When I proposed moving to MMC only, it was with the understanding that Numerai would place very large stakes on internal models to account for MMC being zero sum. In that scenario, users could earn a positive sum as long as the staked internal models earned a negative one. The condition for a positive user sum in this scenario would be that CORR is increased by giving a higher weight to the user meta-model relative to the benchmark models than the weight given in the fixed weight ratio. Essentially a positive sum would require that the users are “contributive” to the benchmark at a particular weight.
Having thought about it some more though, this wouldn’t quite work. The problem being that if benchmark models are consistently burning (condition for a positive sum), participants can earn MMC by inversing them until their effective net stake weight is reduced to the point that they no longer burn and MMC returns to being zerosum for the users. Numerai may not see this as much of a problem because they have observed a net sum from MMC, but we should look at what that sum really is.
First of all we should understand that what might be called “true MMC” will necessarily be zero-sum. True MMC being the stake weighted mean stake derivative of the CORR of a stake weighted mean vector with a target vector. This is fairly easy to demonstrate: https://colab.research.google.com/drive/1sL0aqbT3h0jNka2G-GqBWlv0EsktCHI8?usp=sharing. The current MMC though differs from true MMC in that the meta-model vector is gaussianized before being plugged in. This should be the only difference but if the predictions used to construct the meta-model are transformed differently than predictions used to calculate MMC, that could also affect the sum.
So if gaussianizing the meta-model creates some sum, what is that sum? Well Numerai MMC is still calculating the effect of adding some weight to user predictions, but adding that weight to the gaussianized meta-model rather than the actual stake weighted mean prediction. Less like the effect of adding weight to existing stake but more like adding weight to a different vector from zero. The stake weighted sum derivative (Numerai MMC) is then the effect of adding the stake weighted sum prediction (unguassianized metamodel) to the gaussianized meta-model. Demo: https://colab.research.google.com/drive/1OLdRIEPEzR1KiEOxIKyueWyR-zqKGGyz?usp=sharing. So under this scheme, MMC is positive if the gaussianized meta-model would have more CORR if it were more like the ungaussianized meta-model. Not exactly a great criterion. It may correlate with mean corr somewhat if the gaussianized meta-model has less kurtosis than the ungaussianized meta-model and therefore tends to have a smaller scale of CORR positive or negative, but it still really isn’t measuring mean user performance except maybe incidentally.
Given that we can expect a meta-model constructed from gaussianized predictions to already be fairly gaussian, it’s not too surprising the sum of Numerai MMC is very small. And because it’s also not very meaningful, it really shouldn’t be relied on to determine Numerai’s net payout. Mean CORR is a much better basis for this, which should imply some CORR multiplier per mandatory MMC multiplier and probably more than the 0.5 currently planned. Allowing MMC any impact on sum payout doesn’t make much sense in this light so I suggest using the zerosum true MMC instead.
A zero-sum MMC has a serious benefit in that it doesn’t require a payout factor because the net payout for it is always zero and doesn’t need to be limited by such. Having a payout factor on only CORR implies increasing CORR weight relative to MMC with decreasing total stake (across all users). This allows for a fair CORR to MMC ratio to be discovered by changing total stake in a market sense allowing total stake to find a stable equilibrium point.