A Payout Scheme that Directly Encourages Unique Models

The current payout system rewards correlation with live targets, and loosely rewards originality with mmc. But mmc doesn’t do enough to encourage a wide spread of unique models. This is evident if you look at the largest stakers in the tournament, many of them have a >.8 correlation with the metamodel. Whats worse is that anyone can place a gigantic stake on the example predictions, and be rewarded for 0 contribution to the tournament.

I’ve come up with a mechanism that can mitigate this problem I call the “conformity tax”. The tax works by scaling a model’s payout based on its metamodel correlation and some threshold. It has the effect of increasing the risk of putting a huge stake on a model that is highly correlated with the metamodel.

The code below demonstrates how the conformity tax works

```
import numpy as np
import matplotlib.pyplot as plt
np.random.seed(1)
```

We need to generate simulated payouts that reflect actual tournament data. It is safe to assume real payouts are Pareto distributed, but I am only guessing what the real distribution looks like. They are sorted and ranked in descending order.

```
num_payouts = 1000
largest_payout = 600
a = 2 # distribution shape (not from actual Numerai data!, this is
an assumption)
P = (np.random.pareto(a, num_payouts)) # Normal Payouts
P = np.sort(P) * (largest_payout / np.max(P)) # Rescale distribution and sort
ranks = np.flip(np.arange(num_payouts)) + 1
plt.bar(ranks, P, width=1)
plt.xlabel("Ranks")
plt.ylabel("Positive Payouts")
plt.savefig("payouts.png")
```

```
total_payed_out = np.sum(P)
print(total_payed_out)
```

30192.954174991097

Next we create a distribution of metamodel correlations. Again,

this is only a guess because I haven’t aggregated real data from

the leaderboard.

```
# more assumptions
mean_corr_mm = .6
std_corr_mm = 0.15
# distribution of metamodel correlations
M = np.clip(np.random.normal(mean_corr_mm, std_corr_mm, num_payouts), 0, 1)
count, bins, ignored = plt.hist(M, 100, density=True)
plt.xlim(0,1)
plt.xlabel("Correlation with metamodel")
plt.savefig("corr_with_mm.png")
```

This is the conformity tax. It leaves payouts untouched until they reach a certain threshold. Everything after that is scaled based on the model’s meta model correlation. Models with low meta model correlation are taxed lightly, and models with high meta model correlation are taxed heavily. Notice that losses are untouched. If a model has a high correlation with the metamodel, putting a huge stake on it has diminishing returns, but not diminishing potential losses. This makes it more risky to put a large stake on a model that doesn’t contribute to the metamodel.

```
def conformity_tax(P, M, threshold): #(payouts, metamodel_correlations, threshold) => adjusted payouts
M = np.clip(M, 0, 1)# negative correlation has no effect
P_prime = np.minimum(P, threshold) + np.maximum(P - threshold, 0) * (1 - np.maximum(M, 0))
return P_prime
threshold = 50
payouts = np.linspace(-75, 200, 100)
uncorrelated = conformity_tax(payouts, np.ones(100) * 0, threshold)
slightly_correlated = conformity_tax(payouts, np.ones(100) * 0.2, threshold)
highly_correlated = conformity_tax(payouts, np.ones(100) * 0.8, threshold)
example_preds = conformity_tax(payouts, np.ones(100) * 0.95, threshold)
plt.plot(payouts, uncorrelated, color = "blue", label='mm_corr 0')
plt.plot(payouts, slightly_correlated, color = "green", label='mm_corr 0.2')
plt.plot(payouts, highly_correlated, color = "orange", label='mm_corr 0.8')
plt.plot(payouts, example_preds, color = "red", label='mm_corr 0.95')
plt.xlabel("Payout (NMR)")
plt.ylabel("Adjusted Payout with threshold of 50")
plt.legend()
plt.savefig("simulated_payouts.png")
```

The plot below shows the effect on payouts at various threshold levels.

# conformity tax at various thresholds

P_prime_100 = conformity_tax(P, M, threshold = 100)

P_prime_50 = conformity_tax(P, M, threshold = 50)

P_prime_10 = conformity_tax(P, M, threshold = 10)

```
plt.bar(ranks, P, width=1, color = "red", label = "No threshold")
plt.bar(ranks, P_prime_100, width=1, color = "orange", label = "T = 100")
plt.bar(ranks, P_prime_50, width=1, color = "green", label = "T = 50")
plt.bar(ranks, P_prime_10, width=1, color = "blue", label = "T = 10")
plt.xlabel("Ranks")
plt.ylabel("Positive Payouts")
plt.legend()
plt.savefig("simulated_thresholds.png")
```

The conformity tax also reduces the amount of NMR minted every round. It could potentially replace the current payout factor that taxes everyone and increase the longevity of Numerai’s NMR supply.

(Disregard the hacky-ass labels, I’m a noob at matplotlib)

```
total_payouts = [np.sum(P), np.sum(P_prime_100),
np.sum(P_prime_50), np.sum(P_prime_10)]
plt.bar(range(0, 8, 2), total_payouts, width=1, color = "blue")
plt.xticks(range(0, 8, 2))
plt.xlabel("Normal threshold 100 threshold 50 threshold 10")
plt.ylabel("Total NMR payed out")
plt.savefig("total_nmr.png")
```