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Date: Mon, 19 Dec 94 13:03:06 EST
From: Albin Warth
Message-Id: <9412191803.AA01733@frontier.com>
To: derby@inslab.uky.edu (Horse Racing List)
Subject: Pick3 study
Comments: Derby Message #1773
Sender: owner-derby@inslab.uky.edu
Errors-To: derby-problems@inslab.uky.edu
I've finished (for now) my analysis of pick3 payoffs. I ended up with
240 pick3 sequences, all gathered indiscriminately from recent DRF charts.
Of these, 167 were from HOL and BM; the other 73 from miscellaneous other
tracks including SUF, CRC, and AQU. IMO, the sample is big enough to draw
some "broad brush" results.
I examined pick3 payoffs in relationship to the parimutuel (PM) odds of the
winners of the individual legs. Input for each pick3 sequence was:
- the PM win-pool odds of each winner
- the pick3 payoff (rounded to nearest dollar)
- the base pick3 bet amount
- the win-pool take percent (e.g., 15.33 for HOL, 19% for SUF)
The examination involved four steps:
1) For each sequence, back-calculate the fraction of the win-pool represented
by each of the winners by backing out breakage and win-pool track take. If
we assume that the win pool is efficient, this fraction can be considered the
"natural probability" of a win based on those odds. For example, with "dime"
breakage and win-pool take of 19%, a horse with chart odds of 3.00 (i.e., 3-1)
has a "natural" win probability of 0.20:
1 - (take% / 100) 1 - (0.19)
------------------------- = ------------------- = 0.20
1 + (odds + (breakage/2)) 1 + (3.00 + (0.05))
(In other words, 3-1 at SUF would be 4-1 in horseplayer's heaven, where
there would be no take, no breakage, free coffee, and free programs.)
This is pretty accurate except for very short priced horses, where there
is more room for error in the breakage calculation.
Of course, the assumption that the win-pool is efficient is known to be
incorrect due to long-shot bias. I'll conveniently ignore this for now.
2) Assuming that the three races are independent events, calculate the
probability of three horses with the noted odds winning -- just multiply
the "natural" probabilities together. Using three 3.00-1 shots at SUF,
this would be 0.20 * 0.20 * 0.20, or 0.008 (i.e., 1 in 125.)
Any comments on when this (independence) assumption may not be valid?
In a general sense, one would expect the pick3 payoff (to $1) for these horses
to be $126, before adjustments for the track take on the pick3 bet.
3) Calculate the "edge" on each pick3 sequence by dividing the "natural"
probability by the "PM probabiltity". If the actual payoff in this example
was $126 (for $1), the edge would be 1.00, i.e, neutral. If the actual payoff
was $101 ("PM probabilty" = 0.010), the "edge" would be 0.008/0.010, or 0.80.
If the actual payoff was $251 ("PM probability" = 0.004), the edge would be
2.000 (very good). We want to find odds combinations that yield an edge
higher than 1.
4) "Slice and dice" the 240 results in various ways based on PM odds of the
winners of the individual legs. For each odds hypothesis, I calculated the
*median* edge for both the hypothesis and its complement. For the overall
sample, I also calculated the *average* edge.
Some overall results (all payoffs are to $1):
All 240 sequences
-----------------
average edge = 0.873, median edge = 0.792
average payoff = $307.13, median payoff = $142.33
(The range of "edges" was 2.750 to 0.201; the range of payoffs was $5298 to
$10.50.)
167 California sequences
-------------------------
average edge = 0.861, median edge = 0.798
average payoff = $253.22, median payoff = $138.67
73 Non-California sequences
---------------------------
average edge = 0.899, median edge = 0.790
average payoff = $430.47, median payoff = $179.50
The median edge for California corresponds remarkably well to the actual pick3
take of 20.08%. The median edge for other tracks seems reasonable, with
varying pick3 takes.
The lowest "high" odds in any of the sequences was 5.10-1. In other words,
there were no sequences with all horses 6-1 or higher.
Here are "slices" I made based on odds. I tried to keep each slice relatively
large. The first column is sample size, the second is median "edge".
All sequences 240 0.792
1a) All horses 8-5 or higher 120 0.869
1b) Other 120 0.769
2a) All horses even-money or higher 195 0.813
2b) Other (at least one odds-on) 45 0.764
3a) Two or more horses at 7-2 or higher 128 0.798
3b) Other (two or more less than 7-2) 112 0.791
4a) At least one horse 10-1 or higher 70 0.878
4b) Other (all less than 10-1) 170 0.785
5a) At least one horse 7-1 or higher 126 0.794
5b) Other (all less than 7-1) 114 0.792
6a) At least one horse 12-1 or higher 52 0.778
6b) Other (all less than 12-1) 188 0.892
7a) Two or more horses 5-1 or higher 51 0.704
7b) Other (two or more less than 5-1) 189 0.813
8a) At least one horse 4-1 or higher 197 0.798
8b) Other (all horses less than 4-1) 51 0.778
9a) All horses 3-1 or higher 36 0.825
9b) Other (at least one less than 3-1) 204 0.791
10a) At least one horse 20-1 or higher 21 0.973
10b) Other (all horses less than 20-1) 219 0.791
(Result 10a appears very shaky due to sample size.)
Conclusions
-----------
Well, I didn't find any free lunch here. Only (1b) and (2b) seem to do
significantly worse than track take on pick3 pools. Encouraging are
(1a) -- all horses 8-5 or higher and (4a) -- at least one horse 10-1 or
higher. There seems to be at least a slight "reverse" longshot bias.
It's interesting that the *average* edge based on all samples (0.873)
outperforms just about all of the "slices". This is true even though
there were no sequences with "monster" median edges...
Only five pick3 sequences produced payoffs with an edge greater than 2.000:
ODDS ODDS ODDS PAYOFF/$1 EDGE TRACK
----- ----- ----- --------- ----- -----
1.50 4.50 6.40 509.00 2.755 PHA
4.50 5.60 6.80 1495.00 2.740 SUF
5.10 8.90 20.00 5298.00 2.351 PHA
3.70 4.70 19.80 2481.00 2.314 SUF
2.10 7.70 9.70 978.33 2.002 HOL
There doesn't seem to be anything remarkable about these, especially when
compared with those that produced the lowest edges:
ODDS ODDS ODDS PAYOFF/$1 EDGE TRACK
----- ----- ----- --------- ----- -----
0.70 2.10 4.60 11.33 0.201 HOL
0.70 3.10 4.60 19.00 0.266 HOL
2.10 9.00 14.90 259.50 0.311 BM
1.00 1.20 8.00 23.00 0.320 HOL
0.70 2.30 4.30 20.50 0.330 SUF
Larry K has suggested to me that the race order and morning-line favoritism
(or lack thereof) of the three race winners has a major effect on the pick3
payoffs. I suspect he's right, and might try to analyze this.
Two things to remember: the results are based on the winners, not the losers;
"tax-ticket" results have to be adjusted according to your circumstances.
If you'd like a copy of my input data (it's a text file) or calculation
program (C or Tcl), drop me a note. (I'll be off-line until after the first.)
Would also appreciate comments on my methodology or suggestions for future
experiments.
Best holiday wishes to the derby-listers. (My holiday would have been
a little merrier if my $307 pick3 with The Exeter Man had come through
yesterday!)
-Albin
albin@frontier.com
From owner-derby@inslab.uky.edu Wed Jan 4 10:37:07 1995
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Date: Wed, 4 Jan 95 09:26:05 EST
From: Albin Warth
Message-Id: <9501041426.AA11180@frontier.com>
To: derby@inslab.uky.edu (Horse Racing List)
Subject: Pick3 follow-up
Comments: Derby Message #2076
Sender: owner-derby@inslab.uky.edu
Errors-To: derby-problems@inslab.uky.edu
Thanks to Jetaway Dave for his critique of my pick3 study, and to some
others who have written to me off-list. I think there might be a
misunderstanding of how I calculated the results with respect to average
(mean) vs median "edges". Let me give an example to see if this clears
it up. If it doesn't, please fire back!
The following table shows results from seven pick3 sequences for one day
at HOL. The sequences are sorted in descending order of "edge".
ODDS ODDS ODDS PAID/1$ PROB PROB PROB NATPROB MUTPROB EDGE TRK
------ ------ ------ -------- ----- ----- ----- ------- ------- ----- ---
2.20 3.40 2.10 91.00 0.261 0.190 0.269 0.01332 0.01111 1.199 HOL
7.30 4.40 1.50 217.00 0.101 0.155 0.332 0.00523 0.00463 1.130 HOL
20.70 2.60 7.30 1063.67 0.039 0.232 0.101 0.00092 0.00094 0.973 HOL
3.40 2.10 7.40 173.00 0.190 0.269 0.100 0.00512 0.00581 0.881 HOL
1.50 2.20 3.40 54.33 0.332 0.261 0.190 0.01646 0.01875 0.878 HOL
4.40 1.50 2.20 56.67 0.155 0.332 0.261 0.01344 0.01796 0.748 HOL
2.60 7.30 4.40 163.33 0.232 0.101 0.155 0.00365 0.00616 0.593 HOL
The first three columns are the parimutuel odds of the three winners. The
fourth column is the pick3 payoff to $1. The fifth through seventh columns
are the "natural probabilities" of wins based on the PM odds, calculated by
assuming an efficient win pool and backing out breakage and win-pool take.
The eight column is the "natural probability" of three horses with those
odds winning, calculated by multiplying columns 5-7 together. The ninth
column is the "mutuel probability", calculated from the payoff.
The "edge" is the ratio of "natural probability" to "parimutuel probability".
If there were no take on the pick3, I would expect the edge to be about
1.0. With a 20.08% take in California, I would expect the edge to hover
about 0.7992. The interesting (most profitable) cases for the bettor are
those with edge greater than 1.0.
Using the first line as an example, the actual payoff was $91. Based on the
PM odds, the "natural probability" was 0.01332, which translates to an
"expected" payoff of $76.08. In this case, the actual profit exceeds the
expected profit by a ratio of 1.199.
For these seven example pick3 sequences, the mean edge is 0.915; the median
edge is 0.881.
In the overall sample (now up to 315 sequences), the mean edge was 0.854
and the median edge was 0.791.
Please note that no averaging of odds is done in calculating either mean or
median edge -- each three-race sequence stands by itself. Only the "edges"
go into the calculation.
I would agree that both mean and median results are of interest. Although
the overall mean exceeds the median, the skewing or distortion does not come
(necessarily) from a few abnormally large payoffs, but from those results
with abnormally large edges. (It is true that very large payoffs tend to
produce better edges.) In the mini-sample above, the highest edge happened
to occur for a sequence with relatively low payoff.
Frankly, I calculated median edge rather than mean edge because it was
easier to do. If I post any additional results, I promise to include both
mean and median!
Someone said off-list that the study leads to dismal conclusions. I would
disagree; yes, the edge is depressed by the pick3 takeout, but the average
(mean) results outperform the take significantly, and the median results
track the take pretty closely.
Ken pointed out that the pick3 will almost always outperform the parlay.
This is because the multiplicative effect of the win-pool take is a killer.
For 15.33% take, a parlay will generally return
0.8467 * 0.8467 * 0.8467 = 0.607 of your investment,
for an "edge" of 0.607. For 19% take, it's 0.81 * 0.81 * 0.81. or 0.531,
which is almost as bad as the daily lottery.
I think that the pick3 stands out as a bet in those cases when you can find
at least two races with overlays -- here, the single take of the pick3 and
the multiplicative effect of "natural odds" are the keys.
For example, let's say that your betting line is 3-1 on each of three horses
in consecutive races, and you expect each of these horses to go off at 4-1.
Here, your win-pool edge on each of these horses is 25%.
- A $1 win-pool parlay on these three horses would yield $125 to $1.
- If your betting line is accurate, you would expect the pick3 to come in
1/4 * 1/4 * 1/4 = 1/64 of the time, for a fair pick3 payoff of $65 to $1.
- The expected pick3 payoff (assuming 4.00 PM odds, California takes, and
0.791 median pick3 edge) is about $169 to $1.
In any case, you've still got to beat the take.
-Albin
albin@frontier.com
From owner-derby@inslab.uky.edu Tue Feb 28 10:49:10 1995
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From: albin@tracon.frontier.com (Albin Warth)
Message-Id: <9502281512.AA14968@frontier.com>
To: derby@inslab.uky.edu (Horse Racing List)
Subject: More pick3 data
Comments: Derby Message #4541
Sender: owner-derby@inslab.uky.edu
Errors-To: derby-problems@inslab.uky.edu
Since my original post about pick3 payoffs, I've continued collecting
pick3 data. I decided to stop when I reached 1000 sequences, and overshot
by 1. Thanks to Ned Stoffel for providing data for about 120 of the pick3
sequences, all from Monmouth.
To recap, the "edge" is the ratio of the actual pick3 mutuel odds (payoff
to $1 less the $1) to the "expected" payoff odds. The "expected" payoff
odds are calculated by backing out the take and breakage from the win-pool
odds of each race winner, converting to implied probability, and multiplying.
(The underlying assumption is that the win-pool is efficient, which, of
course, it's not. It would be useful to re-run the data using a formula
that compensates for longshot bias. Anybody have one?)
A simple set of rules that produced an edge greater than 1.0 would be the
equivalent of a free lunch, no handicapping required.
The 1001 sequences come from a number of tracks, with varying takeouts
on the pick3 pool. (The lowest take is for California, at 20.18%.) The
mean edge for all of the sequences if 0.839, which corresponds to an
effective take of 16.1%. This is itself promising, since it's lower than
the pick3 take for any of the tracks.
The calculated edges ran from 4.23 to 0.09, and 241 (24.1%) of the sequences
yielded an edge greater than one.
Extremes...
- 62 of the sequences produced payoffs of more than $1000 (for $1),
with an average edge of 1.28.
- 68 of the sequences produced payoffs of less than $20 (for $1), with
an average edge of 0.539.
As before, I made a number of "slices" of the entire sample. This time,
I looked at the odds of the three horses in each sequence. The "low odds"
correspond to the lowest-price horse in the sequence, "middle odds" to the
next higher-price horse, and "high odds" to the odds of the highest-price
horse.
In all cases, the quoted odds correspond to the tote-board odds. For
example, 2-1 includes 2.00, 2.10, 2.20, 2.30, and 2.40.
I calculated the mean edge, median edge, and mean payoff (to $1) for each
slice. For all slices, the mean edge is greater than the median edge.
Mean edges greater than .870 are (arbitrarily) flagged with a "+", those
less than .820 with a "-". Edges less than .790 or greater than .900 get
double flags.
All Tracks
=================
Count Mean Edge Median Edge Mean Payoff
------------- ----- --------- ----------- -----------
All sequences 1001 0.839 0.765 371
Low Odds Count Mean Edge Median Edge Mean Payoff
------------- ----- --------- ----------- -----------
1-5, 2-5, 3-5 151 0.752 -- 0.705 106
4-5, 1-1 214 0.808 - 0.729 105
6-5, 7-5, 3-2 154 0.885 + 0.810 207
8-5, 9-5 127 0.921 ++ 0.813 287
2-1 143 0.837 0.761 261
5-2, 3-1, 7-2 143 0.850 0.776 524
4-1 and higher 69 0.857 0.758 2204
Middle Odds Count Mean Edge Median Edge Mean Payoff
------------- ----- --------- ----------- -----------
3-2 and lower 126 0.690 -- 0.652 34
8-5, 9-5 89 0.814 - 0.760 62
2-1 157 0.873 + 0.797 96
5-2, 3-1 179 0.851 0.806 143
7-2, 4-1 154 0.846 0.799 262
9-2, 5-1 146 0.926 ++ 0.788 484
6-1 and higher 150 0.840 0.716 1397
High Odds Count Mean Edge Median Edge Mean Payoff
------------- ----- --------- ----------- -----------
5-2 and lower 117 0.714 -- 0.652 27
3-1, 7-2 119 0.804 - 0.764 51
4-1, 5-1 200 0.823 0.768 96
6-1, 7-1 147 0.846 0.792 198
8-1, 9-1 120 0.878 + 0.748 269
10-1 to 14-1 148 0.885 + 0.810 390
15-1 and higher 150 0.904 ++ 0.769 1491
I also made the same "slices" for the Southern California results. Not
suprisingly, the results are somewhat better because of the *relatively*
low pick3 take of 20.18%.
SoCal Tracks only
=================
Count Mean Edge Median Edge Mean Payoff
------------- ----- --------- ----------- -----------
All sequences 354 0.856 0.797 365
Low Odds Count Mean Edge Median Edge Mean Payoff
------------- ----- --------- ----------- -----------
1-5, 2-5, 3-5 54 0.749 -- 0.704 116
4-5, 1-1 78 0.800 - 0.765 104
6-5, 7-5, 3-2 47 0.901 ++ 0.869 239
8-5, 9-5 42 0.980 ++ 0.909 421
2-1 60 0.869 0.799 241
5-2, 3-1, 7-2 45 0.890 + 0.826 543
4-1 and higher 28 0.877 + 0.808 1682
Middle Odds Count Mean Edge Median Edge Mean Payoff
------------- ----- --------- ----------- -----------
3-2 and lower 41 0.711 -- 0.652 34
8-5, 9-5 19 0.738 -- 0.708 56
2-1 58 0.862 0.859 86
5-2, 3-1 63 0.891 + 0.869 152
7-2, 4-1 75 0.874 + 0.806 249
9-2, 5-1 46 0.920 ++ 0.796 543
6-1 and higher 52 0.884 + 0.796 1314
High Odds Count Mean Edge Median Edge Mean Payoff
------------- ----- --------- ----------- -----------
5-2 and lower 31 0.729 -- 0.640 29
3-1, 7-2 38 0.837 0.799 50
4-1, 5-1 85 0.803 - 0.767 89
6-1, 7-1 52 0.911 ++ 0.898 211
8-1, 9-1 50 0.909 ++ 0.768 273
10-1 to 14-1 41 0.843 0.815 335
15-1 and higher 57 0.932 ++ 0.800 1414
Comparison with win-pool parlay
-------------------------------
I calculated the ratio of the pick3 payoff to the corresponding win-pool
parlay payoff for each pick3 sequence...
Mean pick3/parlay payoff ratio: 1.52 (all), 1.49 (SoCal only)
Median pick3/parlay payoff ratio: 1.37 (all), 1.37 (SoCal only)
Of 1001 sequences, 815 pick3 payoffs (81.5%) were higher than the parlay
payoff, and 186 (18.6%) were less or equal. For SoCal only, the ratio
was 306:48.
Win-pool parlays are not the way to go!
Conclusions
-----------
Here's the advice I would give to myself based on the results above:
1) Avoid using any horse at even money or less, unless you include
at least one horse at 9-1 or up.
2) Avoid using two or more horses at less than 2-1.
3) Avoid using all three horses at less than 4-1.
Combining these three rules across the 1001-sequence sample yields a mean
pick3 edge of 0.884, with an average payoff of $575 (for $1) from the 608
sequences that fit the rules. For the 354 SoCal sequences, these rules
produce a mean pick3 edge of 0.895, with an average payoff of $541 (for $1)
from the 226 complying sequences.
Applying one more rule:
4) Include at least one horse in the 6-5 through 9-5 range.
yields an edge of 0.934 from 285 sequences, with an average payoff of
$310 (all tracks), and an edge of 0.925 from 104 sequences, with avg
payoff of $358 for SoCal tracks.
Using these simple rules cuts the pick3 take by almost two-thirds,
*assuming* you (I mean I) picked the winners to begin with. One still
has to find the "overlays" to turn the pick3 into a winning bet.
Obviously, one could construct highly artificial rules that yielded
terrific pick3 edges. (Like, always wheel Best Pal in the anchor leg!)
Important note: these are the results for the winners, not the losers.
It does not necessarily follow that betting all pick3 sequences that
matched the rules would yield a pick3 ROI of .934/.925.
As before, the pick3 data is available upon request: maybe you can find
the free lunch!
-Albin
albin@frontier.com
From owner-derby@inslab.uky.edu Tue Feb 28 10:49:10 1995
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From: albin@tracon.frontier.com (Albin Warth)
Message-Id: <9502281512.AA14968@frontier.com>
To: derby@inslab.uky.edu (Horse Racing List)
Subject: More pick3 data
Comments: Derby Message #4541
Sender: owner-derby@inslab.uky.edu
Errors-To: derby-problems@inslab.uky.edu
Since my original post about pick3 payoffs, I've continued collecting
pick3 data. I decided to stop when I reached 1000 sequences, and overshot
by 1. Thanks to Ned Stoffel for providing data for about 120 of the pick3
sequences, all from Monmouth.
To recap, the "edge" is the ratio of the actual pick3 mutuel odds (payoff
to $1 less the $1) to the "expected" payoff odds. The "expected" payoff
odds are calculated by backing out the take and breakage from the win-pool
odds of each race winner, converting to implied probability, and multiplying.
(The underlying assumption is that the win-pool is efficient, which, of
course, it's not. It would be useful to re-run the data using a formula
that compensates for longshot bias. Anybody have one?)
A simple set of rules that produced an edge greater than 1.0 would be the
equivalent of a free lunch, no handicapping required.
The 1001 sequences come from a number of tracks, with varying takeouts
on the pick3 pool. (The lowest take is for California, at 20.18%.) The
mean edge for all of the sequences if 0.839, which corresponds to an
effective take of 16.1%. This is itself promising, since it's lower than
the pick3 take for any of the tracks.
The calculated edges ran from 4.23 to 0.09, and 241 (24.1%) of the sequences
yielded an edge greater than one.
Extremes...
- 62 of the sequences produced payoffs of more than $1000 (for $1),
with an average edge of 1.28.
- 68 of the sequences produced payoffs of less than $20 (for $1), with
an average edge of 0.539.
As before, I made a number of "slices" of the entire sample. This time,
I looked at the odds of the three horses in each sequence. The "low odds"
correspond to the lowest-price horse in the sequence, "middle odds" to the
next higher-price horse, and "high odds" to the odds of the highest-price
horse.
In all cases, the quoted odds correspond to the tote-board odds. For
example, 2-1 includes 2.00, 2.10, 2.20, 2.30, and 2.40.
I calculated the mean edge, median edge, and mean payoff (to $1) for each
slice. For all slices, the mean edge is greater than the median edge.
Mean edges greater than .870 are (arbitrarily) flagged with a "+", those
less than .820 with a "-". Edges less than .790 or greater than .900 get
double flags.
All Tracks
=================
Count Mean Edge Median Edge Mean Payoff
------------- ----- --------- ----------- -----------
All sequences 1001 0.839 0.765 371
Low Odds Count Mean Edge Median Edge Mean Payoff
------------- ----- --------- ----------- -----------
1-5, 2-5, 3-5 151 0.752 -- 0.705 106
4-5, 1-1 214 0.808 - 0.729 105
6-5, 7-5, 3-2 154 0.885 + 0.810 207
8-5, 9-5 127 0.921 ++ 0.813 287
2-1 143 0.837 0.761 261
5-2, 3-1, 7-2 143 0.850 0.776 524
4-1 and higher 69 0.857 0.758 2204
Middle Odds Count Mean Edge Median Edge Mean Payoff
------------- ----- --------- ----------- -----------
3-2 and lower 126 0.690 -- 0.652 34
8-5, 9-5 89 0.814 - 0.760 62
2-1 157 0.873 + 0.797 96
5-2, 3-1 179 0.851 0.806 143
7-2, 4-1 154 0.846 0.799 262
9-2, 5-1 146 0.926 ++ 0.788 484
6-1 and higher 150 0.840 0.716 1397
High Odds Count Mean Edge Median Edge Mean Payoff
------------- ----- --------- ----------- -----------
5-2 and lower 117 0.714 -- 0.652 27
3-1, 7-2 119 0.804 - 0.764 51
4-1, 5-1 200 0.823 0.768 96
6-1, 7-1 147 0.846 0.792 198
8-1, 9-1 120 0.878 + 0.748 269
10-1 to 14-1 148 0.885 + 0.810 390
15-1 and higher 150 0.904 ++ 0.769 1491
I also made the same "slices" for the Southern California results. Not
suprisingly, the results are somewhat better because of the *relatively*
low pick3 take of 20.18%.
SoCal Tracks only
=================
Count Mean Edge Median Edge Mean Payoff
------------- ----- --------- ----------- -----------
All sequences 354 0.856 0.797 365
Low Odds Count Mean Edge Median Edge Mean Payoff
------------- ----- --------- ----------- -----------
1-5, 2-5, 3-5 54 0.749 -- 0.704 116
4-5, 1-1 78 0.800 - 0.765 104
6-5, 7-5, 3-2 47 0.901 ++ 0.869 239
8-5, 9-5 42 0.980 ++ 0.909 421
2-1 60 0.869 0.799 241
5-2, 3-1, 7-2 45 0.890 + 0.826 543
4-1 and higher 28 0.877 + 0.808 1682
Middle Odds Count Mean Edge Median Edge Mean Payoff
------------- ----- --------- ----------- -----------
3-2 and lower 41 0.711 -- 0.652 34
8-5, 9-5 19 0.738 -- 0.708 56
2-1 58 0.862 0.859 86
5-2, 3-1 63 0.891 + 0.869 152
7-2, 4-1 75 0.874 + 0.806 249
9-2, 5-1 46 0.920 ++ 0.796 543
6-1 and higher 52 0.884 + 0.796 1314
High Odds Count Mean Edge Median Edge Mean Payoff
------------- ----- --------- ----------- -----------
5-2 and lower 31 0.729 -- 0.640 29
3-1, 7-2 38 0.837 0.799 50
4-1, 5-1 85 0.803 - 0.767 89
6-1, 7-1 52 0.911 ++ 0.898 211
8-1, 9-1 50 0.909 ++ 0.768 273
10-1 to 14-1 41 0.843 0.815 335
15-1 and higher 57 0.932 ++ 0.800 1414
Comparison with win-pool parlay
-------------------------------
I calculated the ratio of the pick3 payoff to the corresponding win-pool
parlay payoff for each pick3 sequence...
Mean pick3/parlay payoff ratio: 1.52 (all), 1.49 (SoCal only)
Median pick3/parlay payoff ratio: 1.37 (all), 1.37 (SoCal only)
Of 1001 sequences, 815 pick3 payoffs (81.5%) were higher than the parlay
payoff, and 186 (18.6%) were less or equal. For SoCal only, the ratio
was 306:48.
Win-pool parlays are not the way to go!
Conclusions
-----------
Here's the advice I would give to myself based on the results above:
1) Avoid using any horse at even money or less, unless you include
at least one horse at 9-1 or up.
2) Avoid using two or more horses at less than 2-1.
3) Avoid using all three horses at less than 4-1.
Combining these three rules across the 1001-sequence sample yields a mean
pick3 edge of 0.884, with an average payoff of $575 (for $1) from the 608
sequences that fit the rules. For the 354 SoCal sequences, these rules
produce a mean pick3 edge of 0.895, with an average payoff of $541 (for $1)
from the 226 complying sequences.
Applying one more rule:
4) Include at least one horse in the 6-5 through 9-5 range.
yields an edge of 0.934 from 285 sequences, with an average payoff of
$310 (all tracks), and an edge of 0.925 from 104 sequences, with avg
payoff of $358 for SoCal tracks.
Using these simple rules cuts the pick3 take by almost two-thirds,
*assuming* you (I mean I) picked the winners to begin with. One still
has to find the "overlays" to turn the pick3 into a winning bet.
Obviously, one could construct highly artificial rules that yielded
terrific pick3 edges. (Like, always wheel Best Pal in the anchor leg!)
Important note: these are the results for the winners, not the losers.
It does not necessarily follow that betting all pick3 sequences that
matched the rules would yield a pick3 ROI of .934/.925.
As before, the pick3 data is available upon request: maybe you can find
the free lunch!
-Albin
albin@frontier.com
From owner-derby@inslab.uky.edu Wed Mar 1 01:47:43 1995
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Date: Tue, 28 Feb 1995 19:22:32 +0800
From: djones@Corp.Megatest.COM (Dave Jones)
Message-Id: <9503010322.AA09260@pluto.megaeng>
To: derby@inslab.uky.edu (Horse Racing List)
Subject: More pick3 data
X-Sun-Charset: US-ASCII
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Comments: Derby Message #4596
Sender: owner-derby@inslab.uky.edu
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Albin Warth sezz...
> (The underlying assumption is that the win-pool is efficient, which, of
> course, it's not.
[ Parenthetical pedantry: I think the term may be "calibrated". Is there
a statistics guru in the house? I think "efficient" is a market analysis
term that implies that the win pool is "correct" in some obscurely
undefinable way. Maybe I'm wrong. I had been using the term "consistent"
to mean that horses in a given odds range collectively win at a rate
commensurate with those odds, but I just made that up.]
> It would be useful to re-run the data using a formula
> that compensates for longshot bias. (Anybody have one?)
I use one in my laptop computer exacta-calculator. It's based on charts
from Dr. Z's Beat the Racetrack, or whatever it's called.
> Conclusions
> -----------
> Here's the advice I would give to myself based on the results above:
>
> 1) Avoid using any horse at even money or less, unless you include
> at least one horse at 9-1 or up.
> 2) Avoid using two or more horses at less than 2-1.
> 3) Avoid using all three horses at less than 4-1.
I don't think you'll be able to make those conclusions after you factor
in the longshot bias. I wouldn't be surprised to see the results reversed.
In fact, I doubt that you'll find any magic bullets in high and low odds in
the win-pool. I've noticed that there is enormous variance in the
ratios of win-pool parlay payoff to pick3 payoffs. Is there any way to
predict when the two wagers will be seriously out of whack? When they are, who
is "right"? -- the pick3 players or the win pool players? Who is going
to win in the long run when those odds lines clash? I wrote a
little program that sides with the pick3 players. I feed it the pick3
will-pays displayed after the second leg of the pick3, and it returns an
odds line for the third leg. Is that sensical? It has one thing going for it
at least: The line is made by people who have already picked winners in the
two previous races, so maybe they ain't too stupid. The downside is that
it very seldom produces even one overlay! That may seem odd when you
consider how out of whack the payoffs can be, but when one payoff is out of
whack with the parlay payoff, ALL the payoffs in the third leg tend to
be proportionally out of whack. That consideration lead me to suspect
that the variance is not totally due to action on individual horses.
Some kinds of combinations may get more action than others. More on that
farther down the page.
If you want to take the side of the win pool bettors and the morning
line writers, what you should be looking for is angles that show a flat
bet profit in the pick3. It could be something as simple as linking a
lukewarm favorite to two medium priced horses, but I doubt that anything
that simple has a chance. Short of that, you might try to devise some method
for predicting pick-3 payoffs. If you could come up with a method with an
acceptable correlation coefficient, you might have the beginnings of a
Dr. Z-like system that would work for a while, maybe until you published a
book about it.
I poured over the smaller data-set that you sent me, and ran computer
routines over it. The only angle I came up with that looked at all promising
was to try to bet combinations of "strange bedfellows". Every ticket would
criss-cross short to medium to long prices. In one race you would play a
top favorite or two, in another race only contenders, and in the other
only longshots (live ones, you hope). The abduction argument is that
some people like chalk, others like prices, so neither sort will tend to mix
them on tickets, and thus those combinations will be underbet. Sadly,
further research has not corroborated that theory.
I have doubts that numbers games with win odds and pick3 payoffs
will lead anywhere.
You might try identifying which barns bet big to win on their live ones,
but neglect or misplay the pick3. There you might have something tangible.
Dave
From owner-derby@inslab.uky.edu Thu Mar 2 00:03:19 1995
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Date: 01 Mar 95 23:36:14 EDT
From: "Larry Khirallah"
Subject: RE: More pick3 data
To: derby@inslab.uky.edu (Horse Racing List)
X-Office-To: USENET SMTP
Comments: Derby Message #4637
Sender: owner-derby@inslab.uky.edu
Errors-To: derby-problems@inslab.uky.edu
Albin already commented on Dave Jones comments on the pick-3 data study
performed by Albin. Now, my comments on both sets of comments....
Albin wrote...
>>Conclusions
>>-----------
>>Here's the advice I would give to myself based on the results above:
>>
>> 1) Avoid using any horse at even money or less, unless you include
>> at least one horse at 9-1 or up.
---- I respond: I would be even more demanding and state, 'drop the
include condition and ALWAYS avoid even money or less
horses in the pick-3.'
>> 2) Avoid using two or more horses at less than 2-1.
---- I respond: Totally agree.
>> 3) Avoid using all three horses at less than 4-1.
---- I respond: Be tougher and make that 5-1...
Dave Jones responds:
>I don't think you'll be able to make those conclusions after you factor
>in the longshot bias. I wouldn't be surprised to see the results reversed.
---- I respond: Sorry Dave, but you're way off base here. Albin's
conclusions might not be all that solid (since he doesn't
represent 'losing' pick-3s in his study), But to say
that factoring in longshot bias might reverse the
results is ludicrous. I believe if you look at enough
pick-3 results you will see the opposite, i.e. 'favorite
bias' (in pick-3 play). Albin do you concur??
Dave Jones later states...
>..consider how out of whack the payoffs can be, but when one payoff is out of
>whack with the parlay payoff, ALL the payoffs in the third leg tend to
>be proportionally out of whack.
>
---> I respond: NO WAY! Sorry Dave, but most of the time, ALL the 'out-of-
whack payoffs are NO WHERE near proportional. Nice
recent example (yesterdays pick-3 at Suf): Probable
Pays for the third leg showed the following:
4 horses at 2456
1 horse at 3126 ----> (final win odds 6-1)
1 horse at 9000 (roughly)(final win odds 30-1)
1 horse at 1245 (final win odds 7-1)
1 horse (2 of 3) at 135 (final win odds 12-1)
Of the 4 horses at 2456, there final respective win
odds were 2-1 , 10-1, 5-2, 5-1.
Absolutely no proportionality there. (obviously this is
only 1 example which is fresh in my mind, but I recall this
sort of 'example' as being the norm not the exception)
Dave Jones goes on to say...
> .... what you should be looking for is angles that show a flat
>bet profit in the pick3. It could be something as simple as linking a
>lukewarm favorite to two medium priced horses, but I doubt that anything
>that simple has a chance. ...
I respond: I have found 2 angles which have profitted me over the last
two years rather handsomely (one allowed me to hit that pick-3
mentioned above that returned 1/2 of that 2456 mentioned for
a modest 15 dollar investment). I'm not sure I would call
either an 'automatic flat bet play for profit', as I would
never play every single pick-3 sequence at all tracks with
either (or any specific) method. However, I only use one
or the other angle for every single pick-3 that I 'choose'
to bet. Using that restriction, my pick-3 'edge' as Albin
would call it is well over 2.
> .................... If you could come up with a method with an
>acceptable correlation coefficient, you might have the beginnings of a
>Dr. Z-like system that would work for a while, maybe until you published a
book about it.
---> I respond: I would never write a book about any 'angle' I have found
for many reasons (includding the one you have mentioned).
I do not need any 'justification' other than the money I win.
Dave then finishes with....
>I have doubts that numbers games with win odds and pick3 payoffs
>will lead anywhere.
>
>You might try identifying which barns bet big to win on their live ones,
>but neglect or misplay the pick3. There you might have something tangible.
----> I respond: Both agree and disagree with you here. I do agree
that 'fiddling' with correlations between final post-time
win odds and respective pick-3 payoffs might be a waste of
time. BUT....
Let's change just 1 thing when looking for correlations. Instead
of investigating using final post-time win odds, lets investigate
using ML odds AS PRESENTED in the TRACK PROGRAM. This is one of the
KEYS to both of the angles I use.
Let's digress for a moment, to supply some background to why ML
(track program) odds might be more interesting....
As many on this list have already stated, the majority of track
patrons never buy the DRF. I concur with this statement. Along with
this statement, I think its quite safe to say that the vast majority
do purchase the on-track program (if for nothing else, to know the
numbers and jockeys riding there 'favorite' horses). That means that
the vast majority have the 'program selections' starring them 'right
in the face' at the bottom of each race page. These selections usually
(at least in every program I've ever seen), order the top three ML
favorites. Now, my contention is the following....
In most (if not every) pick-3 pool, these ML program selections are
OVER-represented. In other words, the amount of pick-3 combos utilizing
these 'program selections' in the pick-3 pool is not correctly
proportional to their respective chances of winning, even when considering
their actual ML odds. This, in and of itself, produces overlays when
the UNDER-represented horses win their respective legs.
Unfortunately, other than the profits I have made playing pick-3s
over the last 2 years, I have no data collected to 'prove' these
assertions. But quite frankly, I don't care if you believe them or not,
as I'm not writing to try to convince you of my observations or theories.
Howover, should Albin (or anyone else for that matter) collect a large
sample of data correlating pick-3 payoffs using ML odds (or more
importantly simply a notation of the program selections in each race)
instead of using 'final odds', it might prove me out...
In any case, I will finish here by detailing the aforementioned Suf
pick-3 from Weds 2/2/95 races 3-5.
The winning payoff as stated was 2456.
The individual final win odds (payoff) for each leg winner were:
18.00 / 13.40 / 6.40 Parlay value of (roughly) 386 !!!
Now the ML program odds....
10-1 / 15-1 / 9-2 Parlay value of ML odds 1936
(none were amongst the program selections for their respective race)
Thats just one more to add to the study......
Larry K
From owner-derby@inslab.uky.edu Thu Mar 2 00:17:56 1995
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Date: Thu, 02 Mar 95 00:15:09 EST
From: AJJBSJU
To: derby@inslab.uky.edu (Horse Racing List)
Subject: Re[2]: More pick3 data
X-Mailer: MUSIC/SP V3.1.1
In-Reply-To: In reply to your message of WED 01 MAR 1995 19:50:26 EST
Comments: Derby Message #4639
Sender: owner-derby@inslab.uky.edu
Errors-To: derby-problems@inslab.uky.edu
I don't know if you have considered this but I think you should use
a system of Flags for Data that maybe out of the ordinary.
I and a firend did a rather crude study of Pick3 payoffs at major
Eastern Tracks for 1 month (Last January, using PHI, AQU, GULF,
TAM, SUF, GS) and we tested what a parlay of those 3 winners would be
compared to the pick 3 itself. Our conclusion was that on Average
the Pick3 paid about 25% more than a parlay.
We then broke it down by track, We had better success with AQU because
we had the full charts every day (US college Students won't pay for
a racing form unless we got to the track -- so we used our local paper)
Anyway we began Flagging the pick3's that paid less than a parlay
or lower than the avg.
We found that on certain combos like 3 favorites, or when a consolation
pick3 was paid, or when the outcome of the horses had coresponding
#'s like 1-1-1 or 2-2-2 or 7-7-7 or 7-1-1 or 1-2-3 --- that the payoffs
fell into the categorey of lower than avg. we flaged them to help
explain howthe results were affected.
Anyway at the time it was just for fun and to do something *EASY* as
a class project.
Good Luck,
Tom P
From owner-derby@inslab.uky.edu Thu Mar 2 09:17:33 1995
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Date: Thu, 2 Mar 95 09:05 EST
From: "Jetaway Dave"
Subject: Re: More pick3 data
To: derby@inslab.uky.edu (Horse Racing List)
In-Reply-To: AJJBSJU%SJUMUSIC AT UKCC.uky.edu -- Thu, 02 Mar 95 00:15:09 EST
Comments: Derby Message #4646
Sender: owner-derby@inslab.uky.edu
Errors-To: derby-problems@inslab.uky.edu
>Anyway we began Flagging the pick3's that paid less than a parlay
>or lower than the avg.
>We found that on certain combos like 3 favorites, or when a consolation
>pick3 was paid, or when the outcome of the horses had coresponding
>#'s like 1-1-1 or 2-2-2 or 7-7-7 or 7-1-1 or 1-2-3 --- that the payoffs
>fell into the categorey of lower than avg. we flaged them to help
>explain howthe results were affected.
This is kinda of interesting because right after Albin had posted his
preliminarly pic-3 study, I went to PenNational and the winning pic-3 was
7-7-7. I don't have the program handy, but the parlay 'pay' was around
$600 - $700, while the pic-3 was a miserly $150 +- 25.
Jetaway Dave
From owner-derby@inslab.uky.edu Tue Mar 7 19:49:36 1995
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Date: Tue, 7 Mar 95 19:28:00 EST
From: albin@tracon.frontier.com (Albin Warth)
Message-Id: <9503080028.AA02118@frontier.com>
To: derby@inslab.uky.edu (Horse Racing List)
Subject: Re: More pick3 data
Comments: Derby Message #4880
Sender: owner-derby@inslab.uky.edu
Errors-To: derby-problems@inslab.uky.edu
Dave Jones wrote:
> albin sez..
> > If you decided to play a three-horse win parlay, and picked three horses
> > at random, you're going to get an average return close to:
> >
> > (1 - winpooltake) * (1 - winpooltake) * (1 - winpooltake)
> >
> > For CA's 15.43% win-pool take, this amount
>
> The result would actually be much worse. Yes you can find the claim made
> in several books by "experts" that betting at random loses the track take,
> but it ain't true. Not even close. The actual return will probably be closer
> to 60 percent of money wagered, not 1-take. The culprit is the
> longshot bias. When you pick horses at random, most of your picks will be
> very bad bets. So parlaying three randomly chosen horses will probably return
> about .60^3, or 22%, of your money. It will vary from track to track,
> depending on what percentage of the horses entered at that track
> have no practical chance of winning.
Dave,
You're right, I should have adjusted for the longshot bias. However,
when I used the data from two published studies, the results were not
quite as severe as you predicted.
I found two large race samples, both of which showed (in effect)
actual/expected win ratios for a large number of odds ranges. The
first is the Fabricand study of 10,000 races (93,011 horses) done in
1955-1962 (don't know what tracks); the second is Quirin's study of
all races run in New York in 1971 (19,075 horses). From each of
these, I calculated the average actual/expected win ratio for the
entire sample. The results are quite different:
- For Fabricand, expected/win = 0.865.
- For Quirin, expected/win = 0.956.
Using the Fabricand result and 15.43% take, the expected return on
$1 bet on the "average horse" would be ((1 * factor) - winpooltake)
= (.865 - .1543), or 0.7107. A parlay on "three" average horses would
return .7107 * .7107 * .7107, or 35.9% of your money.
Using the Quirin result, the expected return on $1 bet would be
(.956 - .1543), or 0.7917, and the parlay would return .7917 * .7917
* .7917, or 49.6% of your money.
Without adjusting for longshot bias, the "average" parlay returns approx
61.1% of your investment, based on the take alone.
BTW, Quirin found that $2 bet on all 19,075 horses would return an average
mutuel of $1.58, or a 21% loss, in contrast to the expected return of $1.66
based on the 17% NY win-pool take. This is consistent with the 0.956
calculated from his data.
The big difference between the Fabricand/Quirin results makes me kind of
suspicious of using either in a conclusive way. They're both pretty old.
Anyone know of a more recent study?
-Albin
albin@frontier.com