Who stole my EV?

I wanted to reframe Shannon's Demon which I came across lately from a betting perspective and see what it tells us about Expected Value and staking.

The idea is that I have a bank, say €100.  I have a bet that I can make every day, suppose I am quite a professional and I am easily capable of finding sportsbooks offering 2/1 (3.0, +200) on a true even money chance (2.0, +100).  That is a pretty huge edge right there.  There is no way that I can screw this one up!  (Is there?).

I am going to bet half my bank each time.  Its a big edge, I dont want to waste it.  Also as I half my bank it will prevent me from going bust, it may reduce during a long drawdown but cannot reach zero (in theory!) and is unlikely to go below say €1 - which would double up fast again in good times. And as likely it is to drop below €1, it is equally likely to double up to €10,000s which more than balances out that risk.

So you take your €100 balance and bet half ie €50 @ 2/1 with a 50/50 chance of winning.  Notice that after the first bet you either win or lose, and so your new balance is €50 or €200.  You either half your balance or double your balance. This is true no matter what your balance is as long as you persist with betting half your bank each time.

You are in a situation where 50% of the time you half your bank, 50% of the time you double it.  This is obvious now that long term you will make zero profit continuing this strategy.  You win the first day, you lose the next day, you are back to where you were.  After 100 days, you expect to double your bank about 50 times and halve it about 50 times.  You are back to where you started.

But.. but.. the expected value of the first bet is P[W] x Profit - P[L] x Loss = 0.5 x €100 - 0.5 x €50 = €25. And every bet after this has positive expected value, even more if the first bet wins. Where is this gone? Who stole my EV?  It is still there, but you've picked a distribution of results that sees you expect to break even the overwhelming majority of the time and a small chance of winning a big lottery type win.  All the EV is achieved only when you hit the lottery type scenario for example where you double up 24 times in a row to over €1 billion.  Your chosen staking method is playing around with the median long term profit (in this case €0) vs the average (in this case many €billions).

The solution is to maximise your Expected Growth rather than your Expected Value.  We all already know this intuitely.  As absolutely nobody wanted to bet their entire bank on each wager - the bigger the stake, the bigger the EV i.e. literally maximising Expected Value.  That leaves you a 50% chance you hit ruin on the first day and almost 100% chance you lose a bet (with your entire bank riding on it) within a week. Nobody wanted to do this.

I think some of this kind of stuff gets dismissed as overly theoretical mathematical nonsense. And it probably is at times.  But with a big edge and a small bank roll, and a desire to grow it fast, I genuinely think this is a trap that someone could fall into.  

So just be conscious of the pitfalls of overstaking.  Dont turn a lucrative edge into a breakeven/losing strategy.

I've glossed over most of the detail, so make sure to see the thread linked at the top for a more in depth dive.