Note: This article will be continually updated to take into account game results each day, so continue to check it out to see the latest matrix of outcomes / probabilities.
The final week of the baseball regular season is upon us, and by far the most exciting playoff race right now is for the second Wild Card spot in the NL, where there are three teams currently battling it out for the privilege of facing Zack Greinke and the Arizona Diamondbacks in Phoenix on Oct. 4.
The Colorado Rockies currently have a 2 game lead over the Milwaukee Brewers and a 2.5 game lead over the St. Louis Cardinals, but given that the Brewers and Cardinals will play each other this weekend, exactly how secure is the Rockies’ current position?
In order to visualize all possible scenarios for the final week, I have created a matrix outlining who would win the second NL Wild Card (or if two or more teams would tie) depending on how each of the three teams play down the stretch.
(I will be assuming that the Cubs will win the NL Central – I considered it not worth the additional complexity to add a fourth team to the chart since they have a 99.8% chance of winning the Central, and even if they are swept in 4 by the Cardinals they would still be able to clinch by beating the Reds at home next weekend).
Each column in the matrix represents a record that the Rockies could have over their final 6 games, while each row represents a particular combination of records that the Cardinals and the Brewers could have over their final 7 and 6 games, respectively.
Note that some combinations of Brewers and Cardinals records are not possible, since their weekend series guarantees that their combined record must contain at least 3 wins and 3 losses. This is the reason that, for example, if the Brewers go 0-6, the Cardinals must go 3-4 or better and if the Cardinals go 7-0, the Brewers must go 3-3 or worse.
In the scenarios where one team ends up winning the wild card outright, I have color-coded each cell with a solid color and a single letter (“R” for Rockies, “C” for Cardinals and “B” for Brewers).
And for those scenarios where there would be a tie, I have included a two-color pattern and two or more letters (“CR” for Cardinals and Rockies tie, “BR” for Brewers and Rockies tie, “BC” for Brewers and Cardinals tie, and “BRC” for that rare event where all three teams would tie).
As the graphic shows, the Rockies largely control their own destiny: if they can go 6-0 or 5-1, they don’t have to worry about anyone; if they can go 4-2, they just have to hope either the Brewers or Cards don’t win out (and even then they’re guaranteed at least a tie-breaker game); and even if they go a middling 3-3, they still have a very good chance of making it (and would be guaranteed a tiebreaker game unless one of their opponents wins out).
But if they go 2-4 or worse, the graphic is pretty distressing – the Rockies would be in a difficult position unless the Cardinals and Brewers both slump dramatically in their upcoming series against the Cubs and Reds, respectively, and it would dramatically increase the chance of a zany tiebreak scenario occurring (something that Rockies fans would probably prefer not happen).
Although the graphic shows all possible scenarios, the cells do not represent scenarios that are all equally likely. For example, there are far more ways to go 4-3 than 7-0 in 7 games, so even if you assume the Cards have a .600 true-talent winning percentage, they would have a ~50% chance of going either 4-3 or 5-2 but only a ~3% chance of going 7-0, so keep that in mind when looking at the worst-case scenarios for the Rockies.
In fact, thinking about these probabilities made me and fellow Rockies fan Kyle Douglas curious how likely each of these outcomes actually were, so we took the 538 MLB Probabilities for the upcoming games for each of the three teams and ran a million simulations to compute the joint probabilities of each of the scenarios outlined above.
The following matrices show the joint probabilities in each cell (the chance the Rockies, Brewers and Cardinals finish with any given record), show the Rockies’ marginal probability along the bottom (the chance the Rockies finish with any given record regardless of the other teams), and show the Brewers’ and Cardinals’ marginal probabilities on the right (the overall chance those two teams finish with a given combination of records, regardless of the Rockies).
We also then computed the likelihood of each wild card winner scenario occurring, both in general, and conditioned on a given Rockies record.
Below are all the results of these simulations, which will be continually updated as game results come in, with any scenarios that have now been rendered impossible removed.
Update: Saturday Afternoon, Sept. 30
The Cardinals trotted out another questionable lineup on Saturday and Luke Weaver had a poor start that found the team down 6-0 early, but amazingly, the Cardinals stormed all the way back to win 7-6, eliminating the Brewers and clinching the postseason for the Rockies!
The below matrix shows the situation going into the Rockies Saturday evening game against the Dodgers, and finally we have a matrix that is entirely purple!
UPDATE: Saturday Morning, Sept. 30
On Friday, the Rockies bludgeoned the Dodgers in a convincing win while the Brewers stayed alive by beating a rather lifeless Cardinals ‘AAA’ team.
Despite the Brewers win, the Rockies win is extremely important as it takes away the terrifying option for the Brewers to be able to win the WC outright, and it leaves the Brewers with only a single unlikely path to the playoffs - winning the next two on the road while the Rockies lose the next two at Coors, and then beating them in the tiebreaker Game 163 at Coors.
As the updated matrix shows, there is only a ~6% chance of the tiebreaker occurring, and since even if it did occur the Rockies would have at least a 50-50 chance at winning it, the Rockies have a ~97% chance of making the playoffs at this point.
The Cardinals not playing Dexter Fowler after already announcing that they would be sitting Carpenter and Molina the rest of the season is a bit concerning because it may indicate that they don’t intend to give a good faith effort at winning at all against the Brewers. However, since they were facing Chase Anderson, this was the game in the series that the Cardinals were most likely to lose anyway, and there is at least a silver lining in the fact that the Brewers ended up having to use their closer Knebel who is likely unavailable today.
So as long as the Cardinals decide they don’t want to end their season with an abysmal 1-6 homestand, we should see a better lineup and overall effort today with Luke Weaver on the mound in a favorable pitching matchup (at least on paper) against the Brewers’ Junior Guerra. But if the Cardinals lose again, the Rockies have to either beat Kershaw tonight or enter a suboptimal situation where the Sunday games will decide whether they have to host another Game 163.
UPDATE: Friday Morning, Sept. 29
On Thursday, the Rockies were off, but the Brewers won and Cardinals lost, and the following matrix shows the updated scenarios and their probabilities.
Now that the Cardinals were officially eliminated, all Rockies fans can comfortably root for the Cardinals to beat the Brewers over the weekend, and of the 16 remaining scenarios, 13 of them end with the Rockies winning the wild card outright (a ~90% chance), and 15 of them end with us at least tying the brewers (a ~98% chance).
If the Rockies can just win a single game in the series, they will be in a very comfortable position, and even in the distressing scenario where they get swept by the Dodgers, they would have still have a relatively reassuring ~60% chance of winning and a ~90% chance of at least tying.
Part of the reason for these reassuring numbers is that the 538 probabilities consider the Cardinals heavy favorites on Saturday and Sunday, but I fear they might be overestimating the Cardinals’ chances given that there are players nursing injuries (Carpenter, Wong, Molina and Ramirez), who may now end up sitting out for the rest of the season. At the same time, it would hardly be the Cardinals Way™ to trot out ‘B’ lineups all three games, so hopefully the Cardinals play with pride after their devastating losses to the Cubs and do not completely roll over to another division foe.
And even if 538 is underestimating the Brewers’ chances of winning each game, perhaps the effect is countered somewhat since they may also be underestimating the Rockies’ chances of winning each game against the Dodgers; for example, their model sees Kershaw and pegs the Rockies as significant underdogs without the knowledge that his start is expected to be an abbreviated one, or that they are possibly planning a bullpen game for Sunday.
UPDATE: Thursday Morning, Sept. 28
Wednesday turned out to be the optimal day for the Rockies, with the both the Cardinals and Brewers losing (and thus being eliminated from the NL Central) after the Rockies won earlier in the day.
Below, the updated matrix shows that Rockies now have a ~91% chance of winning the Wild Card outright and a ~99% chance of forcing at least a tiebreaker – the only scenario in which they wouldn’t play beyond Game 162 is if they get swept by the Dodgers while Milwaukee wins tomorrow and then sweeps the Cardinals.
So if the Rockies can just win one game in the Dodgers series, they would eliminate the Cardinals entirely and would force the Brewers to win out just to tie them.
UPDATE: Wednesday Afternoon, Sept. 27
With the Rockies’ 15-9 win against the Marlins in the books, I decided to make a quick update before the Brewers and Cardinals play tonight.
This was a huge win for the Rockies as it takes out the possibility of the 1-5 scenario, and makes it much more difficult for the Brewers and Cardinals to win outright (there is now a ~95% chance the Rockies would at least play a tiebreaker, and no longer a possibility for the Cards and Brewers to tie each other and not the Rockies).
The Rockies won’t play again until the Brewers and Cardinals have each played two more games, and how those four games go will dramatically affect this matrix; if the Reds and Cubs can steal a few, the Dodgers series this weekend will be a lot less stressful.
UPDATE: Wednesday Morning, Sept. 27
Below is an updated matrix taking into account the results of Tuesday’s games (wins for the Brewers, Rockies, and Cardinals) along with using updated 538 probabilities for the remaining games now that the projected pitchers are more certain.
Because all three teams won, the probabilities didn’t change too much (but slightly increased for the Rockies since the season is running out of days), and there are a lot fewer possible outcomes than before.
One surprising result is that if the Rockies go 0-4 (to finish 1-5), there is now a 12.9% probability of a three way tie... In general, the Rockies’ 1-5 and 2-4 scenarios are less favorable since the good outcomes in those scenarios relied on the Brewers and Cardinals not winning much until the weekend.
UPDATE: Tuesday Morning, Sept. 26
Below is an updated matrix taking into account the results of Monday’s games (losses for both the Rockies and Cardinals).
I’m not going to change the scenarios to be in terms of the record in only remaining games – I’ll keep it in terms of records for the 6 or 7 game week-long stretch, just with some possibilities removed since they are no longer possible (e.g. with the losses last night, the Cardinals can no longer go 7-0 and the Rockies can no longer go 6-0).
The Brewers had off, but they gained a lot by seeing both of their opponents lose.
UPDATE: Monday Morning, Sept. 25
Below is the matrix of all probabilities before any games have been played (thus corresponding directly to the original matrix above with all 308 scenarios marked R, C, B, BRC etc.).
One of the reassuring results with these probabilities is the case that the Rockies go 2-4 – according to the simulations, the Rockies would actually still have a ~40% chance of winning the wild card outright and an ~80% chance of playing beyond Sunday.