Kelly Revisited

[Mathcapper]

*** CORRECTION ***

Just realized I mistyped the odds-based formula for the Kelly Criterion. It should read:

edge = (final odds+1)/(fair odds+1) – 1

f = [(final odds+1)/(fair odds+1) – 1]/(final odds)

Sorry for any confusion.

Btw, there was a good example of both the merits and the pitfalls of the Kelly formula Saturday at the Spa.  During the seminar, Jerry said that Tenango looked 50/50 to win the 7th race. The DD Will Pays indicated the horse was going to go off around 3-1, which is exactly where he ended up.

Calculating the expected value and the corresponding Kelly bet,

edge = (final odds+1)/(fair odds+1) – 1 = (3+1)/(1+1) – 1 = 1.0 = +100% (a very big edge)

f = edge/odds = 100%/3 = 33%

So with a 100% edge getting 3-1 odds, one should wager 33% of his current bankroll (assuming the size of the wager doesn’t affect the odds).

The horse never had an anxious moment and paid $8.20. Finding and wagering on horses with big edges like that can increase your wealth pretty quickly using Kelly.

But what if you aren’t as good as Jerry at assessing your edge? What if you identified similar horses that you thought were 50/50 to win but were actually properly priced at 3-1? Then you’d be betting 33% of your bankroll on horses where you really had no edge at all and shouldn’t even be betting.  If you lost 3 races in a row on such horses, which could easily happen when the chances of winning are that low, you’d do severe damage to your bankroll.

That is why it is so important to be able to get your fair odds absolutely right. If you misjudge your edge by even a small amount, you can easily end up overbetting your bankroll to the point of near-ruin.

Making accurate fair odds estimates is much harder to do than predicting final odds accurately, but it is just as important. Most Kelly practitioners make their own fair odds lines and keep detailed records of how well the horses they’ve deemed overlays actually perform with respect to the fair odds they’ve given them. Doing so ensures that they’re not overestimating their perceived edge and consequently overbetting their bankroll.

Rocky

[TGJB]

Have to say its hard to believe the system says to bet a third of your bankroll on one bet. So if he was longer than 3-1, it could have you betting half your bankroll on a 50/50 proposition, albeit a big overlay?

[TreadHead]

Anecdotal evidence to prove a point is fun, isn\'t it?  

First of all, completely agreed that any system that has you betting 33% or even 15% of your bankroll should be set aflame instantly.  Anyone who has spent time playing poker or sports betting will tell you the same story.  No matter how sure you are about odds of success, you have to practice sound bankroll management, and most ppl would say to never have more than 5% of your bankroll in play at any time.

The other part of this discussion that is downright laughable is that there is an absolute right/wrong answer about \"fair odds\" on a horse.  You could put 100 very astute handicappers in a room and undoubtedly get at least 5-10 different opinions on what the \"fair odds\" of a particular horse would be, assuming we are calling 6-5 different from even (which you have to if we are talking mathematical precision of your formula).

Taking subjective concepts and treating them like factual data points is a flashing warning light for a flawed methodology.

[Mathcapper]

That is true, but it\'s not a linear progression, so it\'d have to be a really big overlay.

For instance, with an edge twice as large (fair odds 5-1, +200% edge), Kelly says to bet 40% of your bankroll. At 10-1 (+450% edge), it\'s 45%. Even at 100-1 fair odds, Kelly is just shy of 50%.

Having said that, I agree it seems like a big number. The reason is twofold:

1) The size of the overlay. A +100% edge is pretty rare. Most work out to be much less than that, typically +5% to +50% at best.

2) The low fair odds. This is a danger of Kelly. High probability horses by definition call for large Kelly wagers. That\'s one of the reasons why a lot of people use \'half Kelly\' or even \'one third Kelly,\' especially when dealing with horses they\'ve deemed odd-on propositions.

[Mathcapper]

Tread - you are absolutely right, there is no \"correct\" fair odds line.

The important thing is that your own particular fair odds are performing as they should over time. That is to say, you\'re overlays are winning at the rate you think they should be. This is why it is so important to keep records so that you know you aren\'t overestimating your edge and overbetting your bankroll.

[Mathcapper]

After putting a little more thought into this I confirmed that the Kelly formula does indeed turn out to be the optimal wagering strategy in this case. The key is that it calls for betting a fraction of your current bankroll, not of your original bankroll, so you never really run the risk of ruin, at least not in theory anyway.

I ran a small simulation under which wagers are made only on the rare situation’s like Saturday’s Tenango example, where you have a +100% edge on a 50/50 proposition calling for a Kelly bet of one third of your bankroll. I looked at what would happen if you had a really horrific start and lost the first 6 bets in a row (an extremely unlikely 1.5% chance of occurrence), followed by a more normal 50% alternating win/loss sequence for the next 50 wagers. With a tiny initial bankroll of $1,000, ignoring the effect on the pool, the results would look like this:

1) Wager $333. Lose. Balance: $667
2) Wager $222. Lose. Balance: $445
3) Wager $148. Lose. Balance: $297
4) Wager $99. Lose. Balance: $198
5) Wager $66. Lose. Balance: $132
6) Wager $44. Lose. Balance: $88
7) Wager $29. Win $87. Balance: $175
Wager $58. Lose. Balance: $117
9) Wager $39. Win $117. Balance: $234
10) Wager $78. Lose. Balance: $156
11) Wager $52. Win $156. Balance: $312
12) Wager $104. Lose. Balance: $208
13) Wager $69. Win $207. Balance: $415
14) Wager $138. Lose. Balance: $277
15) Wager $92. Win $276. Balance: $553
16) Wager $184. Lose. Balance: $369
17) Wager $123. Win $369. Balance: $738
18) Wager $246. Lose. Balance: $492
19) Wager $164. Win $492. Balance: $984
20) Wager $328. Lose. Balance: $656
21) Wager $219. Win $657. Balance: $1,313
22) Wager $438. Lose. Balance: $875
23) Wager $292. Win $876. Balance: $1,751
24) Wager $584 Lose. Balance: $1,167
25) Wager $389. Win $1,167. Balance: $2,334
26) Wager $778. Lose. Balance: $1,556
27) Wager $519. Win $1,557. Balance: $3,113
28) Wager $1,038. Lose. Balance: $2,075
29) Wager $692. Win $2,076. Balance: $4,151
30) Wager $1,384. Lose. Balance: $2,767
31) Wager $922. Win $2,766. Balance: $5,533
32) Wager $1,844. Lose. Balance: $3,689
33) Wager $1,230. Win $3,690. Balance: $7,379
34) Wager $2,460. Lose. Balance: $4,919
35) Wager $1,640. Win. $4,920. Balance: $9,839
36) Wager $3,280. Lose. Balance: $6,559
37) Wager $2,186. Win $6,558. Balance: $13,117
38) Wager $4,372. Lose. Balance: $8,745
39) Wager $2,915. Win $8,745. Balance: $17,490
40) Wager $5,830. Lose. Balance: $11,660
41) Wager $3,877. Win $11,661. Balance: $23,321
42) Wager $7,774. Lose. Balance: $15,547
43) Wager $5,182. Win $15,546. Balance: $31,093
44) Wager $10,364. Lose. Balance: $20,729
45) Wager $6,910. Win $20,730. Balance: $41,459
46) Wager $13,820. Lose. Balance: $27,639
47) Wager $9,213. Win $27,639. Balance: $55,278
48) Wager $18,426. Lose. Balance: $36,852
49) Wager $12,284. Win $36,852. Balance: $73,704
50) Wager $24,568. Lose. Balance: $49,136
51) Wager $16,379. Win $49,137. Balance: $98,273
52) Wager $32,758. Lose. Balance: $65,515
53) Wager $21,838. Win $65,514. Balance: $131,029
54) Wager $43,676. Lose. Balance: $87,353
55) Wager $29,118. Win $87,354. Balance: $174,707
56) Wager $58,236. Lose. Balance: $116,471


Comparing the Kelly results to other wagering strategies under this same scenario:

1) Kelly(33% of current bankroll): Min Balance: $88.  Final Balance: $116,471
2) Betting 5% of current bankroll: Min Balance: $734.  Final Balance: $6,699
3) Betting 2% of current bankroll: Min Balance: $886.  Final Balance: $2,288
4) Flat Bet of $50 on each race: Min Balance: $700.  Final Balance: $3,200
5) Half Kelly (16.5% of current bankroll): Min Balance: $335.  Final Balance: $88,641

Even with an almost impossibly horrendous beginning where you lost over 90% of your original bankroll, you still would come out vastly farther ahead in just 50 bets using Kelly. Other percentage or flat betting strategies don’t even come close.

Although it can be extremely volatile, full Kelly is clearly the optimal course of action if you have tested the performance of your overlays and are sure your edge is what you think it is.

The benefits of \"half-Kelly\" are also clearly evident. It captures much of the upside of full Kelly, while greatly reducing the volatility and max drawdown, thereby providing somewhat of a cushion for misjudgements in perceived overall edge or final odds, which is why so many practitioners use it rather than full Kelly.

The other thing to note is how quickly you reach pool size limitations when you have such a big edge like this. With a more typical edge like +10% to +25%, the swings would be much more moderate and it would take much longer to reach pool size limitations.

[TreadHead]

Do you have data that shows the rate at which 33% wager suggestions actually hit instead of making up hypotheticals?  If I were cashing 50% of the bets I made on 3-1 to 5-1 horses, I wouldn\'t need Kelly to show a profit.

[TGJB]

You\'re kinda missing the point. What he\'s talking about has nothing to do with handicapping, only betting. If your handicapping is off no betting system can help you.

[mjellish]

I get the method behind the madness.  But in reality, it\'s not going to work.  Running simulations is one thing. I agree with betting more when you are winning and betting less when you are losing (percentage of bankroll).  But anything that has you betting 33% of your bankroll on any single bet is going to wipe you out.  I could almost guarantee it.

One big assumption here that is not being discussed is that you actually get the odds you think are going to get.  So what happens when your 3-1 horse drops to 8/5 in the last flash after the leave the gate and you have 33% of your bankroll on it?

If you can find and bet with a reliable bookie or offshore site you could maybe eliminate some of this risk, but you are going to lose the vig and they probably have edge already built in anyway, or you may lose bigger when they shut down the site with your bankroll in hand.  The wise money will eat a bookie up on horse racing anyway, and they know that.  There was a place called the Sport of Kings years ago that had a lot of money behind it and would take any size bet on a horse race.  I don\'t remember how long they lasted but it wasn\'t long.

You could maybe just focus on big races with big handles on big days to eliminate some of that last flash stuff.  But even then, I remember putting a pretty big bet on Monarchos in the Florida Derby with 2 minutes to post at 5/2, and he wound up dropping to 8/5 after the gate opened.

I get the theory.  But in practice it won\'t work.  Just my two sense anyway.

[Mathcapper]

mjellish Wrote:
-------------------------------------------------------
> One big assumption here that is not being
> discussed is that you actually get the odds you
> think are going to get.  So what happens when your
> 3-1 horse drops to 8/5 in the last flash after the
> leave the gate and you have 33% of your bankroll
> on it?

mj - I couldn\'t agree more that being able to predict final odds accurately is vital to the Kelly formula - that\'s what got me started on this whole discussion in the first place. As I pointed out in my other posts, it can be done with surprisingly good accuracy by calculating the equivalent win parlays from the DD Will Pays:

Robo Betting

Is Kelly Dead?

Best,

Rocky

[Topcat]

On their opening day, Sport of Kings took an enormous bet on an overnight race at Keystone . . . needless to say, the horse got there, by daylight, paid around 5-2, and it was all downhill, from there.

[mjellish]

Mathcapper,

I went back and read your earlier posts.  

I do think you can often predict approximate Win Odds by using the double pool.  But the problem is they are approximate, and not always accurate.  All it takes is one plunger in the double pool or win pool to shake things up.  So when trying to use a money mgmt. system with a fixed set of rules, it doesn\'t matter if it is accurate 80% or 90% of the time.  You need 100%, and you aren\'t going to get it.  And my gut tells me that if you followed this system, there are going to be too many times where you are going to wind up with 33% of your bankroll or more on a bet (shear lunacy anyway) when after the last flash it turns out you should have had no more than 5%. A junk in = junk out type of deal.  This is exactly why the math whizzes still haven\'t beat the game and probably never will. A true math based system really requires 100% certainty.

That being said, I really like what you have to say, and I think there is a lot of merit to it.  Sound money mgmt. is very important.  Excellent work, and a good discussion.

[Mathcapper]

mj – Thanks for the kind words, glad I’m able to add my little bit to the vast compilation of knowledge you guys have put up on this board over the years.

Believe it or not, the math whizzes actually have beaten the game. Although not much is written about it publicly, what they’ve done is considered to be the most successful professional gambling efforts of all time.

I’ve got a good deal of research on the subject (whatever I could dig up anyway) if anyone’s interested, but the article in the link below called “Horse Sense” provides an excellent synopsis of much of it (the sidebar entitled \"How to Play the Smart Money\" discusses Kelly, although they incorrectly state that you should bet your edge when it should be your edge/odds).

As far as predicting final odds (and more importantly, estimating fair odds), you don’t need to be right 100% of the time, you just need to be sure that your estimates are in-line on average over the long-term.

The example I gave that called for wagering 33% of your bankroll is a very rare occurrence that only comes up when you have a really big edge coupled with a very high probability of winning. As the simulation I ran showed, you never really run the risk of ruin anyway because you’re only betting a fraction of your current bankroll, not your original one. So as crazy as it may sound, following the Kelly formula is still the proper course of action in that case to optimize your bankroll in the long-run.

As my \"worst-case\" simulation showed though, it can be extremely volatile, which is why even the math whizzes use half-Kelly instead of strictly following full Kelly. It’s really a moot point anyway, because that situation only comes along once in a blue moon. A 20% edge on a 5-1 shot calling for betting 4% of your bankroll is much more typical of the kind of wagering you’d end up doing under Kelly.

Contingencies

[Fairmount1]

Have you read Chapman\'s paper I assume?

Did Woods and Benter ever use their programs on U.S. races?--I was unclear on that from the article but didn\'t think they did based on what I read.

[Mathcapper]

Fair - I have read the Chapman paper, yes. It\'s a strictly academic treatise though and the math is pretty deep. Benter\'s follow-up paper called \"Computer-Based Horse Race Handicapping and Wagering Systems: A Report\" is a much more lucid and informative piece readable by us laymen. They can both be found in Efficiency of Racetrack Betting Markets.

From what I understand, Benter has been playing in the North American market for at least a decade now. He has a staff of only 3-4 people and has said himself that a typical team wagers around $1M per day across roughly 150 races, or around $350-$400M per year. Woods never left Hong Kong.

Rocky

Efficiency of Racetrack Betting Markets