Thursday, September 17, 2009
Game Theory and the NL WestPosted by Adam Guttridge
On September 11th, the NL West leading Dodgers began a series against the Wildcard seeking Giants. The NL West standings (and truly the only relevant Wildcard standings) on that day were as follows:
As a Rockies fan, I was scoreboard watching. But wait... who do I really want to win?
Option 1: I root Giants. Besides just plain bragging rights, winning the division instead of the Wildcard provides a great shot at home field advantage in the playoffs, as the Rockies and Dodgers have 2 of the 3 best records in the NL. Also, winning the division ensures you will play at least a somewhat inferior team in terms of W-L record.
Option 2: I root Dodgers. As much as that sentence makes my skin crawl, it has it's benefits. At this point, every Giants loss makes the playoffs a much firmer probability for the Rockies. And for a team who finds themselves in the playoffs as infrequently as the Rockies have (once since '95), there's probably something to be said for taking the lower-risk option.
Let's attempt to quantify some of the values of either proposition. I'm going to start with an assumption; that I have to pick one team at the beginning of the series and stick with it. It may be the case that if the Giants won the first two games, and the Rockies had lost one or two in their series, I'd start to get nervous and want the Dodgers to get them off our tail. But this is an article, and evaluating the game-by-game combinations of the three teams and the range of each their outcomes would be a novel. Hence, I must root sweep. I'm also going to assume that the Rockies record remains 'neutral.'Technically, they could not play 3 games and remain a .574 team. But assuming they would win or lose 2 of 3 games during this time (or be on one end of a sweep) would obfuscate the point; we should only assume they'll remain as good as they have been, and evaluate the Giants-Dodgers series independently.
Let's presume NL West winner has a 85% chance of having home field advantage during the playoffs (Philadelphia would have to make up a handful of games for that not to be the case). In modern MLB, the home team wins roughly 53% of their games. It may be the case that this effect is amplified during the playoffs, but for this evaluation I will remain conservative. Also, the division winner will face a team with a lesser W-L record. If standings remain consistent, this would mean a 96-win team will play a 93-win team. Of course, who is actually the better team is a more complicated answer than W/L record. But it is the safest and best assumption to make. A basic dependent probability model tells us that a .593 team (96 wins) will beat a .574 team (93 wins) 52.3% of the time.
If we combine the effects of home field advantage and an inferior opponent, a binomial distribution predicts that the NL West winning team would win their first playoff series 55.43% of the time (which obviously implies the Wild Card team would win 44.57% of the time). This would suggest that getting to the playoffs as a division winner instead of a Wildcard makes a very large difference.
Now, how would a sweep one way or the other affect the Rockies divisional and Wildcard odds? Let's say winning the division will require at least 95 wins, and winning the Wildcard will require 92. Prior to the series, what were each team's respective odds of getting there?
Just to recap, the three-digit decimals beneath the 95 and 92-win columns are each team's respective odds of reaching those win platforms (using a binomial distribution with their team's pre-series winning percentage as the p term). Even in subsequent graphs, the pre-series winning percentage will determine a team's 'p'. The 'Diff' column is calculated from the Rockies' persective; thus, the first (95-win) diff column there means the Rockies are 40% less likely than the Dodgers to reach 95 wins, and 25.2% more likely than the Giants to do the same. The figures highlighted in red are our most practically important terms here; the Rockies' chances of overcoming the Dodgers for the division (upper left) and their chances of remaining ahead of the Giants for the Wildcard (lower right).
Here's how that graph would change in the event of a Dodgers sweep.
As you can see, the Rockies' comfortable lead of the Wildcard grows all the more secure; they're about 75% more likely to reach 92 wins than the Giants, up from ~69% pre-series. Now, with only 22 games to go, why is the difference so seemingly slight? Primarily because the Giants had a pretty low crack at the WC pre-series. In other words; what are your chances of beating Tiger Woods for 18 holes, if you typically shoot in the mid-90's? Maybe 1%, or something close? Now, what do those chances become if you shoot +6 combined on the first 3 holes? I don't know, maybe .5%? In other words--the raw change in terms of percentage points won't be large if things were already leaning heavily in that direction. The Rockies went into the series a dominating favorite for the Wildcard--they're now a more overwhelming favorite. However, they're much less likely to catch a Dodgers team that has now won 3 of their remaining 21 games. They were 40% less likely than the Dodgers to reach 95 wins pre-series... now they are nearly 60% 'behind.'
Baseball Prospectus 2008 estimated the value of reaching the playoffs to be $40m, on average (and 'average' would imply a team that wins 1.5 series in the playoffs). Just for the sake of argument, let's say the value of getting to the divisional series is $10m, and the value of advancing to the NLCS is another $12.5m, and the value of advancing yet again to the World Series is an additional $20m.
A Dodger's sweep of the Giants would make the Rockies approximately 5.5% more likely to reach the divisional series (.748-.693), and 5.5% of $10m is $550,000. But remember that the NL West winner will have a much better chance at advancing to the LCS than the Wildcard team. This advantage is roughly 5.43% (as calculated above), thus, using the value of advancing to the LCS as $12.5m, winning the division is worth more than winning the Wildcard by $678,750 (5.43% * $12.5m). However, we need to add something; the team who advances to the LCS also has a chance (let's just call it 50%) of also advancing to the World Series, which is worth $20m. The calculus for the value of winning the division outright becomes 5.43% * (12.5 + 10), which is roughly $1.2m. The Rockies, by virtue of the Dodgers sweep, are now 18.9% less likely to overcome them for the division title. This costs them close to $232k.
Thus, the total value to the Rockies of the Dodgers sweeping the Giants would be $319,089 ($550,000 - $231,911).
With a Giants sweep of the Dodgers, the NL West title becomes a tantalizingly distinct possibility for the Rockies. Their odds of overtaking the Dodgers increase dramatically (by 31.5%). The chance of the Giants overcoming them increase, however, by 9.5%. Using the same math as above, the Rockies lose $950,000 versus pre-series by virtue of the fact that making the playoffs is now much less of a sure proposition. Suprisingly, the fact that the Rockies now have a much better chance at taking the division than the Wildcard is only worth $384,451. Thus, the Rockies 'lose' about $565k in exected playoff revenue by the Giants sweeping the Dodgers.
The seemingly small value gained by a Giants sweep actually makes sense when it comes down to it; being ~30% more likely to gain a ~5% adavantage in a series is simply not an enormous value. Given the number of Wildcard teams we've seen go deep into (or win) the playoffs, I think the general rule here--to value entrance over seeding--is valid.
Sure, I took a lot of liberty with the numbers above--a lot of estimation. But this was meant to be an excercise moreso than a precise calculus, and we'd have to really torture the numbers to reverse the conclusions.