Kelly Betting

[PerpetualCzech]

I'm sure this has been discussed here before but for those that haven't seen it, I just came across a simple formula that represents what the Kelly system claims to be your ideal bet size. Here it is:

(C*P - 1)/(C-1)

where,
C- is the coefficient representing your payout amount
P- is how likely your bet will hit (between 0 and 1)

So C for a standard 11-to-10 bet is 1.909090... (It's the number that you multiply your bet amount by to get your *total* amount returned)

Example#1: If you think you can consistently pick 55% winners in the NBA you should bet (1.9090909*0.55-1)/(1.9090909-1) or 5.5% of your bankroll on each bet (assuming you are laying 110 to win 100)

Example#2: If you have an underdog at +400 that you think has a 25% chance of hitting, you should bet (5*0.25-1)/(5-1) or 6.25% of your bankroll on it.

I want to start exploring how to figure out chances of losing a given amount of your bankroll if you use this strategy. Anyone have any ideas on how to approach this deductively? (i.e. without having to resort to simulations?)

 

[dawgball]

I like this concept, but if you multiply 11*1.90909.., you get 21. I must be doing something wrong.

 

[redsfann]

Dawgball-

No, that is correct. If the line is -110, you need to wager 11 to win 10, right? Therefore, your return for that 11 dollar wager if its a winner is 21-- your original 11 and the 10 that you won.
I think thats how it goes--- :shrug:

 

[PerpetualCzech]

Yes, that's right. 21 is the total amount you receive back as a payout, the 10 that you won plus the original 11 that you bet.

 

[yak merchant]

PC,

Let me know what you come up with this thing. I've looked into it several times, and come up with some different outcomes usually depending on length of losing streak. But alot of people swear to the fact that the betting using the KC will end up no where other than broke.

 

[PerpetualCzech]

Like I said, I don't have much of an idea where to start to approach this deductively, but I will try and run a few simulations using Excel and I'll post what I come up with. Not sure it will be worth it though since surprisingly there is not much interest in this.

Off the top though, it's hard how to see how using Kelly makes you go completly broke since you are only betting a small percentage of your bankroll at a time, never the whole thing.

 

[yak merchant]

Well for me 5.5 % of my bankroll is a huge bet, let alone on a 55% shot. Second if you were to really employ KC, you would figure you chances on each bet. If you had proof that you could hit 65% on a certain play then you'd bet.

(1.9090909*0.65-1)/(1.9090909-1) or 26.4%

Miss 4 of those in a row and you are broke.

 

[PerpetualCzech]

Ah, that's where the system is deceiving. The 2nd-4th bets would be 25% of your *current* bankroll, not your original one. If you miss 4 of those in a row you would still have just under 30% of your original bankroll.

Also, keep in mind that missing 4 65% shots in a row happens only 1.5% of the time. But at least this example begins to give us a feel for what it takes to seriously cut into your original bankroll.

 

[PerpetualCzech]

OK, I have run my first set of simulations and am interested in what others think about the results.

Using the scenario from Example#1 that yak merchant and I were discussing, I assumed that we can pick winners at a 55% rate and that we were betting into odds of -110 (risk 110 to win 100). I ran 10000 simulations of betting using the Kelly formula earlier in this thread and counted how many times one of these 2 things happened *first*:

1) Your bankroll doubled in size (i.e. grew to 200 from 100)

or

2) Your bankroll was cut in half (i.e. shrank to 50 from 100)

It happened that 6713 times out of 10000 your bankroll doubled itself first before it was cut in half, for a rate of 67.13%.

Now the question is, is this an acceptable rate of risk??

More simulations to come ...

 

[PerpetualCzech]

OK, I just ran 4 more simulations and got some very interesting results:

Simulation #1: Picking 60% winners at odds of -110

Kelly says you should bet 16% (!) of your bankroll on every bet. 10000 simulations gave a result that bankroll will double 67.72% of the time before it will be cut in half.



Simulation #2: Underdog that will hit 25% of the time at odds of +400

Kelly says to bet 6.25% of bankroll on each bet. Bankroll will double 66.26% of the time first before it will halve itself.



Simulation #3: Favourite that will hit 2/3 of the time at odds of -150.

Kelly says to bet 16.67% of bankroll on each bet. Bankroll will double 69.03% of the time first before it will halve itself.




Simulation #4: Favourite that will hit 70% of the time at odds of -200

Kelly says to bet 10% of bankroll on each bet. Bankroll will double 67.48% of the time before it will halve itself.



It looks like no matter what advantage you are betting with, Kelly seems to be aiming to double your bankroll 2/3's of the time and cut your bankroll in half 1/3 of the time! Again, is this an acceptable risk to take? Personally, my feeling is that this risk is too high. It's the equivalent of getting a 2-to-1 favourite at +200 and betting half your bankroll on it. Interestingly, if you get these terms on a bet, the formula gives out a result that your ideal bet is 50% of your bankroll, so that is a good doublecheck of the simulation results.

(If anyone would like a copy of the spreadsheet that I used to make the simulations, feel free to ask Jack for my email address)

 

[dawgball]

Even though it would be hard for me to use the Kelly system, your thoughts here are interesting. Are there any ways that you can tweak your parameters?

Can you put the target number of doubling your bankroll to 80% to see what size wager you should place?

If a player can pick 55% winners at -110, what should his/her pecentage bet be to have a 80% expectancy to double his/her bankroll before it is cut in half?

I am expecting this number to be around 2-3%.

 

[PerpetualCzech]

Just by the way my spreadsheet is set up, I can't enter the target percentage of doubling, I need to enter the size of the wager instead. If I enter 2.5% as the size of the wager, then these are the results for picking 55% winners at odds of -110:

Out of 5000 runs, 4617 of them doubled the bankroll first before cutting it in half for a percentage of 92.35%. The actual percentage is probably higher for reasons too technical to get into here, but you can definitely take the minimum result to be 92.35%.



So like I said in my previous post, my gut feeling is that Kelly is advocating a higher base bet than I am comfortable with. Dawgball, it looks like you agree. I think the reasons for this are due to the uncertain nature of sportsbetting. Let me show you what I mean by way of example.

To take the basic example which the simulations show, let's say you get a proposition which you feel you have a 2-to-1 chance of hitting and you are offered odds of +200. How much of your bankroll are you willing to risk on this? Kelly says 50%. Depending on the sport that you are familiar with, here are some examples that are similar to this situation:

NHL, Detroit at home -0.5 goals against Buffalo

NBA, Dallas at home on the money line against Minnesota

MLB, Yankees with Clemens at home to an average starter from the Red Sox

NFL, Green Bay at home to the Giants on the money line

Soccer, Manchester United to win at home to Leeds

I doubt that in any of these situations you'd be willing to put half your bankroll on odds of +200. But what about this situation? Say someone comes up to you with a bag of 3 marbles, 2 of which are red and 1 black. He then gives you odds of +200 that you can pull out a red marble. Assuming you can trust the integrity of the game, I think now you are willing to put a much higher amount down than in the situations above, maybe even the full 50% of your bankroll if you know that you can play the game more than once.

The reason for this is that Kelly Betting is designed for situations for which you have perfect information . Anytime as a professional handicapper you assign what true odds there should be on a sporting event, there is always a margin of error that you have to give to it, even if you aren't consciously aware of it. There is always an element of "what does the oddsmaker know that I don't" thrown in that forces us to lower our bet.

The problem with Kelly is that it doesn't take this margin of error into account. It assumes perfect information for the probability of an event, like in the marble example above, which we will never get when we handicap sports.