Slotboom's quiz vs Ed Miller and percent of winning sessions

[Michael Davis]

BS. This statement cannot be correct if you consider the net results of all gamblers unless the EV is actually the same in the high variance and low variance examples.

-Michael

[Mason Malmuth]

Hi Jaquen H'gar:

You wrote:

[ QUOTE ]
Actually, professional gamblers who use risk-aversion, i.e. always considering EV/var rather than simply EV usually perform better over the long term than those who don't.

[/ QUOTE ]

While this statement is true, it's very misleading as written. The true expert gamblers do push their edges far more than less knowledgeable people who do know something about gambling. But on the other hand, they are not completely foolhardy. You may want to look at the discussion of "Non-Self Weighting Strategies" in my book Gambling Theory and Other Topics.

Best wishes,
Mason

[Ed Miller]

While this statement is true, it's very misleading as written.

Furthermore, this sort of reasoning simply doesn't apply to limit hold 'em very often. These "small edge, big risk" issues come up in sports betting and even no limit tournaments, but they really shouldn't factor into your cash limit hold 'em strategy often at all.

People who make lots of decisions to "lower their variance" are usually just making mistakes. It's like the guy that surrenders all his stiffs at blackjack.

[garyc8]

I think people concerned with this issue might do well to also read your discussion (in the same book) of standard deviation, bankroll req., and coefficient of variation.
Knowing how one's bankroll relates to these factors should help one understand how much value risk aversion (or a smaller game) has for them.

[Cazz]

Some math: (It is early so there may be an error or two in here)

If your standard deviation (sigma) is 10 BB for 1 hour, then for N hours the standard deviation is sqrt(N) * 10 BB.

Confidence intervals. Results should be around the expected value (win rate per hour * hours)
----------------------
+/- 1 std. deviation 66.7% of the time
+/- 2 std. deviations 95% of the time
+/- 3 std. deviations 99% of the time

So, over N hours you should expect to win
N * avg +/- sqrt(N) * sigma, 66.67% of the time.
16.7% of the time you will fare worse and 16.7% of the time you will fare better. You can alter this percentage by selecting a larger or smaller range
N * avg +/- X * sqrt(N) * sigma, where X is some scalar.

X = 1 -> 16.7%
X = 2 -> 2.5%
X = 3 -> 0.5%

If your goal was to limit losing session to some percentage of the time, you can select the correct scalar, X. Using X you can then determine the relationship between the average and the sigma.

The lower value on the confidence interval is
N * avg - X * sqrt(N) * sigma
Set this to zero (or >= zero) and solve:
avg >= X*sigma/sqrt(N)

Example: You want limit losing sessions to 16.7% of the time, and a session is 8 hours. Your sigma is 10 BB.
avg >= 1.0 * 10 BB /sqrt(8)
>= 3.54 BB

Also, for every 1 BB your single hour sigma goes up (or down), your average must go up (down) by 0.35 BB (1/sqrt(8)) to keep the losing session percentage the same.
Or for every 1 BB per hour you give up, you have to reduce
the single hour sigma by 2.83 BB.