sure. as it happens i was just looking at this the other day.
suppose you're playing $1 9/6 jacks with a $5500 prog meter, and your
bankroll is $40000. you're dealt A(AQT)7. hold the aces, or draw to
the royal?
someone who's blindly maximizing EV will do the following computation,
or more likely, have their computer do it for them. they'll see that
there are 16215 possible draws off the AA, of which 11559 don't
improve and return $5, 2592 make two pair and return $10, etc:
AA: EV = 1/16215 * (11559*5 + 2592*10 + 1854*15 + 165*45 + 45*125) = $7.68
AQT: EV = 1/1081 * (786*0 + 207*5 + 21*10 + 7*15 + 15*20 + 44*30 +
1*5500) = $7.84
easy, draw to the royal!
but suppose you're a kelly bettor. you want to optimize your rate of
bankroll growth, and you know that the way to do that is by maximizing
log-utility, not flat EV. now you're doing a different computation. if
you hold the AA, 11559 times you're ending up with $40005, 2592 times
you're ending up with $40010, etc:
AA: UTIL = 1/16215 * (11559*log(40005) + 2592*log(40010) + ... +
45*log(40125)) ~= log($40007.68)
AQT: UTIL = 1/1081 * (786*log(40000) + 207*log(40005) + ... +
1*log(45500)) ~= log($40007.51)
easy, hold the aces!
you don't have to go to these lengths to apply this idea in practice.
log(X) is pretty damn close to linear for 40000 <= X <= 40250 ... in
other words, you can ignore the difference between dollars and
log-utility for anything below the royal or other big jackpots.
instead, just replace the face value of the royal (R) with
S*(log(B+R)-log(B))/(log(B+S)-log(B)), where B is your bankroll and S
is your bet size.
in this case, R=5500, B=40000, S=5, and you get a utility-adjusted
royal value of $5154. if you want to play nice by kelly, use that in
your strategy computations instead of $5500.
as expected, you'd want to play more conservatively than this if your
bankroll is smaller (B=20000 leads to a $4860 value), or more
aggressively if your bankroll is larger (B=80000 leads to a $5319
value). but even a very well bankrolled player should be playing a
little more conservatively than straight EV considerations would
suggest.
cheers,
five
···
On Thu, Apr 3, 2008 at 9:47 AM, kcace1024 <cy4873@hotmail.com> wrote:
Many months ago some of the experts here showed me there are other
strategies besides max ev. I did not think any of it would apply to
my play. Recently I have started playing some progressives and I
wonder if there is a way to evaluate increased EV versus increased
variance at strategy break points. Does anyone wait for a certain
amount of increased EV before accepting the increased variance.