1. #1
    Ganchrow
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    Using Kelly to Determine Optimal Hedging Strategy

    (Note: While I've tried to keep this as simple as possible, carefully explaining each step, this isn't a topic for those who aren't already comfortable with gambling mathematics. A basic understanding of the Kelly Criterion and the calculus of differentiation would probably be helpful, too. If anything's unclear, please feel free to ask.)

    Last Saturday, based on purely quantitative criteria, I expressed my opinion that AFC -6 would be a bad bet, yielding an expected return of about -12.2% at -110. This implied that NFC +6 +100 would be a good bet, yielding an expected return of 8.0%.

    Surprisingly, there was at least one person out there who believed me. Let’s call him Peter. Peter put down a fairly sizeable chunk of change on NFC +6 +100. Just how sizeable? Well it turned out to be about a quarter of Peter’s bankroll. His intention had been to substantially reduce his exposure (at a decent sized-profit) after Sunday’s games.

    Well guess what? Fast-forward to today, and the Super Bowl spread is trading at AFC -6½ -109 / NFC +101. Ouch. What’s more, Peter doesn’t even have a view on the game. The only circumstance working in Peter’s favor is that he doesn’t have a pressing need for the funds, and so doesn’t mind having up to 60% of his bankroll locked in until after the game on Sunday.

    So what to do? About 3 weeks ago I gave the following advice to a bettor in another thread:
    Quote Originally Posted by Ganchrow
    Unless you have a view on the game itself or unless you really need to safeguard the at-risk funds just stick with the … position you already have. Without either an actual view on the game or a pressing need to safeguard the money you've already risked, your hedge would be little more than a blind gamble at a house edge [equal to the vig]”
    Although the question had initially been asked with regard to a positive expectation position, it might be reasonable to think that the same advice should also hold in the case of a negative expectation position. So does it?

    Well no. But it doesn't always apply to positive expectation positions either. What I should have explicitly stated in the other thread is that in general this is only true in the case of a risk-averse bettor, for bets that are sufficiently small relative to the bettor's bankroll. In the case at hand, the player has risked 25% of his bankroll on this one bet. This is not a small bet relative to his bankroll. So how much should he hedge?

    Peter has two candidate bets for his hedge (remember that his current position is 25% of his bankroll on NFC +6 +100)
    1. AFC -6½ -109
      [NFC -6½ +101]
    2. AFC -6 -120
      [NFC -6 +107]


    The first hedge candidate (AFC -6½ -109), is offered at 1.87% vigorish. The disadvantage of this bet is that it does not represent a perfect hedge. This is because Peter would be leaving himself exposed (for the size of the hedge bet) if the end result of the Super Bowl were AFC winning by exactly 6. (If you don’t understand how I came up with these figures you might want to check out http://www.sportsbookreview.com/forum/players-ta...ical-hold.html.)

    The second hedge candidate (AFC -6 -120), on the other hand, is a perfect hedge. The vig on that bet, however, is 2.78%, which is considerably higher than the vig on the first candidate bet.

    So it turns out that the solution is really rather straightforward if we assume logarithmic preferences (à la Kelly).

    So to start off we’ll first need to identify the distinct outcome scenarios and then assign probabilities to each of them. It should be apparent that 3 distinct outcome scenarios exist, to wit:
    1. NFC wins straight-up or AFC wins by less than 6
    2. AFC wins by exactly 6
    3. AFC wins by more than 6


    Looking at the market for the first hedge candidate, we see that it implies that the probability of either outcome 1 or 2 occurring is about 48.82%, and that the probability of outcome 3 occurring is about 51.18%.

    Looking at the market for the second hedge candidate, we see that the probability of outcome 1 occurring, conditioned on outcome 2 NOT occurring is 46.97%, and that the probability of outcome 3 occurring, conditioned on outcome 2 NOT occurring is 53.03%. (If you don’t understand how I came up with these figures, you may want to check out http://www.sportsbookreview.com/forum/players-ta...rcentages.html.)

    So, given these probabilities we get set up the following conditional probability equivalence (you'll recall that p(x | y) is the probability of x occuring given that y has occured):
    p(outcome 3 | NOT outcome 2)
    = p (3) / (1 - p(2)), which gives us

    p(2) = 1 - p(3) / p(3 | NOT 2)
    = 1 - 51.18% / 53.03%

    p(2) = 3.49%

    Because we already know that p(3) = 51.18%, this gives us p(1) = 1 - 3.49% - 51.18% = 45.33%.

    So to better visualize what we currently know, we can breakdown each outcome scenario as follows:

    1. NFC wins straight-up or AFC wins by less than 6
      Probability: 45.33%
      Initial bet (NFC +6 +100) wins, paying off at 1/1.
      Hedge candidate 1 (AFC -6½ -109) loses.
      Hedge candidate 2 (AFC -6 -120) loses.
    2. AFC wins by exactly 6
      Probability: 3.49%
      Initial bet pushes.
      Hedge 1 loses.
      Hedge 2 pushes.
    3. Outcome 3: AFC wins by more than 6
      Probability: 51.18%
      Initial bet loses.
      Hedge 1 wins, paying off at 100/109.
      Hedge 2 wins, paying off at 100/120.

    Because we’re assuming logarithmic preferences, the generalized form of our expected utility function would be:
    E(U) = i{p(outcomei) * ln[1 + (outcomei profit or loss)]} for K = 1

    E(U) = ( K/K-1) * i{p(outcomei) * [1 + (outcomei profit or loss)]1-1/K} for K ≠ 1
    where K is the constant Kelly multiplier (so for example K=1 would imply standard Kelly, K=½, would imply half Kelly, etc.), and profit or loss is given as a percentage of the total bankroll.

    So now that we have our utility function, all that's left is figuring out what our variables are, and then setting up the constraints.

    It should be readily apparent that we have two unknowns, the bet size on hedge candidate 1 (call it b1) and the bet size on hedge candidate 2 (call it b2). Note that all bet sizes are given as a percentage of the total bankroll. Our constants are the decimal odds on the initial bet, which is 2, the decimal odds on hedge 1 and hedge 2, which are 1+100/109 and 1+100/120, respectively, the probabilities of each of the three outcomes, which are given above, and lastly the Kelly divisor, K, which we’ll just set to 1 (meaning standard Kelly).

    The constraints are:
    1. b1 ≥ 0
    2. b2 ≥ 0
    3. b1 + b2 ≤ 35% (35% because, as you'll recall, Peter doesn’t want more than 60% of his bankroll tied up. Because 25% of his bankroll is already being used to fund the initial bet, that leaves up to 35% of his bankroll to be used on the hedge).


    (Of course if the maximum bet sizes were small enough relative to the total bankroll, we’d have to include further constraints on b1 and b2, constraining each to be less that the maximum bet, however because this is the Super Bowl, the maximums won’t come in to play for Peter.

    So finally, here’s our objective function:
    E(U) = 45.33% * ln(1 + 25% - b1 - b2) +
    3.49% * ln (1 - b1) +
    51.18% ln (1 - 25% + b1 * 100/109 + b2 * 100/120)

    Which we want to maximize with respect to b1 & b2, subject to b1 ≥ 0, b2 ≥ 0, and b1 + b2 ≤ 35%.

    At this point we need to decide whether we want to proceed computationally or analytically. Either methodology will obviously work. If you wanted to proceed analytically, you’d take the partial derivatives with respect to b1, b2, and the Lagrangians, set each one to zero, and solve for b1 & b2 (making sure not to forget the Kuhn-Tucker conditions for the constraints).

    If you wanted to proceed computationally, you could use any mathematical package, such as Mathematica, or Microsoft Excel Solver to do all the work for you. Solver is probably the easiest route as it’s included for free with Excel.

    So anyway, whatever method you choose, the result is the same. There’s one unique solution that satisfies the constraints (which incidentally, don’t bind). Namely:

    b1 = 22.0602%
    b2 = 0.9120%

    This implies the following results under each outcome scenario:
    1. NFC wins straight-up or AFC wins by less than 6: bankroll increases by 2.03%
    2. AFC wins by exactly 6: bankroll decreases by 22.06%
    3. AFC wins by more than 6: bankroll decreases by 4.00%.


    So what we see here is that under standard Kelly, it’s correct for Peter to eliminate most of his risk using the cheaper candidate (hedge 1), even though it still leaves him the chance of a disastrous reverse middle, and only bet a small portion of his bankroll on the more expensive hedge 2, even though it would provide a perfect hedge.

    The reader should note that if we had used a Kelly multiplier less than 1, what we’d see would have been a somewhat smaller bet on hedge candidate 1, and a somewhat larger bet on hedge candidate 2. With a Kelly multiplier of 0.25 (quarter Kelly), for example, the solution would be:

    b1 = 7.5500%
    b2 = 18.5613%

    implying the following results under each scenario:
    1. NFC wins straight-up or AFC wins by less than 6: bankroll decreases by 1.11%
    2. AFC wins by exactly 6: bankroll decreases by 7.55%
    3. AFC wins by more than 6: bankroll decreases by 2.61%


    So in other words Peter would be trading off a bit of expected value and would be eliminating the possibility of a profitable outcome, so as to reduce his exposure to the “worst-case scenario” (AFC by exactly 6).

    (If anyone's interested, I’ll put together a simple spreadsheet demonstrating this process using Solver. Anyone? Anyone at all?)

    So hopefully, this explains a bit of the mystery behind how to determine optimal hedge quantities. While this example is fairly simple, the same techniques could be used to determine more complicated hedges involving, for example, combinations of straight bets, parlays, and teasers.

    See my simple Kelly Hedge Spreadsheet (a more complicated example is linked to from this post).

  2. #2
    Korchnoi
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    Great post as usual. I guess Peter had risk on last weekends games that he didn't hedge? Do you still stand by last week's analysis of the bet?

    Perhaps Peter should look into SkyBook who offers Colts -6.5 at -112. I know as recently as a few weeks ago they gave you a free 1/2pt as long as it wasn't on/off the 3/7, so this looks like a possibility also.

  3. #3
    Ganchrow
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    Quote Originally Posted by Korchnoi
    Do you still stand by last week's analysis of the bet?
    Yes. I most definitely do.

    Quote Originally Posted by Korchnoi
    Perhaps Peter should look into SkyBook who offers Colts -6.5 at -112. I know as recently as a few weeks ago they gave you a free 1/2pt as long as it wasn't on/off the 3/7, so this looks like a possibility also.
    They're still offering the half-point on wagers up to $500. Actually, Colts -6 -112, based on the Pinnacle numbers I used in the example above, would represent a (slightly) positive expectation bet (0.368%).

  4. #4
    Art Vandeleigh
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    I don't get what you're trying to say.

    Let's say Mr. X has a bankroll of $4,000

    He bet 25% or $1,000 at +100 odds on NFC +6

    This turned out to be a bad bet. To safely hedge at AFC -6 he must lay -120.

    So if Mr. X bets $1090 on AFC -6, he would get back $908

    If AFC win by 7 or more, would lose $1000-$908= $92
    If AFC wins by 6, both bets push
    If NFC wins SU or loses by 5 or fewer, would lose $1090-$1000=$90

    So MR. X is going to lose about 90/4000=2.25% of bankroll.

    What are you trying to optimize here? I'm so not with the program.
    Last edited by Art Vandeleigh; 01-24-07 at 02:07 PM.

  5. #5
    Jay Edgar
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    Quote Originally Posted by Ganchrow

    (If anyone's interested, I’ll put together a simple spreadsheet demonstrating this process using Solver. Anyone? Anyone at all?)
    Yes please!

    (Not that I've ever actually had to sweat a reverse middle or anything . . . .)

  6. #6
    Ganchrow
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    Quote Originally Posted by Art Vandeleigh
    I don't get what you're trying to say.

    Let's say Mr. X has a bankroll of $4,000

    He bet 25% or $1,000 at +100 odds on NFC +6

    This turned out to be a bad bet. To safely hedge at AFC -6 he must lay -120.

    So if Mr. X bets $1090 on AFC -6, he would get back $908

    If AFC win by more than 7, would lose $1000-$908= $92
    If AFC wins by 6, both bets push
    If NFC wins SU or loses by 5 or fewer, would lose $1090-$1000=$90

    So MR. X is going to lose about 90/4000=2.25% of bankroll.

    What are you trying to optimize here? I'm so not with the program.
    In this example we're maximizing the player's expected utility. Putting it another way, we're attempting to find the optimal trade-off between risk (which is bad) and return (which is good).

    If the player were only interested in maximizing return (in other words, if he were risk neutral) the optimal decision would be not to hedge at all. This is because both hedge candidates have negative expected value.

    If the player were only interested in minimizing risk (in other words, if he were infinitely risk averse), he would behave largely as you've described. (An infinitely risk averse player would want the same financial outcome regardless of the game's outcome. He would do this by betting less on AFC -6 -120 than in your example, and more on AFC -6½ -109. So assuming that the player bet $1,000 on NFC +6 +100, the proper hedge for the infinitely risk averse player would be $1,000 on AFC -6, and $86.92 on AFC -6½. This way, regardless of the game's outcome the financial result would be a loss of $86.92.)

    What I demonstrated was how a player with logarithmic preferences (which are the preferences that the Kelly criterion requires of a player for it to be that player's optimal staking strategy) would go about optimally hedging the initial bet, given the two hedge candidate choices.

    The generalized form of the expected utility function (which is given in the initial post) is:
    E(U) = ( K/K-1) * i{p(outcomei) * [1 + (outcomei profit or loss)]1-1/K} for K ≠ 1
    E(U) = i{p(outcomei) * ln[1 + (outcomei profit or loss)]} for K = 1
    where constant K is the Kelly multiplier.

  7. #7
    Ganchrow
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    Quote Originally Posted by Jay Edgar
    Yes please!

    (Not that I've ever actually had to sweat a reverse middle or anything . . . .)
    This is the example spreadsheet I put together as proof of concept. It should be fairly self explanatory (scroll down to fill in the probabilities).

    For the spreadsheet to work you'll need to have solver.xla added as a visual basic reference. If it produces an error when you run it, then you'll need to do this manually. For an explanation of how to do this check out section III ("Hints for Using Solver in Macros") of this link.

    Kelly Multi-Hedge Example

  8. #8
    Dark Horse
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    If the object is to hedge the NFC +6 bet, I would consider an open teaser strategy. If 25% of bankroll is at risk on the Superbowl bet, take five or ten games you would have bet anyway, but now as the first leg of a number of open teasers; the second leg for each is the Superbowl (now down to AFC -0.5). Greek allows two weeks for open teasers to be completed.
    Last edited by Dark Horse; 01-24-07 at 03:55 PM.

  9. #9
    Ganchrow
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    Quote Originally Posted by Dark Horse
    If the object is to get out of the NFC +6 bet
    Technically speaking, the object is to maximize expected utility -- this may or may not entail getting out of some or all of the NFC +6 bet.

    Quote Originally Posted by Dark Horse
    I would consider an open teaser strategy. If 25% of bankroll is at risk on the Superbowl bet, take five or ten games you would have bet anyway, but now as the first leg of a number of open teasers; the second leg for each is the Superbowl (now down to AFC -0.5).
    Unless the implicit juice in teasing from the 6½ to the ½ is less than the juice on the straight AFC -6 bet (which there may or may not be depending on what the Greek charges for the teaser), then this would necessarily be more expensive than and would result in a riskier outcome than (by creating the large middle) what could be achieved with some combination of the two candidate hedges. In other words, it would necessarily be suboptimal for a player with log preferences.

    Of course of it turned out that the teased bet were cheaper (in terms of vig) than either of the unteased bets, then the tease may well be the right play.

  10. #10
    Dark Horse
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    The NFC +100 no juice bet means I'm averaging out around -110 for the hedge, but I'm getting a real shot at winning both sides (with AFC down to -0.5).

    Wouldn't be one teased bet, but a larger number for first leg, until risk on NFC bet was matched. Basically, I would get a hugely improved line for bets I would bet anyway. Losers would be tossed out, and winners go on until risk on NFC bet was matched.

    Edit - In this case, it wouldn't really have to be an open teaser format (normally so for games more than a week away), but could be regular teasers, unless you wanted to include games after the Super bowl.

    I would probably combine angles. For instance, by paying the juice for the AFC -6 -120 to lower my risk (25% of bankroll on one game is a bit steep to me), continuing to play my normal bets, and also playing those normal bets as teasers for a smaller bet size until amount risked on NFC bet was matched. (A teaser for AFC -0.5 and NFC +6 includes my projected outcome for Superbowl, so that would be a nice angle.)

    A mathematical analysis to the right side of the decimal point is just not realistic to me, unless I would do so for all bets. The Lord knows I don't.
    Last edited by Dark Horse; 01-24-07 at 04:35 PM.

  11. #11
    MrX
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    Quote Originally Posted by Art Vandeleigh
    Let's say Mr. X has a bankroll of $4,000
    Why are we bringing me into this?

    -Mr. X

    P.S. Ganchrow, would you please try putting a little thought and substance into your posts. We already have one JJGold around here.

  12. #12
    Ganchrow
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    Quote Originally Posted by Dark Horse
    I'm not sure I understand what you're saying.

    The NFC +100 no juice bet means I'm averaging out around -110 for the hedge, but I'm getting a real shot at winning both sides (with AFC down to -0.5).

    Wouldn't be one teased bet, but a larger number for first leg, until risk on NFC bet was matched. Basically, I would get a hugely improved line for bets I would bet anyway. Losers would be tossed out, and winners go on until risk on NFC bet was matched.
    It's difficult to explain this qualitatively and much easier to explain quantitatively. But before we can do this we'd need to assign probabilities to the teaser. For the sake of simplicitly we can just assume we're dealing with a one-team teaser priced equivalently as a two-team teaser -- this would be decimal odds of 1.3817, which is just the square root of 1.9091. Or we can just say we're taking the AFC on the money line, which would be equivalent to AFC -½.

    So I'll ask you ... assuming that AFC -6½ wins with probability 51.18%, how often would you expect AFC -½ to win? (Paying at odds of 1.3817, it would need to be lower than 70.37% in order for the juice to be higher than on the AFC -6 single. Based on the Pinnacle money line of-250/+230, we'd expect AFC -½ to win with probability 70.21%.)

  13. #13
    Sam Odom
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    Ganchrow is the man!

    btw- I have AFC -5.5 (-110) for 10units and Chi +7 (-110) for 5units

  14. #14
    Dark Horse
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    Ganch, the line is fluid. By the time this thought exchange is finished it could be down to 5.5, and your initial assessment of the NFC line would be correct. lol.

    I'm not sure I understand, because the title of this thread is about hedging, whereas NFC +6 and AFC -6.5 risks getting sided. You are a braver man than I if you're willing to risk 25% of bankroll on that, because this game could certainly end on 6.

    I have Indy winning the game by 2-3 pts, so would have to factor that into my strategy as well. I don't mind paying for a shot at a nice middle, and with the juice involved (+100 on one side and -120 on the other, for a total of -110), that would be far more attractive to me than risking 25% of my bankroll and only breaking even if I 'won'.
    Last edited by Dark Horse; 01-24-07 at 05:11 PM.

  15. #15
    Ganchrow
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    Quote Originally Posted by Dark Horse
    Ganch, the line is fluid. By the time this thought exchange is finished it could be down to 5.5, and your initial assessment of the NFC line would be correct. lol.
    Just take a snapshot. It doesn't need to be exact -- I'm trying to illustrate a concept here.

    Quote Originally Posted by Dark Horse
    I'm not sure I understand, because the title of this thread is about hedging, whereas NFC +6 and AFC -6.5 risks getting sided. You are a braver man than I if you're willing to risk 25% of bankroll on that, because this game could certainly end on 6.

    I have Indy winning the game by 2-3 pts, so would have to factor that into my strategy as well. I don't mind paying for a shot at a nice middle, and with the juice involved (+100 on one side and -120 on the other, for a total of -110), that would be far more attractive to me than risking 25% of my bankroll and only breaking even if my -6.5 'hedge' won.
    No matter how you sound it out qualitatively, based on the scenario outlined above, the fact is that the two solution sets I've provided are indeed the Kelly-optimal hedges for Kelly multipliers of 1 (full-Kelly) & 0.25 (quaretr-Kelly). It doesn't matter whether you believe it or not -- it's still true. If you're so inclined, I welcome you to try to poke mathematical holes in either my methodology or my computations. But if you just don't like the solution, don't blame me, blame either Sir Isaac Newton or Gottfried Leibniz. They "discovered" calculus, not me.

    What I think might be tripping you up are two separate issues. Firstly, you apparently don't agree that the current Pinnacle markets reflect the "best" estimates of the various win probabilities. That's fine, you may very well be right -- but that's just irrelevant to this current exercise. If you believe that the "true" market should be Indy -2½, well that would obviously change the solution. But I'm not looking to set up a different problem for everyone with their own view on the game. That's why right at the top we assumed that Peter had no view.

    Secondly, you just don't have logarithmic preferences. That's clear from the posts you've made in this thread, as well as posts you've made in others. And that's fine, too. Few if any people really do have precisely log preferences (I certainly don't) -- the only reason logs are used so frequently is because they're computationally convenient. But the more the your preferences deviate from logarithmic (and yours may deviate more than mine), the less appropriate the Kelly solution will be for you.

  16. #16
    Dark Horse
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    I'm quite sure that you figured out the right answer mathematically.

    Nevetheless, if the game ends on 6, you're f*cked.


    Unless I'm completely missing the point, which is possible, the suggested teaser approach cost me -110, and gives me a good shot at winning both sides (covering key numbers 1, 2, 3, 4, and 6 for a win and push; including the number 5, these total almost 40% of all mov's! - that percentage would only have to be adjusted to how many times a favorite of less than a TD won SU.). With that much to gain, why would I want to play even, while opening myself up to losing 25% of my bankroll? All to save 5 cents in juice (+100 and -110 for -105, versus +100 and -120 for -110).

    All comes down to why I'm betting; what can I win, and what can I lose. I lose 5 cents more than you if I lose and you win. Logarithmic preferences give you that edge. But I can win 2x 25% of my bankroll (roughly), where you can lose 25% of your bankroll. That's a 75% swing.

    Whatever the best approach, I might also want to bet it live if I was Peter and wanted out. Too many scenarios where a mov of 6 or 7 is out of the door as the game progresses.
    Last edited by Dark Horse; 01-24-07 at 06:47 PM.

  17. #17
    Ganchrow
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    Quote Originally Posted by Dark Horse
    Nevetheless, if the game ends on 6, you're f*cked.
    If the game ends on 6 (3.49% based on the Pinnacle markets I outlined above), then Peter would lose either 22.12% of his bankroll (assuming full-Kelly) or 6.82% of his bankroll (quarter-Kelly).

    That would definitely be a Very Bad Outcome.

    Nevertheless, bankroll fluctuations of this magnitude are not unexpected with Kelly. The full Kelly stake, for example, given a 5% edge and a line of -450, would be 22.5% -- and this bet would lose fully 14% of the time.

  18. #18
    Ganchrow
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    Quote Originally Posted by Dark Horse
    Unless I'm completely missing the point, which is possible, the suggested teaser approach cost me -110, and gives me a good shot at winning both sides (covering key numbers 1, 2, 3, 4, and 6 for a win and push; including the number 5, these total almost 40% of all mov's!
    Based upon the Pinnacle markets from above:
    1. you'd hit your middle (1 - 5) with probability 15.54%.
    2. you'd win the hedge bet and push the original bet with probability 3.49%
    3. you'd win the original and lose the hedge with probability 29.79%
    4. you'd win the hedge and lose the original with probability 51.18%
    So, if you hedged the original bet (call it $100) with your half-teaser (the ML really), such that your loss in scenarios 3&4 would be equivalent, you'd be putting down $142.86 (35.7150% of your bankroll) to win $57.18.

    So this means:
    1. with probability 15.54% you'd win $157.14 (bankroll increases by 39.30%)
    2. with probability 3.49%, you'd win $57.14 (bankroll increases by 14.30%)
    3. with probability 80.97%, you'd lose $42.86 (bankroll decreases by 10.71%)
    The expected utility from all this would be -3.5607%.

    Compare this to the expected utility of the K=1 solution above, which is higher at -2.0501%.

    So in other words, the ML hedge would be sub-optimal for a full Kelly bettor.

    Note however, that at a line of -250 and with a win prob of 70.21%, the juice on the money line (1.70%), is lower than the juice on either of the hedge candidates. This means that we can't rule out the ML bet as hedge candidate 3.

    So if we optimize at K=1 with all three hedge candidates, we'd get 22.1538% ($88.62) on AFC -6.5, 0.7759% ($3.10) on AFC -6, and 0.0726% ($0.29) on the AFC ML. This would result in the following payout set:
    1. with probability 15.54% your bankroll would increase by 2.10%
    2. with probability 3.49% your bankroll would decrease by 22.12%
    3. with probability 29.79% your bankroll would increase by 2.00%
    4. with probability 51.18% your bankroll would decrease by 4.00%

    implying an expected utility of -2.1383%.

    At K=0.25, the solution would be 6.8355% ($27.34) on AFC -6.5, 19.3474% ($77.39) on AFC -6, and 0.0170% ($0.07) on the AFC ML implying:
    1. with probability 15.54% your bankroll would decrease by 1.18%
    2. with probability 3.49% your bankroll would decrease by 6.83%
    3. with probability 29.79% your bankroll would decrease by 1.20%
    4. with probability 51.18% your bankroll would decrease by 2.60%
    So anyway, based on the scenario numbers from above, at both K=1, and K=0.25 it's pretty clear that the AFC ML (or half-teaser) is inferior as a hedge candidate to the alternatives.

  19. #19
    Dark Horse
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    Quote Originally Posted by Ganchrow
    So in other words, the hedge you're illustrating is sub-optimal for a full Kelly bettor.
    I think we may start out with different numbers. I'm getting a much higher probability for a MOV of 1-6. But if your numbers are correct I would simply pay the juice to get out of a bet I didn't like, instead of opening myself up to a pretty big loss just to save some juice.

    Where do I find a full Kelly bettor?
    Right next to the suicide bombers?

    For real, though. I would love to meet or talk to a full Kelly bettor who has proven that the theory translates into reality over the long term.

    Are you a full Kelly bettor?
    Last edited by Dark Horse; 01-25-07 at 12:55 AM.

  20. #20
    Ganchrow
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    Quote Originally Posted by Dark Horse
    I think we may start out with different numbers. I'm getting a much higher probability for a MOV of 1-6.
    Hey, I'm just using Pinnacle's numbers. It's between you and them on this one.

    Quote Originally Posted by Dark Horse
    But if your numbers are correct I would simply pay the juice to get out of a bet I didn't like, instead of opening myself up to a pretty big loss just to save some juice.
    Cool -- but it's still not optimal for either a quarter or a full Kelly bettor.

    Quote Originally Posted by Dark Horse
    Where do I find a full Kelly bettor?
    On blackjack tables and on trading floors to name but two locations.

    Quote Originally Posted by Dark Horse
    For real, though. I would love to meet or talk to a full Kelly bettor who has proven that the theory translates into reality over the long term.
    Kelly is extraordinarily simple. There's no hidden magic to it whatsoever. If your objective is to maximize your expected bankroll growth over time, then (assuming you can fairly accurately calculate your edge on any given bet) you simply can not do any better than full Kelly. It really is elementary mathematics and is "just" a theory in the sense that the theory of gravity or the germ theory of disease are also "just" theories.

    That Kelly does what it claims isn't theory, it's fact. The only 2 real issues are:
    1) Can you accurately calculate your edge on any given bet?
    2) Is your objective to maximize your expected bankroll growth?

  21. #21
    Wheell
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    Ganchrow has it completely right. The issue with running full Kelly is that your edge is rarely what you think it is. If one person handicaps the Bears as a 3.5 point dog and another caps the game as Indy -10 they will make roughly equal bets (assuming full Kelly) but they cannot possibly both be right about their edge. Running half Kelly makes sense when you suspect your edge might be up to 50% less than you would otherwise expect.

    One note: There are people that run anything from double Kelly to 100x Kelly. These people are almost exclusively betting with other people's money. Victor Niederhoffer comes to mind...

  22. #22
    Dark Horse
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    Kelly, who wasn't a gambler, does not account for losing streaks.

    Let's say someone has a 60% winning percentage, and bets four NFL games at full Kelly; i.e. 16% of bankroll for each of the bets. With a 100K bankroll, that is 64K on the line.

    Now let's say our bettor is in the middle of a losing streak. All four bets lose. Long term winning percentage isn't affected, but his bankroll is now down from 100K to 36K.

    But, according to Kelly, this is no big deal. Better luck next time.

    The next week our Kelly follower is back to NFL betting and bets six games for 16% of his bankroll each. Somewhat inconveniently, his bet size is down from 16K last week to only 5,760. This time Lady Luck smiles. In accordance with his 60% winning rate, all six games win! Whooohoooo!!! Unfortunately, he only made 31,418 back (at -110 juice). So even though, after two weeks of betting, he is 6-4, he lost almost 33K. Or about 1/3rd of his starting bankroll.

    How does that optimize profit?!

    I painted a relatively favorable picture for someone who went from losing to winning. If the bettor had gone 0-10 over the two weeks, which sooner or later happens to the best of us, the 100K bankroll would be down to about $1500!!!

    Try getting back to 100K from $1500.

    Full Kelly bettors don't have the first comprehension of streaks. It is an extremely dangerous theory. In my opinion a serious bettor should never make himself dependent on the sequence of his wins and losses.
    Last edited by Dark Horse; 01-25-07 at 02:44 AM.

  23. #23
    Wheell
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    Kelly accounts for losing streaks, winning streaks, and all in between. It isn't Kelly's fault if you either: Don't understand your winning % or don't know how to apply Kelly to non-sequential situations. I might add that to bet 16% of your bankroll on one game your edge must be enormous.

  24. #24
    Dark Horse
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    With a winning percentage of 60%, full Kelly is 16%. As a connaisseur of his theory, surely you know that.

    Please explain how Kelly accounts for a 0-10 losing streak. You have to bet full Kelly.


    From the Wikipedia link:

    Using the Kelly system in practice does have drawbacks. While it guarantees that you will never lose all your bankroll, it does not guarantee that you will not lose money. When a series of serial bets are made the chance of dropping to 1/n of your bankroll is 1/n. Thus you have a 50% chance of at some point losing 50% of your bankroll, a 10% chance of dropping to 10%, and so on.

    The optimum bet for the greatest growth of bankroll is making the full bet suggested by the Kelly criterion, but this produces a volatile result. There is a 1/3 chance of halving the bankroll before it is doubled.....
    Last edited by Dark Horse; 01-25-07 at 03:02 AM.

  25. #25
    Wheell
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    As you wish. Let us assume a stake of $100,000.00 in honor or CRIS adding pennies. Assume a vig of 11 to win 10. Assume a win rate of 60%. First, the odds of going 0-10 are 9535.743164 to 1. Not exactly being hit by lightning, but not easy to pull off either. Second, after going 0-10 you end up at 17490.12. Is this really not where you should end up after going 0-10 with 3-2 odds in your favor at a measly 11-10? I don't mean to be snide but if you end up at 89,000 after going 0-10 with this kind of edge you are not doing a good job of getting your money in with the best of it.

    In the interest of fairness your odds of going 17-0 are 5906.840252 to 1. If you pull that off you will end up making $996,059.39.

    Again, and I can not stress this enough, your edge is smaller than you think, and so half or quarter Kelly stratagies make a lot of sense. So does having a cap on the % of your bankroll that you will bet no matter what you think your edge is.

    Please do not blame Kelly for an 0-10 streak when you thought you were a 3-2 favorite. I my humble opinion, you were more of a 6-5 favorite. In that case, Kelly would only have lost a measly $36,568.22.

  26. #26
    Dark Horse
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    Quote Originally Posted by Wheell
    As you wish. Let us assume a stake of $100,000.00 in honor or CRIS adding pennies. Assume a vig of 11 to win 10. Assume a win rate of 60%. First, the odds of going 0-10 are 9535.743164 to 1. Not exactly being hit by lightning, but not easy to pull off either. Second, after going 0-10 you end up at 17490.12. Is this really not where you should end up after going 0-10 with 3-2 odds in your favor at a measly 11-10?
    If you haven't gone 0-10 with the very same systems that have proven highly profitable, perhaps you haven't been betting long enough. So no, that type of loss, percentage wise, would not be acceptable to me. In fact, it would be an astounding failure.

    Money management wouldn't have to be nearly as conservative as suggested by most pros if losing streaks weren't a reality of gambling. The idea of not going broke lies at the heart of most theories, and is largely ignored by full Kelly bettors (who simply declare that you can never go broke as long as you bet any percentage of your bankroll). For this reason, and in my personal opinion, the key to money management lies in understanding streaks. I would be in favor of betting full Kelly only if a gambler showed he understood the nature of streaks.

    For what it's worth, I have never met a full Kelly bettor in sports betting, nor anyone on-line who admitted to being one. So until we have a club of full Kelly bettors, all of whom have easily joined the billionaire, or at the very least millionaire ranks, I'm inclined to think of this as a fascinating theory only.
    Last edited by Dark Horse; 01-25-07 at 07:43 AM.

  27. #27
    Ganchrow
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    Quote Originally Posted by Dark Horse
    Kelly, who wasn't a gambler, does not account for losing streaks.
    What does that really mean? How might one "account for losing streaks"?

    Quote Originally Posted by Dark Horse
    Let's say someone has a 60% winning percentage, and bets four NFL games at full Kelly; i.e. 16% of bankroll for each of the bets. With a 100K bankroll, that is 64K on the line.

    Now let's say our bettor is in the middle of a losing streak. All four bets lose. Long term winning percentage isn't affected, but his bankroll is now down from 100K to 36K.
    This is not the proper Kelly stake.

    The Kelly stake at a 60% hit rate and -110 odds is 16% if and only if you're making only one bet at a time. If you're making 4 bets simultaneously, then the Kelly stake should be 14.82%.

    After losing those 4 bets our bankroll would have dropped from $100,000 to $40,712.31.

    Quote Originally Posted by Dark Horse
    The next week our Kelly follower is back to NFL betting and bets six games for 16% of his bankroll each. Somewhat inconveniently, his bet size is down from 16K last week to only 5,760.
    Again, your 16% figure is incorrect. The correct Kelly stake given 6 contemporaneous bets at a 60% hit rate and -110 odds is 13.85%. After winning those 6 bets the bankroll would stand at $71,463.01.

    Quote Originally Posted by DarkHorse
    This time Lady Luck smiles. In accordance with his 60% winning rate, all six games win! Whooohoooo!!! Unfortunately, he only made 31,418 back (at -110 juice). So even though, after two weeks of betting, he is 6-4, he lost almost 33K. Or about 1/3rd of his starting bankroll.

    How does that optimize profit?!
    Who says Kelly is supposed to maximize profit? It's not. It maximizes the expected growth rate of a bankroll. (If maximizing expected profit were your only consideration (that is, if you were risk neutral), you'd bet your entire bankroll on the single best bet you could find, and keep doing that until broke or at your target bankroll level.)

    Quote Originally Posted by DarkHorse
    Full Kelly bettors don't have the first comprehension of streaks. It is an extremely dangerous theory. In my opinion a serious bettor should never make himself dependent on the sequence of his wins and losses.
    "Comprehension of streaks?" What's there to comprehend? That they happen?

    The Kelly criterion is a mathematical theory, which I'd be happy to defend on a mathematical level. You, however, are arguing against Kelly viscerally, expressing that it just doesn't "feel" right to you that a bettor could have have a 0.605% chance of his bankroll dropping to 6.89% of its initial level (that's going 0-10, betting first 4 and then 6 contemporaneously). Essentially, your reaction to Kelly is emotional and apparently (willfully?) ignorant of its simple yet elegant mathematical underpinnings. That Kelly does what it says it does is beyond question -- this should be apparent to anyone with a competent high school mathematics background who actually takes the time to read and understand Kelly's original paper.

    If you don't believe the conclusions of Kelly, if you believe Kelly does not represent the strategy that maximizes expected bankroll growth over time, then prove it mathematically -- show where Kelly went wrong. Saying that it doesn't "take into account streaks" is meaningless ... demonstrate mathematically why Kelly's paper is wrong.

    But let me save you little bit of time, here. You won't be able to disprove the Kelly conclusions, because Kelly himself has already proven them in his paper. With all due respect, DH, just because you don't have a complete understanding of Kelly in particular and the expected utility hypothesis in general, doesn't mean that the hordes of quantitative finance professionals who are successfully using it on a daily basis are wrong.

    Quote Originally Posted by DarkHorse
    For what it's worth, I have never met a full Kelly bettor in sports betting, nor anyone on-line who admitted to being one.
    Then you, my friend, simply don't travel in the right circles.

  28. #28
    Dark Horse
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    As far as I'm concerned Kelly is a girl's name and it should stay that way. But I didn't mean to hijack this thread by turning it into a pro or against Kelly argument. So for that I apologize.

    About that stake. Let's have the four games (at 16% each) happen one after the other. A technicality. The risk remains enormous.

    Quote Originally Posted by Ganchrow
    "Comprehension of streaks?" What's there to comprehend? That they happen?
    With all due respect, my friend, there is much to comprehend here, but nothing I would want to share in public, at least not at this point. A year ago I would have been as skeptical as you about this. Not anymore. Streaks lie at the core of betting; they are the heartbeat of gambling.

  29. #29
    Ganchrow
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    Quote Originally Posted by Dark Horse
    About that stake. Let's have the four games (at 16% each) happen one after the other. A technicality. The risk remains enormous.
    Sure. Fine. Except that after going 6-4, the bankroll would have grown by (1-.16)^4*(1+.16*10/11)^6 -1 ≈ 12.46%, rather than having lost about 33% as you suggested. Big difference. It's all about compound interest.

    Neverthless, I get your underlying point. You feel that full Kelly is too risky. And that's completely fair. But just because you don't like the risk characteristics of Kelly doesn't delegitimize it as a staking plan. Kelly is still the betting strategy that maximizes expected bankroll growth. Period.

    Quote Originally Posted by Dark Horse
    With all due respect, my friend, there is much to comprehend here, but nothing I would want to share in public, at least not at this point. A year ago I would have been as skeptical as you about this. Not anymore. Streaks lie at the core of betting; they are the heartbeat of gambling.
    Fair enough. I certainly both understand and respect the need to keep certain information proprietary.
    Last edited by Ganchrow; 01-25-07 at 04:47 PM. Reason: Adjusted bankroll growth for -110 odds

  30. #30
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    If you dog the Kelly method you are just ill informed and don't understand.

    Kelly was an engineer who worked for Bell Labs (at the time probably the most prestigious place for an engineer to work). Nodes in the phone system were expensive, he needed to determine how many nodes to have so you wouldn't make a call and get "all lines are currently busy please try your call later". On the other hand lets not have 100's of these expensive nodes sitting there un used. Gamblers and people in other markets took his idea into their world.

    What Kelly does, using Kelly you will never go broke. Repeat never go broke. Flat bets betting does not afford you this benifit. Secondly you maximize your profits, once again flat betting doesn't do this.

    When you win and lose has no effect on your results. You could lose 45 straight and then win 55 straight and your bankroll would be the exact same place. A straight bettor would not weather an extended losing streak. On the other hand if you won 55 then lost 45 straight your bankroll would be exactly the same. I would be willing to bet the straight bettor would have increased his bet and then given more of his money back.

    The other really great thing about the kelly method is it forces you to approach money management in a disciplined money management form which is one of the biggest weaknesses of most gamblers.

    Lastly, if you are going to implement a kelly method you need to use parlays in your betting when you have events that are going off simultaneously. You use these to help mimic the results of events that went off sequentially. Rough numbers is reduce your straight wager by 30% and use that for a parlay (two team parlay).

    I would recommend a kelly percentage of somewhere in the 4-6% of your bankroll.

  31. #31
    Dark Horse
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    Quote Originally Posted by Ganchrow
    Sure. Fine. Except that after going 6-4, the bankroll would have grown by (1-.16)^4*(1+.16)^6 -1 ≈ 21.30%, rather than having lost about 33% as you suggested. Big difference. It's all about compound interest.


    Starting with 100,000. After four straight losses at 16% of adjusted bankroll, new bankroll: 49,787.
    (using 0.84 x bankroll)

    Starting with 49,787. After six straight wins at 16% of adjusted bankroll (-110 juice), new bankroll: 68,453.
    (using 1.16 x 0.909 x bankroll)

    I just lost -31.5% after going 6-4.
    How do you win +21.3%?

  32. #32
    Ganchrow
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    Quote Originally Posted by Dark Horse


    Starting with 100,000. After four straight losses at 16% of adjusted bankroll, new bankroll: 49,787.
    (using 0.84 x bankroll)

    Starting with 49,787. After six straight wins at 16% of adjusted bankroll (-110 juice), new bankroll: 68,453.
    (using 1.16 x 0.909 x bankroll)

    I just lost -31.5% after going 6-4.
    How do you win +21.3%?
    Sorry. I mistakenly used even odds. My bad.

    At -110, you'd be at $49,787.14 after your four losses.

    After 6 wins, you'd be at (1+.16*10/11)^6 * $49,787.14 ≈ $112,456.43.

    That's bankroll growth of 12.46%.

  33. #33
    Dark Horse
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    So during the six game win streak I profited roughly 19K, while you profited roughly 63K. And we started with the same amount. That's very dramatic.

    Clearly, you would have to be betting a lot more than 16% of your existing bankroll to accomplish this.

    Am I missing something, or are you introducing an element of chasing?
    Last edited by Dark Horse; 01-25-07 at 04:57 PM.

  34. #34
    Ganchrow
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    Quote Originally Posted by Dark Horse
    So during the six game win streak I profited roughly 19K, while you profited roughly 63K. And we started with the same amount. That's very dramatic.

    Clearly, you would have to be betting a lot more than 16% of your existing bankroll to accomplish this.

    Am I missing something, or are you introducing an element of chasing?
    Bankroll at $49,787.14 after four losses.

    Now for the 6 winning bets:
    1. Bet 16% of $49,787.14 bankroll = $7,965.94 to win $7,241.77
    2. Bet 16% of $57,028.90 bankroll = $9,124.62 to win $8,295.11
    3. Bet 16% of $65,324.01 bankroll = $10,451.84 to win $9,501.67
    4. Bet 16% of $74,825.69 bankroll = $11,972.11 to win $10,883.74
    5. Bet 16% of $85,709.43 bankroll = $13,713.51 to win $12,466.83
    6. Bet 16% of $98,176.25 bankroll = $15,708.20 to win $14,280.18
    ending bankroll = $112,456.43

  35. #35
    Dark Horse
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    My bad. Juicy mistake. You didn't apply the juice at first, but I applied it to the entire bankroll. lol.
    Last edited by Dark Horse; 01-25-07 at 06:32 PM.

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