Making Parimutuel Odds With Fibonacci Numbers

Donald A. Swanson

This work describes a subjective method of constructing an odds line from the analysis of a thoroughbred horse race. Symbols are chosen by the handicapper to represent the race analysis. Fibonacci numbers are used to convert symbol combinations into weighted percentages.

1. Symbols

The leftmost column in figure 1 shows the seven symbols used in the method. There are three one-part symbols and four two-part symbols. The left side symbol part is the base (+, N, Ø). The right side symbol part (+, -) is the modifier. Word descriptions are names for the symbols. Each symbol is matched up with a Fibonacci number. Symbols are hand-written with a forward slash "/" as a separator, for example: N- / ++ / Ø+.

Figure 1.
Symbol Word Description Weight Number
++ double plus 21 7
+ plus 13 6
N+ neutral plus 8 5
N neutral 5 4
N- neutral minus 3 3
Ø+ doubtful plus 2 2
Ø doubtful 1 1

2. Analysis Perspectives

Absolute Analysis - Evaluate a single horse in isolation or against todays race conditions. Symbol selection starts with consideration of (+, N, Ø) before moving up or down to the appropriate symbol. The handicapper should mentally move toward N when uncertain.

Relative Analysis - Compare two or more horses against each other. Symbol selection starts with consideration of N versus N before expanding toward the top and bottom. In figure 1 rightmost column the symbols are numbered 1 to 7. The difference between the symbols is the degree of certainty that one horse will outperform the other(s). The range is 0 to 6 degrees.

3. Reverse Reading The Two-Part Symbols

Symbol Word Description
+N positive neutral
-N negative neutral
positive doubtful

Reversing the two-part symbols can be helpful in situations with unknowns. A developing horse might be taking a large rise or drop in class or trying a new distance or surface. A positive reverse read is modified neutral or doubtful depending on the absolute and relative analysis of todays race.

4. Contender Percentage And Confidence Levels

The analysis of contenders versus non-contenders is relative. Non-contenders get a Ø. The handicapper will have some level of confidence that one of the contenders will win the race. Confidence levels can be thought of as moderate, high, or maximum. A symbol subset is chosen to represent these levels, for example: (N, +, ++). If the confidence level is high then the relative analysis is + versus Ø.

cp - contender percentage
nc - number of contenders
fs - field size
wt - weight

cp = (wt * nc) / ((wt * nc) + (fs - nc))

Four contenders in a nine horse field with moderate confidence.
4 / 9 / N
N / N / N / N / Ø / Ø / Ø / Ø / Ø
5 + 5 + 5 + 5 + 1 + 1 + 1 + 1 + 1 = 20 + 5 = 25
cp = 20 / 25 = .800

cp = (5 * 4) / ((5 * 4) + (9 - 4)) = .800

Two contenders in an eight horse field with high confidence.
2 / 8 / +
+ / + / Ø / Ø / Ø / Ø / Ø / Ø
13 + 13 + 1 + 1 + 1 + 1 + 1 + 1 = 26 + 6 = 32
cp = 26 / 32 = .813

cp = (13 * 2) / ((13 * 2) + (8 - 2)) = .813

One contender in a six horse field with maximum confidence.
1 / 6 / ++
++ / Ø / Ø / Ø / Ø / Ø
21 + 1 + 1 + 1 + 1 + 1 = 21 + 5 = 26
cp = 21 / 26 = .808

cp = (21 * 1) / ((21 * 1) + (6 - 1)) = .808

5. Factor Symbols And Combinations

Races with two or more contenders use one to three factors which must be arranged from left to right in order of importance, for example: class, form, distance. Factors can be two-part compounded, for example: form-age, form-surface, distance-surface.

The N symbol can be extended to NN making the neutral base symbols N+, NN, N-. The factor symbols are the right side symbol parts (+, N, -) matched up with Fibonacci numbers 8, 5, 3 respectively. The six factor symbol combinations and word descriptions are shown in the table below.

Symbols Word Description Percentage Doubled
N  N  N neutral 33 / 33 / 34 66 / 34
+  N  N one plus 44 / 28 / 28 -
N  N  - one minus 38 / 38 / 24 76 / 24
+  N  - one plus one minus 50 / 31 / 19 -
+  -  - one plus two minus 56 / 22 / 22 56 / 44
+  +  - two plus one minus 42 / 42 / 16 84 / 16

Weighted percentage calculation for the (+ N -) combination:

8 + 5 + 3 = 16      // factor weight sum
8 / 16 = .5         // weight/sum = percentage
5 / 16 = .3125
3 / 16 = .1875

Factors can be doubled, for example: speed, speed, form making a two factor race. The factor symbol representing the doubled factor must be duplicated in the combination. A single factor can also be tripled (N N N) making a one factor race.

6. Example Race Calculation

Two contenders in an eight horse field with high confidence.

cp = (13 * 2) / ((13 * 2) + (8 - 2)) = .813

Factors: class, form-surface, distance with (+ N -) weight.

8 + 5 + 3 = 16
f1pct = (8 / 16) * .813 = .407   // (weight/sum) * cp
f2pct = (5 / 16) * .813 = .254
f3pct = (3 / 16) * .813 = .152

Each contender gets three symbols one for each factor as shown in figure 2 second column. The symbols can be converted into 1 to 7 numbers for entry into a hand-held calculating device.

Figure 2.
Horse cls / fm-sur / dst 1 - 7 Weights
#1 +  /  N  /  N 1644 13 / 5 / 5
#2 N- /  N+  / + 2356 3 / 8 / 13
13 + 3 = 16                // factor 1 sum
(13 / 16)  * .407 = .331   // #1 (weight/sum) * f1pct  
(3 / 16) * .407 = .076     // #2

5 + 8 = 13                 // factor 2 sum
(5 / 13) * .254 = .098     // #1 (weight/sum) * f2pct                  
(8 / 13)  * .254 = .156    // #2

5 + 13 = 18                // factor 3 sum
(5 / 18) * .152 = .042     // #1 (weight/sum) * f3pct
(13 / 18) * .152 = .110    // #2

Sum the three percentages for each contender.

.331 + .098 + .042 = .471  // #1 
.076 + .156 + .110 = .342  // #2

Sort the final percentages (pct) in descending order before converting into odds.

odds = (1 / pct) - 1

(1 / .471) - 1 = 1.123    // #1
(1 / .342) - 1 = 1.924    // #2  

Select the increment (inc) for rounding off.

if odds < 0.3 then inc = 20
elseif odds < 1 then inc = 10
elseif odds < 2 then inc = 5
elseif odds < 5 then inc = 2
else inc = 1

odds = (int((odds * inc) + .5)) / inc

(int((1.123 * 5) + .5)) / 5 = 1.2  // #1
(int((1.924 * 5) + .5)) / 5 = 2    // #2

Rounded odds are converted into fractions.

n = .5     // numerator
d = 0      // denominator
while int(n)<> n
  d = d + 1
  n = odds * d 
end
print n,"/",d
Finished calculation.
2 / 8 / + cls / fm-sur / dst 1 - 7 +  N  -
#1 +  /  N  /  N 1644 .471 6 / 5
#2 N- /  N+  / + 2356 .342 2 / 1

In the example below the distance-surface factor compound is doubled.

A two factor race.
2 / 10 / + cls / dst-sf 1 - 7 +  -  -
#1 N+  /  N 154 .552 4 / 5
#2 N-  /  Ø+ 232 .213 7 / 2

References

  1. Baker M. (Apr 2009) Converting Decimal To Fractions, www.sitepoint.com, retrieved from http://www.sitepoint.com/forums/showthread.php?608881-Converting-Decimal-to-Fractions&s=bffaeae0b7abc90fc14be8f602ff922b&p=4208775&viewfull=1#post4208775
  2. Cramer M. (1987) The Odds On Your Side: The Logic Of Racetrack Investing, Cynthia Publishing Company
  3. McNeill D. & Freiberger P. (1993) Fuzzy Logic: The Discovery Of A Revolutionary Computer Technology And How It Is Changing Our World, Simon & Schuster
  4. Mitchell D. (1988) Winning Thoroughbred Strategies: With The Right Strategy, You Can Think Like An Investor, Not A Gambler!, William Morrow & Company
  5. Quinn J. (1987) Class Of The Field: New Performance Ratings For Thoroughbreds, William Morrow & Company
  6. Quirin W.L. (1979) Winning At The Races: Computer Discoveries In Thoroughbred Handicapping, William Morrow & Company
  7. Scott W.L. (1984) How Will Your Horse Run Today?, Amicus Press
  8. Scott W.L. (1989) Total Victory At The Track: The Promise And The Performance, Liberty Publishing Company

Measuring Positional Tendency In Thoroughbred Horse Races

Donald A. Swanson

This method uses second call positions of race winners to measure positional tendency at a given distance. Second call position and field size data are retrieved from the result charts. The four furlong position is used for sprints. The six furlong position is used for routes. Maiden races are not representative and should not be used. Calculate figures for each distance and surface. It takes about fifteen races to get a good indication. The user should become familiar with track circumference and start / finish locations.

n = 0    // number of races
repeat
  input 2nd call position (c2)
  input field size (fs)
  n = n + 1
  list(n) = (c2 / fs) * 10
until done
sort the list() in ascending order
average the entire list()
  total = 0
  for i = 1 to n
    total = total + list(i)
  next i
  a1 = int((total / n) + .5)
average the low third of the list()
  total = 0
  lt = int((n / 3) + .5) 
  for i = 1 to lt
    total = total + list(i)
  next i
  a2 = int((total / lt) + .5
average the high third of the list()
  total = 0
  ht = (n - lt) + 1
  for i = ht to n
    total = total + list(i)
  next i
  a3 = int((total / lt) + .5)
display as (a2,a3)a1
  for example: (3,8)5 or [1,5]3

Interpret the figures using the table below.

(+) early  (N) neutral  (Ø) late
Low Third High Third Average
1 (+)
2 (Ø)
5 (+)
6 (Ø)
2 (+)
3 (N+)
4 (N-)
5 (Ø)

References

  1. Brohamer T. (1991) Modern Pace Handicapping, William Morrow & Company
  2. Scott W.L. (1989) Total Victory At The Track: The Promise And The Performance, Liberty Publishing Company

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