> For the complete documentation index, see [llms.txt](https://information-market.gitbook.io/information-market/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://information-market.gitbook.io/information-market/payouts/payout-distribution-on-information-markets.md).

# Payout Distribution on Information Markets

**The goal is to determine the payout function, defined as :** 

$$
f(x) = \text{factor} \times x
$$

{% hint style="info" %}
**x  represents the absolute delta from the correct average probability. The constant factor needs to be calculated to ensure that the total payout equals the total deposit amount.** 
{% endhint %}

## Step 1: Calculating Triangle Areas

For bets within a 3% delta range, the absolute deltas are divided into three categories:

### Delta 2% Area:

Calculated over the interval \[2, 3]:

$$
\int\_{2}^{3} f(x) , dx = \text{factor} \times \frac{3^2}{2} - \text{factor} \times \frac{2^2}{2} = \text{factor} \times \left( \frac{9}{2} - \frac{4}{2} \right) = \text{factor} \times 2.5
$$

### Delta 1% Area:

Calculated over the interval \[1, 2]:

$$
\int\_{1}^{2} f(x) , dx = \text{factor} \times \frac{2^2}{2} - \text{factor} \times \frac{1^2}{2} = \text{factor} \times \left( \frac{4}{2} - \frac{1}{2} \right) = \text{factor} \times 1.5
$$

### Delta 0% Area:

Calculated over the interval \[0, 1]:

$$
\int\_{0}^{1} f(x) , dx = \text{factor} \times \frac{1^2}{2} - \text{factor} \times \frac{0^2}{2} = \text{factor} \times \frac{1}{2} = \text{factor} \times 0.5
$$

## Step 2: Ensuring Total Payout Equals Total Deposit: 

The total payout is the sum of all triangle areas. This sum must equal the total deposit (e.g., 1000 units):

$$
\text{factor} \times (2.5 + 1.5 + 0.5) = 1000
$$

### With all areas: 

$$
\text{factor} \times 4.5 = 1000 \implies \text{factor} = \frac{1000}{4.5} = 222.222
$$

### Without delta 1% bets: 

$$
\text{factor} \times (2.5 + 0.5) = 1000 \implies \text{factor} = \frac{1000}{3} = 333.333
$$

## Step 3: Submitting Information On-Chain

The admin calculates the factor and average off-chain and submits them to the smart contract. The smart contract parameters include:

* Factor: The constant multiplier for payouts.
* Average: The calculated average probability.
* Bets Map: A hashmap with the number of bets in each delta category (e.g., delta0, delta1, delta2).

Example function call:

```
submit(factor, average, delta0NumberOfBets, delta1NumberOfBets, delta2NumberOfBets, ...)
```

{% hint style="info" %}
The Bets Map is required for calculating payouts since individual reward claims cannot determine the number of bets in each category.
{% endhint %}

## Step 4: Calculating Payouts

Using : 

$$
f(x) = x \* e^{2 pi i \xi x}
$$

distribute the pool for each delta category:

### Delta 0 Area:

$$
\text{Payout} = \text{factor} \times 0.5 = 222.222 \times 0.5 = 111.111
$$

### Delta 1 Area:

$$
\text{Payout} = \text{factor} \times 1.5 = 222.222 \times 1.5 = 333.333
$$

### Delta 2 Area:

$$
\text{Payout} = \text{factor} \times 2.5 = 222.222 \times 2.5 = 555.555
$$

Total:

$$
111.111 + 333.333 + 555.555 = 1000
$$

## Adjustments When Bets Are Missing

If there are no bets in the delta 1 category:

### Delta 0 Area:

$$
333.333 \times 2.5 = 833.333
$$

### Delta 2 Area:

$$
333.333 \times 0.5 = 166.667
$$

\
Total:

$$
833.333 + 166.667 = 1000
$$

## Reward Distribution for Claimants

When a user claims their reward, the smart contract calculates the reward based on:<br>

* Category of the bet (delta0, delta1, delta2).
* Number of bets in that category.

Example:

If there are 10 bets in the delta 0 category (payout pool: 555.555):

$$
\text{Individual Payout} = \frac{555.555}{10} = 55.555
$$