Originally Posted by Magister Ludi

A race with an infinite number of horses with an equal amount of money bet on each horse is a perfectly competitive race. If all of the money were bet on a single horse, it is a perfectly uncompetitive race. Most races, of course, are somewhere in between these two extremes.

The following is a formula that quantifies race entropy or competitiveness:

sum ([1/O(i)]^2)/n

where
O(i) = odds of the ith horse
n = number of entries

The larger the value, the more uncompetitive the race.

1. Convert the odds to probabilities (1/o = p)

#***o***p
1***2***1/2=.50
2***3***1/3=.33
3***4***1/4=.25
Total prob = 1.08

2. Normalize the probabilities

1/total prob = 1/1.08 = .926

#***o***p******normalized
1***2***1/2=.50***x.926=.46
2***3***1/3=.33***x.926=.31
3***4***1/4=.25***x.926=.23

Total normalized prob = 1.00

3. Square the normalized probabilities

#***o***p******normalized**norm squared
1***2***1/2=.50***x.926=.46***x.46=.21
2***3***1/3=.33***x.926=.31***x.31=.10
3***4***1/4=.25***x.926=.23***x.23=.05

4. Average the squared normalized probabilities

#***o***p******normalized**norm squared
1***2***1/2=.50***x.926=.46***x.46=.21
2***3***1/3=.33***x.926=.31***x.31=.10
3***4***1/4=.25***x.926=.23***x.23=.05

Average squared normalized prob = (.21+.10+.05)/3=.36/3=.12

The competitiveness index of this race is .12.