Steve wrote:
>I've done a little work with trying to reduce variance, but I realized a
> long time ago that variance itself isn't inherently bad. I personally
> believe that trying to reduce raw variance for its own sake is misguided.
> A player who truly wishes to minimize variance will simply choose not to
> play, since not playing eliminates variance entirely. A strategy which
> minimizes variance would never break a pat hand even when doing so gives
> a huge increase in EV, and I'd call this an example of "good variance"
> that is desirable, as it gives a big reward in exchange for the increased
> uncertainty. The time to reduce variance is when the gain from breaking
> the pat hand is small. So the trick with variance is to avoid it when it
> hurts and embrace it when it helps. I believe that tradeoff is handled
> quite well by a min-risk strategy, which minimizes the probability of
> going broke rather than playing forever without ever having the bankroll
> reduced to zero.
I think a better way of expressing this is that variance is inherently
bad, but it's not the only factor, and a certain balance between high
expected value and low variance is ideal. What you call "good
variance" is when there's enough gain in expected value to justify the
(inherently bad) increase in variance.
But that's my point -- variance almost never exists in a vacuum, it is
usually carbon-bonded to some EV. Whether variance is viewed as
good or bad depends on how much EV comes with it.
Increasing expected value
always helps. Increasing variance always hurts.
That seems obvious on the surface, but it really isn't that simple. There is
one area where variance is definitely good for the player, and that is when
the player is forced to play unfavorable games. If you play an unfavorable
game and reduce variance to zero, then you are guaranteed to lose.
Roulette is a good game for illustrating my point on this, because most all
roulette wagers have the same EV of -5.26%. However, the variance varies
dramatically. Suppose the player's objective is to take a starting bankroll
and either multiply it by 36 or go broke trying. The best way to do this in
Roulette is to place the entire bankroll on a single number, giving the
player a 1/38 chance or reaching the goal.
Now suppose we play using even-money bets on "red" each time, hoping to slowly
build the bankroll up to 36 units while keeping variance as low as possible.
Playing this way, we would need to win 5 bets in a row to build the bankroll
to 32 units, with a probability of (18/38)^5 = 0.023847 = 1/41.93, so using
even money bets gives us a smaller chance just to reach 32 units. Continuing
with a "maximum boldness" strategy (betting just enough to win each time, or
betting the entire bankroll when it is 18 or fewer units) gives an overall
chance of 1/48.33 for reaching 36 units before going broke. Since all
Roulette wagers have the same EV, we can't give EV the credit for making the
difference, it is the higher variance that helps the player in this
situation. This example refutes the claim that "all else [EV] being equal,
lower variance is better".
I'll give one more example. The objective is the same -- turn a one unit
bankroll into 36 units or go broke trying. Player A will play a single
number at the Roulette table and have a 1/38 chance of success. This play
has a terrible EV of -5.26%. We'll let Player B play at the craps table and
use even-money "pass" bets (with no odds) to try to parlay the bankroll up to
36 units. The craps "pass" bet has much better EV of -1.414%, so with much
better EV and lower variance it "must" be the better play, right? Using the
"maximum boldness" strategy at the craps table gives the player a 1/38.90
chance at reaching 36 units before going broke. So, even though the craps
play has *much* better EV as well as lower variance, the roulette play gives
the player a better shot at increasing the bankroll by a factor of 36. In
this case, variance is the player's friend, so much so that it overcomes a
large disadvantage in EV.
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On Sunday 03 August 2008 11:36 am, 007 wrote: