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# Interesting result from Kelly's Criterion

Topic closed. 9 replies. Last post 7 years ago by jwhou.

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 Posted: August 25, 2010, 1:26 am - IP Logged

Kelly's Criterion is where informatics is used in gambling or investing and is for sizing your bet as a fraction of your available capital.   It's actually a logarithmic proof and I'm not sure I'm doing it right, but it works out as:

f = (bp - q)/b

where f = fraction of capital

b = expectations or gamblers odds

p = probability of win

q = probability of loss

Just for a lark, I plugged in the values for tomorrow's 649 jackpot of 3.5 million with simplifications such as ignoring possibility of sharing the prize, ignoring all subsidiary prizes etc.   It says I should use -3.99 times of my available capital.   Basically, it says I should sell tickets, not buy them  and to try to sell four times more tickets than I have money to buy ie.: underwrite the payout myself but hedge it by buying one ticket for every virtual 4 tickets I sell.

Anybody want to buy tickets through me?

I wonder if those internet lotto ticket agents use this?

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 Posted: August 25, 2010, 12:49 pm - IP Logged

jwhou, I didn't know about Kelly's Criterion. What's about?

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 Posted: August 25, 2010, 1:12 pm - IP Logged

jwhou, I didn't know about Kelly's Criterion. What's about?

http://en.wikipedia.org/wiki/Kelly_criterion

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 Posted: August 25, 2010, 3:33 pm - IP Logged

jwhou, I didn't know about Kelly's Criterion. What's about?

The Kelly Criterion was based on Claude Shannon's work in Informatics.   Claude Shannon is the guy that gave us digital electronics, and telecommunications and even genetics.   His last unpublished paper before dying of Alzheimers in 1978 was to be the ultimate solution to the Rubik's cube, something which we've only just been able to sort out but only with Google's server farm as a giant supercomputer.   He also spent many years trying to determine how small can a Unicycle be and still be ridden and cites his greatest achievement as juggling while riding a Unicycle through the halls of MIT.   It's been said that a comparison between him and Einstein would be unfair to Shannon.

Essentially it's the concept of logarithmic or proportional betting.   Sizing your investment by Kelly's Criterion should guarantee that your investment will grow exponentially in the long run.   Bernoulli gave a mathematically identical advice for investment a couple hundred years earlier but very few people knew about it as his was more obtuse.

I just plugged in the Mega Millions numbers into it.   This time I plugged in all the prize levels (those subsidiary prizes are real rip offs) but I still assumed that the Jackpot would not be split.   Works out to f= -492 so we need to sell 492 derivative tickets on the Mega Millions draw for every real ticket that we buy if we want to make money.

I looked up Claude Shannon's and Ed Thorp's book on Amazon and it's \$335 so I'll be reading a book by some unknown person that talks about Shannon's work for \$10 instead.   Maybe the library has Shannon's book.   Claude Shannon and Ed Thorp used his theories in Vegas at BlackJacks and Roulette and used those winnings to found a successful hedge fund company on Wallstreet (Princeton-Newport Partners) which is known for consistently netting a 28% per annum return.

The Kelly Criterion is probably just the tip of the iceberg in terms of what Shannon and Thorp came up with and I'm certain that there are many interpretations of his work in use in the investment world today.

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 Posted: August 25, 2010, 3:54 pm - IP Logged

At how I'm a little surprised at how large a value the equation came up with and it's all from the subsidiary prizes which means no matter how high the jackpot gets, even if it's enough to get the expected return above parity which seems to be the decision point for most people to jump in big time on the lottery, the subsidiary prize structure is so heavily biased against the player that you still can't win in the long run because you can never have enough capital to wait for the big hit.   I was hoping that it would be closer to zero so that monitoring the jackpot or the scratch-off odds would occasionally tip the investment into the positive, that way you could set aside a certain amount of money in a separate account and invest it fractionally by Kelly's Criterion and in the long run it should grow exponentially.

The lotteries are easier for Kelly's Criterion because the odds and payout are known but unfortunately the payout and the odds is heavily biased against the player.   With investments there's the concept of how to evaluate the probability as well as how to evaluate the payout or loss.   There seems to be a few equations that might do the trick if I can just figure out how to use them...

Michigan
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 Posted: August 25, 2010, 3:57 pm - IP Logged

The Kelly Criterion was based on Claude Shannon's work in Informatics.   Claude Shannon is the guy that gave us digital electronics, and telecommunications and even genetics.   His last unpublished paper before dying of Alzheimers in 1978 was to be the ultimate solution to the Rubik's cube, something which we've only just been able to sort out but only with Google's server farm as a giant supercomputer.   He also spent many years trying to determine how small can a Unicycle be and still be ridden and cites his greatest achievement as juggling while riding a Unicycle through the halls of MIT.   It's been said that a comparison between him and Einstein would be unfair to Shannon.

Essentially it's the concept of logarithmic or proportional betting.   Sizing your investment by Kelly's Criterion should guarantee that your investment will grow exponentially in the long run.   Bernoulli gave a mathematically identical advice for investment a couple hundred years earlier but very few people knew about it as his was more obtuse.

I just plugged in the Mega Millions numbers into it.   This time I plugged in all the prize levels (those subsidiary prizes are real rip offs) but I still assumed that the Jackpot would not be split.   Works out to f= -492 so we need to sell 492 derivative tickets on the Mega Millions draw for every real ticket that we buy if we want to make money.

I looked up Claude Shannon's and Ed Thorp's book on Amazon and it's \$335 so I'll be reading a book by some unknown person that talks about Shannon's work for \$10 instead.   Maybe the library has Shannon's book.   Claude Shannon and Ed Thorp used his theories in Vegas at BlackJacks and Roulette and used those winnings to found a successful hedge fund company on Wallstreet (Princeton-Newport Partners) which is known for consistently netting a 28% per annum return.

The Kelly Criterion is probably just the tip of the iceberg in terms of what Shannon and Thorp came up with and I'm certain that there are many interpretations of his work in use in the investment world today.

I am positive you know this but for the benefit of others that haven't got a clue, it only works when you have a positive ROI (rate of return).  Lotteries all have a negative ROI for the player.

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 Posted: August 25, 2010, 4:03 pm - IP Logged

I am positive you know this but for the benefit of others that haven't got a clue, it only works when you have a positive ROI (rate of return).  Lotteries all have a negative ROI for the player.

Which is why the results are negative.   You can still make money of negative results but you need to hedge some short sales or underwrite derivatives.   Unfortunately, the government aint going let you do that without substantial capital and even then they're not going to let you do it with the lotto.

The thing is that there are rare occasions when a Jackpot lottery or a scratchoff lottery has a positive ROI due to the probability of rollover in Jackpot

Michigan
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 Posted: August 25, 2010, 4:08 pm - IP Logged

When the jackpot exceeds the odds, what does Kelly say?  I never plugged it into that situation.

Kelly was (is?) popular with horseracing.  A situation where you can be ahead of the game and can also decide if there are overlays.  When an overlay occurs Kelly gives the optimum bet, however, most of the crowd thinks it is too aggressive.  Too easy to be off on what you consider to be an overlay.  Fractional Kelly is preferred.

LAS VEGAS
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November 22, 2006
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 Posted: August 25, 2010, 4:23 pm - IP Logged

At how I'm a little surprised at how large a value the equation came up with and it's all from the subsidiary prizes which means no matter how high the jackpot gets, even if it's enough to get the expected return above parity which seems to be the decision point for most people to jump in big time on the lottery, the subsidiary prize structure is so heavily biased against the player that you still can't win in the long run because you can never have enough capital to wait for the big hit.   I was hoping that it would be closer to zero so that monitoring the jackpot or the scratch-off odds would occasionally tip the investment into the positive, that way you could set aside a certain amount of money in a separate account and invest it fractionally by Kelly's Criterion and in the long run it should grow exponentially.

The lotteries are easier for Kelly's Criterion because the odds and payout are known but unfortunately the payout and the odds is heavily biased against the player.   With investments there's the concept of how to evaluate the probability as well as how to evaluate the payout or loss.   There seems to be a few equations that might do the trick if I can just figure out how to use them...

Good posting jwhou, Kelly Criteria alive & well @ Vegas Racing Books....

Bye the Bye, for all who are interested, ROI = RETURN ON INVESTMENT and yes player neeeds a positive ROI to overcome the odds long term. Overall very hard to beat negative math expectation with just more conventional math

Happy Hunting.

EddessaKnight

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 Posted: August 25, 2010, 5:04 pm - IP Logged

When the jackpot exceeds the odds, what does Kelly say?  I never plugged it into that situation.

Kelly was (is?) popular with horseracing.  A situation where you can be ahead of the game and can also decide if there are overlays.  When an overlay occurs Kelly gives the optimum bet, however, most of the crowd thinks it is too aggressive.  Too easy to be off on what you consider to be an overlay.  Fractional Kelly is preferred.

Well, the Mega Millions draw is getting close, the cash value is 94.2 million but the odds are 1:175 million.   If you plug in the numbers, you get f=-492 but the contribution from the jackpot prize is only f=-1.9 hence the subsidiary prizes are way too small given the relative odds.  I suspect that you would need a huge  jackpot, much larger than would be necessary for a positive ROI before the Kelly's Criterion will tell you to buy instead of hedge because you would still have to offset the effect of all those smaller prizes which occur more often but I think I may not be attributing the costs of those subsidiaries correctly since those chances basically come free with the ticket if you've included that cost in evaluating for the jackpot (will have to work on that).   Maybe that's why Princeton-Newton is a Hedge fund, there's probably more hedging opportunities then buying opportunities.    Shannon also had a 50/50 portfolio rebalancing strategy that harvests volatility but there' the issue of the transaction costs but I'm sure the stockbroker would love it if you tried the method.

With horse racing, you're betting on inefficiencies in the bookmaker setting the odds that you're betting on hence you're assuming that your information about the odds are better than the bookie's.   I wouldn't want to use Kelly's on the racetrack without identifying and tracking side information sources and analyzing them historically.   That's unlikely with the racetrack as the sources are bound to be intermittent and you won't be able to access the same source for each race.   However with the stock market, you can identify consistent sources such as rating companies, federal interest rate, unemployment rate, consumer index rate, US treasury bond rate, P/E ratio, sale volume, first derivative, second derivative and third derivative values etc.   Indeed, most people use 0.5 as the probability for stock investment rather than track it as a function of side information, this is one of the reasons why people believe it's best for short term volatility but there is a logarithmic function relating current capital to previous capital and probability due to individual surprisal bits.   I think it would be possible to come up with a reasonable approximation of the probability function but I don't know how to address the payout for stocks yet other than to set a increment value that I want to consider as sufficient for a transaction and to base the probability function on such movements with multiple surprisals to account for multiples of that price movement.

To bet a fraction of the Kelly bet is to allow for the possibility of unknown side information and is prudent with all Kelly bets.   The thing is that with lottery, the probabilities are known exactly and the payouts are known with the exception of pari-mutuals, the problem with lotteries is that there are usually no inefficiencies in the prize structure to take advantage of.

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