aa r X i v : . [ m a t h . P R ] N ov KELLY CRITERION FOR VARIABLE PAY-OFF
RICARDO P´EREZ MARCO
Abstract.
We determine Kelly criterion for a game with variable pay-off. TheKelly fraction satisfies a fundamental integral equation and is smaller than theclassical Kelly fraction for the same game with the constant average pay-off. Introduction.
Kelly criterion (see [3]), also called ”‘Fortune Formula”’, is the fundamental tool toensure an optimal positive return when playing with repetition a favorable game orstrategy. It was first proposed by Edward Thorp for the money management of hiscard counting strategy to beat casino blackjack (see [6]). Later he applied the sameideas to the financial markets (see [2]), and L. Breinman (see [1]) proved that it wasthe optimal strategy for long run accumulation of capital.One needs to modify the theory for common situation where the advantage isnot known exactly but is a random variable (see [4]). In this ”‘fuzzy advantage”’situation one needs to be far more conservative (see section 3 in [4]). When we applyKelly criterion to decision making in real situations, we are confronted with a fuzzyadvantage but also a variable reward or pay-off. In this article we study how to modifyKelly criterion when the pay-off is a random variable with a known distribution. Weobtain the same type of conclusion than in [4]: The optimal Kelly fraction is moreconservative than the one in the same game with the constant average pay-off.2.
The classical Kelly criterion.
We assume that we are playing a game with repetition. At each round we risk afraction 0 ≤ f ≤ X . With probability 0 < p < b -to-1, with b ≥
0. This means that if X is our current capital, if we losethe bet (which happens with probability q = 1 − p ) we substract f X to our capital, Mathematics Subject Classification.
Key words and phrases.
Kelly, gambling, variable pay-off.. and if we win, we add bf X to our capital . Thus, the expected gain is E (∆ X ) = pbf X − qf X = ( pb − q ) f X = ( p (1 + b ) − f X . If we play a game with advantage, that is p (1 + b ) >
1, then we expect an exponentialgrowth of our initial bankroll X if we follow a reasonable betting strategy. Weassume that there is no minimal unit of bet. By homogeneity of the problem, asharp strategy must consist in betting a proportion f ( p ) of the total bankroll. In theclassical Kelly criterion, p and b are assumed to be constant at each round. In thisarticle we assume that p is constant but the pay-off b is a random variable. We firstreview the classical Kelly criterion with a constant pay-off that finds f ( p ) in orderto maximize the expected exponential growth. Our bankroll after having played n rounds of the game have is X n = X n Y i =1 (1 + b i f ( p ))where b i = b if we won the i -th round, and ǫ i = − i -th round. Theexponential rate of growth of the bankroll is G n = 1 n log X n X = 1 n n X i =1 log(1 + b i f ( p )) . The Kelly criterion maximizes the expected value of the exponential rate of growth:
Theorem 2.1. (Kelly criterion)
For a game with advantage, that is p ( b + 1) > ,the expected value of the exponential rate of growth G n is maximized for f ∗ ( p, b ) = p − q/b = p (1 + b ) − b . The argument is straightforward. Observe that the expected value is E ( G n ) = E ( G ) = p log(1 + bf ) + (1 − p ) log(1 − f ) = g ( f ) . This function of the variable f has a derivative, g ′ ( f ) = pb bf − − p − f = ( p (1 + b ) − − f b (1 + f b )(1 − f ) , and g ( f ) → f → + , and g ( f ) → −∞ when f → − . Also g ′ ( f ) > g ′ is decreasing, so g is concave. Thus g ( f ) increases from 0 to its maximumattained at f ∗ = f ∗ ( p, b ) = p (1 + b ) − b and then decreases to −∞ . ELLY CRITERION FOR VARIABLE PAY-OFF 3 Kelly criterion with variable pay-off.
This situation arises in a number of practical situations. The original problem thatmotivated the use of Kelly criterion was casino blackjack where the pay-off is constant b = 1. But in other card games, like poker cash, the pay-off is variable. Also sometrading strategies cannot set a predetermined pay-off, for example when speculatingwith a price rebound in a volatile market. The historic of trades of traders present acertain distribution of pay-offs for the successful trades. Therefore in these situationswe cannot consider b constant. We assume that the pay-off b is a random variable witha known non-negative distribution ρ : R + → R + , with ρ ( x ) dx giving the probabilitythat the pay-off is in the infinitesimal interval [ x, x + dx ]. In practice ρ has compactsupport and can be obtained empirically, although the model in [5] shows that thetail is of Pareto type. We do not need to assume that the distribution is absolutelycontinuous with respect to the Lebesgue measure (same proofs).We can use a similar argument as in the previous section in order to determine thesharp fraction to bet ˆ f .To determine when the game is favorable we compute E ( X − X ) = (1 − p ) f X + pf X Z + ∞ b ρ ( b ) db , Thus the condition for a favorable game is(1) p (cid:18) Z + ∞ b ρ ( b ) db (cid:19) > , which (naturally) is the same condition than that of a constant pay-off game wherethe pay-off is the average pay-off: p (1 + ¯ b ) > , and ¯ b = Z + ∞ b ρ ( b ) db. After n rounds, if b i is the pay-off in the i -th round, the expected exponentialgrowth is g ( f ) = E ( G n ) = 1 n n X i =1 E (log(1 + b i f ))= q log(1 − f ) + p Z + ∞ log(1 + bf ) ρ ( b ) db , under the integrable assumption that the density of the pay-off function makes theintegral finite. R. P´EREZ MARCO
Again g ( f ) → f → + , g ( f ) → −∞ when f → − , and g ′ ( f ) = − q − f + p Z + ∞ bρ ( b )1 + bf db . We observe that when f → + , g ′ ( f ) ≈ − (1 − p ) + p Z + ∞ b ρ ( b ) db thus g ′ ( f ) > g ′ is strictly decreasing,so g is strictly concave, and tends to −∞ when f → − . Thus there is exactly onevalue ˆ f = ˆ f ( p ) that maximizes this expression and annihilates g ′ . It is the uniquesolution ˆ f ( p, ρ ) to the fundamental integral equation(2) p Z + ∞ b ρ ( b )1 + b ˆ f db − − p − ˆ f = 0 . Theorem 3.1. (Kelly criterion for variable pay-off )
A game with variablepay-off with a distribution ρ is favorable if p (cid:18) Z + ∞ b ρ ( b ) db (cid:19) > . The expected value of the exponential rate of growth is maximized for < ˆ f =ˆ f ( p, ρ ) < satisfying the fundamental integral equation p Z + ∞ b ρ ( b )1 + b ˆ f db − − p − ˆ f = 0 . Corollary 3.2.
The optimal Kelly fraction ˆ f ( p, ρ ) for a favorable game with variablepay-off is smaller than the optimal Kelly fraction f ∗ ( p, ¯ b ) for the same game withconstant pay-off equal to the average pay-off ¯ b , ¯ b = Z + ∞ b ρ ( b ) db . We have ˆ f ( p, ρ ) ≤ f ∗ ( p, ¯ b ) . We only have equality when the pay-off is constant.Proof.
The function of b , h ( b ) = b b ˆ f , ELLY CRITERION FOR VARIABLE PAY-OFF 5 is strictly concave and therefore, by Jensen’s inequality, we have Z + ∞ b ρ ( b )1 + b ˆ f db ≤ ¯ b bf . So, using the fundamental equation (2)we get p ¯ b bf − − p − ˆ f = − − p − f ∗ − − p − ˆ f ≥ , and the result follows. The case of equality follows from the case of equality ofJensen’s inequality and only occurs for a Dirac distribution. (cid:3) Conclusion:
In a situation when the pay-off is variable one needs to adjust the Kelly fraction ina conservative way.
Remark.
The analysis generalizes to situations where the risk is larger than the fractionwaged. As noted by Thorp, this happens in a leveraged investment in the financialmarkets.
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A new interpretation of information rate , Bell System Technical Journal, , (4),p.917926.[4] MU ˜NOZ GARC´IA, E.; P´EREZ-MARCO, R.; The standard deviation effect , arXiv:math 0006017,2000.[5] P´EREZ-MARCO, R.;
A simple dynamical model leading to Pareto wealth distribution and sta-bility , arXiv:1409.4857, 2014.[6] THORP, E.;
Beat the dealer , Blaisdell Pub. Co, 1962.
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