Question for YOU on Kelly and Bet Sizing

[Robert Fischer]

I want to see what you guys come up with on this question.


I'm in the VIP room and they want to loosen me up, and are offering me a dream deal.

Offering me 3-2 odds on a Coin Flip for as long as I want to play!!

I'm ready to start drinking and start gambling and hopefully impress my lady friend...

There's one catch:

Have to bet on 'Heads' AND there are two types of coins.
"Type A coins" and "Type B coins".

Type A coins = 50% probability of Heads.

Type B coins = 0% probability of Heads. Yea, they are "Tails" on both sides.

Fortunately, 90% of the coins are "Type A coins", and coins are selected at random and must be flipped if drawn.

ROI for Type A Coin = $1.25
ROI for Type B Coin = $0.00
ROI for average = $1.125
Bankroll = $1,000,000

I want to use Kelly Criterion to calculate my bet size.
However if I wager with Kelly based on the $1.125 ROI, I am worried that the 10% of the time when I draw the Type B 'Tails' coin, will expose me to greater risk than necessary.



What would be your bet sizing approach here??
Would prefer if some version of Kelly were used, but all answers are welcome.

[HUSKER55]

I am waiting for my first cup-o-joe.

what a way to start my day!

10 coins
9 heads
11 tails
20 total chances. you can see where I am going with this. For both sides to have an equal chance all coins have to be drawn.


are you sure this guy isn't trying to hustle your girl?

[windoor]

Originally Posted by Robert Fischer

I want to see what you guys come up with on this question.


I'm in the VIP room and they want to loosen me up, and are offering me a dream deal.

Offering me 3-2 odds on a Coin Flip for as long as I want to play!!

I'm ready to start drinking and start gambling and hopefully impress my lady friend...

There's one catch:

Have to bet on 'Heads' AND there are two types of coins.
"Type A coins" and "Type B coins".

Type A coins = 50% probability of Heads.

Type B coins = 0% probability of Heads. Yea, they are "Tails" on both sides.

Fortunately, 90% of the coins are "Type A coins", and coins are selected at random and must be flipped if drawn.

ROI for Type A Coin = $1.25
ROI for Type B Coin = $0.00
ROI for average = $1.125
Bankroll = $1,000,000

I want to use Kelly Criterion to calculate my bet size.
However if I wager with Kelly based on the $1.125 ROI, I am worried that the 10% of the time when I draw the Type B 'Tails' coin, will expose me to greater risk than necessary.



What would be your bet sizing approach here??
Would prefer if some version of Kelly were used, but all answers are welcome.

I too, do not have enough coffee in me yet at this hour, but will give my thoughts anyway.

How many coins are in play?
Is the flipped coin put back in the container ?

I would assume the answers would be many and more and yes.

I do not believe this is a profitable Endeavor unless you use a progression.
We all know (or should know) how risky this is.

Here is some more insight on this type of wagering.

http://www.horseracinggold.com/horse...-article-4.htm

Given the criteria, your win percent should be about 40% or a break even at 3-2 with a flat wager. Expected maximum Consecutive loss's at a 40% hit rate is somewhere around 9 or 10 (I think) or $141. minimum bank needed for every $2 objective. or $439 to cover 12 straight loss's for every $2 of objective. Move the decimals points as you will.

Given a 1,000,000 bank. The objective can not be more that $2000 per hit. Which should be about 80K profit per every 100 plays. This should allow up to 16 straight loss's

Not a believer in Kelly. I always thought of it as a way to go broke more slowly when your selection method is not sound.

If the selection method is sound, then you just need the proper sized bank to support the wager.

Regards,

Windoor

[MJC922]

Only thing I can see as being relevant is the win% and the odds you're getting. If I'm following correctly the win percentage for nine out of the ten flips is 50% and the other ten percent of the time it's zero so you're looking at 45% wins overall at 3-2 odds right?

Kelly says $83,333 per wager with a million dollar bank.

[Robert Goren]

Originally Posted by MJC922

Only thing I can see as being relevant is the win% and the odds you're getting. If I'm following correctly the win percentage for nine out of the ten flips is 50% and the other ten percent of the time it's zero so you're looking at 45% wins overall at 3-2 odds right?

Kelly says $83,333 per wager with a million dollar bank.

My Kelly Calculator says the same amount. My personal opinion that a bet of that size on a coin flip even if you had a million dollars is pure folly. Then there are tax considerations to be dealt with too. Even if you paid off at 3/2 on the spot, the after tax gains would be less if are in an area that taxes gambling winnings. If this were horse racing, there is some chance that you would be taxed on your winning bets and would not be allowed to deduct your losing bets.
This assumes everything is on the up and up. In real life, it is unlikely that would be the case.

[Robert Fischer]

I too need a coffee .

Still reading over this so I will start with a quick one.

Originally Posted by MJC922

Only thing I can see as being relevant is the win% and the odds you're getting. If I'm following correctly the win percentage for nine out of the ten flips is 50% and the other ten percent of the time it's zero so you're looking at 45% wins overall at 3-2 odds right?

Kelly says $83,333 per wager with a million dollar bank.

Exactly right MJC922.

Here's my dilemma - Yes I have an overall average win% of 45% @ 3-2.

The problem is that although I have an overall edge ($2 ROI of $2.25 or $1ROI=1.125), I always have a 10% chance of drawing the "Tails" coin when making each individual wager.

I'm hoping that it's simply a concept that I need to 'grasp'.

Is the right approach simply to average the edge, and wager each flip to that average edge?

Or, will doing so with Kelly be more risky, because of that 0% coin that occurs on approximately every 10 flips?

[GameTheory]

Originally Posted by Robert Fischer

I too need a coffee .

Still reading over this so I will start with a quick one.



Exactly right MJC922.

Here's my dilemma - Yes I have an overall average win% of 45% @ 3-2.

The problem is that although I have an overall edge ($2 ROI of $2.25 or $1ROI=1.125), I always have a 10% chance of drawing the "Tails" coin when making each individual wager.

I'm hoping that it's simply a concept that I need to 'grasp'.

Is the right approach simply to average the edge, and wager each flip to that average edge?

Or, will doing so with Kelly be more risky, because of that 0% coin that occurs on approximately every 10 flips?

I'm assuming you must make your bet before the coin is drawn (otherwise this exercise makes no sense). I'm also assuming the the coins are replaced after you drawn them and can be drawn again, so the percentages stay fixed. Then you just have to use the information you have at that time of your bet, which is that you have a 45% chance of winning your bet at odds of 3-2. The chance of getting the tails coin is built-in to the win percentage, and so it is fairly simple and you bet accordingly.

It becomes more complicated if the two coins both can win, but at different rates and different odds, and/or if you can find out which type of coin is drawn and then make your bet. In the first case you'd have a more complicated problem, but your percentage would still be fixed because the information you have at bet time is always the same. In the second case, your percentage would change with each bet because the information would be different for each bet. So the key is always "what do you know at the point in time you must make your wager" and to use all that information...

[Robert Fischer]

Originally Posted by HUSKER55

are you sure this guy isn't trying to hustle your girl?

Either he is, or she is working with the Casino to keep me gambling .

[Robert Fischer]

Originally Posted by windoor

I do not believe this is a profitable Endeavor unless you use a progression.
...

Not a believer in Kelly. I always thought of it as a way to go broke more slowly when your selection method is not sound.

I'm open to any and all ideas .

I tried your link, but I get a "404 not found" error.

[Robert Fischer]

Originally Posted by GameTheory

I'm assuming you must make your bet before the coin is drawn (otherwise this exercise makes no sense). I'm also assuming the the coins are replaced after you drawn them and can be drawn again, so the percentages stay fixed. Then you just have to use the information you have at that time of your bet, which is that you have a 45% chance of winning your bet at odds of 3-2. The chance of getting the tails coin is built-in to the win percentage, and so it is fairly simple and you bet accordingly.

It becomes more complicated if the two coins both can win, but at different rates and different odds, and/or if you can find out which type of coin is drawn and then make your bet. In the first case you'd have a more complicated problem, but your percentage would still be fixed because the information you have at bet time is always the same. In the second case, your percentage would change with each bet because the information would be different for each bet. So the key is always "what do you know at the point in time you must make your wager" and to use all that information...

Thank you for the advice. All of your assumptions regarding the game rules are correct by the way.

For some reason it is bothering that 10% of the time I will be making a wager based on a 45%chance of winning, - when in reality there is zero chance of that individual wager winning.
This is throwing off my logic.
Kelly seems to be designed to tackle known or "fixed" probabilities and edges.

Obviously this is a simplistic version of a horse racing play.

[Jeff P]

If you can devise some way of identifying which coin is going to be in play ahead of time - you change the 'rules' of the problem. If you are betting on horses instead of coins - think in terms of stuff like track profile and physical condition of the horse.

Short of that, you are 'stuck' with having to make best use of information availble to you at the time you bet.

FYI, if the nature of the game for you really is 45% wins at avg odds of 3/2 then the answer to the question ("Given a $1M bankroll, how much should I bet?") comes down to pool size and not Kelly.

Bet sizes of approx $83k based on Kelly are going to kill your odds.


-jp

.

[GameTheory]

Originally Posted by Robert Fischer

Thank you for the advice. All of your assumptions regarding the game rules are correct by the way.

For some reason it is bothering that 10% of the time I will be making a wager based on a 45%chance of winning, - when in reality there is zero chance of that individual wager winning.
This is throwing off my logic.
Kelly seems to be designed to tackle known or "fixed" probabilities and edges.

It is known, it just has two components. And...it is not like horse racing (too simplified), which is where people screw up in applying it to horse racing because they tend to be looking for "the kelly percentage" to use based on some average when in fact it should be adjusted race-by-race for each situation. In your example, you know the win percentage and the odds -- the stuff about the different coins is irrelevant once you've calculated that. You can't use information you don't have yet to make your bet. It is just like having your horse lose and then saying "well I shouldn't have bet on a loser -- why did I do that?" when of course who the winner and the losers were going to be wasn't determined yet when you made your bet.

[Robert Fischer]

Originally Posted by Jeff P

If you can devise some way of identifying which coin is going to be in play ahead of time - you change the 'rules' of the problem. If you are betting on horses instead of coins - think in terms of stuff like track profile and physical condition of the horse.

Short of that, you are 'stuck' with having to make best use of information availble to you at the time you bet.

FYI, if the nature of the game for you really is 45% wins at avg odds of 3/2 then the answer to the question ("Given a $1M bankroll, how much should I bet?") comes down to pool size and not Kelly.

Bet sizes of approx $83k based on Kelly are going to kill your odds.


-jp

.

Thanks Jeff P,

Unfortunately, my bankroll isn't $1,000,000 yet, and the actual values are more complex with narrow margins, but (aside from the pool-size limits) this is the basic idea.

Pool size is very important.
You want your wagers tied into a percent-of-Bankroll so that you can grow exponentially.
(If you flat-bet the same amount every bet you will grow linearly.)

However - with pool size relatively low in general, I've always believed that you may as well use a conservative form of percent-of-Bankroll. = You are going to reach pool limits in several months if you grow exponentially. Another month or 3 is generally worth it to avoid some risk of ruin.
Still would be nice to clear the fog in my insight here.

Trifecta Mike has sent me Back to School, so I have some reading to do.

[Magister Ludi]

Assuming sampling with replacement:

BR = bankroll = $1,000,000
p = expected win probability
q = expected loss probability
A = net odds
e = (A+1)p-1>0
f* = fraction of BR to invest = e/A
f*BR = amount invested
EV = expected value = ef*BR

p = .45
A = 1.5
e = (1.5+1)*.45-1 = .125
f* = .125/1.5 = .083
f*BR = $1,000,000*.083 = $83,000
EV = .125*.083*$1,000,000 = $10,375

[Robert Fischer]

Seems to be a consesnsus that the Law of Total Probability says that we can in fact basically "average" the win% to come up with an "overall probability".

So, using Kelly for:
Win Percentage: 45% & Odds at 3/2 odds for ALL bets

should give the same results as If we had:

Win Percentage: 50% & Odds at 3/2 odds for 9/10 of bets, and then Win Percentage = 0% at 3/2 Odds for 1/10 of bets ?

Is there a way to simulate these two conditions separately, for a hard-head such as myself?