Flat betting versus the Kelly Criterion - a simulation

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  • MonkeyF0cker
    SBR Posting Legend
    • 06-12-07
    • 12144

    #1
    Flat betting versus the Kelly Criterion - a simulation
    There have been a number of arguments over the merits of the Kelly Criterion in sports betting in the Think Tank. However, there was this post in one of Ganch's threads that had always intrigued me...

    the Kelly criterion as applied to sports betting would be better called the Kamikaze criterion. You can prove it for yourself, and here’s how:

    Here’s what you’ll need, along with at least a half-hour of time:
    1. A hand calculator
    2. Two decks of ordinary playing cards
    3. Lined paper
    4. Pen or pencil
    5. A ‘Thank You’ note to send me after you complete this exercise and realize how much money I've saved you.....

    Pick any size fantasy bankroll to use as your total bankroll. Why not $10,000?

    Thoroughly shuffle the 2 decks of playing cards together and place them face down in front of you. We’re going to turn one card at a time and count it as a win, loss, or tie. Everything 7 through King will be a ‘winner,’ everything 2 through 6 will be a ‘loser,’ the Aces will be ties. With those rules, the double deck contains 56 ‘winners,’ 40 ‘losers,’ and 8 ‘ties.’ That makes an overall 'winning' expectation of 58.3 percent, and that would be a great long term winning percentage against sports betting.

    But that expectation will vary widely as you remove cards from the deck. As you turn the cards and remove them from the remaining deck the deck will turn ‘positive’ or ‘negative'; - that is, if you remove more 'losers' (2's through 6's) from the deck than 'winners' (7's through K's), the remaining deck will offer a higher expectation of 'winning' on the next draw, and vice-versa.

    Figure the sizes of your Kelly bets accordingly. If the first card is a ‘loser,’ there are only 39 losers left in the deck, but still 56 winners. Your winning expectation for the second draw (’bet’) increases to 56 out of 95, or 58.9 percent. If the first card is a ‘winner,’ your winning expectation for the second draw drops to 55 of 95 or 57.9 percent. This, of course, is where the hand calculator comes in.

    Be sure to record whether you won or lost the first bet, and how much you won or lost. As a Kelly bettor, of course, your bet sizes will vary up and down as your winning expectation goes up and down. Go ahead and do this 50-or-so times before reshuffling the deck and starting over. (Don’t do it more than 50 or 60 times without reshuffling. Always reshuffle when you've been through about half the double deck. Don't go through the entire double deck.)

    Remember, according to the Kelly criterion if the deck goes ‘negative’ and you do not have a positive expectation don’t bet anything. Just flip the next card and the next until you do have a positive expectation. Size your Kelly bets exactly as you do against sports, and to make the exercise more realistic, as when actually betting against sports, flip several cards at once. After all, NFL, NBA, MLB and NHL games often go off several at a time and cannot be bet sequentially. You have to lay several bets at once. Try flipping 3 or 4 or more cards at once - maybe even a dozen or so - just like when you're betting on sports.

    After doing another 50-60-or-so observations with the reshuffled deck, it’s time to compare your results using the Kelly criterion against so-called ‘flat’ bets.

    Right here is precisely where Kelly promoters always screw up. Let's say they have 100 actual bets wherein they win, say, 58 and lose 42. That's a great winning percentage of 58%, of course. Now, they'll explain that their basic bet is, say, $100, but if their expectation is higher than such-and-such percentage they risk $120, or $130, or whatever, and if their expectation is even higher than such-and-such they might risk $200 or more.

    Then they compare what they won by using the Kelly system to what they would have won had they been risking only $100 on each of the 100 bets.

    .....Duhhh!.....They risked more money with the Kelly system and they made more money after going 58-42. I hate to burst their balloon, but when you go 58-42, the more money you risk the more money you figure to make. Sorry, boys, but that is not news.

    The only way to fairly compare the Kelly system (or any other progressive betting scheme) to flat betting is to use a flat bet the same size as the average size of all the Kelly bets. That way, you’re risking the same total amount against the same overall won-lost results. No fair risking more money overall with one system than the other. That obviously skewers the results.

    Or, another way to fairly compare betting systems is to keep track of the winnings as a percent of the total amount risked. If Betting System A wins 8 percent of all monies risked while Betting System B wins 12 percent of all the monies risked, Betting System B is obviously better than Betting System A.

    This is precisely what Kelly-promoters choose to ignore. Comparing flat betting against a "1-star, 2-star, 3-star" system, for example, and going 58-42, if all your flat bets are only as big as your "1-star" bets, of course you will win more with the star system. You're risking more and you're winning 58% of your bets.

    All right, back to our double-deck of cards. Time to check the profits from flat betting against the record of the Kelly criterion, and ta-daa! There’s your proof. Using the average size of your Kelly bets as your flat bet, the Kelly loses, and it loses every time. In fact, using most forms of the Kelly criterion, I would be surprised if after 70 or 80 ‘bets’ you are not - for all intents and purposes – broke.

    You can use the same results to compare the "1-star, 2-star, 3-star" system. You don’t have to flip the cards again, you can use the same won-lost progression you got while testing the Kelly criterion. Set your own parameters concerning when to use a "1-star" bet, a "2-star" bet or a "3-star" bet. Perhaps between 55 and 58 percent you could use a "1-star" bet, etc. Of course, when your winning expectation is less than 53 or 54 percent, there is no reason to bet at all. Bet any system - including flat betting - only when you have an acceptable winning expectation

    ----------J.R.Miller "Debunking the Kelly System"


    So what I decided to do was write a simulation to prove/disprove this theory. I have attached an Excel spreadsheet which enables you to do this exact thing, comparing flat betting to the Kelly Criterion. Let me know how many times through the deck (cycles) it takes you to go broke using the Kelly Criterion as the author asserts...
    Attached Files
  • That Foreign Guy
    SBR Sharp
    • 07-18-10
    • 432

    #2
    In the real world winning one bet doesn't make you more likely to lose the next one. Shuffle the selected card back in (or use =rand to allow a custom winning %). I can't put my finger on exactly why but this set up feels like it's a trick.

    Even if it isn't, there's still a problem - betting full Kelly on several simultaneous events is overbetting so what he is doing is no longer Kelly optimal.
    Last edited by That Foreign Guy; 02-17-11, 05:58 AM.
    Comment
    • splash
      SBR Rookie
      • 05-25-09
      • 38

      #3
      If I had known that was written by that moron JR Miller before getting to the bottom, I wouldn't have wasted those minutes of my life reading that.
      Comment
      • djiddish98
        SBR Sharp
        • 11-13-09
        • 345

        #4
        Thanks for posting Monkey - I always enjoy looking at others' code.

        Even though this is beating a beat up dead horse, it would be interesting to add a calculation that compares [return]/[risk] - this would address the issue with....

        "Or, another way to fairly compare betting systems is to keep track of the winnings as a percent of the total amount risked. If Betting System A wins 8 percent of all monies risked while Betting System B wins 12 percent of all the monies risked, Betting System B is obviously better than Betting System A. "

        I did a quick calc, and kelly still comes ahead, as expected.
        Comment
        • Pokerjoe
          SBR Wise Guy
          • 04-17-09
          • 704

          #5
          JR Miller is not completely out of line here.

          This question needs to be answered before deciding whether to use Kelly: is the regression causal, or dilutive. I've written about it before, and won't do much here, but this question needs to be answered before deciding which betting style to use, and the answer isn't clear in sports betting. Previous sports betting results may very well have a causality impact on regression, depending upon the style of handicapping, in which case Miller is right. Right by accident, btw: he doesn't understand the question any better than Kelly worshippers.

          But for example, the change in the CBB totals market this year was caused by previous results. IOW, the deck isn't constantly reshuffled in sports betting. This isn't roulette, this is a market. Markets have memory.
          Comment
          • That Foreign Guy
            SBR Sharp
            • 07-18-10
            • 432

            #6
            You're arguing that the actual totals of college basketball games changed because of people's betting results?

            I am more than willing to accept that the bookie lines changed because of people systematically beating them last season.

            However, your edge is always the amount that your estimate is closer to a perfect estimate than the bookie's estimate is. That edge changing was not because of past wins, it was because the way the bookies made their lines changed. The results of the bets didn't actually reduce your expectation it was arguably the bookie's reaction to those results (they may equally have just got a fancy new model entirely unrelated to your bets).
            Comment
            • trixtrix
              Restricted User
              • 04-13-06
              • 1897

              #7
              Originally posted by Pokerjoe
              JR Miller is not completely out of line here.

              This question needs to be answered before deciding whether to use Kelly: is the regression causal, or dilutive. I've written about it before, and won't do much here, but this question needs to be answered before deciding which betting style to use, and the answer isn't clear in sports betting. Previous sports betting results may very well have a causality impact on regression, depending upon the style of handicapping, in which case Miller is right. Right by accident, btw: he doesn't understand the question any better than Kelly worshippers.

              But for example, the change in the CBB totals market this year was caused by previous results. IOW, the deck isn't constantly reshuffled in sports betting. This isn't roulette, this is a market. Markets have memory.
              i don't even know why i keep responding to post like this except maybe i'm glutton for punishment

              1.) your argument has nothing to do w/ jr miller's argument. his argument is that proportionally bet-sizing corresponding to size of your pre-determined static edge is a losing proposition vs. simply flat betting the same bet size across the board regardless of your pre-determined static edge. your assertion seem to have something (merely guessing b/c it's almost incomprehensible) w/ the fact that your edge w/ the market will fluctuate w/ time, but that is no way a good supporting argument for whether miller's claim is out of line or not

              2.) the above experiment "obv" do have memory since you have to keep track cards and they're not being replaced back into the deck. so what does that have to do w/ your roulette memory vs. no memory argument?
              Comment
              • bztips
                SBR Sharp
                • 06-03-10
                • 283

                #8
                How many times do we have to beat this dead horse???

                Given the assumptions of the Kelly betting system (including the assumption of log utility), it is a mathematical FACT that Kelly maximizes the expected growth of your bankroll. Does this mean you will definitely do better at every point in time with Kelly? No, it maximizes EXPECTED growth. Depending on the "roll of the dice" (or more precisely, the outcomes of the (statistically) random events you bet on), you could do better with flat betting (or some other system) over some finite time horizon. But over a large population of bettors and/or an infinite time horizon, following the Kelly strategy will do better than flat betting. End of story.
                Comment
                • CHUBNUT
                  SBR Sharp
                  • 06-30-09
                  • 321

                  #9
                  betting more on the winners less on the losers is the best system for long term growth
                  Comment
                  • wrongturn
                    SBR MVP
                    • 06-06-06
                    • 2228

                    #10
                    Originally posted by CHUBNUT
                    betting more on the winners less on the losers is the best system for long term growth
                    This best system can only work with using a time machine.
                    Comment
                    • JustinBieber
                      SBR Sharp
                      • 05-16-10
                      • 324

                      #11
                      Pokerjoe is possibly the biggest psuedo intellectual on this forum. That's some hard doing.
                      Comment
                      • tomcowley
                        SBR MVP
                        • 10-01-07
                        • 1129

                        #12
                        Originally posted by That Foreign Guy
                        I can't put my finger on exactly why but this set up feels like it's a trick.
                        Yeah. He determines the flat bet size after the fact. And the exact sequence of bets could end up busting him flat betting even when it wins with kelly if the losses are frontloaded, but he never mentions that.
                        Comment
                        • Flight
                          Restricted User
                          • 01-28-09
                          • 1979

                          #13
                          1) I couldn't figure out what was going on with your shuffle algorithm - seems like it could be simplified to a Knuth shuffle. Also, you should put the Randomize function outside the loop, you should not seed every time, just call Rnd each loop.

                          2) Thanks for the simulator! It is awesome. I could never get my kelly bankroll to reach 0, although I did get it to overflow upperbound(int) several times!
                          Comment
                          • Flight
                            Restricted User
                            • 01-28-09
                            • 1979

                            #14
                            Hmm, although I did just notice that it is straight flat betting and not normalized to the average of the kelly bets as the original author describes. An improvement would be to run a cycle with Kelly bets, get the average of only the placed kelly wagers (don't count the 0s) and then make that the flat bet size.

                            I performed a manual goal-seek by setting flat-bet to 1000 and running 1 cycle about 20 times until I got an average kelly bet of $1000. Kelly still won $23k to $15k
                            Comment
                            • durito
                              SBR Posting Legend
                              • 07-03-06
                              • 13173

                              #15
                              The strategy described by JR Miller isn't possible. You can't select your bet size after the games are over.
                              Comment
                              • That Foreign Guy
                                SBR Sharp
                                • 07-18-10
                                • 432

                                #16
                                After this week, I wish I could, there'd be a couple games I'd set to zero for sure.
                                Comment
                                • Wrecktangle
                                  SBR MVP
                                  • 03-01-09
                                  • 1524

                                  #17
                                  Essentially what Miller is copping to is the fact that he cannot identify the difference between a "normal" play and a "superior" play. Since he can't (and the stats he had posted on his site supported this), he believes that nobody else can either.
                                  Comment
                                  • specialronnie29
                                    SBR High Roller
                                    • 09-19-10
                                    • 140

                                    #18
                                    most of the guys in this thread are replying to the quoted post and not the posted excel spreadsheet which we should all thank monkey for supplying

                                    open the damn spreadsheet. jr miller is wrong, so there is no point debating whether he is right or wrong.
                                    Comment
                                    • FourLengthsClear
                                      SBR MVP
                                      • 12-29-10
                                      • 3808

                                      #19
                                      Thanks to the OP for taking the time to do this.

                                      I am sure most of us know the flaws in J.R Miller's logic but it is always nice to see flawed arguments being debunked.
                                      Comment
                                      • Joe Sharp
                                        SBR MVP
                                        • 06-01-09
                                        • 3011

                                        #20
                                        Thanks for the spreadsheet Monkey!
                                        Comment
                                        • MonkeyF0cker
                                          SBR Posting Legend
                                          • 06-12-07
                                          • 12144

                                          #21
                                          Originally posted by Flight
                                          1) I couldn't figure out what was going on with your shuffle algorithm - seems like it could be simplified to a Knuth shuffle. Also, you should put the Randomize function outside the loop, you should not seed every time, just call Rnd each loop.

                                          2) Thanks for the simulator! It is awesome. I could never get my kelly bankroll to reach 0, although I did get it to overflow upperbound(int) several times!
                                          Didn't realize that I kept Randomize in the loop. Thanks. The algorithm is essentially a Knuth shuffle. It chooses a random card of the remaining cards in the array. I don't know of a way in VBA to remove elements from an array other than iterating through the entire array and copying over the remaining "wanted" elements. However, I don't work with VBA all that much, thankfully.
                                          Comment
                                          • Data
                                            SBR MVP
                                            • 11-27-07
                                            • 2236

                                            #22
                                            MonkeyF0cker, thank you for your contribution. As a token of appreciation, I am going to let you in on the biggest Excel secret ever:
                                            Tools->Options->General->Sheets in new workbook->set to 1.
                                            Comment
                                            • tim0402
                                              SBR Sharp
                                              • 03-18-09
                                              • 492

                                              #23
                                              wish i could understand kelly its too hard.
                                              Comment
                                              • That Foreign Guy
                                                SBR Sharp
                                                • 07-18-10
                                                • 432

                                                #24
                                                ****ing magnets how do they work.

                                                Kelly% = edge / odds

                                                Why it works is harder but unnecessary for most people (do some simulations if you don't trust what all the geeks are saying).
                                                Comment
                                                • LUSabres
                                                  SBR High Roller
                                                  • 02-18-11
                                                  • 231

                                                  #25
                                                  Kelly is a simple equation, if you can't figure it out, don't gamble your money without a financial advisor. Imo, you need some kind of formula when it comes to sports betting and determining how much you place on any given bet. Finding an appropriate formula is tough and requires many weeks/months/years to you're needs and potential growth. Kelly is good, but needs moderation, a divider is definitely needed if you ask me.
                                                  Comment
                                                  • Wrecktangle
                                                    SBR MVP
                                                    • 03-01-09
                                                    • 1524

                                                    #26
                                                    Originally posted by tim0402
                                                    wish i could understand kelly its too hard.
                                                    The equation is simple, but the application is near impossible to do correctly.

                                                    No matter what the "wizards" here in the tank say, 99.99%+ of the sports betting public cannot use Kelly, no matter which "form" they advocate as they cannot correctly calculate the Bayesian Priors.

                                                    And yes, I do include the members of the sect.

                                                    So, you will be much better off to: 1) flat bet with each bet approx 1/100 of your BR 2) don't chase 3) set a workable bankroll at the start of the season and stay within that limit.

                                                    Then you will last all season, probably.
                                                    Comment
                                                    • Dark Horse
                                                      SBR Posting Legend
                                                      • 12-14-05
                                                      • 13764

                                                      #27
                                                      Maximize profit. What a phrase gospel.
                                                      "Very healthy profit you made there. Great job! Why the sad face?"
                                                      "Man, I feel like such a failure for not maximizing my profit."
                                                      " "

                                                      On a sidenote. Parlays, round robins, underdog moneylines.
                                                      Comment
                                                      • That Foreign Guy
                                                        SBR Sharp
                                                        • 07-18-10
                                                        • 432

                                                        #28
                                                        Not being able to calculate edge accurately is why a lot of people use fractional Kelly as this moderates the effect of miscalculating it and also makes it easier to ignore the effect of simultaneous betting.

                                                        Bayesian Priors, really Wreck? Either you're just desperately trying to sound smart by using fancy vaguely appropriate concepts or I'm missing something. I know which you will say, but I'd like to see what someone who isn't trying to impress the gullible has to say.
                                                        Last edited by That Foreign Guy; 02-21-11, 06:22 AM.
                                                        Comment
                                                        • JustinBieber
                                                          SBR Sharp
                                                          • 05-16-10
                                                          • 324

                                                          #29
                                                          Originally posted by Wrecktangle

                                                          No matter what the "wizards" here in the tank say, 99.99%+ of the sports betting public cannot use Kelly, no matter which "form" they advocate as they cannot correctly calculate the Bayesian Priors.
                                                          The intelligent people in the think tank (about 5%) are not 99.99% of the sports betting public though are they?
                                                          Comment
                                                          • Wrecktangle
                                                            SBR MVP
                                                            • 03-01-09
                                                            • 1524

                                                            #30
                                                            Originally posted by That Foreign Guy
                                                            Not being able to calculate edge accurately is why a lot of people use fractional Kelly as this moderates the effect of miscalculating it and also makes it easier to ignore the effect of simultaneous betting.

                                                            Bayesian Priors, really Wreck? Either you're just desperately trying to sound smart by using fancy vaguely appropriate concepts or I'm missing something. I know which you will say, but I'd like to see what someone who isn't trying to impress the gullible has to say.
                                                            Bayesian Statistics is not a vague concept, but it is largely misunderstood even by the bulk of classically trained statisticians, and pretty much completely misunderstood here in relation to Kelly betting. I haven't the time nor inclination, so Google is your friend.

                                                            There are several other threads, largely discounted by the "smart" guys here where Kelly is brought into serious question.

                                                            You "math wizards" are in love with the concept and probably a lost cause, I simply hope that some others might think before jumping into Kelly. But on the other hand, after they burn thru a BR at a torrid rate, it might then give them pause.

                                                            But it's your money, burn it as fast as you'd like or can stand. I care not.
                                                            Comment
                                                            • bztips
                                                              SBR Sharp
                                                              • 06-03-10
                                                              • 283

                                                              #31
                                                              Originally posted by Wrecktangle
                                                              The equation is simple, but the application is near impossible to do correctly.

                                                              No matter what the "wizards" here in the tank say, 99.99%+ of the sports betting public cannot use Kelly, no matter which "form" they advocate as they cannot correctly calculate the Bayesian Priors.
                                                              Hilarious.

                                                              Neither the original Kelly article (nor any of the followups by Thorp and others) ever made a single mention of Bayes, yet Wreck here asserts that the formula can't be used unless you calculate Bayesian Priors.

                                                              If Wreck wants to argue in support of a Bayesian approach to betting, fine. Oh, but wait, he doesn't have either the "time nor inclination" to do so.

                                                              Nice way to make the rounds -- just make a ridiculous assertion, and then cut off any challenge to it right up front.
                                                              Comment
                                                              • masticore
                                                                SBR MVP
                                                                • 07-24-09
                                                                • 1177

                                                                #32
                                                                Doesnt matter what sytem you use, the main rule is: If you play EV+ you will win and if EV- you will lose....
                                                                The only thing between flat betting or kelly or any othersystem is the amount you risk.

                                                                But in the end will you win (or lose) same % of what you risk .

                                                                But you have almost same "formula" when bet more on your 5* then you do on your 1* game....

                                                                Bet more when big EV and bet less with low EV
                                                                Bet more when often hits and bet less when low hit %

                                                                An eample I give you +110 on a 50/50 game...you will bet farly high becuse the hit% will be high
                                                                I give you +2000 on a 20/80 game ...now its higher EV+...but you will bet less because the hit% is low

                                                                But in the long run will win% be the same if you bet flat or some other system.

                                                                regards
                                                                mikke
                                                                Comment
                                                                • Miz
                                                                  SBR Wise Guy
                                                                  • 08-30-09
                                                                  • 695

                                                                  #33
                                                                  Originally posted by tomcowley
                                                                  Yeah. He determines the flat bet size after the fact.
                                                                  Precisely, and this is the first of a few problems with Miller's post.

                                                                  Monkey, very cool how you implemented this in the simulator! Thanks for sharing.
                                                                  Comment
                                                                  • MonkeyF0cker
                                                                    SBR Posting Legend
                                                                    • 06-12-07
                                                                    • 12144

                                                                    #34
                                                                    Originally posted by Data
                                                                    MonkeyF0cker, thank you for your contribution. As a token of appreciation, I am going to let you in on the biggest Excel secret ever:
                                                                    Tools->Options->General->Sheets in new workbook->set to 1.
                                                                    LOL. Cheers, Data.
                                                                    Comment
                                                                    • MonkeyF0cker
                                                                      SBR Posting Legend
                                                                      • 06-12-07
                                                                      • 12144

                                                                      #35
                                                                      Originally posted by bztips
                                                                      Hilarious.

                                                                      Neither the original Kelly article (nor any of the followups by Thorp and others) ever made a single mention of Bayes, yet Wreck here asserts that the formula can't be used unless you calculate Bayesian Priors.

                                                                      If Wreck wants to argue in support of a Bayesian approach to betting, fine. Oh, but wait, he doesn't have either the "time nor inclination" to do so.

                                                                      Nice way to make the rounds -- just make a ridiculous assertion, and then cut off any challenge to it right up front.
                                                                      Yap. That sums it up pretty nicely.
                                                                      Comment
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