Trade deadline season is perhaps the most fun and engaging time of the year for fans, particularly those of us who enjoy playing armchair-GM. Speculating wildly and arguing vehemently about trades is a core part of being a hardcore baseball fan. However, most of our discussions during this time frame are often based in feelings or intuition rather than math. There’s nothing wrong with that, but we can supplement our discussions with some relatively simple math while learning a bit about how one can model win probabilities and playoff odds.
There are two goals we’ll look at; simply reaching the playoffs at all (the topic of today’s article) and winning the World Series once you get there (coming later this week). Either of these goals make sense to aim for; baseball playoffs are fairly random and oftentimes just making the dance is enough, but they’re also not totally random an a more cautious fan might want to hold resources back to make a real World Series push. Once we have some math on hand, we can better understand how trades impact each of these objectives and perhaps bring some numbers to the table next time we argue over trades being worthwhile or not.
First, we lay out the simplifying assumptions for our model:
- We will consider only two teams; our team of interest A, and their primary competitor B
- We will not factor in head-to-head matchups or strength of schedule
- Each team will have three parameters associated with them; true-talent winning percentage (wp_a, wp_b); remaining games (n_a, n_b); and current win total (w_a, w_b).
Note that we’re not setting out to build an optimal model here - many smarter people than me have done so already. Instead, we’re simply looking to build quantitative intuition along with some helpful visualization tools, and the best way to do that is with a simple - but flexible - model.
Earlier in the year, I wrote about using the binomial distribution to model team win totals. To recap briefly, you can think of the binomial distribution as a count of coin flips. It takes two parameters - the number of coin flips and the probability of heads - and returns a distribution over the number of heads you should see in total. In a baseball context, we can treat teams as coins, with flips replaced by games and the probability of heads replaced by true talent winning percentage.
We can visualize two of these distributions in parallel fairly simply. Imagine, for illustration, that team A and team B each have 56.8% true talent winning percentages and 60 games remaining in their season, but that A currently has three more wins than B. The distribution of expected wins for each team is shown below:
Unsurprisingly, a three game lead allows team A to finish with the higher win total in the majority of cases. How much of a majority is it though? What we need is a model that can systematically take in our six parameters and return a probability of how often Team A finishes with the higher win total. You can actually derive a closed form solution for this (and I’m happy to drop this in the comments if you’re curious), but it’s much easier just to use simulations. Instead of doing any fancy math, we’ll just make random draws from the two binomials a bunch of times and count how often Team A’s winning total is higher.
Though a bit inelegant, this solution is quite effective, and we can arrive at an empirical estimate of Team A’s chances at a postseason berth with arbitrary parameter. For instance, let’s imagine another scenario where Team A and Team B are currently tied in the standings. The figure below displays Team A’s chances of winning the postseason spot as a function of the true talent winning percentages of Team A and Team B:
Again, nothing surprising here - the larger the skill gap between your team and your opponent, the more likely you are to win your playoff race. There should also be a saturation effect here; the stronger you are than your opponent, the less valuable each marginal improvement you make is, and vice versa. However, such an effect is difficult to see on this scale.
To get a closer look, let’s create another hypothetical. Here, team B has a fixed 55% true talent winning percentage with 60 games remaining. We’ll vary team A’s winning percentage between 40% and 70% in three different scenarios; A leading in the standings by 5 games, trailing by 5 games, and dead equal. Here’s what those playoff odds look like:
This plot illustrates several ideas. First, we can clearly see the saturation effect, as we’d expect to see. However, the effect is limited to the tails of the curve at winning percentages most contenders are never anywhere close to. This is particularly true in the even standings scenario, suggesting that nearly any typical contender without a large lead in their playoff race is well within the optimal range for adding talent to the team.
Note however that this is not necessarily true when we introduce significant gaps in the standings, as can be seen in the blue (Team A up by 5 games) and red (Team A down by 5 games) curves. Here, the pre-established lead shoves team A out of the optimal band unless it has a winning percentage that’s notably lower/higher than Team B.
Let’s apply this to the Mets, who currently sit at 53-46 with 61 games to go. We’ll consider the odds against both the Phillies and the Braves (in two separate experiments). As an unbiased measurement for true talent winning percentage, we’ll use Fangraphs’ BaseRuns record, which has the Mets, Phillies, and Braves at 52.6%, 47.3%, and 53.7% respectively. Here’s how things stack up currently:
These results line up with our hypotheticals. The Mets have a significant (though not insurmountable, as Mets fans who remember 2007 can attest) lead in the division and have essentially the same talent as the Braves and significantly more talent than the Phillies by BaseRuns. This makes them heavy favorites in either race and, more importantly, places them at the early edge of the saturation region. In other words, any talent the Mets add to the roster given the current standings has less value than it would if they were a little worse either in record, talent, or both.
Does that final line mean the Mets shouldn’t add at the deadline? No, not at all. We won’t connect this model to financial values in anyway (more on that later), but ultimately this decision comes down to your priorities as a fan. Is a 70% shot at the playoffs good enough for you? Cool, no need to add talent. Are you okay with a less-than-perfectly-optimal acquisition? Also great, but you probably think the Mets should be aggressively looking to add and increase their playoff odds.
Remember also that this is only one side of the coin, and we’ll touch on how we can model the chances of winning the World Series once we win a playoff spot. If that’s your priority as a fan, stay tuned.