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. It uses generic symbols matched up with Fibonacci numbers to solve the problem of computational stabilization and the favorite-longshot bias. 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+. Memorize the symbols and word descriptions by writing them down a few times.

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 - Evaluates 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 - Compares 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. Contender Percentage (cp) And Factors

The relative analysis of contenders versus non-contenders is + versus Ø.

Four contenders in an eight horse field (4 / 8).
+ / + / + / + / Ø / Ø / Ø / Ø
13 + 13 + 13 + 13 + 1 + 1 + 1 + 1 = 52 + 4 = 56
cp = 52 / 56 = .929

nc - number of contenders
fs - field size

cp = (13 * nc) / ((13 * nc) + (fs - nc))

Races with two or more contenders use three factors arranged from left to right in order of importance. Factors can be two-part compounded, for example: form-age, form-distance, form-surface, distance-surface. Fibonacci numbers 8, 5, 3 are used for the factor weights.

4. Example Race Calculation

Two contenders in an eight horse field (2 / 8).

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

The three factors chosen are speed, class, form-distance.

8 + 5 + 3 = 16                   // factor weight sum
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 center column.

Figure 2.
2 / 8 sp / cls / fm-dst
Horse Symbols Weights
#1 +  /  N  /  N 13 / 5 / 5
#2 N- /  N+  / + 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
Figure 3.
2 / 8 sp / cls / fm-dst
#1 +  /  N  /  N .471 6 / 5
#2 N- /  N+  / + .342 2 / 1

References

  1. Cramer M. (1987) The Odds On Your Side: The Logic Of Racetrack Investing, Cynthia Publishing Company
  2. McNeill D. & Freiberger P. (1993) Fuzzy Logic: The Discovery Of A Revolutionary Computer Technology And How It Is Changing Our World, Simon & Schuster
  3. Mitchell D. (1988) Winning Thoroughbred Strategies: With The Right Strategy, You Can Think Like An Investor, Not A Gambler!, William Morrow & Company
  4. Quinn J. (1987) Class Of The Field: New Performance Ratings For Thoroughbreds, William Morrow & Company
  5. Quirin W.L. (1979) Winning At The Races: Computer Discoveries In Thoroughbred Handicapping, William Morrow & Company
  6. Scott W.L. (1984) How Will Your Horse Run Today?, Amicus Press
  7. Scott W.L. (1989) Total Victory At The Track: The Promise And The Performance, Liberty Publishing Company

Appendix 1. Factor Symbols And Combinations

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. Figure A1 shows the six combinations and word descriptions.

Figure A1.
Symbols Word Description Percent Doubled Tripled
N  N  N neutral 33 / 33 / 34 66 / 34 100
+  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 -

Factors can be doubled, for example: speed, speed, form-distance. The factor symbol representing the doubled factor is duplicated in the combination. The (+ N N) combination is redundant having the same percentage as the (+ - -) combination. The highest weighted symbol in the combination should be dominant. Each horse will get two symbols instead of three since there are now only two factors.

A single factor can also be tripled. Again the factor symbol representing the tripled factor is duplicated (N N N). Each horse gets just one symbol.


Appendix 2. Objective Factors For Calculating Odds

Scaling Percentage To Symbol Numbers 1 To 7.

pct - percentage from 0 to 1
conv - converted into a 1 to 7 number

conv = int((((pct * 100) + 16.667) / 16.667) + .5)

Win / Place Percentage.

nc - number of entrants or contenders
st() - number of starts this year plus last year
wp() - number of wins and places this year plus last year
wt() - Fibonacci numbers 1,2,3,5,8,13,21

for i = 1 to nc
  wp(i) = wp(i) / st(i)
  conv = int((((wp(i) * 100) + 16.667) / 16.667) + .5)
  wp(i) = wt(conv)
next i

Earnings Per Start Percentage.

en() - earnings this year plus last year
(earnings are rounded to the nearest thousand)
high - highest value found

cap = 180    // eps limited to USD-180k
high = 0
for i = 1 to nc
  en(i) = en(i) / st(i)
  if en(i) > cap then en(i) = cap          
  if en(i) > high then high = en(i)
next i
for i = 1 to nc
  en(i) = en(i) / high
  conv = int((((en(i) * 100) + 16.667) / 16.667) + .5)
  en(i) = wt(conv)
next i

Recent Representative Speed Rating.

scale - scaling increment
sr() - speed rating

scale = 2.0
high = 0
for i = 1 to nc
  if sr(i) > high then high = sr(i)
next i
offset = (high / scale) - 7         
for i = 1 to nc
  sr(i) = int(((sr(i) / scale) - offset) + .5)
  if sr(i) < 1 then sr(i) = 1
  sr(i) = wt(sr(i))
next i

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Copyright © 2011-2014 Donald A. Swanson
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