Mean-Variance optimization trades off expected returns with portfolio variance. The idea is that excess variance is not desirable.
But what if you weren't averse to high variance and you wanted to maximize both expected returns and variance. Has there been any research done on this or concepts similar to this?
As an example of a situation where this might be the case, think of a paper trading competition where there are 100 participants. The winner receives $100, and everyone else gains nothing. Ideally, you'd want your portfolio to be high variance, because in order to win you need to outperform 99 others. If you maximized mean+variance (or mean+std), you would be essentially maximizing the odds that you get above some threshold. Compare this with mean-variance optimization, which might improve the chance you place in the top 10, but not necessarily maximize the chances you get first place.