As described in Fortune's Formula (I hope to remember correctly as I don't have the book in front of me) Shannon came up with this idea.

Pick a stock that doubles or halves everyday (50% chance of that happening) and put 50% of your funds in it leaving the other 50% in cash. Rebalance everyday. You end up making a lot of money off a random walk.

Does it actually work? Shannon says no because the commissions would kill you. But the crucial problem (as noted by the author) seems to find a stock that volatile (without the risk of going bankrupt for good, i.e. game over).

Can you think of anything that  behaves that way? Commodities? Currencies? Art? Stamps? Pink sheet stocks (they do go bust but you can modify the system and diversify if the odds work)? Furs? Umbrellas? (I'm actually serious, I'm trying to think of something seasonal/cyclical/fashionable - well you get the idea)

Shannons idea was later popularized by Thomas Cover, spreading the concept of "universal portfolios" ( also goes by the name of "volatility pumping"), which given a certain number of criteria would beat the best stock in the selection universe in hindsight.

Claude Shannon, Thomas Cover, dig deeper and Mr Simons will show up as well ( or his predecessor...)

Thomas Cover's book Elements of Information Theory has a chapter on information theory and the stock market.

Further interesting articles are found here

Issues are how to 1)optimize the rebalancing process 2) take account of transaction costs (discussed by Bloom and Kalai) 3)be able to both go long and short ( discussed by Cross & Barron)

This french gentleman claims to have implemented such algorithms. I would be very interested in taking a deeper look into them, for sure...

shit, does this mean RFM is gonna start talkin about Kelly?

anyway, if you start with 1 dollar, then, after n days your wealth goes to:

1*(1+.5*x1)*(1+.5*x2)*.....*(1+.5*xn) where xi denotes the random variable of the ith day's return.

if the returns are independent, then the expected wealth after n days is the product of the expectations.  but computing one of those expectations yields:

0.5*(1+.5)+.5*(1-.5*.5)=1.25.

so, your expected wealth after n days is (1.25)^n

for me the big issue is simply the stylized assumption that returns are stationary.

nothing behaves like this stock, other than my blood pressure.

Ah, but could one construct an option portfolio that behaved like [a less volatile] version of this stock, and then trade in and out of that option portfolio . . .

Well, but that option portfolio would have a characteristic that the stock wouldn't have- theta. 

gambler's suicide stategy viz. bet on red (in roulette), if you win, you've doubled up, go home, if you lose, double your bet. Iterate

In the universal portfolio case, you don't go home when you win, and you don't double up when you lose as you always invest a fraction of your assets in every instrument.

Otherwise, the main difference with roulette is the diversity of your investment universe, the lower transaction costs (slippage < house edge), the lower ticksize of bets ( you run out of cash only at the limit ) and the fact that you can stay a little longer at the table with a winning strategy without risking meeting this gentleman with a hammer ready to smash your fingers into "fleshy bone meal"

shavinOccam, I think the Universal Portfolio algorithms have a dominantly contrarian weighting scheme. The weights will "learn" to hold a little more of the outperforming stocks but it takes a very long time and every step before that, any outperforming stock will be pumped out of money used to redistribute in other less well performing assets. That's according to my understanding.

...you may reach such a low level that it will take forever to return to your starting position, no?

Worse case scenario, yes, but doesn't really happen in practice. The drawdowns are nonetheless horrific-

The problem there is really the opposite problem compared to Petersburg - that is, if you keep winning, pretty soon your position is going to be too large compared to the float and the process will defeat itself

Also true in playback with no consideration to slippage and trans costs. In practice though, you need to find an optimal rebalancing rule so that you don't keep churning your portfolio. You lose the optimality in being able to rebalance in the hypothetical near-continuous time and thereby also the compounding effect. The results on final P/L are dramatically lowered so your position never gets that big

The first Thomas Cover papers spurred a whole lot of research in the area, all of them using the same dataset of NYSE stocks, for the purpose of benchmarking. Results of the original Cover algorithm and extensions of his models are available in Borodin and Blum and Kalai.

What I especially like about Borodin is that he plays the strategy backwards in time and still produces ok results. So all is not an effect of the huge bullmarket. Nevertheless there is survivorship bias in the results...

sorry, i mis-read the original post.

Claude Shannon, Thomas Cover, dig deeper and Mr Simons will show up as well ( or his predecessor...)

It's interesting that Berlekamp took out the passage about selling out his interests to precisely Simons.