6 ideas
Because the market portfolio is the sum of the passive holdings and the active holdings, and passive investors hold exactly the market weights, the aggregate active portfolio must also hold market weights. Therefore, before costs, the return on the average actively managed dollar must equal the return on the average passively managed dollar, which equals the market return.
Active managers incur higher fees, research expenses, and trading costs than passive managers, and these costs are subtracted from an identical gross return. So the average actively managed dollar must earn less than the average passively managed dollar net of costs, as a matter of arithmetic rather than empirical observation.
A passive investor holds every security in the market in proportion to its market value, so their portfolio return equals the market return before costs. This precise definition, rather than a loose idea of 'not trading much', is what makes the arithmetic work, because it fixes the passive side's holdings exactly.
View any market as a fixed total split between participant groups: if one group's share and return are pinned down, the remaining group's aggregate return is determined by subtraction. Outperformance by some active managers must be exactly offset by underperformance of other active managers, so claims of beating the market become claims about who the losers are.
Because the conclusion follows from definitions and addition, any study appearing to show active managers beating passive ones on average must be using a mismatched benchmark, an incomplete sample, or unequal weighting. Apparent contradictions signal measurement errors, not a failure of the logic.
The result holds for the average dollar actively managed, weighted by assets, not for the average manager counted equally. Before comparing groups of investors, specify the weighting scheme, because an equal-weighted average of funds can diverge from the dollar-weighted outcome the arithmetic guarantees.