This topic briefs a clear and mathematically strict way to compute an optimal bet parameters in a regular case. The latter means that input parameters don't have their extreme values (e.g. fair odds = 1.01, while the line odds = 5.0), and the minimum stake limitation doesn't have its possible effect.
Unlike simple method based on Kelly criterion, this one allows bettor to use a realistic range of bankroll growth, and to specify confidence probability of reaching the target fund. Two methods give an identical results if the target fund is infinite.
Consider a uniform sequence of n betting rounds, out from which w rounds were successful, and f rounds failed.
As a result of a single successful round, the bankroll will amount to:
where
x - available betting fund (bankroll);
k0 - static portion of fund x;
L - reduced line odds;
Z - factor of fund growth;
As a result of a single failure, the fund will be dwindled to:
w successful rounds and f failures will result in the target fund:
Above formulae allow to express w and f as a functions of x, X, L, n, and k0:
Probability to have not greater than f failures prior to collecting as many as w successes is described by an integral (cumulative) form of Pascal distribution:
where
p - probability of success in a single round;
G - gamma function;
F - hypergeometric function.
It is equal to the negative binomial distribution in case of integral values of w and f. We use the term Macro-probability for P in the context of betting parameters.
Typical functional relationship (P-diagram) between macro-probability P and static portion k0, provided that x, X, L, n, p are fixed parameters, has a maximum corresponding to the least number of rounds required to achieve the target fund, and an optimal value of the static portion:
In order to obtain an optimal bet parameters, Stake Wizard solves a system of equations:
where - confidence probability,
with respect to the indeterminates: number of rounds n and static portion k0. This is done for each bet structure out from thousands possible.
To apply the above formulae, we need to construct a complex bet, and then represent it as a single component abstract bet having certain values of p and L (reduced line odds).
In previous versions of Stake Wizard, bet structures consisted only of independent events. In particular, they did not allow system bets like AB + BC + AC that reduce betting risks. Regular system bets however are far from the optimal bankroll distribution. The following theorem resolves this problem.
Theorem. Maximum possible value of an average factor of fund growth Z is achieved by the optimal distribution of a bankroll amongst 2N - 1 possible simultaneous stakes, and is determined by the following expression:
where
ln Zave - mathematical expectation of ln Z;
pi - probability of i-th event, equal to 1/Fi, where Fi - fair odds;
qi - sportsbook's 'probability' of i-th event, equal to 1/Li, where Li - payout coefficients (aka line odds);
N - number of chosen independent events.
Possible stakes comprise N system bets (combinations) of type (m, N), where m = 1..N. Each system bet contains CmN accumulator bets. In particular, possible stakes include N single ordinary bets, and one N-fold accumulator bet. In general case, optimal system bets are not equipartitioned.
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The proof is trivial for a single event with only one possible stake. In the latter simple case, optimal stake is equal to Kelly's optimum (p - q)/(1 - q). For N > 1, the proof is rather complicated even when N = 2. However using Stake Wizard`s multivariate optimization engine, one can calculate optimal stakes and verify the theorem numerically for the greater N 's.
The above formula means that Zave can be incredibly high, provided that one has found sufficient number of appropriate events (Fi < Li). But the price is tremendous number of stakes.
For example, for 26 events having Fi = 1.5 and Li = 1.7, one should place as many as 226-1 = 67108863 various stakes to achieve 40.2% of an average fund growth in a single round! Practical constraints however, such as minimum stake limitation or the reasonable number of simultaneous stakes, of course reduce the effective value of Zave.
From the viewpoint of conventional betting, there are only two outcomes of a given event: either won or lost elementary bet with probabilities pi and 1 - pi , respectively. Other probability-dependent areas of human activity (e.g. investment management) may involve more outcomes of an event.
In case of multi-outcome events, theorem can be generalized as follows:
where pij - probabilities of j-th outcome in i-th event;
1/qij = Lij - payout coefficients for j-th outcome in i-th event;
Vi - number of possible outcomes in i-th event.
In the simplest case of a single event (N = 1) with a single outcome (V = 1), we would have p = 1, k = 1 - k0 = 1, and Zave = 1/q = L.
Equating Zave and k (total stake portion of a bankroll) of complex bet (N > 1) to those values of a single bet, and solving system of equations with respect to indeterminates p and L
enables to represent multicomponent complex bet as an abstract single bet, and then optimize the value of k = 1 - k0 using macro-probability function.
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