More importantly, it also doesn't take into account that there are investment opportunities that are not as directly (i.e., without intermediary layers which extract some of the returns) available to passive investors, so that the passive/act groups aren't splitting the same universe of alternatives. (E.g., passively investing in any given index fund is on average better than actively investing restricted to stocks in that index, and you can probably even extend that to passively investing in index funds in generally being better than actively investing in investments that directly held by any combination of index funds, but not every investment vehicle in the market is directly held by index funds.)

Is this really true? I think it's only true if you replace average with median (e.g. in [1,2,3,10] only 10 is above the average).

> [I]f all of the "passive" half have a different random subset, then about half will by definition do "above average", and the other will do "below average". While in the "active group", less than half will do "above average" because they pay part of their gain to fund managers.

So, if all the passive investors have pick a random "large enough" subset of stocks, one would expect the distribution to approach the normal distribution, with roughly half to either side of the arithmetic mean.

Similarly, as you point out, on the "active" side, one might see more outliers - and perhaps the selection isn't really uniformly random.

So, it would probably be a stretch to claim that half will be above average "by definition", even if one includes the expected distribution, while note perhaps entirely wrong.