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 6th 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:

Notice that if the end is blanked, TC has a 24% chance. At the beginning of the 6th 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 6th end, the decision is very different. Now, entering the 8th 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 7th end. If the 7th 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 5th and 8th 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 5th through 7th ends in a ten end game (or 3-5th 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 9th End, one down with hammer is a disadvantage greater than any other point in the game. In fact, only the scenarios starting the 9th and 6th 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 (5th and 8th), 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 5th and 8th 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 (5th or 8th 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 5th puts us at a 76% winning probability (Control). Stealing in the 8th an 85% chance (Dominant Control). Blanking 5 and then stealing in the 6th 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 8th 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 5th 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 8th 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 5th or 8th 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 9th 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 5th 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 4th with hammer and have an opportunity to blank, we are more inclined to take this route. If we are one up in the 4th 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 3rd 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 5th and 8th. 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 5th 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 8th end. However, if our opponent ties us in the 6th and we instead manage a blank in the 7th, we are now tied in the 8th 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.

End Game Strategy: BDO Quarterfinal – Howard versus Burtnyk

There were several decisions made during the 6th and 7th 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 6th End: Centre Guard
At this stage, the game is tied 4 -4 and Howard has the hammer. The 6th end is, per my definition, the beginning of the “End Game” (see November 2008 article). Statistics indicate that Burtnyk at the beginning of the 6th 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 6th 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 6th) = 65%

The numbers indicate that holding Howard to a single, in the 6th end, is in fact not much different than a blank. This is NOT the case if Howard is forced to one in the 7th, 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 6th (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 7th 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 6th 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 6th but not the desired outcome in the 7th. Howard should be more inclined to be aggressive in 6 where a single is more advantageous, whereas in the 7th 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 6th 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 3rd and 4th 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.

6th 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!

6th 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 7th end, increasing potential for a blank in the 7th, blanking the 6th 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.

7th End – Playing the Blank with 7 rocks to go
In the 7th 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.

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?

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 5th 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
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?

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.

This graph best shows this trend:

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 6th or 8th).

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 3rd 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
During the Early Game the difference is about 2-3%. It jumps to 5% at the beginning of the 5th 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
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
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”.

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 (9th or 7th). 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 6th 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”.

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 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.

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 don't entirely agree with this approach and will attempt to explain why….in another article.

Blackbook of Curling 2007/08 - Reprints

1.To go for two… or not? Masters of Curling final: Howard vs Ferbey
Ontario’s Glenn Howard squad had a spectacular Seasons of Champions run in 2006-07, going undefeated throughout the Ontario playdowns, and losing only a single round-robin game at the Tim Horton’s Brier and the Ford World Men’s Championship. Earlier in the season, they defeated Edmonton’s Randy Ferbey in the finals of The Masters, one of the four Grand Slam events. An intriguing situation occurred during that final, showing that even the best teams in the world are not always certain of the “correct” strategic decision… even when faced with a common scenario.

 

It is the sixth end of the new Grand Slam format – eight-end games – and there are only two ends remaining. Howard has the hammer, the teams are tied at 3-3 and Randy Ferbey, throwing third stones, has just placed his first stone behind a centre guard. It is hanging on the back side of the button. These are the only two rocks in play.

The first indication from Glenn is for third Richard Hart to throw his lefty out-turn (in-turn for a right hander) and draw to the face of the Ferbey stone, to possibly sit shot-stone. While sitting in the hack, Richard sees another option – a soft hit on the Ferbey stone situated out of the rings with his in-turn, and roll to the open.

After much debate, they agree to hit the Ferbey stone. The decision centers on whether the preference is to aggressively play for a deuce – at a risk of a steal or possible single point scored – or to play conservatively, to increase the possibility of a blank end, in order to keep last-rock advantage in the seventh end. The outcome, at this stage, is unimportant, the question is: what is the correct call?

The dilemma Team Howard faced is very common. With three ends remaining, tied with hammer, what position would I prefer to have going into the second-last end, and what risk should I take in order to get to that position? It is quite fascinating to note that at the highest levels of the sport, much of this decision is made on instinct or “situational analysis”… because, in fact, we can examine the decision using mathematics.

Let’s attempt to analyse the situation based on available statistics and mathematical in-game analysis. To examine the decision, we need to determine all the Expected Results (ER) and Variables (V) to be included and state which are being omitted. ERs are numbers, in percentages, taken from existing statistics. Variables are numbers, in percentages, which are estimated during a game situation. Variables are based on the chance of making a single shot or the expected outcome following a succession of shots during an end. The ERs, Vs and related assumptions I used are listed below. I have indicated in each case which letter is used to represent each ER or V.

Expected Results (ER) with 2 ends remaining:
Odds of winning if tied with hammer (x) = 68.3%
Odds of winning if one down with hammer (y) = 35.2%
Odds of winning if two down with hammer (z) = 14.8%
Odds of winning if three down with hammer (m) = 4.5%

These Expected Results are based on available statistics for four-rock FGZ games played in the Brier, European and World Championships, Grand Slams and other WCT events, Provincial Championships, Olympic Games and Olympic Trials. In using these statistics, we assume skill is relatively equal, conditions are similar and shotmaking ability is similar. These assumptions are clearly debatable, but we disregard any difference in skill level amongst specific teams or events.

Assumptions:
· If Howard attempts to draw a blank will not occur
· Howard will not score 3
· The most Ferbey will steal is a single point
· Richard Hart will not have a complete miss. He will either:
   o Draw and place his stone in the rings (counting first or second)
   o Hit and remove Ferbey stone or remove the guard
· Richard attempts a soft hit and will not roll his shooter out if he removes the Ferbey stone.

Option 1: Draw
With his opponent on the back of the button, a draw around the guard is clearly the aggressive play. Assuming Richard will not miss, two outcomes exist – Howard will sit one or Ferbey will sit one. For each outcome, we can assess the possible outcomes of the ends and assign probabilities to each:

Variables:
Howard sits shot stone:
Score two (a) = 30%
Score one (b) = 60%
Steal (c) = 10%

Howard sits second shot:
Score one = (d) 60%
Steal (e) = 40%
We assume Glenn Howard will have a similar chance at scoring one, regardless of the shot’s outcome. We also assume a deuce is not likely if Richard does not achieve shot-stone position. Both of these assumptions are, as always, debatable, but for our analysis we consider such possibilities negligible.

This appeared to be a difficult shot. Let’s assume Hart will achieve shot-stone position 50% of the time. He might argue this point, but we’ll save that for later analysis.

Win (W) = (.5)(a(1-z)+b(1-y)+cy) + (.5)(d(1-y)+ey))

W = 60.5%

Option 2: Hit and roll
Variables:
Howard sits shot:
Blank (a) = 75%
Score two (b) = 5%
Score one (c) = 15%
Steal (d) = 5%

Howard removes guard, Ferbey sits shot:
Blank (e) = 40%
Score one (f) = 40%
Steal (g) = 20%

We’ll assume Richard will remove the shot stone ¾ of the time (75%).

Win (W) = (.75)(ax + b(1-z) + cy + c(1-y)) + (.25)(ex + fy + g(1-y))

W = 61.6%

It would appear the correct call is the hit – but it is very close. If Howard believes Hart can achieve shot-stone position greater than 50% of the time, the draw would appear to be correct. If in Option 1, for example, Howard “gets shot” 60% W = 62% and at 75% W = 64.2%.

We are using many variables in this analysis (5 and 7 respectively), which can lead to greater variance in our final numbers. For example, with Option 1, if we set d=40% and e=60%, W drops from 60.5% to 57.5%. And because we are examining a situation in which several (in fact seven) rocks have yet to be thrown, this analysis becomes very difficult.

What is perhaps more important than the analysis of this specific situation is the common situation of a close game with two ends remaining. An interesting trend was uncovered when we started digging into available game statistics. Virtually every scenario (tied with hammer, two down without, three down without, one up with, etc.) shows a steady increase or decrease in probability of a Win as you near the final end. For example, tied with hammer reads as follows:

Your chance of winning with hammer increases from being 61% with five ends remaining to 75% in the last or extra end.

However, the scenario of one up without hammer is shown in the next table:

The results show a 3-4% difference both up and down, depending on end completed. The most interesting point we see is the increased chance of winning when one up with hammer with two ends remaining, versus three or one end remaining. 65% versus 62%. Now, commonly a team tied with three ends to play may risk a steal in an attempt to score two. The next table shows the outcome based on two down with hammer:

 

If the team with hammer in the eight end (or sixth of an eight-end game) scores the deuce, their opponent still has a 15% chance of winning and, surprisingly, statistics show a 4% chance when three down with two ends to play.

Yet another interesting indication is that two down with hammer is difficult. A team will win only 24% in this situation after playing the second end in an eight-end game. Useful for a future discussion on the danger of eight end games, but let’s digress…

Statistics would indicate a team with hammer should ensure a blank or score to be in control with two ends remaining, rather than risk a possible steal. Let’s choose variables which represent likely outcomes, based on our approach to strategy in the third-to-last end. For example, if we play aggressively, our chance of a blank is estimated to be 0% and a steal is 30%.

Aggressive starting the 8th (or 6th) end
Blank (b) = 0%
Take 1 (t) = 40%
Take 2 (u) = 25%
Take 3 (v) = 5%
Steal = (s) 30%

Win (W) = bx + t(1-y)+u(1-z)+v(1-m)+sy

W = 62.5%

 

Conservative starting the 8th (or 6th) end
Blank = 20%
Take 1 = 60%
Take 2 = 10%
Take 3 = 0%
Steal = 10%

 

W = 64.6

 

Based on my estimated variables, it appears the conservative approach, with three ends to play, is the preferred strategy. I’d be interested to see what percentages others might estimate for these variables. Once we have enough shot data collected, we may eventually be able to change these numbers from estimates to actual statistics.

 

Going back to our original game situation, Team Howard appeared to be unsure of their desired outcome for the end. If top teams use statistics to analyse both in game shots and late game situations, they can be better prepared to make these decisions while “on the clock”.

 

In case you don’t remember or didn’t see the outcome, Howard took one in the sixth, Ferbey took one in the seventh and Glenn made a great shot in the final end to win the game 5-4.

 

2. Go hard and go home 2007 Brier Final: Howard vs. Gushue
The most interesting – ie. debated – shot call of the 2006/07 curling season was that of Newfoundland’s Brad Gushue during the seventh end of the Brier final versus Glenn Howard. The Gushue team, despite missing the grizzled veteran of their gold medal win at the 2006 Olympics – Glenn’s brother Russ – were on the verge of adding a Canadian championship to their resume. After getting off to a rocky start at 1-3, the young foursome from the St. John’s Curling Club in Newfoundland had reeled off eight straight wins and was now in control, tied with the hammer, with four ends left to play. The critical seventh end came down to a decision of whether to attempt a difficult double-takeout for a possible score of three, or take a single point via the much easier shot of hitting the open stone belonging to the Howard team. The stakes were high as the Ontario team currently sat two, and any miss by Gushue could be disastrous. The reward of a three-shot lead and near certain victory, it seemed, was too great and the decision was made to play the double take-out. Ultimately, crashing on a guard led to a steal of two for Ontario and eventually a Brier and World Championship for the Howard team.

 

Mathematically, how do we analyse this situation, and what was the correct call?

Expected Results (ER) with 3 ends remaining:
1. Odds of winning if tied with hammer (x) = 64.6%
2. Odds of winning if one down with hammer (y) = 38.2%
3. Odds of winning if two down with hammer (z) = 19.8%
4. Odds of winning if 3 down with hammer (m) = 6.9%

 

Assumptions

• Gushue will be successful 100% of the time if he chooses to hit the open stone. Newfoundland will always score 1 point.

Option 1: Hit the open stone:
No variables, assuming 100% success, results in:

Win = (W) = 1-y = 61.8%

Option 2: Double attempt:
Many outcomes appear possible with the double attempt. Choosing the variables and estimating each of them may not be easy. Let’s try to choose most likely outcomes:

 

Successful double and stay, Newfoundland scores three points (a)
Successful double but the third Gushue stone is removed from play and Newfoundland scores two points (b)
Double is missed; Newfoundland scores a single point (c)
Double is missed; Ontario steals a single point (d)
Double is missed; Ontario steals two points (e)

 

Win (W) = a(1-m) + b(1-z) + c(1-y) + dy + ez

 

Recalling the situation, the target stone belongs to Team Gushue. They are attempting to jam it onto a Howard stone (removing it), roll to another Howard stone (also removing it) and keep the shot stone, the target stone and a third Gushue rock at the back of the house all in the rings. In discussions following the game, opinions were varied. Many people believed the double for three may not have been possible, that Gushue at best could score two (possibly spinning the target stone out of play). The outcome made this evaluation more difficult. By wrecking on the front guard, fans were unable to see what would happen if Gushue were to hit the target stone. In future, technology may allow us to study the behaviour of stones and, by mapping location, use a computer to determine the outcome of a shot. This type of analysis is likely possible now… however, the work required would be significant and I am doubtful the various curling governing bodies, would approve. Can you imagine if Gushue were to call a time-out and have his coach, sitting in the stands with a computer, advise them the shot was not possible based on a software analysis?

 

We will attempt to assign some estimates to all the variables and determine a possible outcome.

 

Variables:
Newfoundland scores three points (a) = 10%
Newfoundland scores two points (b) = 20%
Newfoundland scores a single point (c) = 10%
Ontario steals a single point (d) = 50%
Ontario steals two points (e) = 10%

 

Win (W) = a(1-m) + b(1-z) + c(1-y) + dy + ez

W = 52.6%

 

If we believe missing the guard will result in no worse than a steal of one, this analysis gives us a good start. Unfortunately, it is well below the known Expected Result for Option 1, 61.8%.

 

Let’s keep a and e constant at 10% and build a table of possible estimates for further analysis:

Interesting results. For Estimate #3, if Gushue can score 80% of the time, even if he only scores a single point 50%, the call is nearly identical to Option 1. In Estimate #7, if we assume a score of one is not possible but Gushue will score half the time, the shot call will result in two or three points scored (40% and 10%) or a steal of one or two surrendered (40%, 10%) and Gushue wins 58.7%, below Option 1. However, increase the result of a deuce by 10% in Estimate #10 (steal of one becomes 30%) and we are above Option 1 at nearly 63%.

Two other likely extremes:

• Set a, c and e to zero and assume outcome of either a deuce or a steal of one. If Gushue succeeds 60% of the time, W = 63.3%.
• Set b, c, and e to zero and assume either a score of three or a steal of one. Gushue needs to be successful more than 43% of the time to make this the correct call.

The numbers might start to confuse us and the final decision, which at first appeared incorrect, now appears fairly close. The ultimate answer depends on your assessment of how rocks will collide and move around the house, and the ability of Brad Gushue to hit what appears to be a small target on his front stone.

In this scenario, we are less concerned with assessing Brad’s abilities. The final outcome, wrecking on the front guard, was unexpected and likely a low probability. If he attempted this shot repeatedly, under the same conditions, I suspect he would hit the target stone at least 9 times out of 10. What we are more interested in is our estimate of the action on the stones, and who will score what once the dust has cleared. Unfortunately, this will never be known. If someone set the stones up exactly as they were situated we could, perhaps, try to recreate the outcome. If one of our readers decides to attempt this experiment, or if you already have, please let us know. In all seriousness!

Bringing some additional “non-math” analysis into the equation, if you will, is the consideration for the enormity of the moment, and the event. It is the Brier final and the position of being one up without hammer with three ends to play is, while positive, still only 62% headed to victory. Brad was no doubt drawn to visions of three points and a virtual lock on the Brier title (93% in fact). Even with a deuce the Newfoundland rink would be heading to the world championship 80% of the time. Looking back at all of our estimates, we still have Newfoundland winning greater than 50% of the time this shot is attempted. Perhaps it is worth the attempt at the big shot now – in the fourth-to-last end – when the opportunity presents itself, and take pressure off your team in the later ends of a tight game.

Just as in golf, some curlers choose to “grip-it-and-rip it” with an aggressive style, whereas others are tacticians who plod about the course and minimize both risks and mistakes. Personally, I trust the mathematics and believe the shot call Gushue attempted was indeed too risky. Specifically due to doubts about the shot being make-able for three; not because his ability is in question. One point for Howard or Gushue appeared the most likely outcome, with some possibility of a deuce (no better than estimate #2 in table above). It is, unquestionably, up to the skip to come to a final decision. As always, if you use math, it might make the decision clearer.

3. Hitting the post Canadian Olympic Trials semi-final: Morris vs. Stoughton
The use of mathematics to analyse game situations is useful in the later stages of a game and also for critical shot calls during the early ends. The John Morris call during the semi-finals of the 2005 Canadian Olympic Trials is a prime example of an early game decision that proved costly.

After stealing a single in the second end, the Jeff Stoughton team had scored a deuce in the third end to go one up without the hammer. Morris then found himself in deep trouble, facing three Stoughton counters and several guards. The two apparent options available for the Calgarians were to either a draw to force Stoughton to a steal of one point, or attempt a difficult hit between two guards and remove shot stone to count a single.

The port appeared small, but Morris was certain he could thread the needle and succeed in putting the score back to even. The shot wrecked on the front guards and Stoughton was handed a critical steal of three points. Despite a valiant comeback attempt throughout the remaining ends, Stoughton prevailed and advanced to the final, where his Winnipeg foursome eventually lost to Brad Gushue in the finals.

What mathematics can help us determine is: how certain did Morris have to be that he could make the called shot in order for it to be the correct decision?

If John makes the shot 100% of the time, it is the best call. He would much rather be tied without hammer than down two. Using available statistics, we can determine how assured he must be in order for the shot to be the correct decision.

Expected Results (ER) with 6 ends remaining:

1. Odds of winning if tied with hammer (x) = 61.8%
2. Odds of winning if one down with hammer (y) = 42.7%
3. Odds of winning if two down with hammer (z) = 24.3%
4. Odds of winning if 3 down with hammer (m) = 12.7%
5. Odds of winning if 4 down with hammer (n) = 5.4%

 

Assumptions

• The draw is successful 100% of the time. Stoughton will always be held to a single point. The chance of John being unable to draw the full eight-foot rings is considered negligible.
• The hit either results in a single for Morris or a steal of three for Stoughton. The chance of rolling out is negligible.

 Draw:

 No variables to examine, we assume 100% success. This results in:

 

Win (W) = z = 24.3%

  

Hit:

We will choose S as the single variable, indicating a successful shot by Morris.

 

W = S(1-x) + (1-S)n

 Setting W to the result for a draw, .243…

 .243 = S(1-x) + (1-S)n

 We now solve for S

 S = .577

 

The math shows us that if Morris expects to make his shot through the port more than 58% of the time, it is the correct call.

 

What at first appeared to be a foolish call now looks like a reasonable decision, based on statistical analysis. You are welcome to debate the shot call, and outcome, but the ultimate decision rests with John Morris himself, to assess if he can make the shot more than six times in ten attempts.

 

John may have felt his chances were much higher; others would counter it was lower, but the numbers tell us how he can best make the final decision.

 

Given an assessment of the Stoughton team and their abilities, Morris may determine that coming back from two down will be more difficult than 24.3%. Also, a draw is never 100% certain, therefore he may be more inclined to attempt the hit, even if his chances are as little as 50%.