April 8, 2013: Shannon's Demon

If I offered you the chance to play a coin flip game with a 50/50 chance of doubling or halving your money, would you take it?

For a player who gambles his entire bankroll each round, it appears to be a wash. No matter the order of returns, if there are an equal number of heads and tails, the player ends up having exactly as much as he did at the start.

For example...

  • Starting bankroll: $100 
  • Round 1 (heads): $200 
  • Round 2 (tails): $100 
  • Round 3 (tails) $50 
  • Round 4 (heads): $100 

However, for a sophisticated investor this game would represent an incredible profit opportunity. Claude Shannon illustrated this by proposing that an intelligent player would wager only half of their bankroll each round. This seemingly small differences turns the game into a winner.

  • Starting bankroll: $100 
  • Round 1 (heads): $150 
  • Round 2 (tails): $112.5 
  • Round 3 (tails) $84.375 
  • Round 4 (heads): $126.5625 

Converting this game into investment-speak, Claude Shannon proposed a portfolio of 50% coin flip and 50% cash. This portfolio was rebalanced at the beginning of each round. The results of this game are quite profound. It shows that risk reduction has the ability to increase returns by bringing your realized portfolio geometric return closer to the weighted arithmetic returns of the portfolio's components.

Taking the bankroll management strategy in a slightly different direction, we can observe how a diversified portfolio of coin flip games that is rebalanced each round offers a return that is far greater than the weighted performance of the individual games.

(This chart will update automatically every 10 minutes or so, offering a new randomized experiment each time.)

Internalizing the benefits of diversification takes time, but I can’t imagine going back to a concentrated portfolio. Diversification combined with rebalancing offers the opportunity to increase returns while simultaneously decreasing risk. Although the real world is far more complicated than a simple coin flip operation, I think the game can help investors understand how looking at assets in isolation never makes sense. Sophisticated investors only care about what an asset will do to a diversified and periodically rebalanced portfolio. If you enjoyed the topic of this post, try Googling phrases such as “volatility pumping”, “volatility harvesting”, “Shannon’s Demon”, and “Kelly criterion.” Additionally a well done paper on the subject can be found here.

12 comments:

  1. I created an Excel model to test your theory, and I found that by only risking 50% each time you were almost guaranteed to go broke in fewer than 200 rounds, and usually in fewer than 100 rounds.

    However, risking 100% each time allowed for longer game play (typically at least 200 rounds, and often tens of thousands of rounds), and sometimes lead to winnings in the billions, trillions, or even quadrillions of dollars!

    I used Excel's RAND function to serve as the coin flip, and an even number was counted as "heads" and an odd number was counted as "tails". There were not a precisely equal number of heads and tails, but of course neither would there be when flipping an actual coin. But they were statistically close in number (less than 0.1% variance).

    Why do you think my findings are so different from yours?

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  2. I created an excel model to test in the same way. I initially found what you did but I realized that the payoff I was using was flawed (*2 for a win and /2 for a loss) and I found that I was not calculating the rebalance appropriately. Once corrected I repeated the 200 rounds 31 times with phenomenal results each time. I even handicapped the return (based on my market of choices spread costs) and narrowed the payout ratio to 1.1. Even with all of this I still received phenomenal returns (though not as good as a frictionless market of course). What calculations are you using for rebalancing the cash and coinflip "pots"? And what is your payout ratio (*2 for a win and /2 for a loss)?

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  3. Hmmm... your odd/even choice may be the issue. With long strings of odd or even you will get wrecked. This is why in an actual random walk that trends up or down you still need to guess the overall trend correctly for Shannon's Demon to work well. Was your odd/even ratio "trending" heavily and you kept betting in the opposite direction?

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  4. Without looking at the spreadsheets its hard to know why our results were different. Here is the google doc that is driving the charts:
    https://docs.google.com/spreadsheet/ccc?key=0AhyXQ0o4HKEqdHpndzJkMXVHMktQS0xHSDR2aWFMbVE#gid=0

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  5. I see your point about taking risks and gambles in investment. I don't risk one hundred percent of my assets, maybe a half will do. And learning from this post makes me want to think if it would be better to do this alongside with the asset management that I applied for in Australia. Properties on one side and money on the other. I got more information about asset management on this link: http://www.ameri-webs.com/2013/09/10/asset-management-a-better-form-of-money-in-the-bank/

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  6. Hello,

    First, love the posts. Second, I have been pondering this for some time in relation to Kelly Betting. I must caveat that I am not that great at maths, so I apologize if there is an obvious answer that I am missing. Considering the scenario posted above (50/50 chance of doubling or halving your money) how does Shannon's Demon betsizing relate to Kelly Criterion betsizing? The way I see it using the standard Kelly equation f*=(p(b+1)-1)/b you would be told to allocate 37.5% to the coin flip and the remaining 62.5% to cash. Calculations: b=4=2/0.5 and f*=0.375=(0.5(4+1)-1)/4. If one runs the Kelly allocation of 62.5/37.5 instead of the Shannon allocation of 50/50, Kelly is clearly inferior to Shannon (I constructed my own excel sheet). Anyone aware of why Kelly would come up so short here? Thanks in advance for any thoughts.

    Keith

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  7. Good post.

    Just wondering over 50 rounds (could be any finitie number) would you offer (lay) the naive investor lottery (re-invest every round) while betting on the sophisticatic investors (have 50% of capital)?

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  8. The two examples are wrong, they do not make any sense, especially not the second one.

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  9. Hi Bernd,

    This is not an intuitive subject for many. Try it out yourself in excel using the rand() function.

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  10. The coin-tossing model is misleading.
    If there's a guy sitting across the table from Shannon matching Shannon's every bet, the game is heavily rigged in Shannon's favor. When the toss comes up heads, guy B loses ALL of what he ponied up. But then when it comes up tails, guy B's profit only amounts to HALF of what he ponied up.
    If you take the coin-tossing metaphor literally, you'll get confused and you may misunderstand Shannon rebalancing.


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  11. this is nice article, I have been searching about 'Shannon Demon' for a long time and your article and excel sheet explains it in a much simple and better way. However, I think this would only work if the odds of making 2 times of ones investment in a stock are same as the odds of losing half of ones investment, which means its more easier for a stock to go north than south ... Please correct me if I am wrong !!

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  12. You are correct, the sample implies there is 50% chance to have 50% loss (half) of the investment but have the same 50% change to have 100% gain (double) of the investment. Too good to be true in real world. A more realistic sample should use 50% gain to make the overall win/loss amount the same as a "fair" game.

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