To clarify, I am not saying that active managers (or their investors) necessarily beat the market (although if you were the only active investor, you could easily and consistently beat the market). It was Sharpe who claimed that passive investors must outperform active ones based on a 'mathematical imperative.'

My point is that this imperative is incorrect because it relies on an absurd assumption (static market). Once we discard that, it becomes an empirical question rather than a mathematical certainty. In that sense, I find that question interesting: do active investors have positive gross alpha? I have seen papers addressing this, but there seems to be some controversy.