# Baseball Odds Ratio Modeling

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 #123456789
V+40.200.180.150.140.110.090.050.030.01
V+30.330.300.270.230.190.160.110.060.03
V+20.450.450.410.30.330.300.220.150.08
V+10.790.720.670.670.590.540.450.320.18
Tied 1.171.131.131.131.131.131.081.081.08
H+12.331.861.941.942.132.232.703.55
H+23.353.554.004.565.256.6910.1124.00
H+34.565.676.698.0910.1113.2924.0049.00
H+411.5011.5011.5013.2919.0024.0049.0099.00

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 1Double it for every run scoredAdd 10% to the leader every inning

Alternatively, to update the odds ratio for every run scored:

 In the first few innings, multiply by 1.5In the middle innings, multiply 2In 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