assume Mike makes the draw 60%, then x = .4 and Pat only needs to make the runback 40% for it to be the correct call. Further examples show it as a linear equation in that Pat needs to make his shot more often than Mike misses a draw for it to be the correct call. The readers can decide if Pat made the “correct” choice.

Same game, 9^th end, Simmons plays a corner guard? Interesting call (unless of course he was light). Playing as if he prefers to score multiple points at risk of a steal, rather than leave a possible blank open. Linda, any comments on this strategy?
Same game, same end. McEwen’s first stone, he plays a run-back of the centre guard to Simmons rock in the top four. Giving up a deuce here is not good (less than 12% to win). Mike must hold Simmons to one or possibly steal. I like keeping the centre guard in this situation, if only to keep the middle protected. At this late stage, he may have felt there was no good placement in the rings and he may be correct, but see how this decision drives the next decision? Even if he makes the shot (which he did, but was unfortunate the jam) Pat still can make another strong come around (which he did) and put himself in good position for two. On Mike’s last I’d almost prefer a draw to either freeze or the back four foot. As difficult as the shot is, the run-back needs to be a double. The issue here is, a deuce or a three have nearly the same outcome: you are very unlikely to win the game. The interesting analysis here is, Mike deciding to run-back on his first was already setting himself up for another run-back on his second. In essence, by calling it once, Mike is saying “I will make this shot twice” In making that decision, might be wise to determine your odds for making a run-back (ahem) back-to-back. If your chances are 90% of making one, then two in a row is (.9)x(.9) = .81 or 81%. If you only figure 80%, then you drop to 64% to make two in a row.
I grew up a rink rat at the Assiniboine Memorial in the mid 80s, watching Kerry Burtnyk. One of the classiest champions curling has ever seen. Kerry’s loss to Gunnlaugson was heart wrenching. His last rock in 10 needed only to roll an inch, it did not. Then a draw to the full eight turned into a burned rock, ending Kerry’s chances at an Olympic medal. I was hoping to see Burtnyk in Edmonton, but it is not to be. We can only hope he makes another run or two at a Brier bid (a la Werenich in 1995 and 1997).
Back to McEwen vs Gunnlagson, C Final. In the 8^th end, up 6-5, Mike appeared to be in good position, sitting first and second on BJs first rock. They called time out, thought for a while, then peeled a corner guard. I was unclear to their thinking here, other than it seemed like the best idea at the time. It didn’t appear to have much impact on the end. Alternatively, they could have chose a more aggressive play, either draw to open side or play some type of tight guard or draw. The rock they were concerned about (top yellow) did come back to haunt them later in the end. Mike commented they would have to make “a lot of good shots to get two” and, as often happens at this level, they did. The best part of their decision was avoiding 3, so we can’t fault them for that. And as I said above, if Mike had just played the double on his last….
Interesting choice by Jason to play a centre guard when one up in the last end without hammer. Mike doesn’t bite (perhaps Jason wanted him to?) and choses to play a corner guard rather than draw to centre.

Good luck to all in the “Actual” Trials, which I suppose we could now call the “Pre-Olympics”. Final thought…..is it too late for some young team to pick up Russ Howard?

15 Nov 2009 Kevin 0 comments

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New Season Begins and World Cup Final: Koe vs. Howard

Well, the new Curling Season is here and though I’m disappointed in the end to golf season I am anxious for the Olympic Trials. The Olympics themselves, sadly, I’ve never had much interest in. Why is it every four years I’m suppose to get excited about sports which I’ve never cared to watch during the past 1460 days since the last Olympiad? I never tuned in to CBC for skiing during the World Championships in 2008, so why now? If not for Curling, Hockey, and perhaps women’s’ short track speed skating (it’s like a cat fight on skates!), I’d likely pass on the entire thing completely. And don’t even get me started on the Summer Games – days on end of coverage from multiple networks, journalists, etc – and the only thing worth watching is 9.7 seconds of men running and a couple of laps in a pool (which took slightly longer). Better to watch the highlights on the internet after it’s all over.

In any event, we’re back into the “sweep” of things and need to begin.

Shorty Jenkins: Martin vs. Matchett
Gerry Geurts of CurlingZone relayed this shot call to me and he may have not remembered exactly right, but the analysis should still be interesting. In the semi-finals at the recent Shorty Jenkins event, Kevin Martin was one down playing the 7^th, with hammer, against Dale Matchett. Matchett had a rock in the outside rings, perhaps biting eight foot, and there was a slightly off-centre guard. Martin, on his first stone, chose to play for his deuce by drawing around the guard. Remember, this is an 8 end game and only the final end remains. Kevin is forgoing an attempt at a blank, attempting to take two, with an increased risk of being held to one or even giving up a steal.

The chance of winning up one without hammer coming home is roughly 60%. If held to one, Martin’s chances drop to 25%. The difference in outcome between a successful two or held to one is greater here than at any point during a game. Let’s look at each option:

Draw: Assuming the draw takes away any chance of a blank, let’s determine how often Kevin must get a deuce to make this a correct call.

W = d(.6) + s(.25)
Where d = chance of a deuce and s = chance we are held to one. Assuming these are the only likely outcomes (blank and steal of two not likely), we solve for s= 1-d

W = .6d + .25(1-d).

To complete this calculation, we need to compare against the scenario if Kevin hits the stone.
If Kevin hits the open stone, Matchett either hits the Martin stone (trying to roll behind the guard I expect) or simply draws around the guard attempting to force Martin to a single point.

What likely entered Kevin’s thought process “if I hit and stick, Dale will draw around the guard and now I may be forced to one. My chances are better at two if I force the play now.”

When guessing what an opponent may do, we can estimate what the chance is they will make a certain play and evaluate further. For example, let’s start to analyze what happens if Kevin hits open stone.

Assumptions:
Martin will always hit and stick successfully.
If hit and roll succeeds, Martin always draws for one successfully (assuming the roll puts the rock in the back rings).
If hit and roll doesn’t succeed, a blank always occurs.

Let’s first examine what happens is Matchett draws or hits:

Where:
x = Odds of winning if tied without hammer = 25%
y = Odds of winning one down with hammer = 40%
z = Odds of winning if two down with hammer = 12%

Hit: Estimate a roll successful 25% of the time.

W (Martin) = x(.25) + y(.75)
W = .363

Draw: Estimate with a draw Matchett steals 20%, Martin takes one 60% and he gets a deuce 20%…

W (Martin) = x(.6) + (1-y)(.2) + z(.2)
= .294

Matchett should draw (based on our estimates are correct) – but perhaps he will only draw half the time. Then we weigh the chance of winning as:

W (Martin) = .5(.363) + .5(.194) = .33%

Therefore, using this in our original equation:

.33= .6d + .25(1-d) solving for d = .23

Therefore, Martin only has to get his deuce 23% of the time in order for the decision to be correct. Note that if Matchett always plays the draw, then Martin needs to make a deuce even less often (12.5%):

.294 = .6d + .25(1-d) solving for d=.125

Masters Final: Koe vs. Howard
It was an entertaining game last Sunday with Howard winning yet another Grand Slam. This was clearly one that could have gone the other way. An early steal of two had Glenn and his squad battling back. There were some timely misses by the Koe rink, but also some interesting decisions which may have provided them better opportunity to clinch a victory rather than being a bridesmaid for yet another Grand Slam final.

Some observations….

Second End:
Matt Hames in his Curling News blog suggests Koe could have played the in-turn draw instead of leaving Howard an opening for one…
http://curlnews.blogspot.com/2009/10/world-cup-sweeping-rant.html
I would tend to agree. At this stage he’s 2 up and if he does make a poor shot, at worse Howards gets a deuce and Koe’s odds are at 61% tied with 6 ends remaining. A steal would have put the Koe rink at odds of 89% chance to win. That is a chance you want to take during the Early Game, IMHO.

Third End:
Blake throws two draws. One appears 20 feet heavier than the next (according to the assessment by announcer Mike Harris). The sweepers are surprised and there is some discussion that his stones aren’t matched. How can teams at this level not have properly matched stones in a final of this type of event? Situation seemed very strange.

Fourth End:
Koe appears to be sitting third stone in the top eight foot, Howard is first, second and fourth. Kevin calls a hit on his own stone (driving it onto 4^th rock), rolling across the house to then double the Howard stone. If successful, Howard would have a shot to hit and stick and sit two. Koe then would have had a double to force Howard to one. I thought the correct call on his first was the one he played on his final stone. Double the 1^st and 2^nd rocks and roll behind his other rock. If successful, Howard would be left with a choice to either draw around Koe’s top stone or attempt a difficult hit which appeared to be unlikely to allow him to sit two.

Sixth End:
Howard places a centre guard. Please see my articles from January 2009 and March 2009 for the analysis of why this not the “correct” call. I wonder if it was an intentional decision on the part of Howard to choose an alternate strategy, or if he is unaware of the analysis.

Interesting call on Howard’s first. Rather than hit the open stone, he plays into the centre and leaves a (albeit) long run back double for Koe to lie two. This is not a good position to give up a deuce, Howard’s odds of winning would drop to
15% if Koe pulls out the miracle. The flip side is, a steal of one for Howard puts the odds at him winning to 63%. Howard’s call into the rings on his last was clearly an intent to tempt Koe into a big weight shot rather than a draw. He could have instead placed a guard, leaving him a draw – but Koe may have hit no matter what the result. Also, a poor guard could have left a soft double with the inturn and a possible deuce for Koe.

Seventh End:
Blake’s second shot, after much debate, Koe’s team agrees to play a run back. In their position the centre guard is a critical stone to help plug up the four foot in an attempt to force Howard to one. The risk of a deuce is worth the attempt to steal or force a single. Playing the run back was an attempt to take a three out of play – which ultimately succeeded. However, if Blake plays a freeze, Howard may again play a draw, but more likely remove the guard and give Kevin a chance to clear the house on his first shot – or have the option to guard again. If Howard draws, then the play is into the middle with a centre guard and likely a good position for Koe to force a single. Continually attempting run backs is generally counter-intuitive to what a team one up without in the second last end wants to do: force the play to the centre and force the opponent to a single and have a 75% or better chance in the last end – with a risk of giving up a deuce and having a 40% chance coming home. The small chance to steal one, leaving Howard a 12% to win, also supports a draw strategy. In Koe’s defense, playing to avoid a three was perhaps his motive and they were comfortable with that style. It is a case where the risk of a three is minute compared to the great advantage of forcing one or stealing – but sometimes it’s difficult for a team to want to take additional risk if they feel it could take them out of the game. Let’s attempt to analyze the shot call. This involves VERY rough estimates of final outcomes, but allows us to examine how to reach a decision.

Draw:
Koe steals = .1
Howard scores one = .4
Howard scores two = .3
Howard scores three = .2

W (Koe) = 53%

Koe chose the hit, which ultimately resulted in two. Let’s estimate what outcomes may have been most likely:
Hit:
Blank = .1
Koe steals = 0
Howard scores one = .3
Howard scores two = .6
Howard scores three = 0
W (Koe) = 53%.
If Howard scores a deuce more than 60%, then Koe wins less than 53%. The decision appears close. Ultimately, it depends on Koe’s estimation of Howard’s chance at scoring 3 and ensuring that Howard never scores 3 when they attempt the run-backs. I would prefer the draw but it is closer than I had first thought.

Eighth End:
Blake’s first rock, they attempt to come around Howard’s stones staggered in front of the rings. An alternate play would be to double those top stones out and sit 2^nd and 3^rd shot. By playing to the middle it left a greater chance of only a single and increased chance of a steal. This is the style of play I suggested earlier Koe could have chosen in the 7^th, but here the opposite is perhaps true and opening up the play may have increased his chance at a deuce and provided a greater chance at a single if he needed a draw on his last.

A good game and one where a few more made shots on the part of Koe’s rink could have changed the outcome. Whether the decisions we’ve examined here may have had any difference is up to the reader to determine.

Until next month, Happy Halloween!

30 Oct 2009 Kevin 2 comments

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Team Martin Can’t Hold Down the Haggis

What is it that makes sports entertaining? Is it watching the thrill in victory, or the agony of defeat? Do we admire and respect our sports figures for their immense talent and skill, something we do not possess, or for their ability to face the challenge and, sometimes, fail to over come it. Golf, baseball and, similarly curling, present a case for the latter. Do we remember Greg Norman for his heroic achievements at the British Open, or for his complete collapse at the 1996 Masters. The answer is pretty clear. By showing us the same frailty in their inability to overcome the tension of the moment, we see these stars are human. We share the same nervous anxiety when we stumble giving a presentation to a large audience, three-putt from ten feet for a $5 Nassau, or babble incoherently when asking the pretty girl to go on a date.

We still marvel at the feats of Tiger Woods or Michael Jordan, who may lose to an opponent who bests them on a certain day, but never seem to “choke” when the opportunity for greatness is at hand. Jack Nicklaus was the Greatest, but Arnold Palmer and his collapses in the US Open, Masters and PGA, along with his wins, make him “The King”. Something to me is very appealing in experiencing these moments of sport. Moments with a Greg Norman, Jean Van De Velde, Kenny Perry and, yes, a Kevin Martin.

Kevin Martin does not have the records in world competition that anyone would expect from (arguably) the greatest pure curler this country has produced. Kevin Martin beats EVERYONE. Regardless of his team, Kevin has been at the top of Men’s Curling for almost twenty years. He simply Wins. Against every team, at every event….except on the World stage. Is it something more than poor ice, off-days or random chance?

In watching Team Canada’s struggles against Scotland in each game they played, I was trying to determine if it was more Scotland playing great or Canada flat. It was clear Scotland took Canada off their game and disrupted their rhythm. Unlike other games the past two seasons, Kevin never appeared to have a clear sense of his strategy or an ability to dictate the flow of play in each game. In the finals, they looked to have this solved, but the final end proved otherwise.

Most of the discussion in future years, over a cold one at the local rink, will be the call to throw first skip stone away. This shot does appear bizarre and I prefer the option to drive the top yellow stone into the pile – but this is more an opinion versus a fact. Kevin made a very difficult choice, one he must have known would be scrutinized for the remainder of his life and beyond. However, on his last shot, he had two possible options, both likely better than 50%, to win the World Championship. I believe a different choice could have left him a better chance or possibly made Murdoch’s last more difficult, bringing the chance of a hand shake before the final rock. However, several options, such as a guard or a slightly missed hit, could also have left Kevin with less than 50% chance to win. It is my opinion Kevin could do better choosing an alternate shot call. It is a fact that with second’s first stone in this situation: peel the guards. Why does Kevin Martin call Marc Kennedy to play a soft hit and roll into the four foot, when two centre guards sit covering the four foot, in the final end, when tied with hammer? I suspect two things may have led to this critical mistake:

Martin “felt” a need to play a more aggressive end. This could be similar to the Brier final in 1997 against Vic Peters, played during the 3-rock free-guard zone era. I can’t recall the specifics, but the final score was 10-8 and Martin had an option to play a conservative final end, but instead chose to be aggressive.
Kevin strategically “choked”. After months of curling, against the best teams in the world, and the hours of practice and preparation, the culmination of a season comes down to the final end. Similar to Kenny Perry, seeing a two stroke lead with two holes the play, he begins to picture the win. The fatigue combined with the enormity of the moment, leads to a mental breakdown at a critical point. Instead of focusing energies on the job at hand: the shot, the putt or the pitch – just as they would normally - the curler, golfer or ballplayer starts to see themselves winning. In an interview with Scott Hoch during the recent CBS Masters telecast, referring to the missed 2 foot putt from 20 years previous, Scott stated he saw himself in the green jacket and could not focus on what he needed to do, get the ball into the hole. Kenny Perry said in his Saturday press conference, when asked what it would feel like to win, that he wasn’t going to answer the question – he needed to stay in the moment. He did for 70 holes, and then his crisp draws turned into evil hooks and his right hand got twitchy. These golfers have both won many tournaments, just as Kevin has, but when the title that they REALLY want to win, The Masters, was close at hand, they got out of their routine, out of tempo and performed well below their ability.

I obviously can’t say if the latter was the case. Kevin has been in similar pressure situations many times and shown the ability to overcome these nerves. From my vantage point, and my analysis of the three games played, something with team Martin just didn’t seem right when matched against Scotland. They did not look like the same team that plowed through Canada’s best all year. It seemed to be more than simply shot making. I expect Team Martin will prepare, focus and do all they can to ensure next time to execute to their peak ability and with clear thinking, when the “moment” happens again.

Some other notes from the recent Mens’ World Championships:

Thanks to TSN for the extreme close up of the antique measuring device (and the CCA logo) being used for a World Championship. At a critical point in the game, the Swiss were held to a single when their second rock was deemed “tied” with Norway’s. The gaps in the measuring device were similar to those dial kitchen timers from the 60s. Can’t these enormous prices for tickets go to support some digital devices? If not, at least get someone to “sharpie” some extra lines on the dial so this fiasco doesn’t happen again.
No thanks to TSN for constantly missing the leads rocks. Fine, if you need to come back late in the early ends, but in the final end of the final game we missed the first 3 rocks of the end. That’s 18.75% of the end. And the tick attempt in a tied game is perhaps the second most critical shot of the end (other than skips last). Makes it even harder for me to analyze calls if I don’t know what was played. Ray did fill us in on occasion, which is helpful.
In the 3-4 Game, Norway chose not to play a corner guard when one down with hammer and Switzerland came into the rings in the 7^th end. They played out for a blank. My February article examined the interesting position in the 8^th end, where a blank or taking one is essentially the same, but what about the 7^th end? In the 7^th, Norway has a 40% chance to win when one down with hammer. After they blank, their chance in the 8^th is 38%. If they play aggressive and are held to one, they are 35% playing 8th. If this happens in 8, they are 34% starting the 9^th end. Doesn’t appear to be a poor decision. What does 3 do? If Norway takes 3 in the 7^th, they are in Control with 79% chance. A three in the 8^th moves them into Dominant Control and 85% chance to win. I like to stay aggressive, and might prefer to play a corner – but there appears to be some reasoning behind their decision.
Another example of strange events for Team Canada. In the 1-2 game, Martin is down one without playing the 8^th end. John’s rock doesn’t quite move the Scotland stone far enough, and they lay second. On Kevin’s first shot, they elect to guard. I’d suggest this is a position where greater risk could be taken. Kevin may have thought he’d have something on his next shot – but let’s assume he knew he would not and he is playing out the end to force Scotland to one. Two down with and two ends remaining,. Martin will win 15% of the time. If you’re wondering if Kevin’s numbers are better than our average statistics, they’re not. The Martin team is statistically in line with these numbers. If you add in the small chance Scotland manages a deuce in the end, the 15% is actually high. I would have preferred Kevin to play aggressively for a steal, even at the risk of a big end. He needs to steal 44% of the time to make the risk statistically correct. Clearly he felt his chances weren’t that high – or possibly that he’d have a chance with his last shot. Strange that they were unable to see what Scotland’s shot would leave them – similar to the 10^th end in the finals.
In the 10^th end when tied – tick, tick, peel, peel and then, when in doubt, peel some more. Then draw for the win. Anything else, in my opinion, increases your risk of losing. Did you notice that Marc made both his shots called and his Scotland counterpart missed his – and they lost.
Thanks to Scotland for great play and great strategy (long guards, attempting to minimize run-back opportunities for Canada).
And thanks for the drama. In the words of Jim Nantz “ It was a World Finals unlike any other”. Amen.

26 Apr 2009 Kevin 0 comments

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Brier Notes: Martin’s Math Test

During his round robin game with Ontario, Kevin Martin says to TSN: “It’s a real Math Test out there!” Could he perhaps be a fan of this blog? Does he have his own computer programs to analyze data? What goes on in his mind during a game and will he share it with us some day?

Kevin in my opinion appears to make more correct decisions, and in a quicker and more decisive manner, than any other skip. It doesn’t hurt that he and his team have missed very few shots in the last two years either.

I had a great time at the Brier, if only for one afternoon and at the patch later that evening. I’ve finally reviewed all of the Tivo recordings I made from the event, and have started to gather some random thoughts and ideas. I may yet come back to a specific situation again, but for now, below are some non-sensible ramblings.

“It’s a Math Test out there”. Says Kevin Martin to Cathy Gautier of TSN at the 5^th end break of his round robin game with Ontario. Simply one of the finest games I’ve seen in years, perhaps only rivaled by their rematch in the 1-2 game. Condolences to Steve Lobel. That makes it 9 years in a row in our “My Home Province vs Your Home Province Bet”. I’m looking foreword to eating some of those Toronto Pork Chops.

An interesting situation developed in the 9th end. Ontario leads 5-4 and Alberta has the hammer. Forgive me if I’ve stated this before, this is the best time in the game to be one up without. Howard will aggressively attempt to steal (90% chance to win) or force Martin to a single (74% chance to win), at the risk of giving up a deuce, where they still have a 38% chance to comeback and win in the 10^th end. Martin is sitting one, with a Howard stone covering it in the rings. Howard begins placing centre guards and Martin is peeling these guards. For some observers, this could appear a strange situation. Why is Ontario guarding when their opponent is sitting one and they are up one? With 7 stones remaining, John Morris is about to throw a peel, when he heads down to the other end to discuss with Kevin and they instead decide to draw into the rings. Howard actually avoids giving up three, but surrenders a deuce. He is unable to score his deuce in the 10^th end and Alberta wins.

I did not like Howard’s decision to peel on Richard’s first stone and, regardless of outcome, Martin’s decision to draw seemed correct. How might we analyze this situation mathematically?

Let’s assume a peel will result in either a blank or Martin being forced to one. I suspect if that Ontario would have tried a soft hit and roll to split the rings and lie two with Richard’s next shot. This would have forced Alberta to make a double in order to blank. We’ll estimate a blank 80% of the time – i.e. where the rocks were lined up, given three chances, Martin will clear the rings 4 in 5 tries.

W = (1-.74)(.2) + (.38)(.8) = 35.8%

With a come around, there are several rocks to come and difficult for us to determine outcome. Clearly, a blank will not occur, we need to estimate what the odds are Martin is able to score two (let’s assume no three is scored) versus a steal or held to one. We need to come up with Variables, which are estimates of an outcome.

X = Howard steals one
Y = Martin takes one
Z = Martin takes two

W = x(1-.9) + y(1-.74) + z(1-.38)

Initially, let’s assume Howard doesn’t steal. Set W to .358 and y=(1-z) and solve for Z

.358 = (1-z)(.26) + z(.62)

Z = .281

Assuming Howard does not steal, we need to expect a deuce 28% of the time, in order for the draw to be the correct call. Factoring for a possible steal, let’s suggest that at least 1/3^rd of the time we need to score two. Is that the case given Martin’s team, a guard and shot stone? I suspect it is, but that is up to Martin to decide. Adding in even a slight chance for three, and 90% chance to win, it is very favorable to take the risk now. Thinking ahead, Martin would consider Howard’s likely call on Richard’s last stone and his chances of blanking when that occurs.

What should Howard do? Perhaps placing the guard off-centre on the other side of the four foot and giving themselves an open shot at their stone may have provided more options for them. Depending where John’s stone landed, Rich would have a potential double or at least a hit to lie 2^nd and 3^rd with rocks in good position above the four foot. I am more inclined to have Rich hit and roll to lie two and take the chance Martin is unable to double them out for a blank. Worst case is you are 1 up coming home and 62% likely to win. I expect Glenn, like many teams, expect Martin to score his deuce more often than 40% - and this could weigh in his decision to play the 9^th more aggressively than he may have given another opponent. I will continue to argue the case that Martin has no more than a slight advantage over 38% against a team of Howard’s ability. Our study of Grand Slam events supports this argument – but it remains an argument where some will always side against the numbers. Good luck to those who choose luck as the basis for their strategy.

Further evidence that Kevin Martin either reads my articles, studies the math himself, or has the best instinct in the game. On not one but two occasions, Martin chose to bring his first rock into the rings when tied with hammer and without last rock in the 8th end. Martin did this against both Gushue and in the 1-2 game against Howard. Incidentally, in the 1-2 game he placed a centre just the end before in the 7^th, when tied without hammer. This decision was discussed in my January article.
It is the 1-2 game, 8^th end. Howard is tied with hammer. On Kennedy’s last rock, Howard has a single corner in play and Kevin yells down “play for a blank? I don’t know”. They decide to play a tight centre guard and Howard continues his conservative play in the end by peeling. Martin then has John draw around the corner on his first rock. In this situation, Howard can go from a 65% chance if he blanks to an 80% chance by scoring two. A steal puts him at 38%; take one they are 62%. If, however, he blanks, the outcome in 8 of two is 85% compared to 66%. I’m not certain I agree with Glenn’s approach in 7 at that stage. Coming around the centre would have forced the end, increasing a chance for multiple score. The added risk of a steal is not as bad in this end (7^th) as it would be in the next. I’d like to examine this end further, but it’s late and I’m tired. Perhaps another day…
The 3-4 game with Manitoba versus Stoughton. Linda comments in the 8^th end that Gushue wants to force Stoughton to 1, he doesn’t need a steal. Granted he does not HAVE to steal but to imply he is not playing for a steal is foolish. If Stoughton takes one, Gushue has a 15% statistical chance to win. If he can steal, his chance increases to 34%. In the 9^th end, Stoughton chooses to play a run back from outside the rings on his first rock. Linda states he “has to”. I don’t agree has to. In fact, he could play a draw on his first stone. He could have put his rocks in a position where Brad would likely guard again and steal a single, putting Stoughton 1 down in 10 with last rock. The draw could protect Manitoba from what nearly occurred, a steal of two. In the 10^th end, Stoughton made a fantastic shot. But with more time, Gushue may have examined the option of coming into the rings for second shot. An option which, though not eliminating the outcome, may have made Stoughton’s shot either more difficult or only to tie. They both came close to running out of time, something I can’t remember seeing at a game of that level in a very long time.

20 Mar 2009 Kevin 0 comments

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Scotties Decision by Team Canada and Middle Game Strategy

Jennifer Jones’ Team Canada rink made some interesting decisions in their round robin game against Ontario at the 2009 Scott Tournament of Hearts (Scotties). During the 6^th end, Team Canada (TC), one down without hammer, could have chosen a more aggressive route during the latter part of the end. Instead of drawing around their opponent’s stones in front of the rings, they instead chose to hit the Ontario stone towards the back of the rings and play out the end as a blank. Was this the correct decision? Recall:

Statistics for Women’s Curling

Ends Remaining*	Tied with Hammer**	Down 1 with Hammer	Down 2 with Hammer	Down 3 with Hammer	Down 4 with Hammer	Up 1 with Hammer	Up 2 with Hammer	Up 3 with Hammer
1	0.691	0.397	0.145	0.020	0.001	0.865	0.973	0.997
2	0.611	0.371	0.166	0.068	0.018	0.834	0.928	0.975
3	0.602	0.391	0.208	0.083	0.039	0.816	0.882	0.952
4	0.596	0.402	0.216	0.123	0.062	0.763	0.886	0.923
5	0.613	0.433	0.242	0.152	0.086	0.779	0.849	0.917
6	0.600	0.413	0.263	0.163	0.089	0.756	0.854	0.930
7	0.585	0.413	0.280	0.183	0.092	0.766	0.847	0.940
8	0.562	0.444	0.290	0.156	0.123	0.734	0.866	0.927
9	0.556	0.454	0.281	0.176	0.080

* takes into account 8 and 10 end games
** For 1 Ends Remaining includes extra end games

Notice that if the end is blanked, TC has a 24% chance. At the beginning of the 6^th they had a 22%. In fact, their odds get slightly better if the end is blanked.

If they play aggressive and steal, they increase their chances to 40%. Holding Ontario to one is 22%. In fact, a blank appears to be 2% better than actually forcing Ontario to a single point! It appears the risk of attempting a steal at this stage is not necessary.

What about in the next end? If it had been the 7th instead of 6^th end, the decision is very different. Now, entering the 8^th end when 1 down without hammer, the odds are only 18%. Holding Ontario to a single gives a 21% chance. A steal in 7 instead of 6 still leaves them 40% - no change. It would be more tempting to play the final rocks aggressively if it were the 7^th end. If the 7^th were to be blanked and TC is one down without hammer playing 8, it is imperative to either score a steal or force opponent to a single. When entering the final two ends 1 down without hammer, a teams chance drops to 16% - but more importantly, a steal in 9 gains less. A steal in 8 would produce a tie and a 39% chance, but a steal in the 9th to tie the game is only 30%.

The 5^th and 8^th Ends
Recall my previous article on Early, Middle and End Game. The Early Game is the first 4 (or 2) ends. At this stage, most top teams will play aggressive attempting to take control or dominant control as soon as possible. In the End Game (final 3 ends), teams will play the scoreboard more closely, attempting to be tied with hammer or two up without hammer in the final end of a close game. So what is Middle Game Strategy?

Middle Game is the 5^th through 7^th ends in a ten end game (or 3-5^th in an 8 end game). Recalling my previous description, statistical changes start to appear in the middle game. Trends appear for each situation (tied with hammer, 1 up without, etc). So what are some considerations when determining our Middle Game Strategy?

The first decision is whether to continue aggressive play. This is usually the case early in each end, but as the end develops a team will choose shot calls based more on scoreboard and ends remaining before the End game. So to start discussion on Middle Game strategy, let’s start by examining which scenarios are more favorable in the End Game.

I will use a 10 end game from now on, the reduce complication. To transfer this analysis to an 8 end game, just subtract 2.

In the 9^th End, one down with hammer is a disadvantage greater than any other point in the game. In fact, only the scenarios starting the 9^th and 6^th ends have tied without hammer the same (less than 1%) as one down with hammer. In all other cases, one down with hammer is preferred position. If we have the hammer in a close game the ends previous (5^th and 8^th), it allows us to take some additional risk with less penalty for being forced to a single point. Recall a Close game is one in which a team is down one with hammer anytime or tied prior to final two ends.

One down with hammer:
In the 5^th and 8^th ends we can aggressively play for two or three and if we are forced to one, our chances are actually the same as if we had blanked the end. If we instead are one down with hammer in other ends, if our aggression results in being held to one, we are worse than if we blanked the end.

One up without hammer:
Conversely, forcing our opponent to one in this situation (5^th or 8^th end) gains no advantage over blanking the end. The risk of playing aggressive to force our opponent to one, which may result in a deuce, provides no advantage – a steal is required to gain any advantage. Stealing in the 5^th puts us at a 76% winning probability (Control). Stealing in the 8^th an 85% chance (Dominant Control). Blanking 5 and then stealing in the 6^th in fact gives us now an 80% chance. I’d suggest a sensible play is to tempt our opponent into blanking the 5th end and force our position without hammer in the next end. Blanking the 8^th was discussed in our last article and is more open to debate.

Tied without hammer:
If we force our opponent to a single, we again gain no advantage to blanking. However, a steal is a significant advantage over our current position. For the 5^th end, in our previous example (one up without) a steal takes us from a 61% chance to 76%. If we are tied and steal, we move from 39% to 61%. Using our analysis from the last article, switching from 39 to 61 is 50% better, whereas 61 to 76 is only 25% better. For the 8^th end we go from 34 to 65 (91% better). It would appear we are more inclined to attempt to steal in this scenario. However, stealing is always difficult and we risk our opponent making a multiple score. We are forced to be more aggressive because of our position but being one down with hammer gives us an ability to be aggressive without the same risk.

Tied with hammer:
Being forced to one in the 5^th or 8^th end is no mathematically disadvantage. In all other ends, being held to one is a disadvantage over a blank. Again, per situation one down, we can be very aggressive at the risk of being held to one. However, as pointed out above, a steal is very bad for us in this position. In fact, a steal is very bad for us every time in this position. It is most critical in the 9^th where we drop from 74% to 38%.

In every case, having hammer appears to be a greater advantage in this position. We can be aggressive without any risk of being “held to a single” as it is virtually no different mathematically from a blank.

So…what is our finding? In a close game, we’d prefer to have hammer in the 5^th and 8th ends. We may in fact make decisions in the ends previous which will force this situation. For example, if we are tied playing the 4^th with hammer and have an opportunity to blank, we are more inclined to take this route. If we are one up in the 4^th or 7th, we should force our opponent into scoring, even at the risk of a deuce, in order to have hammer in following end. Eight end games become interesting because the advantage exists in the 3^rd end. A team may even choose to tempt a blank if they are without hammer in the first end, to force their opponent to score in the second end, in order to have hammer in the third. This seems drastic and I’d suggest the advantage is not significant enough to “drop an end”. But further analysis might disprove my initial thinking.

Some readers may have noticed that there are two ends between the 5^th and 8^th. This means, without a blank or a steal, if we have hammer in one of these ends, our opponent has hammer in the other end. For example, say having hammer in the 5^th when one down results in a deuce. We are now one up. If our opponent is held to one and then we are held to one, we are now one up without in the 8^th end. However, if our opponent ties us in the 6^th and we instead manage a blank in the 7^th, we are now tied in the 8^th with hammer.

I would not suggest this analysis should be a factor in how a team begins play in and end. The modern game does not allow a team to force a blank end at will. However, as an end develops we may be more inclined to “bail out” of certain ends in order to better position us in ends where we have greater advantage with hammer in a close game.

20 Feb 2009 Kevin 0 comments

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End Game Strategy: BDO Quarterfinal – Howard versus Burtnyk

There were several decisions made during the 6^th and 7^th ends of the 2009 BDO Quarterfinal between Glenn Howard and local favorite Kerry Burtnyk. This month I will attempt to breakdown some of the decisions and determine how math could be applied to each scenario or shot call to support or contradict the final decision.

The 6^th End: Centre Guard

At this stage, the game is tied 4 -4 and Howard has the hammer. The 6^th end is, per my definition, the beginning of the “End Game” (see November 2008 article). Statistics indicate that Burtnyk at the beginning of the 6^th end (in an 8 end game) has a 35% chance to win. (Math whizzes may have already guessed Howard has a 65% chance).

Let’s begin by reviewing the probabilities that Burtnyk will win, based on various outcomes of the 6^th end (percentage chance to win with two ends remaining):

With two ends remaining:
Tied without hammer (Howard blanks 6th) = 34%
Down one with hammer (Howard takes 1 in 6th) = 35%
Down two with hammer (Howard takes two in 6th) = 15%
Up one without hammer (Burtnyk steals in 6^th) = 65%

The numbers indicate that holding Howard to a single, in the 6^th end, is in fact not much different than a blank. This is NOT the case if Howard is forced to one in the 7^th, the difference is 25% (if Howard maintains hammer) to 38% (if he is forced to one).

A steal, however, is a great advantage in the 6^th (65%). In fact, it is the most mathematically advantageous point in the game in which to be up one without. At all other times, chances are 61% or less.

Let’s ask a few questions:

Should Burtnyk play a centre guard or place the first stone in the four-foot?

In the game, Garth Smith placed the first stone in front of the rings. I would suggest this is the incorrect call…and here is why…

If Burtnyk places the first rock in the rings, should Howard play for a blank or attempt to score a deuce?

Howard now has to determine if he will play a corner guard, using hammer aggressively in an attempt to score two (or more) or instead play out a blank. We’ve seen above a blank is identical to a single point. In fact, if Howard plays out for a blank and noses his final stone, there is statistically no difference!

There are few skips at this level that would not prefer to force the issue while they have hammer with 3 ends remaining. The key reason: a deuce results in a high probability of winning. Two up without hammer during the End Game is as follows:

Two Up without hammer wins:
1 End remaining = 90%
2 Ends remaining = 85%
3 Ends remaining = 80%
4 Ends = 80%
All earlier ends = 74-75%

Howard’s only reason to wait to the next end is for a 5% advantage – IF he is able to score a deuce. Is the difference of 85 to 90% enough to make up for the risk of sacrificing an end with hammer? I don’t believe it is, and here is why. Recall my article from Dec 2008 describing a game as Close or one team being in Control or Dominant. A deuce in the 6th or 7^th end results in a Dominant position for Howard.

For those who understand how chip values change in a Poker tournament, a similar analysis can be used here. In a poker tournament, as you collect chips, each additional chip’s “value” is less than those previously acquired. For those interested in understanding this better, read David Sklansky’s “Tournament Poker for Advanced Players”.

Each percentage point in probability of winning becomes less important, the higher your chances are. The difference between having a 50% chance versus 55% is more important that 85% to 90%. Another way to examine this is the advantage in the first scenario is 10% better, but in the latter it is only 5.8%. Some readers may also note the contrary is true. A team that is behind sees a greater benefit from small percentage changes. The advantage of a 15% chance over a 10% probability is 50% better!

If Howard blanks the 6^th and either team scores one in the 7th, the game remains “Close”. He has essentially given up an opportunity to take a Dominant position; with very little risk of Burtnyk taking it at this stage (unlikely Kerry will steal two).

Also, recall that being forced to 1 is fine in the 6^th but not the desired outcome in the 7^th. Howard should be more inclined to be aggressive in 6 where a single is more advantageous, whereas in the 7^th he would then much prefer a blank to being forced to a single point.

So….

Given that Howard would statistically prefer not to blank and Burtnyk is statistically indifferent, Kerry should place the rock in the four-foot. If Howard chooses not to play a corner guard, by not making the correct play he is giving some (however slight) advantage to Kerry. The outcome for Burtnyk is good positioning for a possible steal and reduced chance for a deuce or worse.

However…
All of this analysis does imply that a team prefers to place a stone and then place a guard (after the corner) – rather than having your opponent draw to the four-foot and then corner freezing to their stone (eliminating the placement of a corner guard). Burtnyk may prefer not to have a corner in play and prefers positioning stones to the middle of the rings, even if Howard initially has shot rock. I’d be very interested in the perspective of that discussion.

The 6^th End: Walchuk’s last shot

Burtnyk sits top eight (biting four-foot) and Howard sits to the side, biting the four, guarded by two of Kerry’s stones. Despite repeated rewinding on my Tivo, I could not determine who was shot. Howard also sits third and fourth.
Kerry called for a double run-back by Walchuk. He suggested it was either that or play the guard. The result was removal of the Howard stone and sitting two, but both guards were removed and Howard remained with 3^rd and 4^th shot. This call appeared to be conservative, attempting to lower the chance at 2 or 3 and increase the chances to force Howard to 1. A steal seemed very unlikely at that stage. It appeared a deuce was probable, but Richard missed his next shot. It is my opinion that, given the difficult position if Burtnyk gives up a deuce (15%), keeping the guards and pursuing a more aggressive strategy, with a higher chance at a steal would increase their chances to win. In either case, they were in a difficult spot. Making some very rough estimates of probable outcomes for both calls:

Double Run-back:

Howard take 1 = 60%
Howard take 2 = 30%
Blank = 10%

W = 28.6%

Guard

Howard take 1 = 50%
Howard take 2 = 40%
Burtnyk steal 1 = 10%

W = 30%

These estimates are highly debatable, given the number of rocks remaining. However, on the basis that a three or two has little difference at this stage (see my comments above regarding Dominant position). Kerry should be trying to avoid a deuce at all costs while also trying to steal (not an easy task). It is my opinion (perhaps not his or others) that removing the two guards greatly reduces a chance to steal and may in fact increase the chance for an easy deuce.

6^th End, Burtnyk’s last rock

Kerry had an option to either draw around the centre guard and attempt a steal or remove the Howard stone and possibly roll behind cover and force Howard to a single. This is an interesting scenario. Per above, a steal is a significant advantage, but a deuce is also a huge risk at this stage of the game. A single by Howard or blank has virtually no difference mathematically. In fact, it was statistically irrelevant for Burtnyk to spend extra effort in attempting to roll their rock behind cover – either outcome produces the same mathematic result. Kerry might, however, determine an advantage to be either one down or tied. Also the added chance Glenn might miss his draw for one (however remote) could be considered a slight advantage in this situation. I want to stress that the mathematic analysis does not take into account other factors, ice conditions, opponent, etc, which Kerry may have considered.

Mathematically, how often would Kerry have to make a perfect draw for the steal attempt to be the correct call?

W = x(.65) + y(.35) + z(.15)
Where x = chance of a steal
y = chance Howard takes one
z = chance Howard scores two

We know Burtnyk’s hit results in a likelihood of 35%, set W = .35

.35 = x(.65) + y(.35) + z(.15)
Let’s choose a value for y and solve for x.

We estimate y = .3. That is, Howard will score one 30% of the time

Z = 1 – (x + y) = .7 – x

,35 = x(.65) + .3(.35) + (.7-x)(.15)

X = 28%

Conclusion, if Burtnyk believes Howard will likely take one 30% of the time, he needs to be successful with a steal > 28% in order to attempt the draw. If he actually believes Howard may only score a single half the time, it drops to 20%. If instead you assume no single and either a steal or a deuce, he must be successful stealing > 40%. Ultimately there are many factors, amount of curl and length of guard being most critical.

I found it interesting that, debating whether or not to attempt a steal is an unclear decision. Meanwhile, the very capable commentating team of Mike and Joan (ranking above their TSN counterparts, in my humble opinion) were instead focused on Burtnyk possibly drawing to the back or playing a roll in order to force Howard to one. As we’ve pointed out above, this decision has no statistical impact!

6^th End – Howard’s Last Rock

On Howard’s final stone, assuming Burtnyk’s rock was easily accessibly, should he draw for one or blank the end? As we’ve stated above, mathematically there is no difference. Ice conditions, your opponent and other factors would likely come into play.
I would be tempted to blank, but the decision is not as clear as some might suspect. Many teams could choose to draw for one and they may very well be correct. If your team can hit well and possibly clear the mess your opponent is likely to create in the 7^th end, increasing potential for a blank in the 7th, blanking the 6^th end might be your preference. If your team prefers aggression and is more comfortable forcing the issue without hammer, by all means take 1 in the 6th and attack in 7. Also, you may asses your opponent is stronger with hammer and would prefer to keep it. Or the opposite may be true, and their ability to set-up an end without hammer may be something you wish to avoid.

7^th End – Playing the Blank with 7 rocks to go

In the 7^th end, Burtnyk was faced with another interesting decision. One down with hammer and a corner guard. No other rocks in play. Third’s first stone. Rather than attempt a deuce (and risk being held to one or surrender a steal) they chose to peel and play for a blank.
Blank results in Burtnyk being one down with hammer and one end remaining.

W = 38%

How often would Burtnyk need to score a deuce in order to attempt a draw, rather than blank.

Let’s assume that if they attempt a draw around the corner, a steal or blank will not occur. Either Burtnyk is forced to a single or he scored a deuce.

X = Burtnyk takes 1
Y = Burtnyk takes 2

.38 = x(.26) + y(.62)

x = 1 – y

.38 = (1-y)(.26) + (y)(.62)

y = 35%

Burtnyk will have to score a deuce greater than 35% of the time for the draw (rather than peel) to be correct. Some of our recent analysis indicates that top teams in fact win slightly more than 74% when tied in the final end. Combine this with some small chance of a steal, Burtnyk would need even greater confidence in his chances to score two.

Whew! That’s it for this month. Congrats and good wishes to Team Pahl (Alberta) and Team Lobel (Ontario) in qualifying for their respective provincials. And good luck to all readers of Curl with Math, whether you are chasing your first or fourteenth Purple Heart.

26 Jan 2009 Kevin 0 comments

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Statistics for Womens’ Curling and What is “Control”?

We’ve finally gathered data for Women’s Curling events. This data is taken from 4-rock games played during Provincial, Scotties (i.e. Canadian National Championship), World Championships, Olympic Trials, Olympics and WCT events over the last several years. Unfortunately our sample size is larger (more than double) for the men’s events. However, we have enough numbers to give some indication of general trends and comparisons with the games of their opposite sex counterparts. So what do we find?

Statistics for Women’s Curling

Ends Remaining*	Tied with Hammer**	Down 1 with Hammer	Down 2 with Hammer	Down 3 with Hammer	Down 4 with Hammer
1	0.691	0.397	0.145	0.020	0.001
2	0.611	0.371	0.166	0.068	0.018
3	0.602	0.391	0.208	0.083	0.039
4	0.596	0.402	0.216	0.123	0.062
5	0.613	0.433	0.242	0.152	0.086
6	0.600	0.413	0.263	0.163	0.089
7	0.585	0.413	0.280	0.183	0.092
8	0.562	0.444	0.290	0.156	0.123
9	0.556	0.454	0.281	0.176	0.080

Ends Remaining*	Up 1 with Hammer	Up 2 with Hammer	Up 3 with Hammer
1	0.865	0.973	0.997
2	0.834	0.928	0.975
3	0.816	0.882	0.952
4	0.763	0.886	0.923
5	0.779	0.849	0.917
6	0.756	0.854	0.930
7	0.766	0.847	0.940
8	0.734	0.866	0.927
9

* takes into account 8 and 10 end games
** For 1 Ends Remaining includes extra end games

Tied with Hammer
Within 1-2% of the men until the End Game (last 3 ends). During final ends, winning chances for women are 60, 61 and then 69% in the final end vs. 65, 66 and 75% for men.

1 Down with Hammer
Of all the scenarios, this is most similar to men’s numbers. Usually only 1-2% better chance for women to overcome the deficit of 1 down with hammer to win. Final three ends have same pattern: women’s is 39% to 37% to 39% (38, 35 and then 38% for men).

2 Down with Hammer
Women’s teams are again 1-2% more likely to win in this position except for the final end where data shows a 14.5% chance for a women’s team versus 10% chance for a men’s.

3 Down with Hammer
A 2-4% better chance for women’s team throughout the game.

Down 4 or more with hammer
Shows generally 2-4% increase in chance for women’s team.

1 Up with Hammer
A 2-3% better chance of holding this lead for men’s teams than for women’s.

2 Up with Hammer
Within 2-3 % early on but actually men’s teams hold on to win 4-6% more often from the Middle Game onward, with the exception of the final end where the difference is only 2%

3 Up or more with Hammer
Within 2-3 % early on but actually men’s teams hold on to win 4-6% more often during the Middle Game. The results from the End Game (final 3 ends) are nearly the same – you win nearly every time if you are in this spot.

What does this tell us about Women’s Curling?
Women’s teams tend to have a higher chance of coming back from a deficit and, subsequently, less chance of holding onto a lead. The difference is less noticeable in close games (tied or 1 up) and tends to widen as one team takes a more dominant position. That is, the greater the lead the greater the difference versus men’s teams in likelihood of a comeback. If we believe the ability to throw heavy peel weight successfully is the major difference in women’s and men’s games, then these numbers look much like what we would expect.

The most noticeable and important difference seen is the case of tied with hammer or down two with hammer in the final ends. The difference is about 5%. These numbers provide some support to determine how women’s teams might approach the game differently than men’s teams. For example, if tied with hammer with 3 or 2 ends remaining, a women’s team may be less inclined in blanking to retain hammer than in forcing a score. Tied with hammer is, during these ends, not a statistical advantage over 1 up without. In fact, with 2 ends remaining, women’s teams have a slight (2%) statistical edge in being 1 up without hammer! In the men’s game tied with hammer is an advantage of 3-4% with 3 ends remaining and 1% with 2 remaining.

What is Control?
A common term heard over drinks at curling rinks across the globe is “Control”. “If we make this shot for two that will put us in control”. “You had control from the 5^th end on”. And, the most hated phrase “We had control the whole game and lost it at the end”. So, statistically speaking, what do we think is meant by “Control”?

I propose that there are actually three positions during a game a team can be in. If one is Control it follows that there must be another type of game that is closer than this, which I will call a “Close” game. It also then is reasonable to suggest there is a position where you are even better than in control, let’s call this “Dominant” position.

If we assign the game condition based on a probable outcome:

Close occurs when the odds for a win is no greater than 66% for either team. Another way to say this is the team behind has better than 2-1 odds of winning.

Control exists when one team has greater than 66% but less than 80% odds of winning. This range is between 2-1 odds and 4-1 odds for the team that is behind.

Dominant position is when one team holds a greater than 80% statistical chance of winning. Another way to show this is greater than 4-1 odds of a comeback.

Based on these numbers:
“Control” occurs when team is:

Tied with hammer in the final two ends or extra end.
Up 2 without hammer anytime before two ends remain.
Up 1 with hammer anytime before the final four ends

With three and four ends remaining, 2 up without hammer is right at 79 and 80% respectively. With two or fewer ends left, you are Dominant in this position. When statistically in “Control” Up 1 with hammer, you are between 77 to 80%, with the exception of the third end in a ten end game where your chances are 75%.

“Close” position occurs when a team is:

Down 1 with hammer OR Up 1 with hammer
Tied with hammer anytime before the final two ends

When tied with hammer and 2 ends remaining, chance of winning is exactly 66% - so we could argue whether a team has Control at this stage or it is Close.

“Dominant” position occurs at any other score during the game.

We can start to analyze our pre-game and in-game strategy using the definitions of Control, Close and Dominant. I’d also suggest incorporating the definitions for which section of the game, based on Early, Middle and End Game (from article of Nov 2008, “Is Curling a Battle For Hammer?”). Recall the End Game is the final 3 ends and extra end if required. The Middle Game is the middle 3 ends and the Early game is then either 4 ends for a 10 end game or 2 ends for an 8 end match.

Using this model creates the following 9 game scenarios:
Early-Close
Early-Control
Early-Dominant
Middle-Close
Middle-Control
Middle-Dominant
End-Close
End-Control
End-Dominant

I’d suggest teams could use this model as a way to develop pre-game strategies for how to approach each of these positions. I might even tack a stab at examining these, but that will have to be left for another article…

Statistics for Grand Slams

Watching the Masters a short while ago (except the semi’s which were on the BOLD network –wherever that is), I started to question the statistical basis we are using and see if there are some discrepancies for the Slam events versus the entire dataset for all WCT (including Slams), Olympics, Olympic Trials, Brier, Worlds and Provincials. Our data size is still very small but I wanted to get a general sense if we saw any differences. Tied with hammer in final or extra end is currently nearly 80% for Slams versus 75%. 1 down with hammer in 9 was 39%, close to our baseline of 38%. Down two with hammer was about 12% versus 10%. The only major difference appears to be tied with hammer. The Slams numbers are based on sample size of about 270, compared to over 3000 in our full dataset. Assuming that Grand Slam teams are generally stronger than the other fields; does this mean better teams win more than 75% when tied with hammer? Possibly yes, but difficult to say for certain without a larger sample.

If Martin played Howard a single end game 20,000 times, each team having hammer for 10,000, what do you think the percentages would look like?

26 Nov 2008 Kevin 0 comments

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Is Curling a Battle for the Hammer?

Back during the era of the Grand Old Game; before push brooms, The Ryan Express and Shorty Jenkins ice moved us all to Free-Guard Zone; it was often heard at most every club by players at every level (of play and inebriation) that “Curling is a battle for the hammer”. To some degree this may have been true, though the strategies of teams like Savage and Werenich, Burtnyk and Olson and the Howard brothers opposed this mentality with their aggressive play. Rather than “Take 2 then Give 1”, these Superteams preferred to put constant pressure, take many and then steal until the opponent shook hands.

The question I raise is, in Today’s 4-rock game, is it still possible that Curling can be a Battle for Hammer? In my analysis, I will touch on several areas and provide some data which may help us find and answer.

The Hammer Advantage
Starting with hammer is an advantage. We know this because every time your team wins the toss, you don’t think too long about whether or not to choose stone colour. I expect some Ontario folks will shout out “I remember this one time so and so knew the rocks were bad and chose colour instead of hammer” – but I expect this is not a common occurrence.
The “draw to middle” or pre-selecting number of games in a round-robin where teams get hammer also indicates its importance.
We also have some data which tells us how often the team beginning with hammer wins. Unfortunately not all events keep proper track of who started the game with hammer and our sample size is not as large as we would like. One way to take a sample of data is looking at 10 end games that are tied after 2 ends (i.e. now an 8 end game). Our results show us a 61% winning percentage.
This doesn’t tell us much more than we already know, except to help us understand if 8-end games are fair. That is not, however, the purpose of this article (though we may return to this idea).

The Early Game

I believe Curling, sometimes called “Chess on Ice”, has an Early, Middle and an End Game (no pun intended, honest). End Game is the final 3 ends and extra end if required. The Middle Game is the middle 3 ends and the Early game is then either 4 ends for a 10 end game or 2 ends for an 8 end match. I haven’t chosen these game sections based on my best guess or what I’d like them to be, I’ve looked at the numbers and they reveal something very interesting:

The Odds of winning at the completion of each end during the Early game is nearly equal.

This is a fascinating discovery that not only explains why the Early Game doesn’t end until 6 ends remain in the game, but supports the theory that an 8 end game is competitively equal to a 10 end game. If you extrapolate the numbers, we might be even safe in assuming 12 ends or even 14 would also have the same outcome. The only benefit of a longer game (other than perhaps more beverage sales to the fans) is that the more ends played, the greater advantage a stronger team will have over a weaker team. The analysis behind that theory is not the purpose of this discussion, so back to where we were…

Middle Game
Odds of winning during the Middle Game start to trend in a specific direction.

End Game
Odds of winning in during the End Game trend steeply in one direction. The exception is 1 down with hammer, which drops then rises before the final end.

I have a graph which best shows this trend, but am unable to figure out how to insert it this blog. So, you’ll have to read into this chart to see what I mean….

Score				Ends Remaining
	9	8	7	6	5	4	3	2	1
Tied with hammer	0.602	0.612	0.607	0.619	0.613	0.625	0.648	0.662	0.743
1 down with hammer	0.429	0.421	0.422	0.428	0.393	0.407	0.383	0.347	0.383
2 down with hammer	0.272	0.264	0.262	0.245	0.243	0.200	0.206	0.146	0.104
3 down with hammer	0.147	0.146	0.149	0.129	0.113	0.096	0.068	0.046	0.010
4 down with hammer	0.092	0.067	0.075	0.053	0.042	0.037	0.019	0.010	0.001
1 up with hammer		0.751	0.783	0.770	0.801	0.788	0.835	0.868	0.896
2 up with hammer		0.868	0.899	0.883	0.894	0.930	0.943	0.967	0.989
3 up with hammer		0.882	0.942	0.957	0.957	0.979	0.985	0.993	0.998

Some comments:
Tied with hammer: Winning percentage is within 60-62% until 3 ends remain where it jumps more dramatically to 65%. That is, a tie game has nearly the same statistical outcome for all ends until the End Game is reached (starting the 6^th or 8^th).

1 down with hammer: Maintained around 42-43% until 5 ends remain, where drops to 39% in the Middle Game. It then rises back over 40% and during the End Game, 1 down winning percentage drops to 38%, then down below 35% and back above 38%. This phenomenon was discussed in the article “To go for two… or not? Masters of Curling final: Howard vs. Ferbey” from Black Book of Curling 2007-08.

Two down with hammer shows 27 to 26 % during Early Game than a drop below 25% beginning the Middle Game and a drastic drop to 20% with 4 ends remaining.

3 down with hammer: Stays around 15% until 6 ends remain then drop to 13%, then flat at 11 to 10% with another drastic drop beginning the End Game of below 7%.

1 up with hammer: Stays fairly constant, from 75% then leveling in the range of 77-80% until the End Game begins and it jumps to 84%.

The other scenarios are not as common, but do have some interesting results:

4 down with hammer: Transitions, oddly enough: 9-7-7-5-4-3-2-1-0%. Short answer – don’t be 4 down after the Early Game or you’re screwed.

2 up with hammer: 87-90% until we reach the final stage of the Middle Game (93%).

3 up with hammer: In an 8-end game, anytime you are in this position you shouldn’t lose. In a 10 end game – just get past the 3^rd and it would take a monumental collapse to lose (though we’ve all been there once or twice).

Battle For Hammer?

To try to answer our original question, we need to define what is meant by “Battle for the Hammer”.

Let’s assume “Battle for Hammer” implies a team which starts with hammer wishes to keep it and the opposition is trying to gain that position (tied with hammer). The advantage of tied with hammer is only slight, roughly 60 to 40, until the End Game. Teams who win 60% of their games don’t often place high in the money and they certainly don’t win Briers or Olympic Gold. We don’t gain a substantial position until the final end, where we still lose 1 of every 4 games. By this definition, I’d suggest Curling is not a Battle for Hammer.

Now, if assume we this phrase to mean a battle to gain hammer with the lead, then it could be argued Curling is a Battle for Hammer. 1 up with hammer with 8 ends remaining is the same as tied with hammer at the end of the game (75%)! Having hammer with a lead of 2 or more points is very strong. It is preferred to 3 up without during every stage of a game, except for the final end where both positions are equal.

How Important is the Hammer?

So how do we begin to analyze the importance of last rock? At any time during a game we can determine its statistical value. We’ve determined that leading with hammer is a strong position, stronger than being further ahead without hammer. But by how much? Let’s compare tied and 1 up without, 1 up with and 2 up without, and 2 up with versus 3 up without.

Tied with hammer vs. 1 up without hammer
Often, in a tie game, when a team is forced to 1 we state the opposition has done their job and taken away control. However, stats show us that there is only a small difference between tied or 1 up without

Score				Ends Remaining
	9	8	7	6	5	4	3	2	1
Tied with hammer	0.602	0.612	0.607	0.619	0.613	0.625	0.648	0.662	0.743
1 up without hammer	0.571	0.579	0.578	0.572	0.607	0.593	0.617	0.653	0.617

During the Early Game the difference is about 2-3%. It jumps to 5% at the beginning of the 5^th end, or Middle Game. Then ranges between 0-3% until the last end where it jumps to 12% advantage for tied with hammer.

1 up with hammer vs. 2 up without hammer

Score				Ends Remaining
	9	8	7	6	5	4	3	2	1
2 up without hammer	0.728	0.736	0.738	0.755	0.757	0.800	0.794	0.854	0.896
1 up with hammer		0.751	0.783	0.770	0.801	0.788	0.835	0.868	0.896

When 1 up with hammer you are stronger than 2 up without by 2-4% until 4 ends remain, where it reverses to 1% advantage when 2 up, then back to advantage of 4% to 2% then 0% for the final end.

2 up with hammer vs. 3 up without hammer

Score				Ends Remaining
	9	8	7	6	5	4	3	2	1
3 up without hammer	0.853	0.854	0.851	0.871	0.887	0.904	0.932	0.954	0.990
2 up with hammer		0.868	0.899	0.883	0.894	0.930	0.943	0.967	0.989

Again, up with hammer is slight advantage, usually only 1% with the exception of 7 or 4 ends remaining where it is 3-4%. In real terms, these two positions are essentially equal.

So, what does this mean?
There is clearly not a significant difference in each of these scenarios. In each case having hammer while up is a slight advantage, but usually only 2-4%. Therefore, I disagree with the theory that Curling is a Battle for Hammer. Take, for example, an 8-end game where you have hammer and are held to one in the first end. Instead of having a 60% winning percentage you drop to 58%. Your position is in fact not much different than where you were at the beginning of the game. Much more significant is to have a shot for one and instead give up a steal in the first end, going from 58% to 42%.

Interesting to point out that often when the team without hammer holds the opposition to one in the first end it is perceived they have “won” the end or done their job. In reality, they have only gained a 2% advantage from where they were. More correct perhaps to state they have successfully “avoided” the position of falling behind by two or more.

Next article I will be revealing data on the Women’s Game and also attempt to tackle the question of what is “Control”.

26 Oct 2008 Kevin 0 comments

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What strategy should I employ when one down with hammer in 9th end?

One of the most difficult situations towards the end of a game is one down with hammer playing the next to last end (9^th or 7^th). Statistically, the lowest probability of winning when one down with hammer is in the next to last end (34.9%). In the last end or with two ends remaining it is over 38%. In fact, tied with hammer with two ends to play is 67.5%, only 2.4% higher than if you are one up without!
I have seen every type of play in this situation, from both teams keeping it clean to produce a blank (and a 5 minute end) to every rock in play. So, using mathematics, what is the correct strategy, or at the very least, how do we approach this scenario to be better prepared when it happens?
Recall statistical outcomes for the final end:

Expected Results (ER) in the final end:
Odds of winning if tied with hammer (x) = 74.5%
Odds of winning if one down with hammer (y) = 38.2%
Odds of winning if two down with hammer (z) = 11.0%
So how should we play the end to maximize our chances and overcome our unenviable position? Also, our opponent without hammer can dictate the early part of an end by placing guards, how does this impact our decisions?
Let’s start by examining the numbers. Clearly, the best scenario is to take three (or more). If we score three we have an 89% chance to win. The next best scenario is to score a deuce and have a 61.8% chance in the final end. If we play aggressive and are forced to one, however, we win only 25%. In fact, if the end results in a draw for two and we miss, only scoring one, our chances flip from 61.8% to 25% - a huge difference. A blank is better than scoring a single, leaving us at 38.2%, but still we can expect to lose more than half the time.
Let’s also examine the competition’s strategy. If they read my articles, they know the correct play in this situation is to force the action and attempt a steal or force or a single, at the risk of a deuce.
Let’s assume for now our opponent will place a centre guard.

Option 1:
First, let’s attempt to play a clean end with the expected outcome a blank. After the opposition places a centre guard, we choose to draw to the side. Our opponent will most likely hit our rock and stay in the rings. Assuming we exchange shots and no one rolls their shooter to center, we can remove the centre guard with the 6^th rock of the end. If the opposition splits the rings (most likely) we then are playing out the end trying to make a double in order to blank. If we fail to make a double and our opponent does not roll out, we will be forced to a single point and left with a 25% chance of winning with one end to play.
This option appears to be a losing strategy. No chance for three and not likely two, so our only outcomes are less than 50%. Let’s estimate some outcomes based on this strategy:
Score Three = 0
Score Two = 5%
Score One = 30%
Blank = 60%
Steal = 5%
W = 34%

Option 2:
Let’s try a very aggressive strategy. Come around with our first rock, corner freeze if our opponent does also, and continue drawing or soft taps until an opportunity develops for a “big shot” at scoring multiple points. Let’s again make some rough estimates based on this strategy:
Score Three = 10%
Score Two = 30%
Score One = 40%
Blank = 0%
Steal = 20%
W = 40%
Even if we could successfully blank 100% of the time, we do better playing very aggressive.

These numbers are not completely fabricated; they are based on existing data. For example, as of this writing, during this season (2007/08); Howard, Ferbey and Martin score three 13% of the time across all ends played with hammer. Taking an average of three “average” WCT teams results in 8%. These numbers can be skewed due to the better teams (aka Howard, Ferbey and Martin) being ahead more often and their opponents need to take greater risks – often resulting in big ends during later stages in a game.
Option 3:
Let’s think of a third scenario. We come around the guard, buried in the top eight foot, possibly biting the four foot. Our opponent successfully corner freezes. We now have several options:

Corner freeze to the opponent stone
Draw to the side of the rings
Split the centre guard with a “tick”.

Shot Call 1 will lead us to the scenario in Option 2 above. What about the other two?
2. Draw to side of rings:
If we can sit second stone, our opponent has several options, depending on how all the stones are sitting. In most cases, we could expect he will attempt a hit on the open rock and try to roll to the centre, behind cover, to sit either first or second. If he is successful, we are back to Option 2 (Very Aggressive) above and we are likely behind in the end. We will need a big shot or mistake from our opponent, but the aggressive nature of the end now makes that more likely. If our opponent does not roll successfully, we can attempt a hit and roll. In either case, if we instead choose to remove the guard or run it back, a steal becomes less likely, however a chance for a single increases and a three is highly unlikely. We can expect the end will look more like Option 1 above, with our best outcome a blank or low probability of a deuce.
3. Split the centre guard:
This is the scenario which I don’t recall seeing in a game but appears powerful. If we can split the guard and create two corner guards, our opponent now is left with an unclear decision. Does he put a centre guard back, even though you’re shot stone? Does he peel a single corner, or attempt a double peel (if it is even possible). Does he run his stone onto yours, attempting to lie two? This last option seems the best scenario but, it will leave two guards, an open four foot, and most likely both rocks will not sit perfectly behind cover above the tee line. Even if they do, a corner-freeze is available and with two guards, one which is now yours and could be driven back later on, the advantage appears to be with you. If our opponent gets cautious and elects to peel the guards, we now have options to play for a blank (if we choose), place another guard or move rocks around in the house and attempt our deuce with minimal chance for a steal.

Ultimately, a team must determine which option maximizes a chance for two or three, limiting the times you get a single point AND minimizes a steal by your opponent. Not an easy answer. The difficulty is that a blank, which is preferred to a single, is not a high occurrence if you attempt to score two or three. This specific scenario, one down with hammer with two ends to go, is one of the most interesting in the game and possibly more intriguing than the final end.
What If Our Opponent draws into the rings?
Instead of a centre guard, our opponent puts his first rock in the rings. We can now place a corner guard or hit the rock in the rings. If we call for a guard, our opponent could now choose a centre guard. We then draw around and are back to scenario in Option 2 above, though we are behind in the end. If we hit the stone, we are playing for a blank which, in this case, is very likely. For the team without hammer, assuming we will place a corner guard, this appears to be a stronger play than above. The team with hammer now faces the corner freeze and could have more difficulty getting shot stone.
This decision stems more from a team’s strategy of preferring to sit shot or be positioned frozen to shot stone in order that a shot can be played later in the end. I can see advantages for both cases and will leave it up to the reader to determine what they prefer in this situation.
Conclusion:
Appears there is no clear answer to our original question. It is clear that attempting a blank is a less risky play but provides no upside and most likely results in us winning less than one in three times. Playing aggressive increases our chances, but also creates a complex end; producing many options for both teams and presenting opportunities to stay aggressive or bail out. For fans, it is clearly the most interesting scenario in a game and, for a skip, one that cannot be simply “played by the book”.

14 Jan 2008 Kevin 2 comments

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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:
SHOT SELECTION OPTIONS
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 3^rd 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 3^rd end of a 10 end game is identical to the 1^st 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.
Hit
W (if down two) = z = 26%
Draw
Assumptions
The team with hammer either scores three or is held to one. The chance for a steal or deuce is considered negligible.
Variables
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 disagree with this approach and will attempt to explain why….in another article.

12 Nov 2007 Kevin 0 comments

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Curl With Math
Kevin Palmer takes a look at the numbers behind the roaring game..
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