Re: poker

David C. Ullrich wrote:
On 7 Feb 2006 05:57:10 -0800, matt271829-news@xxxxxxxxxxx wrote:

David C. Ullrich wrote:
On 7 Feb 2006 04:26:54 -0800, matt271829-news@xxxxxxxxxxx wrote:

Robert Israel wrote:
In article <1139282339.260273.161690@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
<matt271829-news@xxxxxxxxxxx> wrote:

[snip]

Thanks for your reply Robert. I confess at the moment I don't really
follow what you are doing here. I have tried the same calculation -
assume that A's strategy is to bet with probability p(a) when his
number is a. The idea is then to find the function p such that A's
expectation is maximised.

No, that's not the idea. The idea is to find the function p(a) that
maximizes the minimum, over all possible strategies for B, of the
expected payoff.

I think this is exactly what I have done. Probably I ought to have said
"find the function p such that A's expectation is maximised **when B
plays optimally** ". B playing optimally corresponds to your "maximise
the minimum". My method is:

1. Assume A adopts the p(a) strategy.
2. Work out B's optimal strategy, assuming that B knows A is following
the p(a) strategy.
3. Work out A's expectation if B adopts his optimal strategy.
4. Choose p(a) so as to maximise this expectation.

Do you still think this is incorrect? If it's just a problem with the
way I'm describing it then take a look at the actual derivation in my
earlier post, and I'd be very interested if you can find any flaw. I
can't at the moment.

Since that's not so easy to find directly, I started
by making some assumptions about the form of the optimal strategies,
which luckily turned out to be correct. The clincher is to show, as
I did above, that

1) if A follows the strategy I found for him (bet with < 1/10 or > 7/10,
otherwise pass), then no matter what B does, the expected payoff is at
least 1/10. This is done by minimizing the expected payoff when A follows
this strategy and B uses probability g(t) to bet when he is dealt t.

2) if B follows the strategy I found for him (bet with > 4/10, otherwise
fold), then no matter what A does, the expected payoff is at most 1/10.
This is done by maximizing the expected payoff when B follows this
strategy and A uses probability f(s) to bet when he is dealt s.

The initial problem I have with understanding your method, which
prevents me from getting further than the first few lines, is that when
calculating the optimal strategy for B you seem to be already assuming
that A's strategy is what you think the answer is. In contrast, I make
no assumption about the form of the function p for this step. It falls
out that B's optimal strategy is to always call if his number is
greater than some number, c, which can be calculated from p in the way
I described.

Note that I haven't had the time to follow the details here - below
I'm assumed that Robert has done the things he says that he's done
above (he's usually an extremely reliable source):

You're missing the point - you seem to be saying that he seems
to have done exactly what he _says_ he did! Here's a valid way
to solve a hard problem (not guaranteed to work, but valid when
it does work):

(i) Assume that the problem has a simple solution (where "simple"
might mean various things.)

(ii) Figure out what the solution is, _if_ there _is_ a simple
solution.

(iii) Show that the simple solution you found in (ii) actually
_is_ a solution to the original problem.


I don't understand your point, and I think you might be
misunderstanding what is going on.

Possibly.

In the model we are discussing, A's
strategy is defined by a function, p, such that A bets on his number,
a, with probability p(a). We need to find p such that A's expectation
is maximised, assuming that B also plays optimally.

Actually I _did_ understand that. I got that far all by
myself some time ago when I first saw the message; I didn't
have time to try to actually work all the details out
(in more detail, p(a) would be a function giving the
probability that A bets if his card is a, q(b) would be
a function giving the probability that B calls if
A bets and B's card is b. Given p and q there is an
expectation E(p,q), which is a certain average over
a and b, and the actual value of the game is then

max_p (min_q E(p,q))

.)

As far as I am
aware, we have no "a priori" knowledge of the form of p,

I'm not aware of any such a priori knowledge either.

and so I do
not understand how any assumptions its properties can be used in the
calculations - unless, of course, it is previously proved that the
optimal p has those properties.

This is exactly where it seems like you're missing the point.
Again, I'm simply taking Robert's word for what he says he proved.
He says "Since that's not so easy to find directly, I started
by making some assumptions about the form of the optimal strategies,
which luckily turned out to be correct."

If we had a priori knowledge of the form of the solution we
could use that. _Another_ valid way to get a valid solution,
if it works, is to start with some possibly false assumptions
about the form of the optimal solution, figure out what
the optimal solution is _if_ those assumptions hold, and
then _show_ that that solution actually _is_ optimal.

Absolutely. The problem I have is that I don't see how the solution
*is* shown to be optimal.

The fact that the "simple" solution is optimal does not
have to be proved _before_ finding the simple solution...

For sure, some "simple" forms for p HAVE been suggested - notably that
p(a) = 1 for a < c or a > d, p(a) = 0 elsewhere, for some c and d to be
determined. This version has been solved, the answer being that c =
0.1, d = 0.7, when A's expectation is 0.1. If we could prove that the
optimal p has this form then we would be done. Did I miss a step where
that was actually proved or something?

He claims that he's proved (1) and (2) above (I say "claims" just
because I haven't looked at the details.) If he actually _has_
proved (1) and (2) then you're missing something.

Question: Do you disagree that he's proved (1) and (2)

No. (I haven't checked the detail of the calculations, but I have no
reason to think they are incorrect, especially given the provenance.)

, or
do you not see how proving (1) and (2) shows that he's done,
assuming he actually has proved them?

Correct. This is *exactly* what I don't understand. I'll reply to
Robert's follow-up and explain there why I don't get it, and hopefully
you can check it out there and somewhere between the two help me
understand why it works.

.

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