bbb Curl with Math

Kevin Palmer looks at different shot options from a mathematical perspective.

Curl with Math

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:
  

 

  1. Corner freeze to the opponent stone
  2. Draw to the side of the rings
  3. 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”.  

21 Feb 2008 Kevin 1 comment

Curl with Math

Curling Myths: Can 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 clean 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:

 

 

 

  1. 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%.
  2. A steal leaves a 43% chance to win, being 1 down with hammer after first end.
  3. 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 (draw into four foot, hit and stay, etc) in early ends does not appear correct.

I would suggest the team with hammer should, generally speaking, be aggressive throughout the first end, attempting to score mulitple points at risk of a steal.  The team without hammer should be aggressive at the beginning of the end (place a centre guard, draw/corner freeze if opponent plays to centre) …but more conservative after the first few stones of the end.  The team without hammer wants  to minimize risk of deuce or higher as the end progresses. 

But some readers might ask, if both teams optimium play is to be agressive, how would a team elect to counter the opponents optimum strategy?  That is a lengthy discussion for another article…

Myth #2: A deuce early is not significant
 

In the CCA document, it is suggested that, when starting without hammer:
 

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

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

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

Hit
W (if down two) = z = 26%
 

Draw
 Assumptions
The team with hammer either scores three or is held to one.  The chance for a steal or deuce is considered negligible.
 Variables
Successful draw (hold team to one) = d
 

W = dy + (1-d)m
 

Setting W to 26%
 

.26 = d(.43) + (1-d)(.26)
 

Solving for d = .395
 

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

 Myth #3: Play conservative against a stronger team
 

It is a commonly held belief that when playing a team that is stronger than yours; play conservative, keep the game close and hope to win a close battle at the end.  The U.S. team in the 2007 World Championship appeared to use this strategy to perfection against Team Canada during the round robin.  I disagree with this approach and will attempt to explain why….in my next article.

 

 

29 Nov 2007 Kevin 4 comments

Curl with Math

Ferbey and Adams Part I: Reprint

This article is a reprint of the 2006/07 Black Book Curling article.  I have just completed some articles for the 2007/08 edition.  I will be starting on some new articles for this site shortly.  We now have statistics for winning percentage for every end, not just the last end.  This allows us to examine many more Shot Call situations than before.  I am also interested in examining the 8-end game versus 10 and some other strategy ideas that may be controversial.  For now, it’s re-runs:

Curling with Math
 It’s the 2005 Brier Final.  Ferbey, appearing in the final game for a 5th straight year, is facing what appears to be a simple decision.  It is the 9th end, and the game is tied.  A miss by the opposing skip, Shawn Adams of Nova Scotia, has left the Alberta rink lying one with no other rocks in play.  It is commonly accepted curling strategy to play for two (or more) when you have the hammer and hold your opponent to one (or steal) without last rock.  In this case, Ferbey would be expected to draw to the open side of the rings and likely have a final shot at two or perhaps three if Adams attempts a freeze and makes a poor shot.  David Nedohin, who throws the final rocks for the Ferbey team, discusses the options with Randy.  Much to the surprise of fans and commentators, they decide to remove their own stone and play out the end for a blank in order to keep the hammer in the 10th end.  The final outcome eliminated any need for second guessing the decision, as Ferbey won - David making a draw to the open four foot in the 10th end for the victory.

The question I immediately asked myself was, is that the right decision?  Not in terms of gut feelings of the moment or using hindsight, but if the game was played 1,000 times or 10,000 or infinite times, what would be the correct play based on mathematics and statistical analysis?  Could this be done and, if so, how valid are the results?

Now, let’s attempt to analyse the situation based on available statistics and mathematical in-game analysis. To examine the 9th end 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 in the 10th end    

  1. Odds of winning if tied with hammer (x) = 75.7%
  2. Odds of winning if one down with hammer (y) = 39.5%
  3. Odds of winning if two down with hammer (z) = 11.7%
  4. Odds of winning if 3 down with hammer (m) = 0.4%

These Expected Results are based on available statistics for 4-rock games played in the Brier, European and World Championships, Grand Slam and other WCT events, Provincial Championships, Olympics and Olympic Trials. In using these statistics, we assume skill is relatively equal, conditions are similar and shot making ability is similar.  This assumption is clearly debateable but we will disregard for now any difference in skill level amongst specific teams or events.Game Situation Assumptions:  

  1. Nova Scotia will attempt a freeze on the rock in the back four (they will not try a double) if Alberta draws to lie two. 
  2. When going for two in 9th, it is assumed that at worse Alberta will take 1.  Odds of a steal for Nova Scotia are considered negligible.

Let’s examine the options Alberta faces, to try for a blank or attempt a multiple score.

Option 1. Blank attempt
 Expected Results:

x = Odds of winning tied with hammer = .757           

y = Odds of winning if one down with hammer = .395             Variables:b =  Odds of blanking = 95%

This number is an estimate a Skip would make based on experience and analysis of the individual shot and subsequent shots to come. Many skips will often evaluate these types of variables at key points during a game. Let’s estimate a team at this level will succeed in blanking 95%.

Win (W) = bx + (1-b)(1-y)           = (.95)(.757) + (.05)(.605) = .7494 or 75%

We have determined that Ferbey can expect to win three of every four times that they choose the questionable call of removing their own stone in an attempt to blank the 9th end. 

Option 2. Attempt to score two or more
 

 Expected Results:

x = Odds of winning tied with hammer = .757           

y = Odds of winning if one down with hammer = .395 z = Odds of winning if two down with hammer = .117m = Odds of winning if three down with hammer = .004Variables:           

 t = odds of taking 1 in 9th = .75           

u = odds of taking 2 in 9th = .20

v = odds of taking 3 in 9th = .5As we did previously, these numbers are an estimate done by a skip prior to calling the critical shot in the game.  Let’s start by estimating that 75% of the time Adams will make a great shot with a freeze and hold Ferbey to 1 (t).  If his shot is not successful, let’s assume that 20% of the time Ferbey scores two and another 5% they score three.W = t(1-y) + u(1-z) + v(1-m)     = (.75)(.605) + (.2)(.883) + (.05)(.996) = .68015 or 68%Assuming that Adams will be successful three of every four attempts, blanking will increase the chance for an Alberta win by 7%.  What at first appears to be a controversial call, is in fact the correct call, if our assessment of Adams’ abilities is correct.Using some other estimates for the variables t, u and v, we can determine when a blank attempt is the incorrect call:    

 

t

 U
 v
 W

.75

.05

.2

70%

.6

.3

.1

73%

.6

.2

.2

74%

.6

.1

.3

75%

.5

.4

.1

75.5%

.5

.3

.2

77%

.4

.5

.1

78%

.3

.6

.1

81%

.2

.5

.3

86%

Once our assessment of Adams’ chance drops to 50%, the decision to blank is questionable.Three other factors can also weigh into the decision:   

 

  1. Our original assumption was that Nova Scotia would not steal if Alberta attempts a multiple score. It is possible (though not likely) that Adams could in fact steal if Alberta attempts a difficult shot for two or three and misses. 
  2.  Consider Ferbey’s confidence in being an “above average” team and in their ability to win if tied coming home.  The Expected Result tied with hammer in the 10th end is nearly 76%, but Alberta may believe their chances are in fact higher. 
  3. By blanking the end, Ferbey keeps greater control over the final outcome.  The Variable t in Option 2 is an estimate of their opponent’s odds of making his final shot of the end. Though they can asses the chances, Alberta cannot impact the final result.  If Adam’s makes the perfect shot, Alberta may have no choice but to settle for 1.  In Option 1, there is no Variable associated with their opponent.

Now, what about these Expected Results and the numbers associated with them? Our sample size of data is still small, only having a few years of 4-Rock Curling in Canada.  As the sample size grows, our confidence in these numbers will increase.  What about evaluation of our team skill and our opponents? Maybe the average Brier or WCT team wins 40% when 1 down, but what if I’m team Ferbey, are we 5% more likely to win?  10%?  What if I’m playing Martin instead of Adams? It would take much more math to examine these questions in detail, and I suspect we do not yet have enough data to develop a clear answer.  We can, however, use the average analysis and apply it to a given situation.  For example, Ferbey may decide their chance is greater than the ER of 76% to win with last rock.  This simply reinforces their decision to attempt a blank.

The use of these numbers is applicable to any competitive team and useful for many game situations.  Examine the scenario of being down one with the hammer in the 10th end, facing a difficult shot for two or a simple draw for one to tie the game and play an extra end.  Math tells us that we should attempt the shot at two, even if we expect to miss 3 out of every 4 times that we try the shot!                

      

 

 

 

 

20 Oct 2007 Kevin 1 comment

Curl with Math

Howard versus Stoughton in 2007 Brier: The 9th End

With all due respect to Linda Moore and Ray Turnbull, the recent game of Howard versus Stoughton during the round robin of the 2007 Tim Horton’s Brier showed how even some of the most knowledgeable don’t understand statistics as it relates to curling.

Howard is one up without hammer in 9th end.  Two rocks remain.  There is a corner guard and a Stoughton rock on the opposite side of the sheet in the back twelve foot.  Richard Hart, the Ontario third, and Glen Howard discuss the options.  Glen prefers a freeze to the back stone, but eventually Richard convinces him to play a draw around the corner guard.  Rich’s argument is their uncertainty of the ice on the freeze attempt – Glen’s concern is the amount of curl he will get with his come-around attempt.
 

Linda’s comments were something to the effect “Howard wants Stoughton to score here and prefers giving up a deuce to a blank”.  Perhaps Linda did not properly express what she meant or in fact, does not recognize the real numbers.
 

Howard would, mathematically, prefer a blank over Stoughton’s deuce.  If, as the announcers stated, Howard preferred a deuce to a blank – why didn’t Stoughton take out his own rock rather than draw for two?
 

Recall the statistics:

Expected Results in the 10th end

  1. Odds of winning if tied with hammer (x) = 75.7%
  2. Odds of winning if one down with hammer (y) = 39.5%
  3. Odds of winning if two down with hammer (z) = 11.7%

If Howard is one up, he should win about 60% and if he’s one down with hammer he will win 40%.  It is clear that a blank is preferred to surrendering a deuce.
 

Then why, might you ask, does he not hit the Stoughton stone out?
 

What the TSN crew perhaps meant to say, or misunderstood, is:
 

Howard is willing to risk a deuce by Manitoba because the chance that he may either steal or force Stoughton to one increases his chance to win.
 

If Howard steals, he wins 88% of the time.  If Stoughton is forced to one, he wins 75%.  The chance that one of these outcomes may happen, overcomes the difference between 60 and 40 for one down versus one up. 
 

Let’s examine the Ontario decision and also determine if Richard or Glen had the correct call.
 

From my previous article, “One up in Nine without Hammer” if we expect to give up a deuce roughly less than half the time, we should draw around the guard.  I assume we all agree Glen can be expected to make a successful draw (where Stoughton cannot score two) more often than 50%.  Because a steal is less likely (corner guard instead of centre guard) – he may need to succeed more often, perhaps 60% (see previous article for more precise numbers). I’d suspect Glen’s odds of making this shot are as high as 80%, depnding on ice conditions. 

Referring to the article “Aggressive Play in 9th end”, where no guard was available, I roughly determine that Howard needs to make either a freeze or “near-freeze” and avoid a deuce at least 80% of the time.
 

Based on these numbers, our rough estimate to attempt a come-around is the correct shot.  
 

Let’s examine a little closer…
 

Hit:
Assuming Stoughton will blank 95% of the time Howard attempts a hit, we know from previous articles:
 

W        = 61%
 

Come-around:
Assume a steal will not occur.
 

a = Stoughton takes one point = 75%
b = Stoughton scores two points = 25%
 

W        = ax + by
            = 67%
 

Freeze
Assume a steal will not occur.
 

a = Stoughton takes one point = 50%
b = Stoughton scores two points = 25%
c = Stoughton blanks = 25%
 

W        = ax + by + c(1-y)
            = 63%
 

This shows us mathematically that Richard’s call is correct.  If you include the slight chance at a steal, we could also expect it more likely with the come-around than the freeze.
 

A few observations:
 

  1. Blank is not likely with the come-around.  A freeze attempt makes a blank very possible.
  2. A deuce is, perhaps in Glen’s opinion, more likely to occur with a missed come-around.  Though I’ve used 25% in both calculations for a deuce, Howard may determine he is more likely to give up the deuce on a come-around and would prefer the freeze in order to add the possibility of a blank.
  3. Ice conditions – Glen was not comfortable with the ice and with the shot and that can have an impact (though not mathematical).  In hindsight, appears to be the case, Glen put his draw through the rings.

Assuming acceptable ice conditions, I would play the come around every time.  What would you do?
 

Extra-end
 Why do some players and possibly announcers believe it is preferable to be one down with hammer?
 Statistics show that one up without wins roughly 60% of the time.  Why would skips choose to be one down with hammer instead?  I have two theories:
 

  1. They don’t know it’s 60%…and more importantly
  2. They want the last rock.

Top skips believe last rock puts the outcome in their hands.  What they fail to realize, I suspect, is that very often the end results in no opportunity at two and, sometimes, a very difficult shot for one in order to force an extra end.  In the example above, Howard needed to make a double on his last shot to score one and had no realistic chance at two as the end developed.
 

My belief is the preferred position is actually one up without hammer in the 9th end.
 

Four rock free-guard zone allows the team without hammer to force the play.  The risk of a three exists, but I suspect the opportunity to steal or force your opponent to a single, and greatly increase your chance to win, is much higher.  You are able to aggressively play the end with the risk of giving up a deuce and still having a 40% chance to win in 10. 
 

We have not yet derived the statistics to analyse 9th end results, but once we do I will be examining the numbers to determine if they support or dispute my theory.
.

 

 

 

 

06 Mar 2007 Kevin 7 comments

Curl with Math

Ferbey and Adams Part II. Extra-End: Martin’s decision in 9

 Extra-End: Martin vs. Koe - Provincial Finals
 

Let’s start with the “Extra-End” this month….

There were very dramatic events in the recent final game to determine Alberta’s 2007 Men’s Provincial Champions. Kevin Martin took the game in heart wrenching fashion when Blake MacDonald, skip for the Koe rink, missed a draw in 10 and another in the extra end to hand him the title. 
An interesting moment is Martin’s decision in the 9th end.  Kevin was down 2 with hammer, 7-5, and Koe had a single rock in the twelve foot.  With their last stone, the Martin team opted to draw for a single point rather than blank and keep the hammer in the 10th.  Feedback I’ve read on Curlingzone forums is that Martin felt taking two on the rather straight ice would be more difficult than stealing a single.  What is the math in this situation?
  Expected Results:
Odds of winning if two down with hammer = .117
Odds of winning if one down without hammer = .103
 The mathematical difference is very small, less than 1.5%.  This makes 2 down with or 1 down without as nearly a coin-flip decision.   In this case, Martin weighed in the impact of ice conditions and determined a steal was more likely.  The numbers indicate that when this situation occurs, a team should not quickly assume a blank the correct call.
And now to our article….

Ferbey and Adams Part II

This article continues the discussion from “Curling with Math”, included in the 2006/2007 Black Book of Curling, available from Curlingzone at www.blackbookofcurling.com. 
Let’s start by repeating the background from that article:

It’s the 2005 Brier Final.  Ferbey, appearing in the final game for a 5th straight year, is facing what appears to be a simple decision.  It is the 9th end, and the game is tied.  A miss by the opposing skip, Shawn Adams of Nova Scotia, has left the Alberta rink lying one with no other rocks in play.  It is commonly accepted curling strategy to play for two (or more) when you have the hammer and hold your opponent to one (or steal) without last rock.  In this case, Ferbey would be expected to draw to the open side of the rings and likely have a final shot at two or perhaps three if Adams attempts a freeze and makes a poor shot.  David Nedohin, who throws the final rocks for the Ferbey team, discusses the options with Randy.  Much to the surprise of fans and commentators, they decide to remove their own stone and play out the end for a blank in order to keep the hammer in the 10th end.  The final outcome eliminated any need for second guessing the decision, as Ferbey won - David making a draw to the open four foot in the 10th end for the victory. 

Specifically we used math and statistics to analyse the decision by Ferbey to attempt a blank end in 9 in order to secure last rock coming home.  An assumption made from that article was that Adams will always attempt to freeze to the Alberta stone, if Ferbey chooses to draw and lie two. 

Our intention in this article is to:
1.      Determine the correct choice for Adam’s shot, if Ferbey attempts a multiple score.
2.       Re-examine the Ferbey decision based on a weighting factor of Adam’s decision. 

First, 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 in the 10th end

  1. Odds of winning if tied with hammer (x) = 75.7%
  2. Odds of winning if one down with hammer (y) = 39.5%
  3. Odds of winning if two down with hammer (z) = 11.7%

  4. Odds of winning if 3 down with hammer (m) = 0.4%

These Expected Results are based on available statistics for 4-rock games played in the Brier, European and World Championships, Grand Slam and other WCT events, Provincial Championships, Olympics and Olympic Trials. In using these statistics, we assume skill is relatively equal, conditions are similar and shot making ability is similar. 

Situation 1. Adams Decision
Game Situation Assumptions:

  1. Alberta chooses to draw and lie two.  Nedohin’s first stone lands on the tee-line in the eight foot, higher than shot stone, leaving a possible freeze, double or hit and roll.
  2. If Adams attempts a hit, he will not miss both rocks.  Alberta cannot score greater than two.
  3. When going for two in 9th, it is assumed that at worse Alberta will take 1.  Odds of a steal for Nova Scotia are considered negligible.
  4. If we miss the double, Alberta will always score two.  The chance they will miss is considered negligible.

Expected Results:
x = Odds of winning tied with hammer = .757
y = Odds of winning if one down with hammer = .395
z = Odds of winning if two down with hammer = .117
m = Odds of winning if three down with hammer = .004  

Variables:
a =  Odds of forcing Alberta to 1 with a freeze = .75
b = Odds of Alberta scoring 2 = .2
c = Odds of Alberta scoring 3 = .05
d = Odds of double = .75
 

These numbers are estimates based on evaluation of where the rocks may have been positioned.  We have assumed Ferbey places the second stone on the tee-line, higher than shot stone, making a double possible. Specific location will impact our estimate of the ability to make a specific shot.  If we evaluate our chances of making a double the same as the freeze (75%):


Freeze
W = ay + bz + cm = 32%
  Double
 

W = d(1-x) + (1-d)z = 21%
 

Based on the analysis, the freeze is the correct call.  If we are 100% sure of making a double, we are still only 24% likely to win. If we estimate our chance of the freeze as 50%, W= 24.5% and the decision is much closer if the double is 100%
 

Situation 2: Weighing the options
In the Ferbey decision, we assumed that Nova Scotia will always attempt a freeze. Now that we have analysed their options, let’s use a weighting factor to determine and re-examine Alberta’s options.
We are Alberta, faced with an option to blank or attempt a multiple score in the 9th end. If we expect to make a perfect shot 50% of the time and leave little option for a hit and roll, then Adam’s will have no choice but to choose a freeze.  But, perhaps 50% of the time our shot leaves an easy double.  Let’s estimate that when this happens, 10% of the time Adam’s will freeze, 40% attempt a double.
 
If Alberta also estimates Adams is successful 75% of the time with his freeze, we score a deuce 20% and three points 5%, resulting in a win percentage of 68%.  Using results from Situation 1 above:
W = .5(.68) + .1(.68) + .4(.79) = .72%
What is very clear is our desire for Adams to choose to hit. This increases our statistical probability of winning.  What also comes into mind is the possibility of enticing Nova Scotia to attempt the double, by intentionally placing the rock in a better position for them to choose this option. 
  This leads us into another application of statistical analysis, where we choose shots which may appear to be the incorrect call, but in fact are intended to lead the opposition into choosing an incorrect shot call in order to increase our chance of winning.   For now I will leave it to the reader to ponder similar examples.

13 Feb 2007 Kevin 1 comment

Curl with Math

Aggressive play in 9th End

Here’s a scenario that was described to me by a friend of mine.

One down playing the ninth end and our hero, throwing Red stones, has the hammer. His opponent is throwing his final stone, Blue. There are no rocks in play except for one Red stone at the back of the eight foot. The Blue skip is about to throw his last rock. We would expect him to hit and stick, hoping our hero noses for one or, at worse, blanks and remains one down coming home. Instead, the Blue skip chooses to attempt a freeze. This shot call appears to conflict with common curling strategy, but, using math, what is our analysis of this play?

Expected Results in the 10th end

  1. Odds of winning if tied with hammer (x) = 75.7%
  2. Odds of winning if one down with hammer (y) = 39.5%
  3. Odds of winning if two down with hammer (z) = 11.7%

Option 1: Hit
 
Let’s assume the odds of Blue missing the hit as negligible.  Either he rolls out and a blank is assured or, Red rolls out most of the time for a blank.  To simplify, let’s state the odds of a blank are 95%.
 
W        = (1-y)(.95)+x(.05)
            = .6126
 
Option 2: Freeze
 
Possible outcomes after Blue’s shot: 

  1. If Blue makes the perfect freeze, our hero is forced to draw for one. 
  2. If the freeze is off slightly (call it a “near-freeze”), the deuce is not likely but Red will be able to blank by blasting both rocks and rolling out as well. 
  3. If the freeze attempt is too heavy or light or fails to cover part of the Red stone, a chance for two becomes possible.

Blue needs to initially estimate his chances of making the freeze.  Let’s estimate this and other Variables:

f = Odds of the freeze, Red must draw for 1 = 25%
n = Odds of near-freeze = 50%
m = Odds of a miss = 25%

Assumptions:
Initially, let’s try to simplify our analysis as much as possible…
·        If Blue freezes, Red will always score 1
·        If Blue misses, Red will always make the shot for two. 
 
Further math is then required for the “near-freeze”.  Let’s start by estimating that, after a near-freeze, Red will always attempt a hit. Variables as follows:
 
b = blank = 70%
t = take 1 = 20%
s = steal = 10%
 
W        = fx + my + n(b(1-y) + tx + s(1-z))
            = .6196
 
Very close to our original result.  This would imply that, based on our estimates, either decision is correct.
 
Using some other values for b, t, s
 

b

 t s W
.95 .05 0 59%
 .8 0.2 0 61%
 0.7 0.2 0.1 62%
0.5 0.4 0.1 63.5%
0.5 0.3 0.2 64%

Not very drastic changes, even if Blue steals 20% and forces Red to one 30%, we only increase our chances 3% over hitting.  Given these numbers are very close, let’s use 62% (our original estimate), and make changes to the original variables on Blue’s shot. 

 

f

 n M W
0 0.5 0.5 53%
 0.15 0.35 0.5 54%
 0.25 0.25 0.5 55%
0.25 0.5 0.25 62%
0.3 0.4 0.3 61%
0.5 0.25 0.25 64%
0.5 0.1 0.4 60%
0.5 0 0.5 58%
0.6 0 0.4 61%
0.6 0.2 0.2 66.5%
0.6 0.3 0.1 69%
0.7 0 0.3 65%
0 1 0 66%
0.2 0.8 0 68%
0.1 0.8 0.1 64.5%
0.2 0.6 0.2 63%

(Variables:  b = 0.7, t = 0.2 and s = 0.1)
 
What does this show us?  If Blue can make the freeze 60%, even if Red scores a deuce the other 40% of the time, our decision is equal to hitting.  If, however, Blue gives up a deuce at least half the time, they are making the wrong call. 
 
However, making a perfect freeze, even at the top level, is very difficult. Our original estimate of 25% is likely high. I suspect the chance of a near-freeze is greater than 50% and the chance of a deuce is the greatest Variable in our decision. Notice if f = 0.2, n = 0.8 and m=0, we jump to 68%, but if we add 10% chance of a deuce we drop down to 64.5%.  It is my opinion that Blue must be confident they will not allow Red to score a deuce more than 20% of the time, in order for the reward to be greater than the risk.
I suspect that Blue was not aware of the actual statistics for 1 down in the 10th with hammer.  Many competitive curlers believe (incorrectly) the number to be greater than 40%.   I know many players who felt that it was 50% or slightly higher – prior to Curlingzone statistics being available.  At first looking at these numbers, I myself was surprised at the results seen in 4 rock play.
 
For what it’s worth, in this example Blue made a perfect freeze, Red was held to one and lost in the 10th end.
 
Extra End:
Let’s go back to our original Assumptions: 

  • If Blue freezes, Red will always score 1
  • If Blue misses, Red will always make the shot for two. 

We simplified in order to do a quick analysis.  If we dig deeper, let’s create some additional scenarios.
 
If Blue freezes, 95% Red scores one and 5% Blue steals.
If Blue misses, 90% Red scores a deuce and 10% Red scores one.
 
W        = f(.95x + .05(1-z)) + m(.9y + .1x) + n(b(1-y) + tx + s(1-z))
 
Using our original values, f=0.25, n=0.5, m=0.25, b=0.7, t=0.2, s=0.1
 
W = 63%
 
Only 1% higher than our original analysis.  Our assumptions appear to be accurate for our purposes.
 
Extra-extra End:
 
Some readers have mentioned that, though interesting, these studies are not practical for game situations.  I disagree, and here’s why: 

  1. Many game situations are similar and occur often enough that “off-ice” analysis can prepare a team better for these game situations.  The article here is one example.  This is similar to how poker players practice decisions away from the table.
  2. A coach could have a computer with a spreadsheet available to run  scenarios and a time-out can be used to discuss.
  3. Players can make these decisions on ice and, with some work and practice, can examine even more complex situations.

Let’s re-examine this article doing quick “in-head” analysis on the ice, per comment number 3.
 
Blue skip thought process…
  ¼ of the time I give up a deuce.  If I do, I win 40% so ¼ of 40 is 10.
  ¼ of the time I force Red to one and win 75% so ¼ 75 is about 19.
  Half the time I make a near freeze. Usually Red will blank, and I win 60%, but sometimes we can   steal or hold them to one, let’s say we win 64%. Half of 64 is 32.
  10+19+32 = 61%
 That’s nearly the same as hitting, but:
   a.      I have confidence and think a deuce is less that ¼, therefore I will draw
   b.      I don’t trust the ice or my weight, I will hit because a deuce is likely greater than ¼.
 
Some may argue “but I knew that already, I draw if I can make it!” 
 
In poker, an amateur player may make the same decision as the world class player on whether to raise or call.  However, the ability to work through the process and make more correct decisions and know why they are making them, is the difference between an average player who breaks even or loses, and a world class player who wins all the money.  I believe this analogy applies not only to curling but many other areas in sport, the stock-market, and in life.

 

 

08 Jan 2007 Kevin 2 comments

Curl with Math

One up in Nine without hammer

Its down to the final two rocks of the end. You are Team A and do not have last rock advantage. You have a one point lead and the house is open except for your opponent’s stone in the eight foot at the tee-line and a center guard.

What should you do and, more importantly, what analysis can you do to determine the correct decision?

Assumptions:

  1. The guard is halfway from the 12 foot to the tee-line. It is your rock.
  2. Ice conditions are ideal.
  3. If we attempt a hit we will not miss the rock completely (flash).

Expected Results (ER) in the 10th end:

Odds of winning if tied with hammer (x) = 75.7%
Odds of winning if one down with hammer (y) = 39.5%
Odds of winning if two down with hammer (z) = 11.7%

Clearly we have three options:

  1. Hit and stick.Variables
    b = Blank = 95% 

    We estimate that if we attempt to hit and stay, the chance we roll out or the chance our opponent hits and rolls out successfully to be 95%.

    W = b(1-y) + (1-b)x = 61%

  2. Hit and attempt a roll behind the guard.Variables
    c = Team A steals one point = 5%
    d = Blank = 70%
    e = Team B takes one point = 25% 

    We estimate that 70% we miss hit and roll enough that our opponent is able to blank. Of the 30% we make the hit and roll, we steal 5% of the time and Team B draws for one 25% of the time.

    W = c(1-z) + d(1-y) +ex = 66%

  3. Attempt to draw behind the guard.Variables
    t = Team A steals one point = 25%
    u = Team B takes one point = 50%
    v = Team B scores two points = 25% 

    We estimate that half of the time we can hold our opponent to one, 1 in 4 attempts we steal one point and the other quarter of the time Team B scores a deuce.

    W = t(1-z) +ux + vy = 70%

Based upon our estimates, we have determined that our best choice is to disregard our opponent’s stone and to draw around the center guard.

Lets disregard Option 1 for now and compare Options 2 and 3, using different values for the variables.

Hit and Roll
c d e W
0 0.95 0.5 61%
0.05 0.95 0 62%
0 0.75 0.25 64%
0.05 0.7 0.25 66%
0 0.5 0.5 68%
0.1 0.5 0.4 69%
0.2 0.5 0.3 71%
0 0.3 0.7 71%

 

Draw
t u v W
0.1 0.2 0.7 52%
0.05 0.35 0.6 55%
0.2 0.2 0.6 56.5%
0 0.5 0.5 58%
0.25 0.25 0.5 61%
0.5 0 0.5 64%
0.2 0.4 0.4 64%
0 0.75 0.25 67%
0.3 0.4 0.3 69%
0.25 0.5 0.25 70%
0.5 0.25 0.25 73%
0.75 0 0.25 76%
0.5 0.5 0 82%

With a hit and roll, we need to allow a blank no more than 50% of the time to have at least a 68% chance to win the game.

With a draw, if we expect to avoid a deuce 75% of the time, we range from 67% to 76%, depending on our estimate to steal a point.

The draw is the correct call, based upon our assessments of the situation. If, however, we expect to give up a deuce half of the time or more, the hit appears to be the correct call.

Now lets consider if the hit and roll is an overly difficult option, with the rock in the 12 foot and back from the tee-line. The option appears clear to draw behind the guard as we are only 61% likely to win (option 1). However, we still need to insure that the draw gives our opponent a deuce less than 50% of the time in order to justify this call. What this tells us is that the decision is based less on where the rock is and more upon our confidence in making the draw around the guard.

What if the guard is our opponent’s stone, and is closer to the rings? Now, the added chance of a run-back for two points may sway our decision as well.

I will leave the exercise of evaluating that shot call up to you, the reader.

Till next time…

29 Nov 2006 Kevin 3 comments