The "paradox" arises from a highly plausible miscalculation of the expected gain from a swap.
    Let a > 0 and b > 0 be the ammounts of the cheques initially given to Alf and Bert. We know that either b=a/2 or b=2a with 50-50 probability.
    Bert is offering Alf a "double or halve" bet with 50-50 odds, which seems tempting.
    Alf's expected gain for swapping is (1/2)(2a - a) + (1/2)(a/2 - a) = a/4 > 0 , whereas his expected gain for not swapping is zero.
    Likewise Bert's expected gain must be b/4 > 0.
    But since this situation is symmetrical the expected gains must be equal, from which we can deduce that a=b, which we know to be false. A mistake has clearly been made.

    Since the situation is completely symmetrical, there is no reason to swap. Indeed, if it was in Alf's interests to swap, then, post swap, it would equally be in his interest to swap again. And again. And again.
    The correct way to calculate Alf's expected gain for a swap is as (1/2)(M - M/2) + (1/2)(M/2 - M) = 0 where M = Max{a,b}.
    Both Alf and Bert have an averaged expectation of 3M/4 regardless of whether they swap or not, and can realise this if they agree to pool and share.