Conglomerability says that if you have an event E and a partition {Ri : i ∈ I}
of the probability space, then if P(E∣Ri) ≥ λ
for all i, we likewise have
P(E) ≥ λ.
Absence of conglomerability leads to a variety of paradoxes, but in
various infinitary contexts, it is necessary to abandon
conglomerability.
I want to consider a variant on conglomerability, which I will call
independence conglomerability. Suppose we have a collection of events
{Ei : i ∈ I},
and suppose that J is a
randomly chosen member of I,
with J independent of all the
Ei taken
together. Independence conglomerability requires that if P(Ei) ≥ λ
for all i, then P(EJ) ≥ λ,
where ω ∈ EJ
if and only if ω ∈ EJ(ω)
for ω in our underlying
probability space Ω.
Independence conglomerability follows from conglomerability if we
suppose that P(EJ∣J=i) = P(Ei)
for all i.
However, note that independence conglomerability differs from
conglomerability in two ways. First, it can make sense to talk of
independence conglomerability even in cases where one cannot
meaningfully conditionalize on J = i (e.g., because P(J=i) = 0 and we
don’t have a way of conditionalizing on zero probability events).
Second, and this seems like it could be significant, independence
conglomerability seems a little more intuitive. We have a bunch of
events, each of which has probability at least λ. We independently
randomly choose one of these events. We should expect the probability
that our randomly chosen event happens to be at least λ.
Imagine that independence conglomerability fails. Then you can have
the following scenario. For each i ∈ I there is a game
available for you to play, where you win provided that Ei happens. You
get to choose which game to play. Suppose that for each game, the
probability of victory is at most λ. But, paradoxically, there is a
random way to choose which game to play, independent of the events
underlying all the games, where your probability of victory is strictly
bigger than λ. (Here I
reversed the inequalities defining independence conglomerability, by
replacing events with their complements as needed.) Thus you can do
better by randomly choosing which game to play than by choosing a
specific game to play.
Example: I am going to uniformly randomly choose a positive
integer (using a countably infinite fair lottery, assuming for the sake
of argument such is possible). For each positive integer n, you have a game available to you:
the game is one you win if n
is no less than the number I am going to pick. You despair: there is no
way for you to have any chance to win, because whatever positive integer
n you choose, I am infinitely
more likely to get a number bigger than n than a number less than or equal
to n, so the chance of you
winning is zero or infinitesimal regardless which game you pick. But
then you have a brilliant idea. If instead of you choosing a specific
number, you independently uniformly choose a positive integer n, the probability of you winning
will be at least 1/2 by symmetry. Thus
a situation with two independent countably infinite fair lotteries and a
symmetry constraint that probabilities don’t change when you swap the
lotteries with each other violates independence conglomerability.
Is this violation somehow more problematic than the much discussed
violations of plain conglomerability that happen with countably infinite
fair lotteries? I don’t know, but maybe it is. There is something
particularly odd about the idea that you can noticeably increase your
chance of winning by randomly choosing which game to play.
