16: Down 2 with or tied without?

In Monday’s semi-final of the Ashley HomeStore Curling Classic, Glenn Howard trailed Brendan Bottcher 4-3 going into the seventh end. With Bottcher sitting three on Howard’s final shot, Glenn elected to try and give Bottcher one instead of drawing for one himself. He was choosing to go into the final end down two with hammer instead of tied without.

This has become something of curling conventional wisdom among the best teams at this point. We discussed it here when Kevin Koe made this choice against Jeremy Harty in Banff last month and now we have Howard doing it against Bottcher.

As fate would have it, the anonymous owner of the Curling Clips YouTube account recently posted a couple of cases involving John Shuster either discussing his preference for the strategy or actually executing it in separate games of the 2019 World Championships. What’s interesting is that in both cases it seems like there hasn’t been a prior team discussion about the scenario. This becomes a problem because it’s clear in both videos that Chris Plys is most definitely *not* on board with the idea that down-2-with is better than tied-without.

While the data is clear on favoring being up-one-without over down-one-with, an investigation into being tied-without vs. down-two-with is trickier. The former situation is not very sensitive to team skill. Regardless of the level you play at, being one-up-with gives you a 55-60% against an equal opponent. However, the latter is very sensitive to team skill.

If you’re not playing against one of the best teams in the world, then you don’t need to worry about this issue. Being tied without is better than being 2 down with. But if you’re playing against one of the very best teams on the planet, then it’s worth considering. At least, according to the people who play the game. But let’s dive into the data and if they’re right.

As we did with the up-one-without/down-one-with situation let’s break down the possible paths to winning in each scenario:

Down 2 with hammer, we win if we:
– Score 3 in the last
– Score 2 in the last and steal in the extra

Tied without hammer, we win if we:
– Steal in the last

We can convert this into a useful inequality. We should prefer being down-2-with if:
Chc(3) + Chc(2) * Chc(steal) > Chc(steal)

Because the chance of stealing is on both sides, this simplifies to:
Chc(3) > (1 – Chc(2)) * Chc(steal)

And knowing that (1 – Chc(2)) is the same as
Chc(3) + Chc(1 or worse),
we can further simplify to:
(1 – Chc(steal)) * Chc(3) > Chc(1 or worse) * Chc (steal)

What’s neat about our work here is that we can use it to find the break-even steal percentage that separates the preference of being down-2-with from being tied-without. I’ll save you further derivation because nobody wants to read derivations, but here is the break-even formula:

Chc(steal) = Chc(3) / (Chc(3) + Chc(1 or worse))

The next step is to come up with estimates for each of those terms: the chance of scoring 3, scoring 1 or worse, and stealing. For a first stab at this let’s look at all games over the past two-plus seasons between top 100 teams (those with a rating of 10 or better in my ratings).

Event               Men     Women
Score 3            10.4%     9.7%
Score 1 or worse   55.9%    55.8% 
Steal              20.7%    26.6%

Plugging the data from the first two rows into our formula, the break-even steal percentage for the men is 10.4/(10.4+55.9), or 15.7%. For the women, it’s 9.7/(9.7+55.8) or 14.8%.

The steal percentage listed in the table is quite a bit higher than those figures. For most games involving top 100 teams, it’s clearly better to be tied-without than down-2-with. So the math says in most cases it’s better to be tied without hammer. But that’s very dependent on the chance of stealing. As we’ve talked about in recent weeks, the very best teams are quite a bit better than 20.7% in steal prevention when it’s a tie game in the final end.

As a rough estimate, maybe the top 25 to 30 men’s teams figure to be better at preventing steals than the break-even value, while perhaps only the top five or so teams meet the threshold for the women’s game.

Therefore it makes sense to prefer to be down-2-with against the top-25 men’s teams and the top-5 women’s teams. Well, not so fast. Against better teams, it’s even harder to get three in the final end. And so our break-even point changes. For instance, here’s the data from the past two seasons, if we limit it to games between teams rated 11.5 or better (roughly top-25 teams).

Event               Men     Women
Score 3             7.4%     9.4%
Score 1 or worse   59.2%    56.6% 
Steal              15.6%    24.5%

Using this data, the break-even point for men drops to 11.1% and for women it’s 14.2%. And those numbers make the strategy a no-go against almost any team. Over the past two seasons, the top five men’s teams allowed a steal 11.7% of the time and for the top five women’s teams that number rises to 14.0%. Both numbers nearly match the break-even point.

One disclaimer in that last table is that the data for scoring 3 or scoring 1 or worse comes from just over 50 cases. That’s not enough to have high confidence that those percentages will be stable going forward. But that said, it makes sense that against better teams it would be more difficult to score 3 with hammer and more likely hammer scores 1 or does worse. And that’s what the limited data shows.

My conclusion for now is that being down-2-with is defensible, but far from irrefutable against the very best teams. And with more data, it’s possible we’ll find out it’s not irrefutable at all.

That’s not a very satisfying conclusion because I’d like to pick a side in the Shuster-Plys dispute. I can say that Shuster’s assertion that being down-2-with is “way, way better” is certainly a stretch. It’s defensible (but again, not irrefutable) in the first clip against Koe, but less so in the second clip when he’s playing Yuta Matsumura, who was ranked 31st headed into World’s according to my rankings.

Howard’s case is another one where choosing down-2-with was less defensible. In order to put himself in that situation he had to make a shot that carried a risk of giving up 2 and being down 3 going into the final end. That risk, while small, surely tilted the probabilities towards taking his one and going to the eighth tied-without. Furthermore, the ice at the Penticton Curling Club probably wasn’t as reliable as the ice you’d get in an arena for a major event — Howard’s previous shot actually picked — which would further reduce the value of hammer.

The wacky thing is that Howard won the game but in the way he was trying to avoid. His nose-hit attempt rolled inside a bit, forcing him to take one and go to the eighth needing to steal against Brendan Bottcher, maybe the best in the world right now at preventing steals in this spot. But when Bottcher’s draw on his final shot came up three feet short, Howard won and improbably added a data point in favor of being tied without hammer against one of the best teams in the world.

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