# Formalizing Print Success Rate and Reward

Here's a breakdown of the mathematical formula for calculating the print success rate and reward:

Variables:

p: Print completion rate (a decimal between 0 and 1, representing the percentage of successful prints)

f: First-attempt success indicator (1 for success, 0 for failure)

R_max: Maximum reward as a percentage of order value (e.g., 0.005 for 0.5%)

R_min: Minimum reward as a percentage of order value (e.g., 0.002 for 0.2%)

p_min: Minimum completion rate threshold for minimum reward (e.g., 0.8)

order_value: Value of the order in tokens

### Formula

I(p >= p_min) is the indicator function, returning 1 if p >= p_min (eligible for minimum reward) and 0 otherwise.

Reward:

#### Explanation

The formula captures the key aspects:

If successful in first attempt (f=1), get the maximum reward (f * R_max).

If not successful in first attempt (f=0):

If completion rate >= threshold (p >= p_min), get the minimum reward ((1 - f) * p * R_min).

If completion rate < threshold (p < p_min), no reward (I(p >= p_min) = 0).

The reward is then calculated by multiplying the success rate by the order value.

### Example

p = 0.95 (95% completion rate)

f = 1 (successful in first attempt)

R_max = 0.005 (maximum reward of 0.5%)

R_min = 0.002 (minimum reward of 0.2%)

p_min = 0.8 (minimum completion rate for minimum reward)

order_value = 100 (order value in tokens)

#### Calculations

success_rate = 1 * 0.005 + (1 - 1) * 0.95 * 0.002 * 1 = 0.005

reward = 0.005 * 100 = 0.5 tokens

#### Output

Success rate: 0.005 (or 0.5%)

Reward: 0.5 tokens

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