Hypothetical Wagering Question

[JohnGalt2341]

Let's say you were on a Game Show and you made it to the final round. In this round you have 10K to play with. The game is rolling a 100 Sided Die. If the Die lands on numbers 1 to 55 you Win. If it lands on 56 to 100 you Lose. Their is no juice in this game. It's all even money. You can bet as much or as little as you want per roll. You get 10 rolls total. After 10 rolls what ever money you have left you get to keep. What would be the optimum percentage of your bankroll to bet on each roll? Thanks in advance for any and all responses.

[donjuan]

Is the 10k all you have to your name?

[JohnGalt2341]

Is the 10k all you have to your name?

No... lets say the person playing makes 40K a year. They are doing ok and they aren't looking to hit the Lottery. However, they do want to buy a new car. Mathematically... what would be the optimum amount for them to bet on each roll so they can come out ahead 10K+?

[donkson]

All-in each roll

[mjespoz]

55% win prob at even money? Full Kelly would be to bet 10% of your bank at each roll.... Not sure that will get you $10k up in 10 bets though.

[mr.inpak]

all depends on your financial situation if you need the money you would bet the minimum if you were well off you could bet the entire amount on one roll

[babastar]

Hypothetical Wagering!!!!!could be interesting

[Warwick44]

John Galt: (<-- Who ARE you?) *hehe*

As to your question...you need to offer a little more information, unless I've misunderstood. Firstly, what is won with each roll? Is it some multiple of the number on the die? Or is the payoff the same so long as the number is 55 or less?

Clearly if it's the latter, then any roll you take is going to have 55% odds of landing on any of those numbers. And given the expected value of any roll of the dice is 50.5 (<-- remember there is no zero...so the expected value is NOT 50), you have a built in 4.5% advantage, and should simply bet the entire 10k on the first roll if losing it all is of no consideration.

HOWEVER, if the goal here is to calculate what value you would have to keep betting to give yourself even money odds at staying in the game and building a bankroll, then you would bet $4,784.70, which @ 1.045% payoff would be $5,000, which would keep you at a theoretical 50/50 odds of winning on each successive roll. (detailed as --> ((10,000 /2) / 1.045) = 4,784.70, which @ 4.5% advantage pays 5k)

So you just keep betting 4.5% less than 50/50 odds to keep yourself in the game. That of course, is subject to "bad runs", and will not necessarily keep you from losing it all after multiple rolls. I dont have Excel in front of me, but if I did I could work out a further optimization toggled out to whatever standard deviation one were comfortable with. Of course the more conservative you get in terms of sigma risk (say, beyond 1 sigma), the more you will have to roll many times.

I'm guessing your question is more advanced than this though, so please clarify. The answer gets more complex, and includes a game theory overlay to understand the optimal number of rolls, which would be partially determined by what each roll yielded -- IF the payoff is non-linear.

I can answer that with more info.

[Firefox14]

John Galt: (<-- Who ARE you?) *hehe*

As to your question...you need to offer a little more information, unless I've misunderstood. Firstly, what is won with each roll? Is it some multiple of the number on the die? Or is the payoff the same so long as the number is 55 or less?

Clearly if it's the latter, then any roll you take is going to have 55% odds of landing on any of those numbers. And given the expected value of any roll of the dice is 50.5 (<-- remember there is no zero...so the expected value is NOT 50), you have a built in 4.5% advantage, and should simply bet the entire 10k on the first roll if losing it all is of no consideration.

HOWEVER, if the goal here is to calculate what value you would have to keep betting to give yourself even money odds at staying in the game and building a bankroll, then you would bet $4,784.70, which @ 1.045% payoff would be $5,000, which would keep you at a theoretical 50/50 odds of winning on each successive roll. (detailed as --> ((10,000 /2) / 1.045) = 4,784.70, which @ 4.5% advantage pays 5k)

So you just keep betting 4.5% less than 50/50 odds to keep yourself in the game. That of course, is subject to "bad runs", and will not necessarily keep you from losing it all after multiple rolls. I dont have Excel in front of me, but if I did I could work out a further optimization toggled out to whatever standard deviation one were comfortable with. Of course the more conservative you get in terms of sigma risk (say, beyond 1 sigma), the more you will have to roll many times.

I'm guessing your question is more advanced than this though, so please clarify. The answer gets more complex, and includes a game theory overlay to understand the optimal number of rolls, which would be partially determined by what each roll yielded -- IF the payoff is non-linear.

I can answer that with more info.

What a beautiful mind.

[Flight]

For regular middle class people without a large bankroll, play kelly

10% stake each roll = $1k
expected profit 1% for each roll, total expected value is ~11K

http://www.sbrforum.com/betting-tools/kelly-calculator/

This is probably less than you would expect - most of us on here are gamblers and if we know we have an edge like that we would go all in on every roll.

In other words, it's all about where you want to set your risk of ruin.

[donjuan]

John Galt: (<-- Who ARE you?) *hehe*

As to your question...you need to offer a little more information, unless I've misunderstood. Firstly, what is won with each roll? Is it some multiple of the number on the die? Or is the payoff the same so long as the number is 55 or less?

Clearly if it's the latter, then any roll you take is going to have 55% odds of landing on any of those numbers. And given the expected value of any roll of the dice is 50.5 (<-- remember there is no zero...so the expected value is NOT 50), you have a built in 4.5% advantage, and should simply bet the entire 10k on the first roll if losing it all is of no consideration.

HOWEVER, if the goal here is to calculate what value you would have to keep betting to give yourself even money odds at staying in the game and building a bankroll, then you would bet $4,784.70, which @ 1.045% payoff would be $5,000, which would keep you at a theoretical 50/50 odds of winning on each successive roll. (detailed as --> ((10,000 /2) / 1.045) = 4,784.70, which @ 4.5% advantage pays 5k)

So you just keep betting 4.5% less than 50/50 odds to keep yourself in the game. That of course, is subject to "bad runs", and will not necessarily keep you from losing it all after multiple rolls. I dont have Excel in front of me, but if I did I could work out a further optimization toggled out to whatever standard deviation one were comfortable with. Of course the more conservative you get in terms of sigma risk (say, beyond 1 sigma), the more you will have to roll many times.

I'm guessing your question is more advanced than this though, so please clarify. The answer gets more complex, and includes a game theory overlay to understand the optimal number of rolls, which would be partially determined by what each roll yielded -- IF the payoff is non-linear.

I can answer that with more info.

I'm not sure I've ever seen a more complex answer that was so completely wrong.

[donjuan]

No... lets say the person playing makes 40K a year. They are doing ok and they aren't looking to hit the Lottery. However, they do want to buy a new car. Mathematically... what would be the optimum amount for them to bet on each roll so they can come out ahead 10K+?

You need to better define what your bankroll is and your objectives are. However, if you only consider 10k your bankroll the mathematically correct answer is Kelly.

[George7904]

Kelly says to bet 10% of your networth on this situation. Since the max play the 1st roll is 10k, one should bet 10k if they are worth more than 100k. If they win the 1st roll, the limit is now 20k. One should bet the whole 20k, if their networth is 200k or more. This will continue until you lose, or the max bet is more than 10% of your worth. At this point, you should bet 10% of your net worth.

Basically, the answer is to bet the max or 10% of your net worth, whichever is less for each roll.

[JohnGalt2341]

All-in each roll

Even though I am pretty certain that you are joking I would be willing to bet that around 10% of the general population would do exactly this if they were actually on The Game Show. After a few rolls it is very likely that they would have $0 and go home with nothing.

[JohnGalt2341]

55% win prob at even money? Full Kelly would be to bet 10% of your bank at each roll.... Not sure that will get you $10k up in 10 bets though.

I'm not trying to get up 10K in 10 bets. I'm trying to go home with between 10K and 11K. I could bet $0 on every roll and walk away with 10K. I would be fine with that... but I want to know the optimal betting strategy. If I had an infinite number of rolls I would likely only bet between 1 to 3% of my Bankroll on every bet. Eventually I would have millions if the Show allowed me to roll for 24 hours straight. The thing that worries me about Full Kelly is if I bet 10% of the starting 10K and I happen to lose my first 4 rolls... even if I win my next 4 rolls I would still be down $400(or 4% of my starting bankroll). To me it just seems too aggressive. Am I wrong? I'm no expert... which is why I started the thread.

[JohnGalt2341]

John Galt: (<-- Who ARE you?) *hehe* As to your question...you need to offer a little more information, unless I've misunderstood. Firstly, what is won with each roll? Is it some multiple of the number on the die? Or is the payoff the same so long as the number is 55 or less? Clearly if it's the latter, then any roll you take is going to have 55% odds of landing on any of those numbers. And given the expected value of any roll of the dice is 50.5 (<-- remember there is no zero...so the expected value is NOT 50), you have a built in 4.5% advantage, and should simply bet the entire 10k on the first roll if losing it all is of no consideration. HOWEVER, if the goal here is to calculate what value you would have to keep betting to give yourself even money odds at staying in the game and building a bankroll, then you would bet $4,784.70, which @ 1.045% payoff would be $5,000, which would keep you at a theoretical 50/50 odds of winning on each successive roll. (detailed as --> ((10,000 /2) / 1.045) = 4,784.70, which @ 4.5% advantage pays 5k) So you just keep betting 4.5% less than 50/50 odds to keep yourself in the game. That of course, is subject to "bad runs", and will not necessarily keep you from losing it all after multiple rolls. I dont have Excel in front of me, but if I did I could work out a further optimization toggled out to whatever standard deviation one were comfortable with. Of course the more conservative you get in terms of sigma risk (say, beyond 1 sigma), the more you will have to roll many times. I'm guessing your question is more advanced than this though, so please clarify. The answer gets more complex, and includes a game theory overlay to understand the optimal number of rolls, which would be partially determined by what each roll yielded -- IF the payoff is non-linear. I can answer that with more info.

Thanks for your reply. To answer some of your questions.. What you Win on each roll has nothing to do with the # of the Die. It only has to do with if it lands between 1 to 55 or 56 to 100. For instance... Lets say on the first roll I bet $500 and I roll a 34. I Win! I now have $10500. On the second roll I bet $700 and I roll an 83. I lose. I now have $9800.

I'm still sort of unclear as to what your answer is. Are you saying to bet 4.5% on each roll? To me that sounds about right. Although I still think it might be on the aggressive side.

Let's assume you had a thousand rolls instead of just 10... Do people still recommend Full Kelly? If you happen to lose 10 straight(which I've done on more than one occasion) you would lose 65% of your bankroll. I don't like the idea of that.

[mjespoz]

Hey JohnGalt,

OK, I misunderstood your "Mathematically... what would be the optimum amount for them to bet on each roll so they can come out ahead 10K+?" to mean that you wanted to make $10K on your $10K "bankroll".
"The thing that worries me about Full Kelly is if I bet 10% of the starting 10K and I happen to lose my first 4 rolls". If the win prob really is 55% then the probability of losing your first 4 rolls is only 4.1%. Your probability of WINNING your first 4 rolls is 9.15% (more than twice as likely!).

If full Kelly is too volatile for you (it is for me too) then choose a fractional Kelly that suits your risk profile.

Cheers,
mjespoz

[andywend]

JohnGalt,
Since nobody asked what should be a pretty obvious question, I'll ask:

Is somebody offering you the deal you mentioned (allowing you to bet as much as you want at even money on a bet that has a 55% chance of winning)?

If yes, I would sure like to get in on that deal.
If not, why create a thread like this and ask the question in the first place?

[Warwick44]

JohnGalt,

Back after a long weekend away. YES, your read on what I was saying is indeed what I was saying. The corrollary to applying the break-even expected value percentage, is to apply that percentage as a bet size. But even that is only a theoretical answer, subject to the practical reality of whatever money management you are actually comfortable with.

I understand the base logic behind Kelly system, but I have never looked at it in detail, so I cant speak to some of the other comments on here. I'd only say that betting 10% of bankroll on any one bet would be make for very volatile returns given the small percentage advantage you're getting. If this were simulated running 10% bet size my guess is it would end in $0 due to bad runs in a surprising number of instances...probably within 1 sigma.

[Warwick44]

Just back from a long weekend and reading through this thread and I caught your comment DonJuan. Maybe it's raining and Monday, so where normally I'd ignore it, this morning I'm salty.

Is giving a long-winded response to someone asking for an opinion on something...on a forum...NOT why we're here?

I love that not only do you go out of your way to deride someone else, but that the entirety of YOUR point was to mumble "fully Kelly" along with the words "mathematically correct,"...while of course offering not a single digit of mathematical reasoning.

10% of bankroll on every bet? Setting aside your "mathematically correct" line, offered of course with NO MATH, I kinda wonder whether you'd be better off spending more time learning and testing, and less time making snarky comments on this forum...


"He who knows little, quickly tells it."

[bztips]

W44, donjuan did give the correct answer: Kelly (could be full Kelly or something less depending on your risk aversion). The "missing math" that you're complaining about is on your end -- you need to read up and understand the math behind Kelly, which yields the maximum expected growth in bankroll (conditional on the underlying assumptions about utility and risk aversion).

[Warwick44]

bz,

I'm pretty comfortable with my answer. Actually, very comfortable. And if you read my comment to JohnGalt, then you'd see I make no claim to Kelly knowledge beyond the basic idea. But that has no bearing on calculating probability.

I do find it interesting, the amount of acrimony on here, and "dj's" or YOUR need to attack another posters point.

Whatever the details of Kelly's system, it doesnt work on the basis of MAGIC. So...when/if/ever you can offer an explanation for describing a given probability, using Kelly as a framework, I'll be open to seeing it. "Explanation" is going to involve some actual numerical digits coming from your smartazz tapping fingers.

Otherwise..."GO GET YOUR SHINEBOX".

[Indecent]

The responses you are getting may stem from the fact that you aren't answering the question as asked. Your response was based around this goal:

HOWEVER, if the goal here is to calculate what value you would have to keep betting to give yourself even money odds at staying in the game and building a bankroll

Your answer may have been correct for that goal, but it's not what the OP was asking.

Whatever the details of Kelly's system, it doesnt work on the basis of MAGIC. So...when/if/ever you can offer an explanation for describing a given probability, using Kelly as a framework, I'll be open to seeing it. "Explanation" is going to involve some actual numerical digits coming from your smartazz tapping fingers.

As far as not being familiar with Kelly, you could probably clear that up pretty easily with a search here or elsewhere. If you had shown you actually did some of your own research on Kelly, you might have gotten some help if you had a specific question. In terms of proving it, there is far too much material on the subject that is easily available to care about someone who hasn't helped themselves to it, especially when Kelly does all the work proving it for us/you.

Simply put, you're wrong. If you want to know why, stop acting like a dbag, do some of your own research, and if at that point it's not clear why you are wrong, ask a real question. People like to help, people hate wasting their time. Your attitude doesn't make it clear that you aren't a waste of time.

[Warwick44]

OK, I'll put up one more post on this and then let it go...because I'm really not interested in getting into a pissing contest for the love of offering a helpful comment.

As I said, I'm not up on Kelly beyond the basic premise. I don't have a huge interest in optimized wagering because my approach in sports is purely about building my own fundamental value estimates for lines and then looking for value where I can find it. Period.

If I were abettor at pure-chance games, where wager-theory and optimization is literally what it's all about, I'd have the interest to devote the time to looking carefully at Kelly.

Kelly may provide a probabilistic framework (I'll take your word on that). But you dont need that (in MY opinion) to calculate the probability of Galt's original question. I will say, though, that 10% bets is something I will definitely take the other side of. Whether that's what Kelly prescribes, or if the 1 or 2 people who said this are incorrect in asserting so, I wouldn't know.

If you've got a passion for Kelly system that's fine. I spend my time focused in a different manner (again, line value). I'm not saying there isn't value to Kelly, nor have I anywhere in this thread.

I simply took issue with the comments belittling my comment...particularly where it came offered with little more than a blanket statement. IF indeed, Kelly prescribes 10% wager size for Galt's original question, then I'd say I don't agree with it.

Common sense, and a lot of experience in applying probability to make money in both sports betting an markets, tells me this isn't what that system prescribes...or that the idea that it does, is being misinterpreted -- particularly where unlimited bankroll does not exist.

This is pretty much the limit of my desire to argue about it, especially if it isn't going to stay friendly.

[Blax0r]

Just back from a long weekend and reading through this thread and I caught your comment DonJuan. Maybe it's raining and Monday, so where normally I'd ignore it, this morning I'm salty.

Is giving a long-winded response to someone asking for an opinion on something...on a forum...NOT why we're here?

I love that not only do you go out of your way to deride someone else, but that the entirety of YOUR point was to mumble "fully Kelly" along with the words "mathematically correct,"...while of course offering not a single digit of mathematical reasoning.

10% of bankroll on every bet? Setting aside your "mathematically correct" line, offered of course with NO MATH, I kinda wonder whether you'd be better off spending more time learning and testing, and less time making snarky comments on this forum...


"He who knows little, quickly tells it."

donjuan can be quite blunt, but he's right. Check out the paper; the proof for the kelly criterion isn't that bad.

[Warwick44]

Given how long it's been around (1950's?), I'm sure there's a lot of merit to it. Of course the merit of anything is partially in applying it correctly...

My interest is really just sports, so I wonder if people use Kelly for that, or only for pure-chance games with no human element. I see SO MANY comments on SBR about "chase systems" and did some snooping with Google and the majority of what I found is pure fantasy...con men hyping "systems" they say never lose, that kind of thing. But from what I read I gather these people weren't using Kelly-style logic, just incidental short runs of games.

If anyone has used Kelly successfully to wager on sports I'd be curious to hear how they apply that. The cynic in me sees little more than short, random runs where I've read about "chasing" games.

[bztips]

All of your posts make it abundantly clear that you have no idea what you are talking about.

1) Kelly is not, and never was intended to be used to help "calculate probability". It is a money management strategy that assumes you already KNOW your probability of success. This fundamental fact seems to have escaped you; and contrary to your statement, you obviously do NOT "understand the base logic behind the Kelly system".
2) Its has nothing to do with whether you're looking at "pure chance games" or not.
3) It explicitly does not assume unlimited bankroll.
4) It is NOT a chase system.

As others have already stated, you would be best served by putting just a little effort into reading and research, before posting any further nonsense.

[donjuan]

LOL at making an argument against Kelly without even understanding what it is. Since you appear to be lazy I'll be nice and give you a link.

http://forum.sbrforum.com/handicapper-think-tank/29841-maximizing-expected-growth-kelly-criterion-part-ii.html

Read it and then get back to me.

P.S. Calculating the expected value of the die is pointless and idiotic. It literally has nothing to do with the problem and then going on to say you have a 4.5% edge is also completely wrong. Your edge is 10%.

[Warwick44]

BZ,

I'm goning to skip the childish, and go right to the factual.

1. Actually, Kelly was never intended to be a money management strategy. It's simply been adapted for that by guys ages ago. It was created to find the optimal travelling path for traffic through the phone system. That much...I do know.
2. Optmizing bets on anything which has external factors OTHER than pure chance, is ABSOLUTELY not the same as for games which are nothing but pure chance. This comment fails the common sense test.
3. I dunno what it assumes regarding bankroll.3
4. Not once did I imply it's a chase system...merely questioned aloud whether people who use chase systems, employ some sort of Kelly logic in doing so.

[Warwick44]

"Calculating the expected value of the die is pointless and idiotic. It literally has nothing to do with the problem and then going on to say you have a 4.5% edge is also completely wrong. Your edge is 10%."

I had to show this to some guys looking over my shoulder...the laughter goes on and on. It is one of the single most foolish quotes anyone who is within 10 miles of anything quantitative, could ever make.

You're in the Hall of Fame with that one. *LOL*

[donjuan]

Cute. Go on, show your work.

[MonkeyF0cker]

"Calculating the expected value of the die is pointless and idiotic. It literally has nothing to do with the problem and then going on to say you have a 4.5% edge is also completely wrong. Your edge is 10%."

I had to show this to some guys looking over my shoulder...the laughter goes on and on. It is one of the single most foolish quotes anyone who is within 10 miles of anything quantitative, could ever make.

You're in the Hall of Fame with that one. *LOL*

(1*0.55)+(-1*(1-0.55)) = 0.1

Hall of Fame? Your team of experts is a team of nimrods.

[Warwick44]

Sorry, after ALL THAT. All the name calling, and blustery everything...THAT is what you came up with? I see your problem now.

You are confusing the net difference between the two outcomes (55% - 45% = 10%)...with what Galt originally asked in his question. Which was, what would the bet size be. Just because the NET ADVANTAGE between the two outcomes is 10%, does NOT mean that 10% is the bet size you should make in maximizing your capturing this advantage.

I'm astounded that this is what you thought. To demonstrate, imagine an example where you had a 75% NET ADVANTAGE between the two sides. (87.5% - 12.5%) By your logic, you would make a first bet of 75% of your bankroll...which in practical terms would be a binomial outcome, where you'd be crippled after one loss...and all but done after a second.

The idea should be, at what bet % can you MAXIMIZE your 10% advantage (<-- thank you Albert Einstein), while still staying ahead of bad runs.

This is why...if you read, I said it can be even further tweaked via sigma adjusting, based on what your risk tolerance is.

I'm done here, as I think I WAY overestimated the level of conversation going on here.

But yes, 55-45 does indeed equal 10%. Congratualations...you passed 7th grade. *confetti*

This last bit is the best laugh yet. We're walking out in 3 minutes, and will drink the first beer in dedication to you. *L*

[Warwick44]

Oh and by the way...go back to the first message and you can see how (frankly, simply) it is calculated. The key aspect is...there is no zero on his 100 dot die. AND, that you need to use max 1/2 of the original bankroll to make one bet, such that you are guaranteed of having at least one more crack at capturing your 10% odd (thanks again demonstrating how to fin that 10% *s*) Hence, the 5,000 handle used.

ps YES DonJuanita, calculating the odds on a die is important.

pss So we're clear, I have no criticism of Kelly. Might even look into it. Don't confuse my return-heckling of your angry misunderstanding, with a criticism of a concept which is old and well-used. I just gather I'm hearing the Kelly logic from the wrong 3 people...

[donjuan]

LOL what an epic fail. Please stop using terms you don't understand the meaning of.

At even money, your Kelly stake is always equal to your edge. The math behind it proves it but you're clearly too lazy to even read the article I linked. Full Kelly may seem too aggressive to you and many others, but it is the mathematically optimal stake for optimal growth of a bankroll which is what the original question was.