1. The 24 Base-Out States & Run Expectancy (RE24)
At any discrete point in an inning, the game resides in exactly one of 24 possible base-out states
(8 base configurations multiplied by 3 out states: 0, 1, or 2). Tom Tango, Mitchel Lichtman, and Andrew Dolphin
formalized the Run Expectancy matrix by analyzing millions of major league plate appearances.
For instance, starting an inning with nobody out and nobody on yields an expected run total of approximately 0.48 runs in modern run environments.
However, loading the bases with nobody out increases expected inning runs to 2.29 runs. With two outs, that exact same bases-loaded configuration
drops to 0.75 runs, demonstrating how each subsequent out steepens the offensive hazard curve.
Run Expectancy Delta:
ΔRE = RE(State_end) - RE(State_start) + Runs_scored_on_play
2. Calculating Win Expectancy (WE)
While Run Expectancy evaluates the expected number of runs to cross the plate before three outs are made,
Win Expectancy (WE) converts that run potential into the ultimate objective: game victory.
WE accounts for:
- Current score differential: Whether the batting team leads, is tied, or trails.
- Inning and half: The 9th inning features asymmetric boundaries where the home team can immediately terminate the game with a walk-off run, while the away team must finish the frame and defend.
- Run distribution modeling: Postseason run environments compress variance. Bullpen specialization, higher average fastball velocity, and shorter pitcher leashes reduce the expected value of late-inning rallies.
3. Win Probability Added (WPA) & Leverage Index (LI)
Win Probability Added (WPA) measures the change in Win Expectancy caused by an individual play:
WPA = WE(After Play) - WE(Before Play)
Because not all situations are created equal, Tom Tango introduced the Leverage Index (LI).
LI scales the critical nature of an encounter relative to the average plate appearance across all baseball games (defined as LI = 1.0):
- Low Leverage (LI < 0.85): Blowouts or early innings with low scoring threat.
- Medium Leverage (0.85 ≤ LI < 1.5): Standard competitive innings.
- High Leverage (1.5 ≤ LI < 3.0): Tense late-game situations with tying or winning runs on base.
- Extreme / Postseason Clutch (LI ≥ 3.0): Elimination-game scenarios, 9th inning bases loaded, where a single pitch shifts win probability by 40% or more.
Empirical 24-State Run Expectancy Reference Table
| Base State |
0 Outs (RE) |
1 Out (RE) |
2 Outs (RE) |
Run Prob (0 Out) |
Run Prob (2 Out) |
| Bases Empty |
0.48 |
0.25 |
0.10 |
27.1% |
7.1% |
| Runner on 1st |
0.86 |
0.51 |
0.21 |
41.8% |
12.8% |
| Runner on 2nd |
1.07 |
0.66 |
0.32 |
61.2% |
22.1% |
| Runner on 3rd |
1.35 |
0.95 |
0.35 |
83.5% |
25.7% |
| 1st and 2nd |
1.44 |
0.88 |
0.43 |
63.4% |
23.0% |
| 1st and 3rd |
1.78 |
1.13 |
0.48 |
86.1% |
27.5% |
| 2nd and 3rd |
1.96 |
1.38 |
0.58 |
85.2% |
26.2% |
| Bases Loaded |
2.29 |
1.54 |
0.75 |
86.1% |
32.4% |