# Beta-Weighted Distribution

Yamata's optional beta function rewards both stake size AND timing, creating incentives for early market participation when enabled by the market creator.

### How Beta Works

* **Before Market Opens** (t < tOpen): Maximum beta weight (betaOpen, typically 5.0)
* **During Market Period** (tOpen ≤ t ≤ tClose): Beta decreases linearly from betaOpen to 1.0
* **At Market Close** (t = tClose): Beta = 1.0 (no timing bonus)

### Mathematical Formula

<figure><img src="https://3208287650-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FShV7r2hdrp8YWYffPtEU%2Fuploads%2Fkqt2DDQKHJqPaLjOrKsQ%2FScreenshot%202025-11-07%20at%2013.47.30.png?alt=media&#x26;token=c21b10de-13b4-40a7-ae40-5b965d7bd603" alt=""><figcaption></figcaption></figure>

### Numerical Example

Market Parameters:

* betaOpen: 5.0
* tOpen: Jan 1, 2025 00:00
* tClose: Jan 2, 2025 00:00
* tResolve: Jan 2, 2025 01:00

### Commitments:

<table data-header-hidden><thead><tr><th width="66.5234375"></th><th></th><th></th><th></th><th></th><th></th></tr></thead><tbody><tr><td><strong>User</strong></td><td><strong>Time</strong></td><td><strong>Outcome</strong></td><td><strong>Amount</strong></td><td><strong>Beta</strong></td><td><strong>Weighted</strong></td></tr><tr><td>Alice</td><td>Dec 31, 23:00</td><td>Germany</td><td>$20</td><td>5.0</td><td>$100</td></tr><tr><td>Bob</td><td>Jan 1, 00:00</td><td>France</td><td>$300</td><td>5.0</td><td>$1,500</td></tr><tr><td>Carol</td><td>Jan 1, 06:00</td><td>Germany</td><td>$50</td><td>4.0</td><td>$200</td></tr><tr><td>Dave</td><td>Jan 2, 00:00</td><td>Brazil</td><td>$100</td><td>1.0</td><td>$100</td></tr></tbody></table>

**Resolution**: Germany wins

### **Payout Calculation:**

* Total losing stakes: $300 + $100 = $400
* Platform fees: $10
* Profits to distribute: $390

### **Winners' Share:**

* Alice's share: 100/(100+200) × $390 = $130 profit + $20 stake = $150 total
* Carol's share: 200/(100+200) × $390 = $260 profit + $50 stake = $310 total

**Notice**: Alice committed less money but received a higher profit ratio due to earlier timing and higher beta weight.