From: wallace@dtc.hp.com (David Wallace)
Newsgroups: rec.gambling
Subject: Re: Vegas World info wanted
Summary: Refutation of previous analysis
Date: 12 Oct 92 16:14:13 GMT
Article-I.D.: news.Bw0nrr.7Mq
Organization: Hewlett-Packard Design Technology Center, Santa Clara, CA

A few days ago, I posted a "value-based" analysis of the double-down bet at
Vegas World, that basically concluded that you should use real money to
double down a casino action (CA) chip bet whenever q, the probability of
losing the bet ignoring pushes, was less than v, the value of a CA chip,
and double down with CA chips the rest of the time that a double-down is
called for.  The value v was to be determined by looking at the remaining
bets one would make with CA chips (only), and computing an average value
for the CA chip (I used .48 as an estimated value of v, without doing the
actual computation).  Others, including Steven Jacobs, have analyzed the
use of real money to double down a CA chip bet as if it was a pure "triple
down."

I have since carried on an off-line discussion with Steven Jacobs about the
subject, and have concluded that I was wrong.  Here is the example I
constructed to discriminate between the two models, which wound up refuting
the "value-based" analysis.

Consider the following "Urn Game": an urn contains 3 black balls, 4 white
balls, 1 green ball, and 4 red balls.  A player places a bet, and a ball is
drawn from the urn.  Red is an "instant loser": the player immediately
loses the bet.  Green is an "instant winner": the player is immediately
paid even money for the bet.  If a black or white ball is drawn, the player
is not told which it is, but is offered the opportunity to "double down" by
placing an additional bet equal to the initial bet.  Then the ball is
revealed and both bets are settled: white wins and pays even money, black
loses.

The number of each kind of ball has been carefully chosen such that:
the initial bet has negative expectation, the "double down" bet has
positive expectation, a simple bet+double down still has negative
expectation, a bet+a true "triple down" would yield an even game,
and the "value-based" model I posted earlier calls for a double down
with CA chips, not real money (q=.429, v=.417, so q>v, whereas the
"value-based" model claims we should use real money when q<v).  Because a
true "triple down" would yield an even game, the "triple down" analysis
would call for a pseudo triple down by betting real chips on the double
down bet.  Thus this example discriminates between the models: they call
for different strategies on the double-down bet.  Which yields better
results?

Assuming that each ball is drawn once over twelve hands, betting CA chips
on the double-down bet loses 2 CA chips when a black ball is drawn
(3 times), wins 2 real chips but gives up 2 CA chips when a white ball
is drawn (4 times), wins 1 real chip and gives up a CA chip when a green
ball is drawn (1 time), and loses 1 CA chip when a red ball is drawn
(4 times).  Total over 12 hands: 9 real chips won, 19 CA chips given up.

On the other hand, the "triple down" strategy of betting real money on
the double-down bet loses 1 CA chip and 1 real chip when a black ball
is drawn (3 times), wins 2 real chips and gives up 1 CA chip when a
white ball is drawn (4 times), wins 1 real chip and gives up 1 CA chip
when a green ball is drawn (1 time), and loses 1 CA chip when a red ball
is drawn (4 times).  Total over 12 hands: 6 real chips won, 12 CA chips
given up.

Since the second strategy converts the CA chips at a 50% rate while the
first converts at a 47.4% rate, the second strategy is superior.
Furthermore, since it converts at the predicted "triple down" rate
of 50%, I conclude that this strategy is indeed equivalent to a true
triple down, at least for the urn example.  Finally, if instead of
using v as the threshold for the "value-based" analysis I use v', the
average value of a CA chip when CA chips are also used to double down, I
find that there are no solutions in positive integers to the number of
balls of each type in the urn that will satisfy all the constraints
specified above: the requirement that q>v' plus the requirement that a true
triple down be an even game implies that the number of black balls is
larger than the number of white balls, so that the double down bet no
longer has a positive expectation.

Conclusion: the "triple down" analysis is correct, mine was wrong.  The
error was using one value of v when deriving a strategy that implied a
different value for v.
-- 
Dave W. 	(dwallace@dtc.hp.com)
Newsgroups: rec.gambling