[FoRK] penny pitching or bit bashing?

Dave Long dl at silcom.com
Fri Mar 5 11:05:13 PST 2004

> > A) what is the optimal fraction of
> >    one's stake to bet per flip?
> > B) what is the expected return?

Per information theory: 2% and 0.04%

If one knew one had a 100% chance of
a win, then a 100% bet would be best;
likewise, if one knew one had only a
50% chance, then 0% is the right bet
to make (think of Getty's statement:
"If I wanted to gamble, I would buy
a casino"), and it turns out linear
in between the two.

1/ Expectation Value leads us astray;
one might think it would be better to
make 100% bets than 2% bets,
    1*(.51-.49) > .02*(.51-.49)
but remember that it only takes one
loss over a series of trials to wipe
out the aggresive bettor.

2/ That 0.04% expected return is why
it's difficult for arbs to drive any
slightly disequilibrated market into
perfect equilibrium, and why I don't
think any of us will ever notice the
difference between a practical coin
flip at 51:49 and a theoretical coin
flip at 50:50.


:: :: ::

1000 trials* of 200 rounds,
51% to win twice one's bet

bet%    median  mean    sketch
0       100     100     ----------

No risk, no return.

1       103     104     _--------~
2       104     107     _--------~

Taking on additional risk
(up to that information-
theoretic optimum) yields
additional likely return.
5        95     121     ___---~~~~|
10       54     161     _______--/

Going over that optimum will
increase returns on average,
yet make it much more likely
that one registers a bottom-
line loss in any given trial.

25        0.7    96     ___________|
50        0       0     ___________|

And with enough lack of risk
aversion, one can get to the
gamblers' ruin; even average
returns show losses.

(although, like the lottery,
there are still big wins in
a few trials.  The luckiest
trial in that sample of 25%
bets had 125x the mean, and
in the 50% bets the luckiest
wound up with 919x the mean)

* even 1000 trials isn't
sufficient to get a stable
value for the bets of 10%
and greater.

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