From: daryl@oracorp.com (Daryl McCullough)
Subject: Re: Games with Nonmeasurable Sets
Organization: ORA Corporation
Date: Wed, 4 Nov 1992 19:19:56 GMT

In article <thompson.720739075@daphne.socsci.umn.edu>,
thompson@atlas.socsci.umn.edu (T. Scott Thompson) writes:

>I am puzzled by the implicit definition of "certainty" in the
>discussion of these problems.

If the probability of an outcome is 1, then the outcome is said to be
certain (or almost certain to be more precise).

>Based on the original game and the followup versions it appears that
>the "paradox" arises from the fact that the sets in which either
>player wins are nonmeasurable.  This leads me to question the sense in
>which each player is "certain" that she will win.  The answer to this
>question appears to be that each player knows that the outer measure
>of the set in which she wins is one (and the inner measure of its
>complement is zero).

You are right, that before the game starts, the probability for each
player that she will win is undefined. However, I am talking about the
probabilities each player calculates *after* she is dealt a card. For
definiteness, let me assume that player 1 is randomly dealt a card, and
then player 2. Let's then look at how the probabilities work from the
point of view of player 1.

There are three stages in the game: Stage 1: neither player has been
dealt a card, Stage 2: player 1 is dealt a card, and Stage 3: player 2
is dealt a card.

At Stage 1, player 1 tries to calculate her odds of winning as
follows: The probability of player 1 winning = the measure of the set
of all pairs <x,y> such that LT(x,y) = undefined, since this is not a
measurable set. At this point, player 1 has no confidence that she
will win.

At Stage 2, player 1 has been dealt a card labelled with the real r1.
She is allowed to reassess her odds of winning based on the
information that has been dealt r1. Just as in the case with the game
"High Card Wins" after being dealt the King of Spades, she is allowed
to recompute her odds based on this additional information. She now
computes her odds as follows: Let C(r1) = set of all reals r2 in [0,1]
such that LT(r2,r1). Then

     Probability of player 1 winning after being dealt r1
     = Probability that player 2 will be dealt a card greater than r1 in
       the ordering LT
     = 1 - Probability that player 2 will be dealt a lower card than r1 =
     = 1 - measure of C(r1)
     = 1

(Since C(r1) is a countable set, it has measure 0).

So, while the probability of player 1 winning starts off undefined, it
changes to probability 1 after seeing what card she was dealt.

>If I were to be told "the probability that you have won is no less
>than zero and at most one" then I don't think that I would be very
>"certain" that I had won.

After seeing your card, you calculate that your probability of winning
is 1. Period. Inner and outer measures agree that the measure of a
countable set is always zero, and complements of countable sets have
measure 1. So assuming "probability 1" means "certain", then you
should be certain you will win. Of course, the fact that your opponent
will also calculate 1 as her probability of winning, there is
something strange about identifying "certain" with "probability 1".

Daryl McCullough
ORA Corp.
Ithaca, NY