Quote:
Originally Posted by Ganchrow
A more accurate way to simply state Kelly would be that a bettor should set his "to-win" amount to equal his edge.
For example, if you had an edge of 5% (what you're referring to as an expected value of 105%) you'd bet the following: - odds of +100: bet 5% of bankroll
- odds of -110: bet 5.5% of bankroll
- odds of -200: bet 10% of bankroll
- odds of +200: bet 2.5% of bankroll
You'll note that in each one of these cases a win would net you the same 5% of bankroll.
As to what you refer to as "risk-of-ruin", I'm actually not sure how this makes sense in the context of Kelly. One frequently speaks of a Kelly multiplier, such that a Kelly multiplier of 1/2 would imply a bets of (approximately) half of the full-Kelly stake, and in general a kelly multiplier of κ would imply bets of (approximately) κ × the full-Kelly stake. But of course this has nothing to do with risk-of-ruin in any traditional sense.
Actually, even from an arithmetical perspective, I'm a bit perplexed by your risk-of-ruin coefficient. You state, "Kelly says you should always multiply your bankroll x expected value x the risk of ruin you are willing to accept," and then curiously continue with "since I use 1% as risk of ruin, the risk of ruin falls away and you are left with expected value", which if I hadn't seen you write elsewhere I'd assume to be a typo.
Using Kelly or (much more typically) some fraction thereof, can frequently be a valuable risk management tool for the advantage player. However, what you've described, while perhaps (depending on how you explain your risk-of-ruin coefficient) close to Kelly for bets at odds near even, will drastically diverge as odds lengthen or shorten away from that point. |
I see what the problem is. First, expected value and odds are not the same thing. Just because the book is offering +$400 does not mean that you will receive $400 for every $100 bet. Suppose Dallas vs New England has Dallas at +$400 moneyline. Just because you bet $100 on Dallas does not mean you will receive $400. Even if you could make this bet over enough number of trials to enter into the law of large numbers it does not mean you can expect $400 for every $100 bet. The odds the book gives do not reflect the true expected value. If this event occurred over enough trials to be in the law of large numbers the outcome as a percentage would not equal the odds given by the bookmaker. I know you know this. So perhaps I am using the term expected value differently than you are.
The true expected value is the winnings or losses you would have if you bet at these odds over a large number of trials. Let's say that you calculated that if this game was played 1 million times, Dallas would win the game 23% of the time. To make it simple we will use 100 games for the calculations. So, for 100 games Dallas would go 23-77. If you bet on each of these games you would end up winning $9200 on the 23 games @ +$400 and losing $7700 on the 77 games @$100. So, the profit would be $1500. Betting $100 on 100 games is $10,000 at risk, netting $1,500 means you got back $11,500. The expected value is then $11,500 for $10,000 bet or 15%. So, if your "guess" that Dallas would win this matchup 23% of the time is correct and a book is offering the matchup at +$400 then the expected value is 1.15, not 4. Of course I would NEVER bet 15% of my bank on any one bet, I don't care what Kellly says, which brings me to my second point.
Second, determining the raw kelly number is only the first step. Now we have to look at variance, and use variance to calculate risk of ruin. I am only willing to tolerate a risk of ruin near zero, so I am going to lower the kelly number quite a bit. Kelly does not care about variance, Kelly would let you take your bankroll down to 1% of its original size right before you doubled it. I would have a heart attack.
So, I want to use a bet size small enough that risk of ruin is 1%.
So, the unit sizes I give are the true expected value plugged into kelly but then using a risk of ruin of 1% to get a bet size that will keep variance very low.
This is standard practice in blackjack because the variance in blackjack is so ridiculous.