If You Could Remove One Opposing Team's Player in a Game, How Would You Do It? Part 2

This post is a continuing analysis of the "Tranquilizer Theory": what if you could remove one opposing player for the remainder of the game, at any point in the game? When would you use it? Who would you use it on? Part 1

Previously I only looked at using the tranquilizer on the best player on the opposing team at the beginning of the game and how it would affect your team's win probability. Now I'll look at how when you use the tranquilizer affects your odds of winning, assuming you wait to use it during the game. One critical component of this, however, is that you must be in the lead late in the game for your team to gain the largest advantage (by taking away the game-changing play that would have led the losing team to come back). Intuitively, this makes sense: when you're trailing you have more to gain; thus, as the leading team, you have more to lose.

MLB:
Baseball was the first sport to be heavily scrutinized statistically, and way back in 2002 someone came up with a metric called the Leverage Index, which is described as such:

Based on the various game state (inning, score, men on base, and outs), we can determine the chances of winning at any point in the game. We can also determine how much swing (leverage) those situations present.

Using this idea we're able to look at how win probability changes play-by-play in baseball, such as in this case for the Chicago Cubs' collapse against the Florida Marlins in the 2003 NLCS. In this extremely high leverage situation (the Cubs could have clinched their first trip to the World Series since 1945), the biggest play-by-play swing was 36.3%. This is actually much lower than in some other sports (as we'll see), but is still much bigger than the effect of even the best starting pitcher over a whole game.

Verdict: Use in-game

NFL:
Football had one of the largest full-game differences when the starting quarterback was removed. Even so, late in-game situations easily trump this full-game shift in win probability. Take this past year's Super Bowl: Jermaine Kearse's incredible catch increased the Seahawks' win probability by 48% (according to Prediction Machine), which is even larger than replacing Peyton Manning with a backup QB (36.65%).

Verdict: Use in-game

NCAAB:
Basketball is obviously going to be the sport where canceling out a play late in the game has the largest effect on the ultimate outcome. Thanks to KenPom's in-game win probabilities (which actually are based on the idea of leverage in baseball), I can directly quantify how large this effect can actually be. The largest possible swing occurs when a team leading by 2 loses on a 3-point buzzer beater. The leading team's probability goes from close to 1 to 0 in a single play. 

I found a few instances of this, including one that will forever be fresh in my mind: the Austin Rivers buzzer beater in 2012. Per KenPom, North Carolina had approximately an 82% chance to win before that shot, which then went to 0 in an instant. Another buzzer beating game-winner, South Carolina State over Maryland Eastern Shore in 2015, had an even more dramatic effect: it dropped Maryland Eastern Shore from 90% to 0 on the last play of overtime.

Verdict: Use in-game

NBA:
NBA should be similar to college basketball, but I still wanted to get a feel for how large this effect can be in comparison. So, I went to one of the most dramatic shots in the history of the league: Ray Allen's game-tying 3 in Game 6 of the 2013 NBA Finals. We can see how large a swing this gave the Heat thanks to Prediction Machine. In this case his game-tying 3 upped Miami's odds by 24.1%. Had he missed that shot, the Spurs would have won the NBA Finals, so this specific situation certainly carries a lot of weight. If we remove that aspect, I can identify at least 2 reasons why this specific difference is not as large as the NCAAB example:

1. This shot tied the game, it wasn't a game-winner
2. NBA players are so good that the a priori probability took into account the (relatively) high chance that one of Miami's players would make a shot to tie the game

Even so, late in-game play-by-play changes can be much larger than the effect of any one player (such as Allen's teammate at the time, LeBron James) over the whole game.

Verdict: Use in-game

In all cases the late-game swings could be greater than the effect of the other team's best player, so it makes sense to wait to use the tranquilizer in that potentially high-leverage situation. However, this is a gamble: what if you never get close enough to have one single play have a large effect? What should you do if the other team uses their tranquilizer on YOUR best player before the game starts? Should you wait, or should you even the playing field right away? In my next post I'll look at these strategic questions and "what if?" scenarios.