Bet sizing - Kelly Criterion

General Discussion
#3

I’d also be interested in hearing other thoughts on this. Unfortunately I think it gets complicated very quickly. Our current bet sizing strategy is not sophisticated. We have a few heuristics we use to ensure we don’t ever have too many units in play at once. Often we’ll run the scripts that spit out the suggested bets and reduce the Kelly fraction if the total units seems too high. Not a science at all though.

You’ve probably already seen this as it’s one of the first google hits, but it’s an OK article on the simultaneous Kelly (for independent bets though…). Has a calculator at the end. Using your Oli Farr example it doesn’t make any adjustments (at least to 2 decimal places) for the fact that the bets are simultaneous, probably because they are very low probability and the edge is small. But it’s not addressing your concern which is that the bets are correlated.

Haha yes sometimes this happens without us noticing until after the bets are placed. Last week we bet twice on Cantlay against Finau in R3. Normally if we catch this we would reduce the stake size.

In the top 40 markets I assume that a correlated Kelly adjustment would increase your wagers (because they are negatively correlated) whereas in the Oliver Farr case presumably the optimal wagers are reduced. It’s also true that to maximize EV the correlation is irrelevant… as expectation is an additive operator… but for optimizing bankroll growth it seems really complicated.

#4

I’ve experimented with a few different sizings but finally came up with a good system. Try this one:
Size your wager so that (Wager amount + Win amount) = 1% of bankroll

So if your bankroll is $1000, you want (Wager amount + Win amount) = $10

Once you’re at that size, just bet all 3. So you would bet $0.12 on the Top 5, $0.30 on the Top 10, and $0.70 on the Top 20.

If you follow this rule you won’t run out of money and your bankroll will thank you.

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#5

I cant help too much because I’m pretty new to this but this simple equation helps get close to optimal bet sizes on simultaneous independent events (I realize that your bet is correlated though)

P minus P~

P=Probability of winning

p~ = 1/Decimal Odds

The above calculation assumes the bets being placed simultaneously are approaching 100% of the bankroll.

I like it because it keeps me from betting too much which was a big mistake I made in the beginning!

I assume that in a correlated event the bet size is just reduced so apply your Kelly Factor accordingly.

E.g. Betting 10 people to come in the Top 5?

#6

I started out betting Kelly and found that it didn’t work because I ran out of bankroll and I was overestimating my edge over the book. Even if you are pumping in +EV plays you get crushed if you’re overbetting longshots.

Reducing your wager size so that the stake + win amount = 1% of bankroll worked pretty well for me.

#7

It’s a scaled version of the Kelly Criterion that Datagolf uses though, isn’t it? I haven’t fully figured it out but it is 10% for most areas, 3% for make/miss cut and 8% for other areas. I have seen 15% at other times particularly for some in-play outrights.

A strategy focusing on outrights and Top 5s would have issues I feel. Focusing on more likely outcomes with edge (round and tournament matches, Top 20/40 etc.) seems a safer bet. We are dependent on the bookmakers for the mathcups and it is very unusual to find any edge on more fancied players in the place market though.

Pat Perez at the Schwab is an example. I had backed him in each of the Top 20/30/40 market and until the 67th hole or so it looked like all would pay out. By the 72nd, he missed out on them all. I still have no idea how I should have backed those 3 markets that he had a similar edge in all. My staking strategy definitely over exposed me to a collapse such as that.

#8

What were the odds on Perez for those markets?

For example, if Perez was +200 for Top 40, +300 for Top 30, and +500 for Top 20, I would have played it this way:
0.34 units to win 0.68 units for Top 40
0.25 units to win 0.75 units for Top 30
0.17 units to win 0.85 units for Top 20
Total risked: 0.76 units to win 2.28 units
(1 unit = 1% of bankroll)

So your total risk amount on Perez is less than 1 unit which should avoid overexposure.

Betfair exchange prices
#9

Perez for Schwab odds in European format
Top 40 Bet at 3.4, DG valued at 2.99 - 0.57 units
Top 30 Bet at 4.5, DG valued at 4.2 - 0.2 units
Top 20 Bet at 7.5, DG valued at 6.85 - 0.14 units

Total bet size was 0.91 units but in sizing the bets I treated each as independent events and scaled the Kelly criterion to 10%.

Reviewing the bets, the Top 30 edge probably wasn’t sufficient - I probably shouldn’t have bet at that level regardless. Still though, unsure whether to continue to bet in this fashion or not. It does work out as well obviously - I had KH Lee in Top 20/30/40 markets for Byron Nelson for instance. Just trying to figure out the maths on long term strategy.

#10

It looks like Datagolf is using ~10-20% of Kelly amounts for their wagers. With the lower amounts for the big longshots and the higher amounts for the 50-50 matchups.

Also very few wagers are for more than 1 unit. The exceptions are the big chalk wagers that have a big edge. For example a 4% edge on a -300 wager is HUGE and probably warrants a 2 unit wager or so.

#11

In addition to the 1- unit on Perez in the outrights, would you extend beyond 1% considering the matchup bets that involve Perez? Are matchup bets involving Perez factored in the 1%? Thanks.

#12

I probably would play the matchup independently and ignore the T20/T30/T40 exposure.
I don’t really keep track of how much exposure I have on each player, I just keep the bet sizes small and trust that I won’t be overexposed. Sometimes I do go over 1 unit but rarely do I go over the Kelly%.

#13

@DanielSong39 a flaw in only using wager and win amount is disregarding the edge. Even the same player will have different edges at top5, top10 and top20 markets due to a book not accurately pricing these events

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#14

Good point, but I have to live with the flaws. I have a very short window to get the wagers in and I often find that the numbers disappear within a minute or two. Speed is the most important factor when it comes to locking in wagers.

It’s not unusual to have 10-20 plays in the T20 market and have 5 of them disappear while I’m in the middle of creating my betting ticket.

#15

I am a bit late to this discussion, but just wanted to add a few things:

I can recommend this document on the theory of fractional betting by Ed Thorpe. The whole thing is worth reading, but pages 19 and 20 are of particular relevance to this discussion. In them, Thorpe walks through finding the optimal bankroll fraction to risk in the following situations:

  • two simultaneous (rather than sequential) bets with independent outcomes

  • two simultaneous bets with dependent (correlated) outcomes (with a known covariance matrix!)

To find them we solve the following maximization problem:
max g(f) = E[log(Xn)]

That is we are trying to find

f1 , f2 that satisfy:

d g(f1,f2) / f1 = 0

d g(f1,f2) / f2 = 0

(these should be partial derivatives, but I cannot figure out how to type-set them, but you get the picture.)

You can find the answers in the linked document if you are interest, it is just a bit cumbersome to type, but the basic intuition is that:

  • for independent, simultaneous bets, you bet a little less than Kelly, but not much less. If you are employing a fractional Kelly scheme (betting something like half-Kelly) you do not need to really adjust anything at all.
  • for dependent outcomes you bet anywhere from half-Kelly to half your bankroll(!!) depending on if they are positively or negatively correlated (as has been pointed out earlier in this discussion)

Does this generalize to more than two bets? That is, to calculate the optimal bet size, do we just need to pairwise correlation of each of the bets (or if you prefer, the covariance matrix)? Though it is perhaps not obvious, the answer is no.

To solve the optimization problem stated above, you need to be able to calculate the expectation at each feasible point, which requires the entire joint probability distribution for all of the bets under consideration, which is reserved for the double-secret-triple-platinum skratch-tools-members.

With that lengthy (and basically pointless) preamble in mind, I think that when it comes to golf betting there are two examples that are particularly instructive, and may provide rules-thumb-that cover a large portion of the instances where this really matters. It is often the case that DG will have a disagreement with a sportsbook over a particular players skill on a given week. These will often present as:

  • +EV to-win, top five, and top ten bets, top twenty bets
  • +EV matchups and 3balls

These are qualitatively a bit different from each other, in that “the event that Cameron Young wins the US Open” is a strict subset of “the event Cameron Young finishes in the top 5 at the US Open” (and so on), where “Cameron Young beats Kevin Kisner in a matchup” is correlated with "Cameron Young beats Jordan Spieth and Max Homa " in a three ball, but it is not a subset of it.

Top 5s and Top 10s, etc

Consider for a moment a situation where there is a +EV top 5 bet and +EV top 20 bet on the same player. These seem pretty correlated, so what should we do-- bet half the usual amount on each? Here it becomes important to consider at a more granular level the source of your disagreement with the sportsbook. In particular it is useful to observe that when you place a bet on someone to finish in the top 20, you can think about it as placing two separate bets

  • One bet on the player to finish in the top 5 (at whatever odds offered by the book)
  • One bet on the player to finish in position 6-20, with odds implied by Implied Probability of top 20 - Implied Probability of top 5

Consider the following example, which is from some random tournament from November of last year. DG had Oliver Bekker top 5 and top 20 as +EV. The left-most column is a 1k bet on Bekker to finish top 20, and the next two columns break it down into the equivalent bets implicitly placed simultaneously on a top 5 and a 6-20.

Top 20 Top 5 Finish 6-20
Book Probability 0.33 0.10 0.23
DG Probability 0.42 0.12 0.30
Betsize $1000 $300 $700
Payout $3000 $3000 $3000

The crucial thing here is that a bet on a top 5 and a bet on 6-20 have negative correlation, so you do not have to size down your bet from your typical fraction kelly scheme. Something you might do in this case then would be to:

  1. Figure out how much you want down on the top 6-20 bet. If the answer is zero, bet zero on the top 20 bet.
  2. Place the top 20 bet that corresponds to this amount
  3. Figure out how much you want down on the top 5 bet. Since you have already bet on a top 5 (implicitly) only bet the remaining amount to get you to the desired fraction of your bankroll.

Matchups and 3 Balls
Here I will say that if you are not interested in simulating the outcomes and solving the optimization problem stated above, and are looking for a rule of thumb I would say treat these events as highly correlated and do the following:

  1. Pick the bet with the bigger edge.
  2. Calculate the fraction of your bankroll you would wager on just this bet– call this f1
  3. Go to the smaller edge, and find f2
  4. Take max(f1,f2). This will be the total amount you risk. Suppose without loss of generality it is f1.
    Find new_f1, new_f2 that satisfy:
    new_f1 + new_f2 = f1
    new_f2 = f2/f1 * new_f1

And watch the money pile up baby!

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#16

A good rule of thumb is to just bet everything at ~10-20% Kelly and take your chances

Sometimes it makes sense to bet half your bankroll but those are scalp situations where you absolutely cannot lose both wagers

#17

I would say that may be a good rule of thumb for generally trying not to exceed whatever fractional Kelly scheme you are trying to implement.

But, taking your edge as given, and assuming that you are trying to stick as closely to your pre-defined fractional-Kelly strategy as possible when faced with correlated bets, it is not a good rule of thumb.

The idea behind managing your bankroll with a fractional Kelly scheme is to not “take your chances,” but rather to be thoughtful about:

  1. What your objective is
  2. How you achieve this objective

A scheme that ignores the dependence between wagers is going to perform much worse in the medium and wrong run versus one that closely approximates the desired fractional Kelly scheme. Even if you are trying to bet .1 or .2 Kelly, you need to make adjustments (sometimes large ones) when placing correlated wagers, as pointed out in my previous post.

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#18

Another interesting post. In your top 5 / 20 example, another thing to consider is the correlation in model error. If we are off on a player and the bet is actually negative EV, this will likely be true for all bets we’ve placed on them (top 5, 20, etc.). I’m not sure whether this would affect the analysis that much because changes to EV (e.g. suppose the bets are actually -1% EV vs 5% EV) doesn’t actually affect your range of outcomes much – it changes the mean slightly, obviously, but doesn’t affect the tails much, which is the more important region of the distribution for this type of analysis.

#19

That is a great point— the above analysis is assuming the edge is known with certainty, rather than a (possibly correlated) random variable.

Another thing to consider in practice is that it may be that situations where a book just gets a players top 5 odds wrong but not his top 20 (or the reverse) may present better betting opportunities (as they represent a mistake, rather than a genuine disagreement about a player’s skill).

#20

@OldSchoolScout this is super informative post, thank you! I’ve often struggled with how to properly reduce risk when betting on same player to top5/10/20. I had a couple related questions and wanted to get yours (or anyone else’s) opinion. The example uses 2 bets, top 20 and top 5, but often we have +EV bets for winner, top5, top10, top20. What’s a reasonable process to get the optimal bet size when using 3-4 bets on same player? Also on step 3 you wrote “only bet the remaining amount to get you to the desired fraction of your bankroll”. Is this remaining amount to get you to the desired win amount or bet amount? Intuitively I would think win amount but I cant confidently claim that is correct. Thanks!

#21

I’ve sized the wagers so that the risk amount + win amount = 0.5% of bankroll

Then I fire away with as many +EV bets as I can without worrying about overexposure

This seems to work pretty well

#22

I am glad you found it helpful.

Also on step 3 you wrote “only bet the remaining amount to get you to the desired fraction of your bankroll”. Is this remaining amount to get you to the desired win amount or bet amount? Intuitively I would think win amount but I cant confidently claim that is correct

Sorry about that-- I should have been a bit more clear here: I meant,

  1. calculate the fraction of your bankroll that you would want on the top 5 if it were the only bet offered, and you were not worried about correlation.
  2. subtract the amount you have implicitly placed on it through the top 20 bet.
  3. Bet this remaining difference (your original desired amount - the amount you already have implicitly bet) on the top5

My day job is preventing me from working through an example at the moment, but if faced with top 40, top 20, top 10, top 5, to win:

  1. Start with the top 40
  2. Use the odds offered on the other bets to break it into 20-40,10-20, 5-10, 2-4, 1st
  3. Use the procedure I outlined above
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