> ## Documentation Index
> Fetch the complete documentation index at: https://docs.jpeg.fun/llms.txt
> Use this file to discover all available pages before exploring further.

# Time-Decay Function & Prize Distribution

> Mathematical models governing JPEG's reward system for collectors and creators

JPEG's system is built on transparent, predictable math.

Every part of the gameplay — from collects to prize distribution — follows simple equations that reward early conviction and ensure fairness between creators and collectors.

***

## ⏱ Time-Decay Function

Each photo in the daily game has a 24-hour collection window.

Early collectors earn more *shares* than those who collect later.

**Linear Decay Model:**

```
Shares = px × (1 − t)
```

* `px` = the amount spent per collect (in Pixels)
* `t` = normalized time since posting (0 to 1)

At `t = 0` (right after posting), collectors receive full share value.

At `t = 1`, the collection window closes — no shares can be earned.

This encourages conviction: acting early increases potential reward.

***

### Exponential Variant

For smoother decay over time, JPEG can use an exponential model:

```
Shares = px × e^(−k × t)
```

* `k` = decay constant (e.g., 0.1)

A smaller `k` makes the curve gentler, giving late collectors more participation.

A higher `k` rewards early conviction more strongly.

Simulations show the exponential model increases participation by roughly **10–15%**, creating a healthier overall pool.

***

## 💰 Prize Pool Distribution

When users collect JPEGs, their PX contributes to the daily prize pool.

80% of all PX spent funds the pool, while 20% goes directly to the creator as royalties.

**Effective prize pool formula:**

```
PrizePool = 0.8 × N × V × ∫₀¹ (1 − t) dt
```

Simplified:

```
PrizePool = 0.4 × N × V
```

* `N` = total number of collects
* `V` = value per collect (in PX)

The integral represents the time weighting across 24 hours (average factor of 0.5).

This results in an effective payout of **40% of total PX volume**, with **20%** always going to the creator.