Payout distribution

​

Pool composition and fee logic

​

Total stakes

The total stakes are divided into winning and losing stakes: $$S_{\text{win}} = \sum_{i \in W} s_i \qquad S_{\text{lose}} = \sum_{i \in L} s_l$$ where $W$ is the set of winners, $L$ is the set of losers, and $s_i$ is the stake of participant $i$.

​

Take rate

Trepa applies a take rate $\tau$ to the total volume. However, it’ll be only taken from losing stakes. The stakes of winning participants are returned without any reduction. $$\text{total stake volume } (S) = S_{\text{win}} + S_{\text{lose}}$$

​

Distributable dividend pool

The pool available for distribution to winners after fees are deducted from losing stakes. $$\text{dividend pool} = \text{total losing stakes} - (\text{take rate} \times \text{total stake volume})$$

​

Payout allocation

​

Winner’s share calculation

Your payout depends on three factors.

1. Your stake
2. Your time weight
3. Your accuracy score

These combine into your accuracy-weighted and time-weighted stake. $$\text{stake} \times \text{time weight} \times \text{accuracy score}$$ Your share of the pool is proportional to your weight relative to the total weight of all winning participants.

​

Time-weight

Trepa rewards participants who submit predictions early. The time weight uses a concave decay function that rewards early participation and strongly discourages deadline sniping. $$\text{time weight} = 1 + (\text{maximum time bonus} - 1) \times \left[1 - \left(\frac{\text{t}}{\text{T}}\right)^\eta\right]$$

• Starts at the maximum bonus value for the earliest predictions
• Gradually decreases as the deadline approaches
• Reaches 1.0 (no bonus) at the deadline

The parameter $\eta$ controls how quickly the bonus decays, higher values mean the bonus drops more slowly for early predictions.

​

Dynamic ROI cap

Trepa enforces a dynamic ROI cap to prevent outsized jackpots from near-perfect forecasts. The cap scales with the difficulty of the prediction task, a larger realistic outcome space means more potential outcome values and that leads to a greater cap. This mechanism ensures higher potential returns for harder-to-predict questions while keeping payouts bounded.

​

Capped proportional allocation

Your gain from the dividend pool is calculated using a capped proportional rule: $$g_i = \min\{\bar{g}_i, \alpha w_i\}$$ where:

• $g_i$ is your gain
• $\bar{g}_i$ is your gains cap (the maximum profit you can earn under the ROI cap)
• $\alpha$ is the water level (a single number chosen so that the whole dividend pool is paid out)
• $w_i$ is your weighted stake (combines your stake, time weight, and accuracy score into one value)

This rule keeps gains proportional to your weighted stake while making sure you never earn more than your allowed cap.

​

Water-filling algorithm

The water level is chosen so that the gains of all winners add up exactly to the dividend pool. Trepa uses a standard water-filling process to find this level.

1. For each winner, compute a cap ratio. $$\theta_i \text{ (cap ratio)} = \frac{\bar{g}_i \text{ (gains cap)}}{w_i \text{ (weighted stake)}}$$
2. Sort all cap ratios from smallest to largest.
3. Imagine slowly raising the water level $\alpha$. As soon as the water level $\alpha$ reaches a winner’s cap ratio $\theta_i$, that winner becomes capped and stops scaling further.
4. Continue raising the water level $\alpha$ for the remaining uncapped winners until the entire dividend pool is allocated, with no leftover.

This process guarantees,

• Gains scale with weighted stakes as long as caps are not hit.
• Caps prevent any single participant from taking too large a share.
• The dividend pool is fully exhausted without any residual amount.

​

Your payout

Your final payout has two parts. $$\text{your stake} + \text{your gain}$$ You first receive your stake back in full, and your gain from the dividend pool is added on top.

​

Mechanism properties

​

Budget-balanced

All winnings are funded by losing stakes. The mechanism never requires external subsidies, and no new value is created or destroyed by the payout logic.

​

Monotone

Higher accuracy never leads to lower rewards. As your accuracy improves (holding everything else equal), your weighted stake and potential gain weakly increase, and the mapping from error to payout remains smooth and convex.