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Re-run: Which Games Are the Most Important In a 7-Game Series?

Posted by Neil Paine on May 10, 2011

Since we're in the thick of the playoffs, it seems appropriate to revisit this post from last June regarding the importance of each game in a best-of-7 series:

We can try to quantify [the relative importance of each game] by looking at the potential swings in each team's probability of winning the series based on the outcome of a given game. Let's establish a scenario where two morally .500 teams are playing each other in a 7-game series; the home team in any game has a 60% chance of winning (60% traditionally being the NBA's home-court advantage), and the away team has a 40% chance. At the beginning of a series in the 2-2-1-1-1 format, the team playing Game 1 at home has a 53.2% probability of winning the series (go here for the formulae I used to arrive at these numbers). If that team wins Game 1, their probability of winning the series suddenly increases to 66%, a boost of 12.8%, and if they lose, their probability drops to 34%, a decrease of 19.2%. Since there are only two possible outcomes in any game (win or loss), we can say that the average swing in series win probability for the home team in Game 1 is +/- 16% (12.8% plus 19.2%, divided by 2).

Do this for both teams and every possible situation in a 7-game series, and you can establish which games produce the biggest swings in series win probability:

Game# Home W Home L Home SerW% Swing w/ W Swing w/ L Avg Δ
Game 1 0 0 0.532 0.128 -0.192 0.160
Game 2 1 0 0.660 0.132 -0.199 0.166
Game 2 0 1 0.340 0.122 -0.182 0.152
Game 3 2 0 0.843 0.100 -0.150 0.125
Game 3 1 1 0.539 0.154 -0.231 0.193
Game 3 0 2 0.207 0.100 -0.150 0.125
Game 4 3 0 0.942 0.058 -0.086 0.072
Game 4 2 1 0.693 0.163 -0.245 0.204
Game 4 1 2 0.307 0.141 -0.211 0.176
Game 4 0 3 0.058 0.038 -0.058 0.048
Game 5 3 1 0.904 0.096 -0.144 0.120
Game 5 2 2 0.552 0.208 -0.312 0.260
Game 5 1 3 0.144 0.096 -0.144 0.120
Game 6 3 2 0.760 0.240 -0.360 0.300
Game 6 2 3 0.240 0.160 -0.240 0.200
Game 7 3 3 0.600 0.400 -0.600 0.500

Clicking the table header to sort by the "Avg Δ" column, we see that the most critical games are as follows:

Game# Home W Home L Home SerW% Swing w/ W Swing w/ L Avg Δ
Game 7 3 3 0.600 0.400 -0.600 0.500
Game 6 3 2 0.760 0.240 -0.360 0.300
Game 5 2 2 0.552 0.208 -0.312 0.260
Game 4 2 1 0.693 0.163 -0.245 0.204
Game 6 2 3 0.240 0.160 -0.240 0.200
Game 3 1 1 0.539 0.154 -0.231 0.193
Game 4 1 2 0.307 0.141 -0.211 0.176
Game 2 1 0 0.660 0.132 -0.199 0.166
Game 1 0 0 0.532 0.128 -0.192 0.160
Game 2 0 1 0.340 0.122 -0.182 0.152
Game 3 2 0 0.843 0.100 -0.150 0.125
Game 3 0 2 0.207 0.100 -0.150 0.125
Game 5 3 1 0.904 0.096 -0.144 0.120
Game 5 1 3 0.144 0.096 -0.144 0.120
Game 4 3 0 0.942 0.058 -0.086 0.072
Game 4 0 3 0.058 0.038 -0.058 0.048

Obviously, Game 7 is the most crucial possible game in a 7-game series; it carries (on average) a +/- .500 swing in series p(W), simply because everything comes down to one game and one game only. However, a Game 5 tied at 2-2 is only the third most crucial game in a 7-game series -- a Game 6 when the home team is leading 3-2 is actually even more important, because it represents that team's best opportunity to close out the series (and people scoffed when the Celtics said they were treating Game 6 of this year's Orlando series like it was "their Game 7").

In general, situations where the team with fewer home games in the series can grab a decisive home victory (Game 6, up 3-2; Game 4, up 2-1; etc.) are among the most critical in a series with this format. Meanwhile, predictably at the bottom are the 3-0/0-3/3-1/1-3 games, suggesting that those types of elimination games aren't really "crucial" at all -- you're basically going to lose the series anyway, no matter whether you win or lose the game. Also, we see that Game 3 of a 2-0 series (no matter which team leads) is relatively unimportant as well when it comes to determining the series' ultimate outcome.

Of course, right now we're in the middle of the NBA Finals, and any basketball fan will tell you that the Finals employ a 2-3-2 format, not the familiar 2-2-1-1-1 from the rest of the playoffs. How does this change the relative importance of each game?

Game# Home W Home L Home SerW% Swing w/ W Swing w/ L Avg Δ
Game 1 0 0 0.532 0.128 -0.192 0.160
Game 2 1 0 0.660 0.132 -0.199 0.166
Game 2 0 1 0.340 0.122 -0.182 0.152
Game 3 2 0 0.843 0.100 -0.150 0.125
Game 3 1 1 0.539 0.154 -0.231 0.193
Game 3 0 2 0.207 0.100 -0.150 0.125
Game 4 3 0 0.942 0.058 -0.086 0.072
Game 4 2 1 0.693 0.163 -0.245 0.204
Game 4 1 2 0.307 0.141 -0.211 0.176
Game 4 0 3 0.058 0.038 -0.058 0.048
Game 5 3 1 0.856 0.144 -0.216 0.180
Game 5 2 2 0.448 0.192 -0.288 0.240
Game 5 1 3 0.096 0.064 -0.096 0.080
Game 6 3 2 0.840 0.160 -0.240 0.200
Game 6 2 3 0.360 0.240 -0.360 0.300
Game 7 3 3 0.600 0.400 -0.600 0.500

Sorting once again by the average potential change in series p(W) for each game, these are the most crucial games in a Finals-style 2-3-2 format:

Game# Home W Home L Home SerW% Swing w/ W Swing w/ L Avg Δ
Game 7 3 3 0.600 0.400 -0.600 0.500
Game 6 2 3 0.360 0.240 -0.360 0.300
Game 5 2 2 0.448 0.192 -0.288 0.240
Game 4 2 1 0.693 0.163 -0.245 0.204
Game 6 3 2 0.840 0.160 -0.240 0.200
Game 3 1 1 0.539 0.154 -0.231 0.193
Game 5 3 1 0.856 0.144 -0.216 0.180
Game 4 1 2 0.307 0.141 -0.211 0.176
Game 2 1 0 0.660 0.132 -0.199 0.166
Game 1 0 0 0.532 0.128 -0.192 0.160
Game 2 0 1 0.340 0.122 -0.182 0.152
Game 3 2 0 0.843 0.100 -0.150 0.125
Game 3 0 2 0.207 0.100 -0.150 0.125
Game 5 1 3 0.096 0.064 -0.096 0.080
Game 4 3 0 0.942 0.058 -0.086 0.072
Game 4 0 3 0.058 0.038 -0.058 0.048

As was the case with the 2-2-1-1-1 format, Game 7 is easily the most critical matchup in any 2-3-2 series. Also note that Game 6 is once more the 2nd-most important game, but this time when the home team is trailing 3-2 (in the 2-2-1-1-1 format, Game 6 was at its most crucial when the home team led 3-2). The rest of the list is similar, with one notable exception being that a Game 5 with the home team leading 3-1 is far more important in the 2-3-2 format. This dovetails with my simulations of the Finals before the series began, which found that if the Celtics were going to win the championship, the most likely length of the series would be 5 games, thanks to the stretch of 3 consecutive home games.

Using these measures of relative importance, will be theoretically possible in the future to give weight to individual production in each playoff game, crediting performances more or less based on how crucial the game was in determining the series outcome.

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24 Responses to “Re-run: Which Games Are the Most Important In a 7-Game Series?”

  1. Neil Paine Says:

    So for the Heat-Celtics series, the importance of each game would be:

    Game 1: 0.160
    Game 2: 0.166
    Game 3: 0.125
    Game 4: 0.176
    Game 5: 0.120

    The Celtics blew the most important game of the series and lost 21% from their series win probability. Ugh.

  2. Jirka Poropudas Says:

    Interesting analysis.

    Yet, wouldn't it be better to study the EXPECTED Delta instead of the AVERAGE Delta?

    Thus, one could include the unequal winning probabilities of the home team and the visitor into the analysis.

  3. David Hess Says:

    Jirka,

    The expected delta will be 0, as the odds of a home team winning are already built into the pre-game series win odds. For example, in game 1:

    Home team win: +0.128 swing * 0.6 p_win = +0.0768
    Home team loss: -0.192 * 0.4 p_loss = -0.0768

  4. Neil Paine Says:

    #2 - That's a good point. Here's what we get when you give a home win 60% likelihood of occurring rather than splitting everything 50-50:

    2-2-1-1-1 series:

    Game# Home W Home L Home SerW% Swing w/ W Swing w/ L Exp Δ
    Game 7 3 3 0.600 0.400 -0.600 0.480
    Game 6 3 2 0.760 0.240 -0.360 0.288
    Game 5 2 2 0.552 0.208 -0.312 0.250
    Game 4 2 1 0.693 0.163 -0.245 0.196
    Game 6 2 3 0.240 0.160 -0.240 0.192
    Game 3 1 1 0.539 0.154 -0.231 0.185
    Game 4 1 2 0.307 0.141 -0.211 0.169
    Game 2 1 0 0.660 0.132 -0.199 0.159
    Game 1 0 0 0.532 0.128 -0.192 0.154
    Game 2 0 1 0.340 0.122 -0.182 0.146
    Game 3 2 0 0.843 0.100 -0.150 0.120
    Game 3 0 2 0.207 0.100 -0.150 0.120
    Game 5 3 1 0.904 0.096 -0.144 0.115
    Game 5 1 3 0.144 0.096 -0.144 0.115
    Game 4 3 0 0.942 0.058 -0.086 0.069
    Game 4 0 3 0.058 0.038 -0.058 0.046

    2-3-2 series:

    Game# Home W Home L Home SerW% Swing w/ W Swing w/ L Exp Δ
    Game 7 3 3 0.600 0.400 -0.600 0.480
    Game 6 2 3 0.360 0.240 -0.360 0.288
    Game 5 2 2 0.448 0.192 -0.288 0.230
    Game 4 2 1 0.693 0.163 -0.245 0.196
    Game 6 3 2 0.840 0.160 -0.240 0.192
    Game 3 1 1 0.539 0.154 -0.231 0.185
    Game 5 3 1 0.856 0.144 -0.216 0.173
    Game 4 1 2 0.307 0.141 -0.211 0.169
    Game 2 1 0 0.660 0.132 -0.199 0.159
    Game 1 0 0 0.532 0.128 -0.192 0.154
    Game 2 0 1 0.340 0.122 -0.182 0.146
    Game 3 2 0 0.843 0.100 -0.150 0.120
    Game 3 0 2 0.207 0.100 -0.150 0.120
    Game 5 1 3 0.096 0.064 -0.096 0.077
    Game 4 3 0 0.942 0.058 -0.086 0.069
    Game 4 0 3 0.058 0.038 -0.058 0.046

    It actually doesn't change the order of importance, though.

  5. Neil Paine Says:

    #3 - I used the absolute value of each delta in both the 50-50 and 60-40 calculations.

  6. Ben Says:

    Lebron's performance positively correlated with importance of games so far...

  7. David Hess Says:

    Ooooops, my bad.

  8. Jirka Poropudas Says:

    Neil,

    Thanks for the additional calculations -- even though the order of importance remains unaffected. (Glad to note that I came to the same conclusion on my own spreadsheet.)

    Keep up the good work!

  9. Neil Paine Says:

    The next step is to determine the Exp Δ of the "average game" in a 7-game series, state each game's Exp Δ relative to that average, and use that to weigh individual performance in each game based on the relative importance of the game. So like Ben said in #6, LeBron's poor Game 3 would carry less weight than his excellent Game 4.

  10. Jason J Says:

    I found this fascinating last year, and when we lost at home it pretty much crushed my faith in the team. Perkins's knee and the Lakers' extra 20 free throws notwithstanding.

  11. WanderingWinder Says:

    Interestingly, the frequency you'd expect the winner of game 1 to win the series is .66, using Neil's assumptions (.6 *.66 for the home team winning, .4*(1-.34) for the road team winning), which is a little bit lower than the historic percentage if my memory serves me, but not THAT much, considering that you wouldn't expect the teams to be all that evenly matched all that often.

  12. Dwight Howard Says:

    hey neil, unrelated, but what is up with this media bullshit about how pau gasol was the reason the lakers lost (not kobe, who shot 105 for the series)?

    I love to watch the lakers' bigs (especially bynum, who has emerged as an elite defender and a ridiculously efficient offensive force - imagine how much better the lakers' offense would be if they ran it through the post tandem!), but I find Kobe frustrating as hell because (even though he still manages to be efficient on a ridiculously high usage) watching someone use 35% of a team's possessions is so boring/obnoxious.

    this is compounded by the extent to which kobe constantly receives copious undeserved media praise from national outlets and can do no wrong. it's almost like players like kobe and d-rose are so overrated that i start hating them so much that I forget they are really good players.

    assume two options, a post-up wing like Kobe or Dirk, and a post-up big like Bynum or Howard, have ORTG and usage equal... is it more useful (more winning) to use possessions in the post rather than on the perimeter? the assumption behind this would be that using possessions close to the basket is more likely to result in the PFs and Cs, the key help defenders on most teams, getting into foul trouble and thus hurting their team defense.

  13. Sebastian Says:

    The analysis if flawed. Game 1 is the most important... To be most accurate, the most important game is always the current one, or the next one if between games.

  14. Sebastian Says:

    *is* flawed.

  15. Roby Says:

    Game 4 is the most important game. Open an shut.

    If you are trailing 2-1, you better not lose, because it all but impossible to come back from down 3-1. You are done.

    If you are ahead 2-1, you really want to win game 4 to go ahead 3-1, because it almost guarantees that you win the series. Moreover, you will likely win without having to play 7 games.

    Game 4 has the potential to set you up for the coup de grace. I guess the reason the numbers don't quite reflect the importance of game 4 is because sometimes it isn't a kill shot, it is just a reset button. Tied 2-2, its back to a near 50/50 proposition.

    But ask any player, which game they would most like to win, (besides the clincher) they would always choose game 4.

    Game 4 is the most important because it is guaranteed to mean life and death (figuratively) for one of the teams.

  16. Sebastian Says:

    You have to win three games before the clincher, to win the series. None of those are less important than the clincher. If you show up at game four down 0-3, it probably won't make much difference if you manage to win game 4. The most important game is always the current game. That starts with game 1. You can only retro-actively analyze these things from a perspective outside the contest - that is the logical flaw in the analysis above, and the mistake most people make when debating this.

  17. Neil Paine Says:

    I don't think you understand what I did here, Sebastian. I'm just looking at the impact in series win probability of a given outcome in each "game state" (W-L for each team and game location). All calculations are being done from the perspective of an observer before the game.

  18. Neil Paine Says:

    To clarify: this entire exercise is inherently being done retroactively for past games, but you can also judge how important the current game is relative to those past games.

  19. Sean Says:

    The one you're playing TODAY.

  20. Sebastian Says:

    Neil,

    I understand the analysis. I don't understand the application (outside of ways that it could be used in certain forms of wagering). What purpose does it serve?

  21. Neil Paine Says:

    OK, it's a little like leverage index in baseball... If I can say that Game 4 had 1.3 times as much impact of series win probability as Game 1, then I can weight production in Game 4 1.3 times as heavily as Game 1, etc.

  22. Sebastian Says:

    Ok... I don't think there is anyway to discuss this without sounding like an @$$, but here goes...

    If player X and his teammates had won games 1-3, then his production in game 4 wouldn't be "worth" 1.3 times as much as game 1. Player X didn't know, while he was half @$$ing it in game 2, what the series scenario would be in game 4. How much weight do you give to production in a losing effort? The analysis might be statistically consistent, but I'll keep measuring production by weight of gold (rings). I understand the modern metrics explosion does yield some actionable intel for players, coaches, general managers, etc. It also gives fans something else to argue about... And it certainly sounds like a dumb cliche to say the most important game is the one right now, but there is a reason cliches become cliches...

    Consider another sport - baseball, and how managers decide the starting rotations. Look at the specific example of the 1960 World Series. Run an analysis of weighted production on the players in that series. Now pretend it is the day of game 1 and you are Casey Stengel and have to decide who your starting pitcher should be in game 1. Does the after the fact analysis change anything about what Stengel should have done?

    What kind of actionable intel does an analysis like the one above give you. Does it matter who had the best 'weighted production' after the fact? I'd argue that what matters is that a complete idiot would have sense enough to start Whitey Ford in game 1, if the only thing he understood about sports was the cliches.

    I don't mean to discount the effort or the other ways in which this type of analysis can be used (looks like a great way to take money from your friends by betting on some combination of the outcome of a given game and the outcome of a series), or the value of different types of metrics in general, but the single most significant thing I see coming out of this approach is adding another way to make excuses for supposed 'big game' players that don't win because they don't understand the concept of teamwork. Lebron's entire career of performances "positively correlates" with being the new Wilt Chamberlain...

    Taking future uncertainty out of the equation when considering the relative value of a player's contribution describes an impossible circumstance and makes analysis of a player's production less significant, not more... It tells you more about who the biggest ham is, rather than the best player. Take the average delta for every possible game scenario, on a per game basis, and only consider the games that have to be played (1-4)... now which game is the most important? I'll take the 5 players that have the best weighted production calculated on that basis.

  23. Neil Paine Says:

    I don't think it was ever claimed that this was actionable from a team's POV. WPA in baseball isn't especially actionable, either -- they're value metrics, not ability metrics. The distinction is important:

    http://gosu02.tripod.com/id11.html

    Looking backwards, you can say how important a given situation was in determining the final outcome, just as you can say a 2-run double in the 9th inning of a 1-run game was more important than the same 2-run double in the 5th inning of a 12-3 blowout. I don't really see why this is controversial.

  24. WallyBoy Says:

    Related to what Roby said earlier, game four does appear to be psychologically and statistically important. Does anyone know if there is any real world data on game 4? That is, since almost any series is over after 3-0 (expect for the NHL for some reason...), then most series are 2-1. What percentage of the time does the 'tying' team (making it 2-2) win the series? What about the team that goes up 3-1? Anyone?