> Subsequently, how can the passive investor possibly match market returns without w being 0, or the passive investor knowing exactly how the active investors will allocate their holdings? That information lies in the future - one only needs to go though the exercise of simulating market returns on a discrete time basis to realize that the passive investor cannot possibly allocate to achieve exactly market returns.
Because all he has to do is do absolutely nothing and he achieves that. The passive investor doesn't make choices, he buys a portfolio that has all the components of the index in the proportions at that point. Then the active traders trade among themselves and the prices move. And now the passive investor still holds all the components in the index at the proportions at t=2 because he hasn't traded, only the assets have changed value. You can observe this in practice. Index funds track their index incredibly well.
Because all he has to do is do absolutely nothing and he achieves that. The passive investor doesn't make choices, he buys a portfolio that has all the components of the index in the proportions at that point. Then the active traders trade among themselves and the prices move. And now the passive investor still holds all the components in the index at the proportions at t=2 because he hasn't traded, only the assets have changed value. You can observe this in practice. Index funds track their index incredibly well.