Injury System

At the end of each block, every player who appeared in games during that block has a chance of suffering an injury. The probability is calculated as:

```
P(injury) = Base Rate x Usage Factor x Block Factor
```

### A.1.1 Base Rate

The base injury rate reflects a player's inherent injury risk based on their Strat-O-Matic injury rating:

```
Base Rate = 0.010 + (Effective Rating x 0.0125)
```

Where:
- **Effective Rating** = Injury Rating for position players
- **Effective Rating** = Injury Rating + 1 for pitchers

Pitchers receive +1 to their effective rating because they are inherently more prone to injury than position players. Approximately half of all pitchers have a 0 injury rating on their Strat card.

| Injury Rating | Position Player Base Rate | Pitcher Base Rate |
|---------------|---------------------------|-------------------|
| 0 | 1.00% | 2.25% |
| 1 | 2.25% | 3.50% |
| 2 | 3.50% | 4.75% |
| 3 | 4.75% | 6.00% |
| 4 | 6.00% | 7.25% |
| 5 | 7.25% | 8.50% |
| 6 | 8.50% | 9.75% |

### A.1.2 Usage Factor

The usage factor adjusts injury probability based on how heavily a player has been used relative to their expected workload:

```
Usage Factor = e^(alpha(U - 1.1) + gamma(U - 1.1)^3)
```

Where:
- **U** = Actual Usage / Expected Usage
- **alpha** = 0.2 (linear coefficient)
- **gamma** = 4 (cubic coefficient)
- **e** = 2.71828 (Euler's number)

**Expected Usage** is pro-rated based on games played:
```
Expected Usage = (MLB Season Usage x Games Played) / 162
```

The formula is designed so that:
- At normal usage (U <= 1.1), the factor stays near 1.0
- Above 110% usage, risk increases exponentially
- At 150% usage, risk roughly doubles
- At 200% usage, risk increases approximately 5x

### A.1.3 Block Factor

The block factor accounts for survivor bias - players still healthy late in the season have demonstrated durability:

```
Block Factor = 1 - (beta x B / 14)
```

Where:
- **beta** = 0.2
- **B** = Current block number (1-14)

| Block | Block Factor |
|-------|--------------|
| 1 | 0.986 |
| 7 | 0.900 |
| 14 | 0.800 |

When a player is injured, the duration (in blocks) is determined using a durability-adjusted exponential distribution.

### A.2.1 Durability Adjustment

Players with higher MLB usage have demonstrated durability and tend toward shorter injuries:

```
Adjusted lambda = lambda_base x (Durability Ratio)^theta
```

Where:
- **lambda_base** = 0.8
- **theta** = 0.4
- **Durability Ratio** = Actual MLB Usage / Baseline Usage

**Baseline Usage Thresholds:**
| Player Type | Baseline |
|-------------|----------|
| Position Players | 300 PA |
| Starting Pitchers | 75 IP |
| Relief Pitchers | 25 IP |

### A.2.2 Duration Probabilities

The probability of each duration follows an exponential decay:

```
P(duration = d) is proportional to e^(-Adjusted lambda x d)
```

Where **d** = duration in blocks (1 to 14).

Probabilities are normalized so they sum to 100%.

### A.2.3 Late-Season Compression

To avoid injuries that would have players returning in the final 1-2 blocks of the season (when rosters expand and the injury would have minimal impact), probabilities are compressed:

```
Compression = (Max Return Duration / d)^zeta
```

Where:
- **zeta** = 3
- **Max Return Duration** = Remaining Blocks - 2

Compressed probability mass is redistributed to the max return duration, creating a spike of players returning just before the season ends.

### A.2.4 Season-Ending Injuries

If the rolled duration exceeds the number of blocks remaining in the season, the injury is classified as **season-ending**. The player is placed on the 60-day IL and cannot return until the following season.

This approach ties season-ending probability naturally to the duration formula - longer base durations and injuries occurring late in the season are more likely to result in season-ending outcomes.

| Parameter | Symbol | Value | Purpose |
|-----------|--------|-------|---------|
| Base rate constant | - | 0.010 | Minimum injury probability |
| Rating coefficient | - | 0.0125 | Per-rating-point increase |
| Usage linear coefficient | alpha | 0.2 | Linear usage penalty |
| Usage cubic coefficient | gamma | 4 | Exponential overuse penalty |
| Block reduction | beta | 0.2 | Late-season survivor adjustment |
| Duration decay rate | lambda | 0.8 | Base duration distribution |
| Durability exponent | theta | 0.4 | Usage-based duration adjustment |
| Compression exponent | zeta | 3 | Late-season return compression |
| Position player baseline | - | 300 PA | Durability reference |
| Starting pitcher baseline | - | 75 IP | Durability reference |
| Relief pitcher baseline | - | 25 IP | Durability reference |