Postseason Odds

It’s postseason time. The wild-card series are done, and four teams have had their hopes dashed already. Somebody once described the MLB postseason as deciding who wins a marathon with a game of Paper, Rock, Scissors, which has always felt like an apt description. Nothing against the format, it’s plenty exciting, but the statistician in me enjoys the elegance of most soccer leagues - the team with the most points at the end wins.

But we’re not here to change formats, but to forecast their outcomes. How likely is it that your team will move on? First, we need to consider who they’re playing. If a .540 team is playing a .510 team, we know that they won’t win .540 of the time, but it will also be higher than .510. We can use Bill James’ Log5 method to get the probability of winning a single game. Let’s look at predicted win probability for different combinations win percentages.

library(hrbrthemes)
library(gt)
library(gtExtras)
library(tidyverse)

set.seed(202509)

log_5 <- function(team_a, team_b){
  (team_a - (team_a*team_b)) / (team_a + team_b - (2 * team_a * team_b))
}

odds <- seq(from = .450, to = .600, by = 0.01) %>%
  as_tibble()

odds_data <- odds %>%
  rename(team_a = value) %>%
  crossing(odds %>% rename(team_b = value)) %>%
  mutate(team_a_win_odds = log_5(team_a, team_b))

odds_data %>%
  ggplot(aes(x = as.factor(team_b), 
             y = as.factor(team_a), 
             fill = team_a_win_odds, 
             label = round(team_a_win_odds, 2)))+
  geom_tile()+
  geom_text()+
  theme_ipsum()+
  theme(legend.position = 'none',
        panel.grid.major = element_blank(),
        panel.grid.minor = element_blank() )+
  ylab('Your Team Win %')+
  xlab('Opposing Team Win %')+
  scale_fill_gradient2(
    low = "firebrick",
    mid = "white",
    high = "dodgerblue4",
    midpoint = 0.5
  )

Firstly, we note that when teams have identical win percents, the expected outcome is a coin toss. We also see that the better the opponent, the harder it is to win. Hopefully this is not a surprising finding.

This is all well and good for a single game, but, as you’ve undoubtedly noticed*, the postseason is not a single game, but a series of games. It’s tempting to use R’s dbinom function to model a series, but that will only be approximately correct. That assumes all games are played - i.e. what are the odds of winning 2 of 3 games. But that would also include a third game, even if one team won the first two games and hence the series. This is, as the kids would say, problematic.

*Oh, you’re clever.

We’ll build our own function to handle that. It will stop once either team has won the series. This lets us better model our expected outcomes. Let’s model the ALDS series with the Tigers (.537 win %) vs. the Guardians (.556 win%). The Log5 model gives the Tigers a 48.1% chance of winning a given game, so we’re fairly close here. We could adjust these percentages based on late season performance, expected wins based on runs scored/allowed, etc, but for simplicity we’ll use the entire season.

sim_series <- function(series_length = 5, odds = .5){
  series <- replicate(10000, rbinom(series_length, 1, odds))
  
  wins_needed <- ceiling(series_length / 2)
  
  series_sum <- series %>%
    as_tibble() %>%
    mutate(trial = row_number()) %>%
    pivot_longer(
      cols = -trial,
      names_to = "sim_id",
      names_prefix = "V",
      names_transform = as.integer,
      values_to = "outcome"
      ) %>%
    arrange(sim_id) %>%
    group_by(sim_id) %>%
    mutate(n = row_number(),
           w = cumsum(outcome),
           l = n - w) %>%
    mutate(series_over = w >= wins_needed | l >= wins_needed,
           series_over2 = cumsum(series_over)) %>%
    filter(series_over2 == 1) %>%
    mutate(won_series = ifelse(w > l, 1, 0)) %>%
    select(n, w, l, won_series)
  
  series_sum
}

sims <- tibble(odds = log_5(.537, .556)) %>%
  mutate(series = map(odds,\(code) .f = sim_series(series_length = 5, odds = code))) %>%
  unnest(series)

sims %>%
  summarise(won_series = mean(won_series))

# A tibble: 1 × 1
  won_series
       <dbl>
1      0.465

The model says the Tigers wins 47% of simulations, so not quite a coin flip but fairly close. It’s probably a bit lower since we’re not account for the Mariner’s home field advantage. Let’s look at all the potential series breakdowns. This is from Detroit’s viewpoint, so W is DET wins and L is SEA wins.

sims %>%
  group_by(w, l) %>%
  summarise(n = n()) %>%
  ungroup() %>%
  mutate(p = n / sum(n)) %>%
  gt() %>%
  gt_theme_espn() %>%
  fmt_percent(columns = p, decimals = 1) %>%
  fmt_number(columns = n, decimals = 0)

w	l	n	p
0	3	1,353	13.5%
1	3	2,038	20.4%
2	3	1,961	19.6%
3	0	1,058	10.6%
3	1	1,800	18.0%
3	2	1,790	17.9%

Seattle is slightly more likely to sweep, or just plain win in any series length. But again, it’s close to a coin flip. Would it be better for the Tigers if the series was a different length?

sims_length <- c(3, 5, 7) %>% 
  as_tibble %>% 
  rename(series_length = value) %>%
  mutate(odds = log_5(.537, .556)) %>%
  mutate(series = map2(series_length, odds, sim_series)) %>%
  unnest(series)

sims_length %>%
  group_by(series_length) %>%
  summarise(won_series = mean(won_series)) %>%
  gt() %>%
  gt_theme_espn() %>%
  fmt_percent(columns = won_series, decimals = 1)

series_length	won_series
3	47.1%
5	46.7%
7	45.3%

While it’s a small margin, they’d be better off in a 3 game series. Randomness rules in the playoffs, and randomness thrives in small numbers, whereas in a longer series true talent is more likely to win out.

As one final aside, it’s worth noting that this is based on the assumption that the actual win percents of each team represent their talent. If we instead look at their Pythagorean Expectations (DET .543, SEA .546), which is based on runs scored and allowed, we get to 49.5%.

sims_py <- tibble(odds = log_5(.543, .546)) %>%
  mutate(series = map(odds,\(code) .f = sim_series(series_length = 5, odds = code))) %>%
  unnest(series)

sims_py %>%
  summarise(won_series = mean(won_series))

# A tibble: 1 × 1
  won_series
       <dbl>
1      0.487

Since the teams had similar offensive numbers (DET 4.68 runs per game, SEA 4.73) and defensive numbers (DET 4.27 RA per game, SEA 4.28), it’s not unreasonable to say that, at the surface, they’re evenly matched, though the Tigers don’t have the star power of Raleigh and Rodriguez and are sorely missing Olson and Jobe in the rotation. That said, Skubal should be able to start twice, although start two would be in a game five if necessary. I’d consider Seattle the slight favorites for this series, but it’s certainly not unwinnable for the Tigers.