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.

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Editor’s note: This post contains considerable math. If you’re the sort interested merely in conclusions, scroll to the bottom.

Introduction
In my previous article, I developed a model for calculating the WAR threshold for qualifying offers, the threshold above which it makes sense for teams to hand out a qualifying offer to a player. In order to keep the math to a reasonable level, I limited myself to linear and exponential functions to model the probabilities that a player accepts the offer, declines the offer, and signs with another team. In this article, I extend the model to use a more rigorous class of functions called sigmoids. This should yield a better model, while still keeping the math to a reasonable level.

Analysis
Sigmoids are a class of functions that have “S” shapes, and which asymptotically approach two y-values as x approaches negative infinity and positive infinity. The most well known sigmoid is S(X) = 1 / (1 + exp(-X)) :

This function satisfies the fundamental requirements of the P(rejects) function, that P(rejects) approaches 0 for low values of X, and approaches 1 for high values of X. We can generalize this function to S(X) = 1 / (1 + exp(-(a+bX))). Note that the previously graphed function is the special case of this generalization where a=0 and b=1.  In an intuitive sense, “b” controls the sharpness of transition between 0 and 1, and the value of X where S(X) = 0.5 is X = -a/b.

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We recently went through the 2015 qualifying-offer season, the basic facts of which Five Thirty Eight’s Rob Arthur provided a helpfully summarized back in November. In that piece, Arthur asserts that “Each offer is essentially a bet that the player… will be worth 2 or more WAR in the coming year.” He adds that “the math works out so that teams tender offers to almost every remotely deserving free agent [and] [w]ithout fail, those free agents refuse them.” Arthur was writing before the deadline for players to accept their offers, and for the first time this year, there were players who accepted their qualifying offers (Brett Anderson, Colby Rasmus, Matt Wieters).

While teams may have been very liberal in giving out qualifying offers in the past, now that there is a precedence of players accepting the offers, teams will likely need to be more cautious about giving out offers in the future. This article attempts to build a model that determines at what WAR threshold it makes sense for teams to give out qualifying offers.

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Every June, about 1500 players from the United States, Canada, and Puerto Rico are selected in the amateur draft in June. Some of these amateur players are drafted out of high school, while others are drafted from two-year and four-year colleges. The majority of these players will elect to sign a professional contract, thus ending their amateur careers, and beginning their professional careers in minor league ball, with hopes and dreams of making it to the majors. Only a fraction of these players will eventually make it to the highest level of professional baseball.

This article will look at an unlikely group of MLB feeder colleges: US News and World Report’s top-25 national universities.

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This is a follow-up to my previous post, “Predicting Secondary Market Prices for ALDS/NLDS Tickets”. Now with a complete set of price data from 2011 to 2015, I’ve amended and refined my previous model for ALDS/NLDS ticket prices. I’ve also been able to build additional models to predict prices for ALCS/NLCS and World Series ticket prices in the future.

Before I go further, I’d like to thank Chris from TiqIQ. Chris was nice enough to give me TiqIQ’s complete set of price data from 2011 to 2015 for each year’s playoff teams. Needless to say, without his help, this study could not be completed.

The new set of data is superior to the previous data I collected from TiqIQ’s blog for the following reasons:

• It takes into account all the transaction values, instead of only the transactions at the time the TiqIQ blog posts were written; and
• It only includes playoff games that were actually played, instead of all possible playoff games (which include prices for games that may not be played); and
• We have values for each individual game, instead of only an average value for the whole regular season and the whole ALDS/NLDS.

As before, the statistic that is predicted is the average price of the tickets for each playoff series. Because the final game of each series (Game 5 in the ALDS/NLDS and Game 7 in the ALCS/NLCS and World Series) is guaranteed to be an elimination game for both teams, it commands a premium compared to the other games, so I excluded that data in calculating the average value.

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So far this October, we have been treated to some great playoff games.  Most of us watched these games at home in our living room, or perhaps at the local sports bar.  A select few of us have had the chance to watch the games live at the stadium.  Due to the high demand for playoff tickets, most teams conducted some type of lottery to determine who gets to purchase tickets at face value.  Those who aren’t lucky enough to win the lottery can still get into the stadium by purchasing tickets on the second-hand market.

Perhaps unsurprisingly, there is a lot of variation in the second-hand cost of playoff tickets between the different teams.  There are many factors that go into this variation, including regular season ticket cost and how recently the team has gone to the playoffs, among other things.  For example, the Cubs have had the highest playoff tickets prices this year, which is due to the fact that they have a passionate fan base, and that they haven’t been to the playoffs since 2008.

I wanted to see if it would be possible to create a model that predicts the cost of ALDS/NLDS ticket prices based on other factors.

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This week’s article is related to last week’s on a similar topic — and, much like that first article, begins with the premise that you’re the manager of major league club (congratulations!) and your team has qualified for the playoffs.

In that first piece, I argued that, under some conditions, you would slightly increase your chances of winning the ALDS/NLDS by employing the “save the ace” strategy, where you start your number-two pitcher in Game 1 and your ace in Game 2. Whether you choose this strategy or the conventional one of starting your ace in Game 1 and your number-two in Game 2, last week’s article assumed that all of your starting pitchers are rested and at your disposal at the beginning of the LDS. If you were fighting a close division or wild-card race (or perhaps both of these) at the end of the season, however, then this likely would not be the case, since you would have stuck to your regular five-man rotation until possibly even the final game of the regular season.

So this week, we will look at starting pitching in the final regular reason game, tiebreaker games, and wild-card game.

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Last year Clayton Kershaw had an absolutely dominating season, which culminated in him winning both the NL Cy Young and MVP awards. When the playoffs started, I remember feeling a little bad for the Cardinals, who were facing the possibility of having to face Kershaw twice within the five-game NLDS. Of course, the Cardinals defied the odds by defeating Kershaw twice en route to winning the series, showing that anything is possible in a short series. I wonder, though, if they couldn’t have improved their odds of winning the series had they matched up their starters a bit differently.

In a five-game Division Series, most managers start their ace in Game 1, assuming that he is available to pitch on regular rest. That will optimize a club’s chances of winning Game 1. However, I’ve wondered if that does not necessarily optimize that same club’s chances of winning the series. Would a team be able to, under some circumstances, optimize its chances of winning the series by using their #2 starter in Game 1, and starting the ace in Game 2?

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