Proebsting's paradox

[Joshua Ulrich]

Has anyone given any serious thought to if / how Proebsting's paradox
might apply to optimal f? I just ran across the Wikipedia entry this
morning:
http://en.wikipedia.org/wiki/Proebsting%27s_paradox

Best,
--
Joshua Ulrich  |  FOSS Trading: www.fosstrading.com

[RVince]

Interesting Josh.

The second equation in the Wikipedia article uses the expected values
of the logs of returns, whereas the first value uses the binomial
formula. Things break down into apples and oranges there, as I have
pointed out. The binomial formula will always give you a fraction
(0<=f<=1) whereas the expected value of the logs (which IS the Kelly
Criterion) not necessarily so. The Paradox arises from assuming the
Kelly Criterion yeilds an optimal fraction. Because if you are dealing
with an optimal fraction, and the sum of the probabilities of gain is .
5, then the most the optimal fraction can be is .5 (at the most
miniscule loss outcomes) The only time the optimal fraction is 1 is
when there are no possible losses.

This is inconsistent with Thorp's response to the "Paradox."

This paradox presented in the Wikipedia article is an artifact of the
fact the the Kelly Criterion yields an optimal leverage factor -- not
an optimal fraction. This is discussed in the IFTA article published
last year (I think a pdf of it is available through my website) and
discussed further in a forthcoming book on to asses games.

[Henry Patner]

A Solution to Proebsting's Paradox or 'How to Skim a Bettor if You Must'

Eduardo Zambrano 

Cal Poly-Department of Economics
May 25, 2009

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1409843

[BBands]

I find it hard to imagine that this paradox would ever be a problem in
the real world. Am I missing something?

John

[RVince]

The problem is this: People think they are discussing a "fraction"
when in fact they are discussing numbers that are 0 to infinity,
hardly a fraction at all. Let's say one of those numbers, however, is
< 1, giving the illusion of being a fraction. But it is NOT a
fraction, and likely the optimal fraction (as given by the Optimal f
formula) being the absolute most one should ever wager
(asymptotically, as the number of plays approach infinity) is far less
than this dissillusioned "fraction" as returned as the solution to
Kelly's Criterion.

See page 3, just before section 3, of the articel of Zambrano cited
here by Henry Patner:

"Moreover, Thorp showed that if a gambler is offered 2 to 1 odds, then
4 to 1,
then 8 to 1, and so on, the Kelly criterion would have you eventually
bet your
entire wealth,thus exposing the bettor to a risk of complet eruin of
exactly 50%,
just as if he was risk neutral. This appears to challenge the view
commonly held
of the Kelly criterion keeping the investor away from any risk of
ruin."

As I said, I said in last years paper in the Ifta 2011 Journal
(available at ralphvince.com) the maximum value one can see for the
growth-optimal fraction equals the sum of the probabilities of the
winning scenarios. In the immediate context of the article, that would
be .5. Under no circumstances in a fair coin toss -- regardless of
payout, does the optimal fraction exceed .5.

I do not assume to be the mathematician any of these guys are -- but
I stand by this with absolute certainty. So no, this faux-"paradox" is
certainly NOT a problem in real life. The problem is people are
working with something that is different than what they think it is.

-Ralph Vince

[BBands]

On Wed, Jun 22, 2011 at 2:13 PM, Henry Patner <hpa...@gmail.com> wrote:
> I'd be keen to hear of any implementations that manage position size through
> time as interim odds change during the holding period.

First I am mostly interested in practical portfolios, not paradoxes or purity.

Second, it has been at least 20 yeas since I traded ideas of this
sort, so my thoughts may be moot to start.

Third, I suspect that there is not sufficiently comparable data for
phase three trials to compute the odds properly. On the other hand,
there is probably enough relevant risk-arbitrage data to have a go at
it.

So, If I wanted to do this, and I don't, I'd only do it in a portfolio
context. 30 positions would probably be enough to get diversification
high enough to create a reasonably smooth equity curve. Each slot
would be 3.3% and I'd allocate half that capital to each initial
position and then size using f. That would leave enough room to more
than double the bet size if need be and of course to cut it as far as
you would like. I would not use the largest loss as the divisor of f
-- likely to be an inconvenient 100% -- but the average of the largest
losses as we are in a portfolio context. No position slot could leach
capital from another, but oversize slots would contribute capital to
other slots. Finally, I'd consider at least a partial market neutral
overlay.

In my opinion that would get you into space where you could work
successfully without worrying about paradoxes, a space where the risk
of ruin would be vanishingly small, yet the risk taken high enough to
generate returns worthy of the effort.

Best,

John

[RVince]

John,
It's clearly a viable solution, and one that, in my humble opinion,
acknowledges the fact that if you aren't prepared to survive getting
smoked when correlations reduce to 1, you really are not prepared. So
it can be said from what you have written here that you ARE prepared
for that -- and that if you lose, and correlations have all conspired
to 1 against you, you are staring at a 50% drawdown. Though not hardly
insignificant, it is worst case -- a situation where most of your
peers have perished, a situation that permits you to find that point
pushing the pedal as hard as possible, while being prepared for that
worst-case. Ralph Vince

On Jun 23, 9:54 am, BBands <bba...@gmail.com> wrote:

[RVince]

I had the great pleasure of meeting one of our list members, Henry
Patner, for a very early meeting Friday morning at the Stamford
Station of the New Haven line. Not the poshest of places, but who
cares when you are meeting a guy like Henry!

One of the (many) things he got me thinking about had me revisit the
so-called "Proebstring's Paradox." I'm not so sure this is a paradox
after all, having revisited it. I'm not so sure I understand it,
because it seems (to me) the solutions are obvious from what we
already know.

The Wikipedia article (http://en.wikipedia.org/wiki/Proebsting
%27s_paradox) explains this as "For example, if a 50/50 bet pays 2 to
1, Kelly says to bet 25% of wealth. If a 50/50 bet pays 5 to 1, Kelly
says to bet 40% of wealth.Now suppose a gambler is offered 2 to 1
payout and bets 25%. What should she do if the payout on new bets
changes to 5 to 1? "

Are these:
A. Subsequent bets?
B. Simultaneous bets?
C A single bet whose odds change throughout the time the bet is
placed, and the result cast?

If A, the answer is clearly .25 then .4 -- to maximize expected
geometric growth asymptotically you would treat each bet as being
something that will be infinitely repeated. This should seem clear
enough to most.

If B we can readily find the answers to maximize expected geometric
growth with .179 and .387 for the 2:1 and 5:1 wagers respectively
(Yes, when bets occur in a multiple, simultaneous sense, they can and
often do, if the quantities are great enough and the joint
probabilities allow for it, sum to greater than 1. That's ok, PROVIDED
the future comports to the joint probabilities table)

if C, the answer is the same as A for the same reasons. In other
words, if you have the chance to change your bet size throughout the
course of the wager, and new odds are known, you are best to amend
your bet to the new odds offered.

Upon further examination of this -- I don;t see any "paradox" at all
-- what am I missing? Ralph Vince