(Subtitle: Which Game Should You Win? Part 3)

Three years ago, my friend Andrew pitched in to the blog and asked

which game in the playoffs was most worth winning. The results were a bit inconclusive, but from it he developed a database of all playoff outcomes since 1943, so a year later I looked at the dataset again and developed Markov-style chains of

playoff odds based on different positions in the playoffs.

Now that it's playoff season again, people are naturally interested more than normal in hockey and I recently overheard someone comment that, though a series was currently at 2-1 for wins, the next team to win was undoubtedly going to win the series.

Good thing I have this handy database of all playoff outcomes ready, because that immediately intrigued me as to how likely it actually is that, at any given point, the next team to win a game will win the series overall. This is perhaps another way of asking the same question as before - how much does
this upcoming playoff game matter to the grand scheme of things?

Before looking into the historical data, though, it's worth doing the math to see what the odds would be if the human element were removed (with all games having a 50/50 chance of going either way, and all games being independent). Obviously, if a best-of-seven playoff series is tied at 3-3, then the next game winner is guaranteed to win the series, so that's an easy starting point.

From there, it's not too hard to work backwards to figure out the rest of the odds. If a series is at 3-2, then there's a 50% chance that the leading team wins (which would give them the series win, and a 100% chance therefore of winning the series), and a 50% chance that we get to a 3-3 position, where the chance of the trailing team being the overall winner is again 50%. Overall, that makes the chance that the next game winner will be the series winner (50%*100%)+(50%*50%)=

**75%**.

If we continue this way, then we can generate this table of values. For all following graphs, the 'home team' is the team that has home town advantage for the first two games:

So what's not surprising here is that the odds that the next team to win will be the series winner are always above 50%. That makes sense, because no matter what the position is beforehand the winner is improving their overall odds of winning the series. What's more interesting is how little games tend to matter when the series is lopsided.

Of course, games aren't all independent or aren't all 50/50 toss-ups. Historically, home teams win 54.5% of games, so let's see what happens if we recreate this table with that factored in. It's a bit more complicated, but essentially the same analysis as before, to get this table:

Here we start to see the effects of the playoff structure and the pattern with which it allocates home games to different teams. For instance, when the original home team is up 3-0, the upcoming game almost doesn't matter at all, but the situation isn't quite the same if the original away team is up 3-0. Similarly, both 3-2 game situations have different values. This can be perhaps more easily rationalized - if the original home team is up 3-2, then the upcoming game is going to be in their opponent's home town, which makes it more likely that that other team will win, but if they do then it's tied coming back home, so that's less of a big deal. On the other hand, if the original away team is leading 3-2, they're more likely to win this upcoming game 6, and can lock the series up right there.

Of course, this is all fun and games from a theoretical point of view, but what's actually been happening in real playoff series? Here we go:

This is definitely more interesting! Here we have a clear outlier from the theoretical projections from before, where the 'least important' game is game 5 when the original away team is up 3-1. At this point, the original home team would be playing back at home, but would be down by such a significant deficit, resulting in a situation where they end up with a fairly high 'last hurrah' win rate, before ultimately losing the series 2-4.

On the other hand, there's a surprisingly high predictive score for whoever wins the game after the original home team gets up 1-0, at 74% (8% higher than what you'd expect in a coin toss scenario). I imagine this indicates that the original home team is likely to win their first game, and that if the original away team can't bounce back then the series is likely sorted out by that point (at least, in harder-to-quantify matters than you'd expect).

So the answer to the question 'which playoff game is the most important' remains a solid "it depends", but now you have three different ways of looking at the question. Use them wisely, and enjoy the

2017 playoffs!