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

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

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

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.
   

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

Statistics for Women’s Curling

Ends Remaining*	Tied with Hammer**		Down 1 with Hammer		Down 2 with Hammer		Down 3 with Hammer	Down 4 with Hammer	Up 1 with Hammer	Up 2 with Hammer	Up 3 with Hammer
1	0.691		0.397		0.145		0.020	0.001	0.865	0.973	0.997
2	0.611		0.371		0.166		0.068	0.018	0.834	0.928	0.975
3	0.602		0.391		0.208		0.083	0.039	0.816	0.882	0.952
4	0.596		0.402		0.216		0.123	0.062	0.763	0.886	0.923
5	0.613		0.433		0.242		0.152	0.086	0.779	0.849	0.917
6	0.600		0.413		0.263		0.163	0.089	0.756	0.854	0.930
7	0.585		0.413		0.280		0.183	0.092	0.766	0.847	0.940
8	0.562		0.444		0.290		0.156	0.123	0.734	0.866	0.927
9	0.556		0.454		0.281		0.176	0.080
* takes into account 8 and 10 end games
** For 1 Ends Remaining includes extra end games

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.    
 

2. The 6^th End: Walchuk’s last shot
   

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

Double Run-back:
 

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

W = 28.6%
 

Guard
 

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

W = 30%
 

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

3. 6^th End, Burtnyk’s last rock
   

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

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

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

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

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

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

Z = 1 – (x + y) = .7 – x
 

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

X = 28%
 

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

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

4. 6^th End – Howard’s Last Rock
   

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

5. 7^th End – Playing the Blank with 7 rocks to go
   

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

W = 38%
 

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

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

X = Burtnyk takes 1
Y = Burtnyk takes 2
 

.38 = x(.26) + y(.62)
 

x = 1 – y
 

.38 = (1-y)(.26) + (y)(.62)
 

y = 35%
 

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

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

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

Statistics for Women’s Curling

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

           
 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 “Close” position occurs when a team is:

• Down 1 with hammer
• 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?

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

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

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

The Early Game 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Battle For Hammer?

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

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

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

How Important is the Hammer?

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

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

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

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

1 up with hammer vs. 2 up without hammer
 

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

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

2 up with hammer vs. 3 up without hammer
 

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

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

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

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

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