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. 

1. “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 play a guard 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.
 

2. 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.
   
3. It is the 1-2 game, 7^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…
   
4. 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.
   

Statistics for Women’s Curling

           
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.

1. 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.    
 

Statistics for Women’s Curling

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….
 

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
 

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
 

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”.