Goal: Determine a robust way to issue payouts for a trading tournament.

If a trader generates P total profits (positive) from participating in a trading tournament, we’re looking for an adjustment factor α to calculate an adjusted profit that we can use to pay out the trader’s proportional share of a fixed dollar prize pool.

$$ P_{adj}=\alpha P $$

Problem: Tournament payouts are always positive but trading profits on markets are not, so users are incentivized to make higher variance and more concentrated bets than they would otherwise.

Solution: Combine a number of heuristic factors to adjust for market coverage, temporal coverage, and volatility.

The first heuristic factor is market coverage. We want to encourage traders to bet across the entire range of markets in the tournament, and penalize traders for having their profits come from only a few markets. If there are n markets in the tournament and p_i is the trader’s profit on the ith market, the factor will be 1 if the trader made an equal amount of profit on every market and 1/n if they made all their profit on one market:

$$ m = \frac {\sum_i { \begin{cases} \min({\frac{p_i}{P/n}}, 1) &\text{if } p_i \gt 0\\ 0 &\text{otherwise} \end{cases}}} {n} $$

Similarly, the second heuristic factor rewards users for making a steady profit across the duration of the tournament and penalizes last-minute betting. The factor t is the percentage of days the trader makes an profit at least their daily average. It’s 1 when the trader makes an equal amount of profit each day, and 1/d if they make all their profit on one day. Let d be the number of days the tournament lasts, pi_i be the daily pnl on the ith day:

$$ t = \frac {\sum_i { \begin{cases} \min({\frac{\pi_i}{P/d}}, 1) &\text{if } \pi_i \gt 0\\ 0 &\text{otherwise} \end{cases}}} {d} $$

The last factor is just the volatility of daily pnl. We want to penalize traders for making highly risky or very correlated bets.

$$ \sigma = \text{std dev of daily profits} $$

Put them all together and you get the overall adjustment factor. We first calculate the geometric mean of the market and temporal coverage factors to give us something like the fraction of “valid” profits. We then divide by the volatility (adjusting for the d days of the tournament).

$$ \alpha =\frac{\sqrt{mt}}{\sigma\sqrt{d}} $$

The final adjusted profit P_adj thus looks like something like the Sharpe ratio adjusted for tournament coverage.

How this payout method works in practice remains to be seen…