[Phys-l] game theory problem (was: probability)

[John Denker <jsd@av8n.com>]

On 06/28/2010 06:35 AM, Carl Mungan wrote:
> The following problem comes out of Boas. I have a solution on my 
> website. From time to time, someone will email me and say my solution 
> is nonsense. In any case, it's a cute problem:
>
> ---
>
> You're sitting across from a dealer. He shuffles a single deck of 
> cards and deals you two cards face down. He then looks at them 
> without showing them to you. Consider the following three distinct 
> scenarios:
>
> 1. He tells you nothing.
> 2. He tells you, "You've got at least one ace."
> 3. He says, "Wow, you've got the ace of spades."
>
> For each of these three scenarios, what is the probability that if 
> you now turn over the two cards you'll find that you've got two aces? 
> IOW, what odds would you take to bet on it?


1) The web-site version of the problem
   http://usna.edu/Users/physics/mungan/Scholarship/TwoAces.pdf
is ill-posed.  It can be solved if you make some additional
assumptions.  The solution given there makes some assumptions
that are (a) not very clearly stated and (b) IMHO not particularly
plausible.

For one thing, Curtis O. and John M. are quite right to bring
up the Monty Hall issue.  To say the same thing another way,
it very much depends on whether the "observer" (as he is 
called on the web site) is your friend or your adversary.
The "dealer" (as he is called above) is presumably your
adversary.

Also, in problems of this sort, it is conventional to assume
that all three scenarios apply to any given instance of the 
game, and that the three scenarios are disjoint.

If we assume a friendly observer and disjoint scenarios, then
in scenario (i) the probability of winning is zero, because
if you had two aces your friend would have told you you had
at least one.  Similarly in scenario (ii) your probability 
of winning is one in 49 not one in 33, since you have to win 
without using the ace of spades, and if you did have the ace 
of spades your friend would have told you.

Let's be clear:  Silence is not the same as zero information.
In general you can can obtain information from the curious 
incident of the dog in the night-time
  http://en.wikipedia.org/wiki/Silver_Blaze
and in this case from the the curious silence of the observer.

  In contrast, the analysis on the web page seems to assume
  highly non-disjoint scenarios.  For example, in scenario (i) 
  the apparent assumption is that scenarios (ii) and (iii) are 
  forbidden _a priori_ in this instance of the game, and hence
  it was entirely futile for the observer to look at the cards.  
  You are assumed to obtain zero information from the curious
  silence of the observer.  I find this assumption unrealistic
  and unconventional.

  Semi-similarly the evident assumption in scenario (ii) is 
  that scenario (iii) is forbidden _a priori_ in this instance
  of the problem.  Again this seems unrealistic and unconventional.



2) The email version of the problem (above) is even more severely
ill-posed.

Note the stark contrast:
  a) the _probability_ of drawing this-or-that hand from 
   a shuffled deck, with possibly additional information
   from an unbiased and/or friendly oracle, versus
  b) the _odds_ that I would require to play the game
   against an adversary who might be bluffing or worse.

The question first asks for the probabilistic probability 
and then asks for the game theoretic odds.  These are not
the same.  The question is self-inconsistent because it 
equates the two.

The probability calculation on the web site is one thing,
but a proper game theory analysis is something else entirely.
When you ask me to calculate the odds for playing against
an adversary, we move instantly into the domain of game
theory.

In the email version of the game, there are two sources 
of information, namely the dealer's statements about the 
cards, and the odds he offers you.  I assume he looks at
the cards before deciding what odds to offer, which is
what is suggested by the wording of the problem (but not
explicitly stated).

For starters, the most elementary game theory analysis would
look at things from the dealer's point of view.  He will
not offer you good odds unless he thinks you are going to
lose, and since for him it is a game of perfect information,
he *knows* you are going to lose.  So for him, the entire
game consists of bluffing!  The only way he can win is to
offer me odds I will accept under situations where he
*knows* I am going to lose.  Conversely in situations
where he *knows* I'm going to win, he should offer me
trivial odds of 1:1 or worse, so that I will not accept.

The game does not mention any "ante" or "pot" so there is
no way I could calculate pot odds or anything else that
might tempt me to play this game for any odds whatsoever.
It is clearly a heads-you-win tails-I-lose proposition.

If the dealer is required to call out the ace of spades
if it is present, the question should be explicit about
this rule.  The analysis on the web site is inconsistent
with this rule.

If the dealer is required to call out one or more aces
if present, the question should be explicit about this
rule.  The analysis on the web site is inconsistent with
this rule.

If the odds are fixed before the dealer looks at the
cards, the question should be explicit about this rule.

If the dealer is in cahoots with the player, the question
should be explicit about this.  It changes the game from
a no-win proposition to a no-lose proposition.

====================

This is obviously a made-up problem _pretending_ to be a
real-world problem.