Kelly criterion mathematical proof

Math derivation of the Kelly formula for optimal investment

In a succession of risky sports bets, using the Kelly criterion theoretically leads to the highest bankroll, higher than with any other long-term strategy. This Kelly's formula yields the theoretical maximum return over all bets.
This mathematical formula calculates for each sports bet, depending on odds and estimated probability of winning, the fraction of bankroll to wager. A calculator can here calculate this formula automatically.
This formula is generally quoted and used without much detail.
This article aims at exploring the mathematical details of the formula, and proving it (general idea of this whole website). So let's go for hte math proof of Kelly Criterion for sports betting.

Kelly criterion mathematical formula

Details on this mathematical formula and its use are on the previous page.
As a reminder, the Kelly criterion is the mathematical formula

f ^* = p − q/c − 1

where

f ^* is the optimal fraction of one's bankroll to bet
p is the estimated probability of victory
q = 1 − p is the estimated loss probability
odds of c

Mathematical proof of Kelly's formula

The study and complete proof of the Kelly criterion formula can be found in his bibliography, notably "A New Interpretation of Information Rate" then, by E.O. Thorp, one of his successors, in "Optimal Gambling Systems for Favorable Games".
We give here a simplified version, somehow less rigorous, and adapted to sports betting (Kelly's version considers more general risky investments), in order to better understand this formula and how it works.
First of all, as we used for example to obtain a simplified formula for surebet detection, for odds of c we denote t the net odds associated, that is

c = 1 + t

For example, odds of c = 2.2 are associated with net odds of t = 1.2: if I bet for example 10 euros, then I earn 10 × 2.2 = 22 euros, that is 10 × 1.2 = 12 euros in addition to my 10 euros bet.
This is what appears in Kelly's formula: t = c − 1

Variation for gain or loss

Assume we have a capital, or bankroll, W (for Wealth in Kelly's jargon) of which we bet a fraction f.
For example: a bankroll of W = 20€ and of which we bet f = 10%, i.e. a wager of m = W×f = 2€ at odds of c = 1.6 (also for example).
Excluding my betting, my remaining bankroll is

W_m = W − W×f

thus

W_m = W(1−f) = 18€

if I lose, I lose my stake and my new bankroll is therefore simply and only W_m.
Bankroll is therefore in this case multiplied by (1−f).
if I win, I have bet m = Wf = 2€ at odds of c = 1.6 and therefore the profit is then W_g = m×c = Wfc = 2×1,6 = 3.2€.
My new bankroll is therefore calculated by
W' = W_m + W_g = 18€ + 3.2€ = 21.2€
More generally, and using the net odds t such that c = 1 + t
W' = W_m + W_g = W(1−f) + Wfc = W(1−f) + Wf (1+t) = W(1+ft)
The bankroll is therefore in this case of victory, multiplied by (1+ft).

To sum up, when I lose, my bankroll is multiplied by 1−f whereas when I win, my bankroll is multiplied by 1+ft.

Growth rate on several successive bets

We now focus in calculating and obtaining the best increase in capital on a set of bets and losses/gains: long term growth.
Let's say for example that I bet T times successively in total among which I win N times and lose M times, with of course the relationship T = N+M.
The order of victories and losses does not matter and for each victory I multiply my bankroll by 1+ft and for each defeat I multiply by 1−f. All in all, starting from bankroll W₀, with N wins M loss, my bankroll becomes

W_T = W₀(1+ft)^N(1−f)^M

The geometric average growth rate r is the constant rate at each bet and which yields the same overall variation, that is to say such that

W_T = W₀ (1+r)^T

We therefore get, by comparing and identifying these last two algebraic expressions

(1+r)^T = (1+ft)^N(1−f)^M

the by taking the n-th root:

1+r = (1+ft)^N/T(1−f)^M/T

Now, coming back to the meaning of the parameters, the fraction N/T is the ration between the number N of bets won and the number T of bets. This ratio tends towards the probability p of victory for a large number T of bets.
Likewise the other fraction M/T is the ration of the number of loss and tends to q = 1−p.
We therefore have found the asymptotic average geometric growth rate, that is to say for a large number of bets, when T tends to infinity,

1+r = (1+ft)^p(1−f)^q

r = (1+ft)^p(1−f)^q − 1

Graph: growth rate displayed

The curve representing the geometric average growth rate found above versus the fraction f of the bankroll wagered is plotted below. Parameters can be adjusted: estimated probability of victory and odds, so as to observe a maximum value (the value of which can be compared with that provided by Kelly's calculator).

Variation du taux de retour r en de la part de capital misée f

Optimum detection and calculation

We go on with the mathematical proof, and now seek to calculate exactly the fraction f ^* which yields the bankroll maximum growth rate.
We thus consider the geometrical growth rate expression r found previously as a function of the invested fraction f. Mathematically, to find the variations and extreme values (maximums and minimums), we calculate the derivative function

r' = dr/df = pt(1+ft)^p−1(1−f)^q − q(1+ft)^p(1−f)^q−1

or, factoring by the expression of r

r' = r pt/ 1+ft − q/ 1−f

We then seek the critical value f ^* such that this derivative is zero, that is

pt/ 1+f ^*t − q/ 1−f ^* = 0 ⇔ pt(1−f ^*) = q(1+f ^*t)

We then remember that probabilities p and q of loss ang gain are linked by p = 1−q and the previous expression to be then rewritten

pt(1−f ^*) = (1−p)(1+f ^*t)

or, after some algebraic developments:

f ^*t = pt − (1 − p) = pt−q

thus finally

f ^* = p − q/t

or also, remembering that c = 1+t ⇔ t = c−1 we find at last the expected mathematical formula

f ^* = p − q/c − 1

Numerical simulation and comparison

Kelly's formula therefore gives the optimal proportion of bankroll to bet for maximum overall return. On the next page, Simulations and comparisons, with or without the Kelly criterion, are carried out over the long term. A numerical illustration of the effectiveness of the Kelly criterion.

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