How likely is a win for a baseball team that is several runs ahead of its opponent after a few innings? One might build a mathematical model, using methods of probability and statistics. Imagine that run-scoring happens at a random rate, for instance, like radioactive atomic decay. The fewer innings remaining, the less likely it is for a team to overcome a deficit by lucky fluctuation. Other parameters could take into account the relative strength of two clubs, their individual rate of scoring, home-field advantage, etc.
Alternatively, there's the data-driven approach. Greg Stoll's "Win Expectancy Finder" takes inning-by-inning information from ~175,000 major league games played between 1903 and 2020, and tabulates the win percentage for each team. Define the odds ratio R = W/L if the odds of the home team winning are W:L. (A probability of P represents odds of P:(1-P), so the conversion formula is odds ratio R = P/(1-P) and contrariwise P = R/(1+R). Thus, for example, a probability of 50% = 0.5 = odds of 1:1 = odds ratio 1; alternatively, a probability of 80% = 0.8 = odds of 4:1 = odds ratio 4.) From Stoll's tables, here are the approximate odds ratios for each inning if the visitors "V" or home team "H" are ahead:
| Inning # | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|
| V+4 | 0.20 | 0.18 | 0.15 | 0.14 | 0.11 | 0.09 | 0.05 | 0.03 | 0.01 |
| V+3 | 0.33 | 0.30 | 0.27 | 0.23 | 0.19 | 0.16 | 0.11 | 0.06 | 0.03 |
| V+2 | 0.45 | 0.45 | 0.41 | 0.3 | 0.33 | 0.30 | 0.22 | 0.15 | 0.08 |
| V+1 | 0.79 | 0.72 | 0.67 | 0.67 | 0.59 | 0.54 | 0.45 | 0.32 | 0.18 |
| Tied | 1.17 | 1.13 | 1.13 | 1.13 | 1.13 | 1.13 | 1.08 | 1.08 | 1.08 |
| H+1 | 2.33 | 1.86 | 1.94 | 1.94 | 2.13 | 2.23 | 2.70 | 3.55 | |
| H+2 | 3.35 | 3.55 | 4.00 | 4.56 | 5.25 | 6.69 | 10.11 | 24.00 | |
| H+3 | 4.56 | 5.67 | 6.69 | 8.09 | 10.11 | 13.29 | 24.00 | 49.00 | |
| H+4 | 11.50 | 11.50 | 11.50 | 13.29 | 19.00 | 24.00 | 49.00 | 99.00 |
This suggests that the logarithm of the odds ratio (= ln(P/(1-P)) for winning changes with runs and innings, gaining ~0.6 per run scored and ~0.1 per inning for whoever is ahead. That crude fit suggests a rough-and-ready rule-of-thumb:
| Begin with odds ratio 1 Double it for every run scored Add 10% to the leader every inning |
Alternatively, to update the odds ratio for every run scored:
| In the first few innings, multiply by 1.5 In the middle innings, multiply 2 In the final innings, multiply by 3 |
Or perhaps when late in the game invert this approach and work backward from final outcome certainty? Hmmmm – much to ponder!
(cf Square Root of Baseball (2005-05-13), Baseball Odds (2007-04-21), Baseball Expected Runs (2015-04-16), ...) - ^z - 2021-11-29