## "Must-Win" Games

“This is a must-win game…” Well, unless one team has a 3 wins already no game in a 7-game MLB, NBA, or NHL series is technically ‘must win.’ But certainly some games are more crucial than others in terms of a team’s chances of winning the series.

For example, the Rangers currently have a 2-0 advantage over the Capitals in the first round of the NHL playoffs. The difference in Series Win Probability (SWP) between being down 0-3 and down 1-2 tells us exactly how critical this Game 3 is. Based on a symmetric binomial distribution (that is, a coin flip--each team has an equal chance of winning each game), the SWP of being down 0-3 is 0.0625 and the SWP of being down 1-2 is 0.3125. The potential change in SWP (∆SWP) for Game 3 is 0.25.

The difference in SWP for Game 2 however was larger. The SWP(1-1) is 0.5. And the SWP(0-2) is 0.1875. The ∆SWP for Game 2 was 0.3125. Game 2 therefore had more leverage than Game 3 will. It was about 20% more crucial (.31 vs .25).

Continuing on this path, the ∆SWP for all Game 1s is SWP(1-0) vs SWP (0-1) which is also 0.3125. So both Game 1 and Game 2 were both more critical than Game 3. But this is only because the situation of being 0-2 is already fairly dire.

Ironically, the must win situation of 0-3 yields a ∆SWP of only 0.125. Again, this is because 0-3 is very dire. A team is pretty close to elimination anyway. Winning Game 4 when down 0-3 still only buys a team a .125 chance of winning the series while losing the game would make it zero.

The most critical situation is Game 7 of a 3-3 series . The 3-3 situation features a ∆SWP of 1.0—the winner goes from 0.5 probability to 1.0 (certainty) in a single game, while the loser goes to zero.

The next most critical games are Game 6
of a 3-2 series and Game 5 of a 2-2 series. All Game 6s are 3-2 and yield either a 4-2 (1.0) or 3-3 (0.5) result for a difference of 0.5. Game 5 of a 2-2 series is just as critical. Being up 3-2 yields a SWP of 0.75, while (symmetrically) being down 2-3 yields a SWP of 0.25, the difference being 0.5.

Here is the full table:

 Game Situation Leverage Game 1 0-0 0.3125 Game 2 1-0 0.3125 Game 3 2-0 0.25 1-1 0.375 Game 4 3-0 0.125 2-1 0.375 Game 5 3-1 0.25 2-2 0.5 Game 6 3-2 0.5 Game 7 3-3 1

Notes:

1. This method ignores home ice/court/field.
2. It also assumes teams are evenly matched. Empirical observations of teams that comeback from 0-3 deficits will be less frequent than predicted by the theoretical average because teams down 0-3 tend not to be evenly matched. I think a symmetric binomial model (coin flip) is sufficient because we're looking at the 'typical' leverage for the various series situations and not necessarily for particular match-ups.
3. You can also use this table for 5-game series. A 5-game series is no different from a 7-game series that starts tied at 1-1. To find the leverage of a game in a 5-game series, take the current record and add 1 win for each team. For example, the leverage for a 5-game series that's at 2-1 is identical to that of a 7-game series that's 3-2 (0.5).