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

# Binary markets

# Binary Market Fees

For standard two-outcome markets (Team A vs Team B), SportToken charges a fee that is mathematically equivalent to the optimal Kelly criterion bet sizing.

## The Key Result

**The fee percentage F we charge equals the optimal Kelly fraction f\* of vault capital to risk.**

```
f* = F
```

This means our fee structure is mathematically optimal for bankroll management.

***

## Full Mathematical Proof

### Goal

We want to show that the fee percentage F we charge on a bet is equal to the optimal percentage f\* of our bankroll that we should wager according to the Kelly criterion.

### Step 1: The Kelly Criterion

The Kelly criterion is given by:

$f^* = \frac{p - q}{b}$

Where:

* **p** = probability of winning
* **q** = 1 - p (probability of losing)
* **b** = profit multiplier for the odds offered

*Example:* If offered odds of 0.4, then b = 0.6/0.4 = 1.5

In general, if offered odds x:

$b = \frac{1 - x}{x}$

### Step 2: Setup

Suppose we charge a fee of F, where we claim F = f\*.

Let the user's gross bet amount be G. Then:

$\text{Actual bet amount (after fee)} = (1 - F)G$

* If we (the vault) **win**: we gain G
* If we **lose**: we must pay out $(1 - F)G(1 + b)$

### Step 3: Implied Odds Calculation

We calculate implied odds based on risk vs potential win:

$\text{Implied odds} = \frac{\text{risked amount}}{\text{total win}}$

The amount we risk:
$(1 - F)G(1 + b) - G$

The total win:
$(1 - F)G(1 + b)$

Thus:

$\frac{(1 - F)G(1 + b) - G}{(1 - F)G(1 + b)} = \frac{(1 - F)(1 + b) - 1}{(1 - F)(1 + b)} = \frac{b - F(1 + b)}{(1 - F)(1 + b)}$

### Step 4: Implied Probabilities

Let p\* be our implied probability of winning and q\* be implied probability of losing.

From the above:

$p^* = \frac{b - F(1 + b)}{(1 - F)(1 + b)}$

$q^* = \frac{1}{(1 - F)(1 + b)}$

The implied odds multiplier becomes:

$b^* = \frac{q^*}{p^*} = \frac{1}{b - F(1 + b)}$

### Step 5: Applying Kelly Criterion

Returning to Kelly:

$f^* = \frac{p - q}{b^*}$

Since b = p/q, we substitute:

$f^* = \frac{p - q}{\frac{1}{b - F(1 + b)}}$

$f^* = (p - q)[b - F(1 + b)]$

$f^* = (p - q)\left[\frac{p}{q} - F\left(1 + \frac{p}{q}\right)\right]$

$f^* = p - \left(p - Fp - Fq\right) = F$

### Conclusion

$\boxed{f^* = F}$

**The fee percentage F we charge is exactly the optimal Kelly fraction f\* of our bankroll to risk per game.**

***

## What This Means in Practice

1. **Optimal Risk Management** - The vault never over-exposes itself on any single bet
2. **Fair Pricing** - Users pay fees proportional to the actual risk their bet creates
3. **Long-term Profitability** - Kelly sizing maximizes long-term growth while avoiding ruin
4. **Dynamic Adjustment** - As vault exposure changes, fees automatically adjust to maintain optimal sizing

***

## Rebates: When You Help the Vault

The same Kelly logic works in reverse. **When your bet reduces vault risk, you earn a rebate instead of paying a fee.**

### How Rebates Work

If the vault is exposed on Side A and you bet on Side B:

* Your bet offsets existing risk
* The vault's expected loss decreases
* You receive a rebate proportional to the risk reduction

### Rebate Calculation

```
Rebate Rate = (Exposure Before - Exposure After) / Vault Assets
```

The rebate uses the same linear averaging as fees:

* Start rate: Current imbalance / Vault
* End rate: New imbalance / Vault
* Your rebate = Bet Amount × (Start + End) / 2

### Example: Earning a Rebate

**Setup:**

* Vault: \$100,000
* Current exposure: \$2,000 on Team A (2% imbalance)
* You bet \$1,000 on Team B at +150 odds

**Calculation:**

* Your to-win: \$1,500
* This offsets \$1,500 of Team A exposure
* New imbalance: \$500 (0.5%)
* Average rebate rate: (2% + 0.5%) / 2 = 1.25%
* **Your rebate: $1,000 × 1.25% = $12.50**

Your net fee = System fee (0.3%) - Rebate = $3.00 - $12.50 = **-\$9.50** (you earn money!)

### Rebate Scenarios

| Market State | Your Bet    | Result                               |
| ------------ | ----------- | ------------------------------------ |
| Heavy on A   | Bet on A    | Pay fee (adding risk)                |
| Heavy on A   | Bet on B    | **Earn rebate** (reducing risk)      |
| Balanced     | Either side | Small fee (minimal imbalance change) |
| Heavy on B   | Bet on A    | **Earn rebate**                      |
| Heavy on B   | Bet on B    | Pay fee                              |

### Why Rebates Matter

1. **Better odds than sportsbooks** - When you take the underbet side, your effective odds improve
2. **Market efficiency** - Rebates incentivize balanced betting, reducing vault risk
3. **Transparent value** - You see exactly how much you're earning for helping the vault