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Thursday, January 17, 2008

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

One of the most difficult situations towards the end of a game is one down with hammer playing the next to last end (9th or 7th). Statistically, the lowest probability of winning when one down with hammer is in the next to last end (34.9%). In the last end or with two ends remaining it is over 38%. In fact, tied with hammer with two ends to play is 67.5%, only 2.4% higher than if you are one up without!

I have seen every type of play in this situation, from both teams keeping it clean to produce a blank (and a 5 minute end) to every rock in play. So, using mathematics, what is the correct strategy, or at the very least, how do we approach this scenario to be better prepared when it happens?

Recall statistical outcomes for the final end:

Expected Results (ER) in the final end:
Odds of winning if tied with hammer (x) = 74.5%
Odds of winning if one down with hammer (y) = 38.2%
Odds of winning if two down with hammer (z) = 11.0%

So how should we play the end to maximize our chances and overcome our unenviable position? Also, our opponent without hammer can dictate the early part of an end by placing guards, how does this impact our decisions?

Let’s start by examining the numbers. Clearly, the best scenario is to take three (or more). If we score three we have an 89% chance to win. The next best scenario is to score a deuce and have a 61.8% chance in the final end. If we play aggressive and are forced to one, however, we win only 25%. In fact, if the end results in a draw for two and we miss, only scoring one, our chances flip from 61.8% to 25% - a huge difference. A blank is better than scoring a single, leaving us at 38.2%, but still we can expect to lose more than half the time.

Let’s also examine the competition’s strategy. If they read my articles, they know the correct play in this situation is to force the action and attempt a steal or force or a single, at the risk of a deuce.

Let’s assume for now our opponent will place a centre guard.

Option 1:
First, let’s attempt to play a clean end with the expected outcome a blank. After the opposition places a centre guard, we choose to draw to the side. Our opponent will most likely hit our rock and stay in the rings. Assuming we exchange shots and no one rolls their shooter to center, we can remove the centre guard with the 6th rock of the end. If the opposition splits the rings (most likely) we then are playing out the end trying to make a double in order to blank. If we fail to make a double and our opponent does not roll out, we will be forced to a single point and left with a 25% chance of winning with one end to play.

This option appears to be a losing strategy. No chance for three and not likely two, so our only outcomes are less than 50%. Let’s estimate some outcomes based on this strategy:

Score Three = 0
Score Two = 5%
Score One = 30%
Blank = 60%
Steal = 5%

W = 34%

Option 2:
Let’s try a very aggressive strategy. Come around with our first rock, corner freeze if our opponent does also, and continue drawing or soft taps until an opportunity develops for a “big shot” at scoring multiple points. Let’s again make some rough estimates based on this strategy:

Score Three = 10%
Score Two = 30%
Score One = 40%
Blank = 0%
Steal = 20%

W = 40%

Even if we could successfully blank 100% of the time, we do better playing very aggressive.

These numbers are not completely fabricated; they are based on existing data. For example, as of this writing, during this season (2007/08); Howard, Ferbey and Martin score three 13% of the time across all ends played with hammer. Taking an average of three “average” WCT teams results in 8%. These numbers can be skewed due to the better teams (aka Howard, Ferbey and Martin) being ahead more often and their opponents need to take greater risks – often resulting in big ends during later stages in a game.

Option 3:
Let’s think of a third scenario. We come around the guard, buried in the top eight foot, possibly biting the four foot. Our opponent successfully corner freezes. We now have several options:

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

Shot Call 1 will lead us to the scenario in Option 2 above. What about the other two?

2. Draw to side of rings:
If we can sit second stone, our opponent has several options, depending on how all the stones are sitting. In most cases, we could expect he will attempt a hit on the open rock and try to roll to the centre, behind cover, to sit either first or second. If he is successful, we are back to Option 2 (Very Aggressive) above and we are likely behind in the end. We will need a big shot or mistake from our opponent, but the aggressive nature of the end now makes that more likely. If our opponent does not roll successfully, we can attempt a hit and roll. In either case, if we instead choose to remove the guard or run it back, a steal becomes less likely, however a chance for a single increases and a three is highly unlikely. We can expect the end will look more like Option 1 above, with our best outcome a blank or low probability of a deuce.

3. Split the centre guard:
This is the scenario which I don’t recall seeing in a game but appears powerful. If we can split the guard and create two corner guards, our opponent now is left with an unclear decision. Does he put a centre guard back, even though you’re shot stone? Does he peel a single corner, or attempt a double peel (if it is even possible). Does he run his stone onto yours, attempting to lie two? This last option seems the best scenario but, it will leave two guards, an open four foot, and most likely both rocks will not sit perfectly behind cover above the tee line. Even if they do, a corner-freeze is available and with two guards, one which is now yours and could be driven back later on, the advantage appears to be with you. If our opponent gets cautious and elects to peel the guards, we now have options to play for a blank (if we choose), place another guard or move rocks around in the house and attempt our deuce with minimal chance for a steal.

Ultimately, a team must determine which option maximizes a chance for two or three, limiting the times you get a single point AND minimizes a steal by your opponent. Not an easy answer. The difficulty is that a blank, which is preferred to a single, is not a high occurrence if you attempt to score two or three. This specific scenario, one down with hammer with two ends to go, is one of the most interesting in the game and possibly more intriguing than the final end.

What If Our Opponent draws into the rings?
Instead of a centre guard, our opponent puts his first rock in the rings. We can now place a corner guard or hit the rock in the rings. If we call for a guard, our opponent could now choose a centre guard. We then draw around and are back to scenario in Option 2 above, though we are behind in the end. If we hit the stone, we are playing for a blank which, in this case, is very likely. For the team without hammer, assuming we will place a corner guard, this appears to be a stronger play than above. The team with hammer now faces the corner freeze and could have more difficulty getting shot stone.

This decision stems more from a team’s strategy of preferring to sit shot or be positioned frozen to shot stone in order that a shot can be played later in the end. I can see advantages for both cases and will leave it up to the reader to determine what they prefer in this situation.

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”.