While watching a myriad of different sporting events in college, my friends and I would often discuss a theory that would make any sport more interesting and complex: what if you could remove one opposing player for the remainder of the game, at any point in the game? In other words (and this is the PC version), what if you had a tranquilizer, with one dose, that could take out someone on the other team at any time? When would you use it? Who would you use it on?
The second question is a lot easier than the first, so let's assume we use the tranquilizer right when the game starts, so the opposing player would be out for the entirety of the game. Your goal then would be to remove the player who would most likely have the largest positive impact on the game for their team (and we're talking in the general case, for an average team. In some cases this choice would be obvious based on your opponent).
Basketball
Basketball is pretty straightforward thanks to a metric called Win Shares, which aims to estimate what its name implies: how many wins each player contributes. So you would use the tranquilizer on the player with the highest Win Shares per 48 minutes (the length of an NBA game). This WS/48 then directly estimates the change in win probability by taking out the opposing team's best player. The average WS/48 for the top 30 players in the NBA last season (I'm using this as a proxy for the best player on each team) was 0.195, and the average for the rest of the league was 0.069. The difference between these two numbers is 0.1252, which translates to a 12.52% change in win probability.
Verdict: Player with highest Win Shares per 48 minutes
Difference in Win Probability: 12.52%
Football
It seems obvious that the player with the largest impact is the quarterback, but I went ahead and gauged Football-Reference's "Approximate Value" metric to check this. For active players, quarterbacks have the largest average value and the largest total value, but I was surprised to see the prevalence of defensive ends on this list. So we go with the quarterback. We don't have a statistic like WAR in baseball (which we'll get to shortly) to approximate how much of an impact replacing the starting QB with their backup would have on a game, but thankfully Prediction Machine did an analysis on this very topic last season. Per their numbers, replacing the starter with the backup is worth -3.12 points per game on average. If the teams were even before the tranquilizer was used, this translates to a 12.47% change in win probability.
Verdict: Quarterback
Difference in Win Probability: 12.47%
Baseball
Baseball is one of the hardest sports to apply this strategy to, since each individual player only marginally affects the overall game. We had some discussion about using the tranquilizer on the manger, but FiveThirtyEight found that only a select few managers definitively affect their team's performance. For players, WAR (i.e. Wins Above Replacement) is a metric designed to determine how many wins a player contributes relative to a generic replacement (a AAA minor league player). There is some debate as to how this should be calculated, but for this post I'll be using Baseball-Reference's version. Typically, position players' WARs are higher than pitchers'; however, this is because they play a lot more games. On a per game basis, the starting pitcher has a much larger impact on each game.
This trend is evident in the pregame betting lines posted in Las Vegas for certain teams and certain pitchers. The Los Angeles Dodgers are a very good team, and they also have one of the most dominant pitchers in MLB, Clayton Kershaw. In all non-Kershaw started games, the average implied win probability (per the Vegas moneyline) for LA has been 58.32% this season. In Kershaw started games, this rises to 67.37%. This is perhaps the most extreme case of a single player's impact in all of baseball, but removing him from the game would result in roughly a 9.05% change in win probability.
Verdict: Starting Pitcher
Difference in Win Probability: 9.05% (high end)
Hockey
Hockey has a similar statistic to Win Shares called Point Shares, which aims to estimate how many points each player contributes. In NHL, teams get 2 points for a win and 1 point for an overtime loss. Instead of per minute, I calculated Point Shares per game and did the same process as I did for basketball. The average PS/G for the top 30 players in the NHL last season was 0.130, while the average for the rest of the league was 0.034. The difference between the two is 0.097, but this isn't a direct translation to win probability. An OT loss is equivalent to half of a win, so we divide this WS difference by 1.5. This gives us 0.0645, which gives an analog to win probability equal to 6.45%.
Verdict: Player with highest Point Shares per game
Difference in Win Probability: 6.45%
Soccer
Soccer is probably the hardest of these five sports to apply the tranquilizer idea to and has the fewest statistics as well. The best estimator I can offer is plus/minus, which is simply how many goals are scored while a specific player is on the pitch minus how many goals are conceded during that same time frame. This is also used in basketball and hockey, and aims to quantify the effect a player has that might not show up in the box score. With a variety of leagues around the world, there's not much to go on even for this simple of a statistic. The best I can offer is an international-level view thanks to a FiveThirtyEight article that concluded Lionel Messi, the best player in the world, would boost an average team's win probability by 15% (this doesn't take into account ties). That same article showed Brazil's loss of Neymar in last year's World Cup dropped their win probability in each game by 5%, and Neymar is one of the best players in the world. Unfortunately, for the case of soccer, the tranquilizer idea isn't very conclusive.
Verdict: Best player on the other team (not much of a conclusion sorry)
Difference in Win Probability: From 5% to 15% (best-in-the-world scenario)
Overall, basketball and football definitely separate themselves as the two sports in which the tranquilizer idea would be the most effective (in the general case, when you're not facing Messi). In football it's not surprising that the quarterback has the most influence for a single player; however, it is interesting that basketball is roughly equal in this effect on win probability, when basketball has 5 players on the court for each side and football has 11.
This post only covers which player you should take out given that you do it at the beginning of the game. In future posts I aim to analyze whether using the tranquilizer at a specific moment could have a larger impact, and also how things would be affected if both teams could use this strategy at the same time.
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