Pirates and Expected Wins

The Tigers-Pirates game on Wednesday got rained out, resulting in a doubleheader yesterday. Tarik Skubal was scheduled to pitch Wednesday, while Paul Skenes was scheduled to go Thursday. Naturally, the delay led to a call for the pitchers - two of the best in the game - to face each other.

Neither manager budged. From a fan standpoint, it’s somewhat disappointing, but it’s easy to see why AJ Hinch and Don Kelly did what they did. (Skubal was apparently given a choice and stuck with game 1.) Kelly stuck with Andrew Heany as his game 1 starter, and the Tigers didn’t announce their game 2 pitcher before the game. Both managers know that you’re more likely to win with your ace on the mound, and if you put him against another ace you may not be maximizing his value.

To prove this, we’ll simulate the doubleheader 10,000 times. We’ll use data from before the game - Skubal’s RA9 was 1.99, Skenes 2.06, Heaney 3.33. Since the Tigers game 2 pitcher wasn’t announced before the series, we’ll use the team run allowed per game of 3.70.* The Tigers have scored an average of 4.84 runs per game, and the Pirates have 3.18. We’re mixing measures somewhat here, but for a quick back-of-the-napkin model it’s fine.

*We don’t want to cheat and pretend Don Kelly knew more before the game than he did. Maybe he knew, but while he was popular in Detroit I doubt they tipped him off.

We’ll refine our expected number a bit by using the league average of 4.31 runs (see methodology here - it’s soccer, but we can make it work here). So, for instance, if the Tigers are facing Skenes and they have 4.84 runs per game, that’s 1.13 over league average. Meanwhile, Skenes has an RA9 of 2.06, which is 0.48 of league average. So we get 1.13 * 0.48 * 4.31 = 2.31. We’ll use 2.31 to when simulating Poisson results for the Tigers.*

*Our model is ignoring all sorts of important stuff - home field advantage, who’s batting, bullpens, etc. Building that out is overkill for my purposes here, but worth noting this is an extremely basic model.

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

set.seed(2025)

get_expected_runs <- function(rs, ra){
  #sqrt(rs*ra)
  lg_runs_per_game <- 4.31
  (rs / lg_runs_per_game) * (ra / lg_runs_per_game) * lg_runs_per_game
}

simulate_games <- function(home, away, n = 10000){
    
  home_rs <- rpois(n, lambda = home) 
  away_rs <- rpois(n, lambda = away)
  
  home_w <- sum(home_rs > away_rs)
  away_w <- sum(away_rs > home_rs)
  
  home_w / (home_w + away_w)
}


game_stats <- tribble(~'scenario', ~'tigers_starter', ~'pirates_starter', ~'tigers_exp_ra9', ~'pirates_exp_ra9',
                     'Actual',  'Skubal', 'Heany', 1.99, 3.33,
                     'Actual', 'TBD', 'Skenes', 3.70, 2.06,
                     'Skubal v Skenes', 'Skubal', 'Skenes', 1.99, 2.06,
                     'Skubal v Skenes', 'TBD', 'Heany', 3.70, 3.33) %>%
  mutate(tigers_exp_rs9 = 4.84,
         pirates_exp_rs9 = 3.18) %>%
  mutate(tigers_xrs = get_expected_runs(pirates_exp_ra9, tigers_exp_rs9),
         pirates_xrs = get_expected_runs(tigers_exp_ra9, pirates_exp_rs9))

sim_games <- game_stats %>%
  rowwise() %>%
  mutate(tigers_exp_wp = simulate_games(tigers_xrs, pirates_xrs),
         pirates_exp_wp = 1 - tigers_exp_wp)

sim_games %>%
  select(scenario, tigers_starter, pirates_starter, tigers_xrs, pirates_xrs, tigers_exp_wp, pirates_exp_wp) %>%
  gt() %>%
  fmt_number(columns = 4:5, decimals = 2) %>%
  fmt_percent(columns = 6:7, decimals = 1) %>%
  gt_theme_espn() %>%
  cols_label(tigers_starter = 'Tigers Starter',
             pirates_starter = 'Pirates Starter',
             tigers_xrs = 'Tigers Expected Runs Scored',
             pirates_xrs = 'Pirates Expected Runs Scored',
             tigers_exp_wp = 'Tigers Expected Win %',
             pirates_exp_wp = 'Pirates Expected Win %')

scenario	Tigers Starter	Pirates Starter	Tigers Expected Runs Scored	Pirates Expected Runs Scored	Tigers Expected Win %	Pirates Expected Win %
Actual	Skubal	Heany	3.74	1.47	87.9%	12.1%
Actual	TBD	Skenes	2.31	2.73	41.6%	58.4%
Skubal v Skenes	Skubal	Skenes	2.31	1.47	69.6%	30.4%
Skubal v Skenes	TBD	Heany	3.74	2.73	66.6%	33.4%

So, if we put Skubal up against Heany, Detroit wins that 87.9% of the time.. If Pittsburgh started Skenes instead, their odds of winning drop to 69.6%, which seems extreme but also not unreasonable. However; this would mean that the Pirates odds of winning game 2 would drop from 58.4% to 33.4%*. The Tigers have a much better offense, so they’re generally favored regardless, though Skenes gives the Pirates their best shot.

*We can check our assumptions against the more sophisticated Fangraphs model - game 1 had Tigers’ starting win expectancy at 73%, against our 87%. Game 2 was 51% against our 39.6%, so our model is not great, but decent enough.

Really, each team wants to know how to maximize their odds of winning both games. This is fairly straightforward - we simply add the win % for each scenario (Skubal v Skenes against what actually played out). So for reality, the Tigers expected win percent for Skubal v. Heaney was 87.9%, while it was only 41.6% for TBD v Skenes. So they’d expect 1.29 wins overall. Totals are:

sim_games %>%
  group_by(scenario) %>%
  summarise(tigers_exp_wp = sum(tigers_exp_wp),
            pirates_exp_wp = sum(pirates_exp_wp)) %>%
  gt() %>%
  gt_theme_espn() %>%
  fmt_number(columns = 2:3, decimals = 2) %>%
  cols_label(tigers_exp_wp = 'Tigers Expected Wins',
             pirates_exp_wp = 'Pirates Expected Wins')

scenario	Tigers Expected Wins	Pirates Expected Wins
Actual	1.29	0.71
Skubal v Skenes	1.36	0.64

Under option B (Skubal v Skenes), the Pirates expect 0.64 wins, vs 0.71 if they pitched Skenes against Skubal. Obviously, teams can only win games in whole numbers, not decimals, but our goal is to see what’s more likely. And our simulation suggests that the Pirates had the optimal strategy, getting 0.07 more wins. Not a lot, but enough to make it worth their while.

End of the day, Tigers won game #1 with Skubal on the mound, and the Pirates took game two with Skenes starting, though the game went into extra innings. I’d have preferred a sweep, but the Tigers are up 9.5 games in the Central and have the best win percentage in baseball, so they’re still just fine.