How to Think of Postseason Contention, Elimination

We’re at that time of the regular season during which most teams are making a final push to clinch a spot in the postseason. Some teams (such as the Cubs) have basically already clinched a spot and some others (such as the Twins) are already mathematically eliminated. Most teams fall somewhere in between.

Many baseball fans will look at the standings every day in September. If they see their favorite team is leading its respective division, they’ll hope that, for the rest of the regular season, that team win will more games than anyone else in the division, thus allowing that team to become the division champion. This is guaranteed.

If they see, on the other hand, that their favorite team is not leading its respective division, they will check the number of games remaining and the number of games by which their team is behind the division leader. If the number of games remaining is greater than the number of games behind, then they can hold out hope that their team can win the division by winning all its remaining games, while the division leader loses all their remaining games. Unfortunately, this is not always guaranteed.

The type of error is made not only by Average Joe sports fans, but also professional sportswriters.  This article will describe these tricky scenarios in which teams are eliminated from postseason contention.

Before I go any further, I should point out that the “Baseball Elimination” problem that we’re analyzing here is very well documented in the academic math and computer-science communities. For those readers with advanced math and CS backgrounds, I’d recommend discussions by Kevin Wayne and Jeff Erickson. They explain everything in more detail than the present article. The goal of this article is to break the problem down so that people without these advanced backgrounds can get an appreciation of this topic, too.

Necessity and sufficiency

Let’s start with a quick lesson in logic. Two commonly used adjectives in logic are “necessary” and “sufficient.” A broad definition for “necessary” is “if condition A is false, then condition B is false”. Similarly, “sufficient” means “if condition A is true, then condition B is true.”

Here’s an example to illustrate these concepts. Suppose we are trying to determine if a person is eligible to vote in the upcoming US presidential election. Two necessary conditions are that:

• The person is at least 18 years old; and
• The person is a citizen of the USA.

This means that, if a person is not at least 18 years old, or if a person is not a citizen of the USA, then he/she cannot vote in the upcoming US presidential election. However, even if he/she meets both of these conditions, there is no guarantee that he/she can vote.

A sufficient condition for a person to vote is that he/she is registered to vote in the USA. If a person is registered to vote, then he/she will be able to vote. In fact, voter registration is both a necessary and sufficient condition for voting.

(Note: This is a friendly reminder for all eligible people to register to vote!)

Applying this logic to baseball standings

Let’s get back to baseball. Suppose we look at a team A that is N games behind the division leader B, and both teams A and B have M games remaining to play. A sufficient condition for team A to be eliminated from division-champion contention is that N>M. What this means is that, if N>M (the number of games team A is behind is greater than the number of games remaining), then team A can’t finish as the division champion, because even if

• Team A wins all its remaining M games; and
• Team B loses all its remaining M games

team B will still have more wins than team A.

If team A is in second place, then it is also a necessary condition that N>M for team A to be eliminated. This means that if N>M is not satisfied, then it’s possible that:

• Team A wins enough games; and
• Team B loses enough games

in order for team A to finish as the division champion. One way this could happen is if team A wins all its remaining M games, and team B loses all its remaining M games, although there are other ways this could work out, too.

If team A is in third place or worse, then having N>M is not a necessary condition for team A to be eliminated from division-champion contention. Even if team A wins all its remaining M games, and team B loses all its remaining M games, there could be situations where team A does not finish as the division champion, depending on how the other teams’ games play out. This is where the tricky scenarios come into play.

Illustration of a tricky scenario

Here is an illustration of a tricky scenario using a hypothetical season.  Let’s assume that the AL East standings looks like this going into the last weekend of the season:

Let’s analyze the possibility that the Red Sox finish as the AL East division champion. The Red Sox trail the Yankees by three games (N=3), and both the Red Sox and Yankees have three games left to play (M=3), so therefore the sufficient N>M condition is not satisfied. Perhaps the Red Sox still have a chance.

One might look at this situation and argue that, since the Red Sox play the Yankees in a final three-game series, if they defeat the Yankees in all three games, they will tie the Yankees at 90 wins. Unfortunately, that isn’t good enough, because since the Blue Jays and Orioles already each have 89 wins, and they play each other in a final three-games series, one of those teams will be guaranteed to finish with at least 91 wins, which is more than the maximum of 90 wins the Red Sox can record. Therefore, even though the sufficient N>M condition is not satisfied for the Red Sox, they’re already eliminated from AL East division-champion contention.

Solving this problem with a computer program

The situation above can be analyzed manually by hand, as with only three games left per team, there are only so many ways that the season can play out. However, if there were three weeks of games left per team, then it’s impossible to analyze manually.

Fortunately, some very smart people have written computer programs (described in the previous external links) to analyze these situations. How this program is written is outside the scope of this article, but suffice it to say that these programs compute how many games each team can possibly gain on the division leader, factoring in the current standings and the remaining schedule to be played. In addition to determining which teams are eliminated from division-champion contention, these programs can also be used to determine which teams are eliminated from wild-card contention.

In our previous discussions, we have assumed that each team has the same M number of games left to play. In reality, this is often not the case. Fortunately, these programs are smart enough to take into account the cases where each team has a different amount of games left to play.

Although these programs are quite complex, one fascinating thing is that, once the eliminated teams have been computed, it is always possible to construct a relatively simple argument using only grade-school arithmetic to explain why each team has already been eliminated (similarly to what we did above with the Red Sox).

A site created by engineers at the University of California and called RIOT baseball uses these programs to compute the status and clinching numbers for each team, updated daily.

Conclusion

Ideally, a fan will have his/her favorite team in first place going into the final stretch of the regular season, and will not need to think about these tricky scenarios. But for most fans, it is important to keep these tricky scenarios in mind, in order to understand the true predicament that their favorite team is facing in order to make the playoffs.