Mathematical Model

1.BET ticket generation and emptying mechanism

BET Tickets requires the following features:

1.Pass-Ticket: After staking in the prediction market, the user can participate in the prediction bets for all events without having to purchase additional participation rights or trade the prediction results. Tickets are therefore non-consumable in the event dimension, with tickets for separate events being independent of each other. The number of tickets does not decrease for each update of a prediction within an event.

2.Non-tradable: Tickets do not support transfers and are not tradable.

3.Fair Game: Includes generation and emptying mechanism.

a.Generation mechanism.

The amount of ticket output needs to be related to both the amount of money staked by the user and the length of time staked, similar to the Defi liquidity mining mechanism, it could otherwise lead to cheating. For example, if only the amount of money is considered and not the time of stake, then there would be users who only stake a large amount of money before the predicted result is revealed, in order to get a large number of tickets and then get the money back immediately after the result is revealed. These cheated funds do not actually contribute to the prize revenue of the prediction market.

b.Emptying mechanism

If tickets were produced and never consumed, the number of tickets would only increase, resulting in unfairness to subsequent users. Therefore tickets need to be emptied periodically, with the emptying period matching the end of the prediction event.

In summary

Tickets represent the weight of a user's vote in a prediction event, which is generated block-by-block from the creation of that prediction event based on the amount of money staked by the user in the platform. The user's tickets for each prediction event are independent and do not affect each other.

1.Tickets represent a user's voting weight in a prediction event, which is based on the amount of money staked by the user in the platform and is generated block by block from the time that prediction event is created. User's tickets for each prediction event are independent and do not affect each other.

The number of "actual tickets" of a user in a prediction event $n_{}$ = $\sum_{i=1}^{n}{LP_{i}Value*StakingBlockAmount}$ ，where $LP_{i}Value$ is the dollar value of the user's stake, and $StakingBlockAmount$ is the number of blocks of the user's stake. The user can deposit or withdraw different amounts of funds in separate batches, and the LP token will increase or decrease accordingly, while the length of staking time will vary for each fund. To simplify calculations, the v1 version of the tickets output rule is 1 ticket per $1 staked per 1h (1h is the approximate time, measured by the number of blocks).

2.Tickets do not support transfers and are not tradable.

3.At the end of the prediction event, users with correct predictions will share the prize in proportion to the number of tickets they hold.

4.At the end of the prediction event, the tickets of all users in the event are cleared.

2.Early Bird Incentive mechanism

The earlier the user stakes, the higher the uncertainty of winning. In order to motivate users to make their predictions early, instead of waiting until the end of the event when the prediction is likely to be obvious, an "early bird incentive multiplier" is set.

1.The "early bird incentive multiplier" is proportional to the time until the end of the predicted event when the user stakes, and is a simple quadratic linear function. The earlier the user stakes, the larger the multiplier.

If the initial value of the "early bird incentive multiplier" for an event is $E_{0}$ , the value of the "early bird incentive multiplier" at the moment of staking is $E_{}$ , the event creation time is $t_{0}$ , the prediction stop time is $t_{1}$ , and the moment of staking is $t_{}$ , then the "early bird incentive multiplier" for an event at the moment of staking is $E_{}$ = $E_{0}-\frac{E_{0}-1}{t_{1}-t_{0}}\times(t-t_{0})$

2.When distributing the winning prize, the user's number of "Actual Tickets" will be multiplied by this incentive multiplier to determine the final number of "Prize Tickets".

3.The "early bird incentive multiplier" can be set to 2 decimal places.

4.The initial value of the "early bird incentive multiplier" can be set varying for different prediction events, initially in the range of 1-10.

3.Prize distribution mechanism

Prediction prizes distribution principle:

• The prizes will be distributed once a week. Tentatively, the prize will be awarded every Friday at 0:00 GMT (i.e. GMT+0).

• 90% of the prize pool for each period is allocated to the user who has correctly predicted the prediction event, in proportion to the number of "Prize Tickets" the user has got in that event.

• 3% of each prize pool is used to buy back the Better governance tokens, and the use of the bought back governance tokens is governed by the community DAO.

• 7% of the prize pool in each period will go into the total prizes pool in the next period and will not be allocated in the current period.

The user prediction prize distribution formula, similar to the Defi mining distribution mechanism:

1. Since 7% of the platfor's prize pool for each period goes into the total prize pool for the next period, the total platform prize for the first $i_{}$ period is $P_{i}$ = $P_{i-1}\times0.07+P_{i}^{'}, whereP_{i-1}$ are the total platform prize for the period $i_{}-1$ and $P_{i}^{'}$ is the total platform financial lending revenue for the periodi_{}
2. A prediction event is allowed to span multiple draw cycles (e.g., last 3 months), and the prize allocation percentage for that event for the period $i_{}$ is $R_{i}$ = $W_{i}\div\sum_{i}^{n}{W_{i}}\times100\%$ , where $W_{i}$ represents the prize allocation weight for that event, note that it is in the form of weight, not a percentage.
3. The total prizes of a prediction event: $P_{}$ = $\sum_{i=1}^{n}{P_{i}\times0.9\times R_{i}}$ , where $P_{i}$ represents the total prize of the platform for the period$i_{}$ , $R_{i}$represents the percentage of the prize distribution of the event for the first $i_{}$ period.
4. The number of "Actual Tickets" of a user in a prediction event $n_{}$ = $\sum_{i=1}^{n}{LP_{i}Value*StakingBlockAmount}$ , where $LP_{i}$ Value is the dollar value of the user's stake and $StakingBlockAmount$ is the number of blocks staked by the user's assets.
5. The number of "Prize Tickets" for a user in a prediction event is $N_{}=n_{}\times E$ , where $n_{}$ is the number of "Actual Tickets" of the user in a prediction event, and $E_{}$ is the "early bird incentive multiplier" of at the moment of the user's participation in the staking.
6. The mathematical formula for the prizes that user A receives for a correct prediction in a prediction event is $P_{a}=P_{}\times\frac{N}{\sum_{i}^{n}{N_{i}}}$ ， where $P_{}$ are the total prizes for the prediction event,N $N_{}$ _{} is the number of "Prize Tickets" of the user in the prediction event and ${\sum_{i}^{n}{N_{i}}}$ is the sum of the number of "Prize Tickets" of all users who got the prediction correct in the prediction event.

4.BET liquidity mining distribution mechanism

User A receives a BET liquidity mining reward after predicting the market staked assets, the amount available in each block is $B_{a}=B_{}\times\frac{V_{a}}{V_{TVL}}$ , where $B_{}$ is the total number of BET rewards for the platform in each block, $V_{a}$ is the value of the user's staked assets and $V_{TVL}$ is the total staked asset value of the platform.

5.APY Calculation formula

The user benefits come from 2 components, including:

1) Prediction prize when the prediction is correct.

2) BET prize earned from liquidity mining.

5.1.Prediction Prize APY

Since the prediction behavior of users is difficult to assess accurately, the APY data of their prediction prize can only be roughly estimated. To estimate the APY of the prediction prize, the following conditions are assumed to hold.

1. User A will participate in all prediction events on the platform and has an 80% win rate.

2. The prizes for all prediction events on the platform are equally distributed.

3. Each prediction event is predicted correctly by 50% of tickets and incorrectly by 50% of tickets.

4. All users participate in all prediction events at the same time (avoiding the effects of ticket start output time and "early bird incentive multiplier").

Then the total predicted prize available to user A in period $i_{}$ = where $V_{a}$ is the user's staked asset value, $V_{TVL}$ is the total staked asset value of the platform and $P_{i}$ is the total platform prize for the first $i_{}$ period. In this way, the user can receive a return equivalent to 1.6 times the return of the underlying lending and aggregator protocol. Therefore, it is straightforward to calculate the estimated prediction prize APY $APY_{1}$ = overall APY of the platform's underlying lending protocol * 1.6.

Since the underlying lending and finance protocols include multiple assets and have different APYs, the platform's overall APY for the underlying lending and finance is the weighted APY of each asset.

5.2.BET Reward APY

Users are rewarded with BET liquidity mining after staking in the prediction market, with an annualized rate of return $APY_{2}$ =$\frac{B_{}\times Price_{BET}}{V_{TVL}}\times N_{block}\times100\%$, where $B_{}$ is the total number of BET rewards for the platform in each block, $Price_{BET}$ is the price of BET, $V_{TVL}$ is the total staking value of the platform, and $N_{block}$ is the total number of blocks in a year.

In summary,

The platform's overall $APY_{}$ = $APY_{1}$ + $APY_{2}$