FURTHER ASPECTS OF GAMBLING WITH THE KELLY CRITERION. We consider two aspects of gambling with the Kelly criterion. First, we show that for

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1 FURTHER ASPECTS OF GAMBLING WITH THE KELLY CRITERION RAVI PHATARFOD *, Monash University Abstract We consider two aspects of gambling with the Kelly criterion. First, we show that for a wide range of final results for a series of games, the Kelly bettor would be worse off compared to a flat bets bettor. Secondly, we consider, for the Kelly bettor, situations similar to the gambler s (or his opponent s) ruin in the classical gambler s ruin problem. Here the end points are a reduction of the gambler s capital to a fraction, or its growth to a certain multiple, of its original value. Keywords: Kelly criterion; win to action ratio; fractional Kelly; halving and doubling 2000 Mathematics Subject Classification: Primary 91A60 Secondary 60G40 * Postal address :School of Mathematical Sciences, Monash University, Clayton, Vic 3168, Australia. Address: ravi.phatarfod@sci.monash.edu.au

2 1. Introduction. Over the past few decades, there has been a considerable amount of interest, particularly in the gambling community, in what is known as gambling with the Kelly Criterion, (Kelly (1956)). The underlying assumption in this system of gambling is that we have a gambler with a finite capital who is playing a series of games that are favourable to him. The Kelly betting criterion is for the gambler to bet a fraction f of his capital at each stage. The fraction that maximizes the expected exponential growth, G(f), of the capital is known as the optimal Kelly fraction, denoted by f *. For example, let us suppose the gambler may win a unit amount for each unit wagered with probability p (> 0), so that the game is favourable with mean gain p q > 0, where the probability of loss is q = 1 p. It is known (see, for example, Rotando and Thorp (1992)) that f * = p q. In this paper we consider only the simple gambling game described above. Breiman (1961) showed, among other things, that for values of f < f c where f c ( 0) is the solution of G(f) = 0, the expected value of the gambler s capital increases exponentially to infinity almost surely, with the maximum rate when f = f *. As there are few gambling games favourable to the gambler, there was, initially, only an academic interest in the idea. It was only when Thorp (1966) showed that the game of Blackjack could, under certain circumstances, be made favourable to the gambler that the idea captured wide interest; see Wong (1981), and Griffin (1981). It was seen that Kelly betting had two advantages over flat betting. First, as Breiman (1961) showed, for f < f c, the probability of the gambler being ruined is zero. Secondly, as Phatarfod (2007) showed, for any f < f c, the expected value of the capital at the end of a series of games with Kelly betting is greater (marginally or substantially, depending on the gambler s edge) than that for flat betting when the flat bet is equal to the initial Kelly bet. We shall show here that these advantages of Kelly 2

3 betting over flat betting come at a price. Wong (1981), whose primary interest was in the application of Kelly betting to the casino game of Blackjack, demonstrated that Kelly betting has two disadvantages over flat betting. These disadvantages are related, although they are not quite the same. The first disadvantage is that, for the case when the number of games won is equal to the expected value, the net gain of the Kelly bettor is about half of that of the flat bettor, for the same initial capital, the same number of games and the same edge. For example, suppose we have p = 0.505, q = 0.495, (a typical case in Blackjack) so that f * = Suppose, the initial capital is $1000 and the gambler plays 200 games. If the gambler wins the expected number of games, namely 101, his net gain can be calculated as $ On the other hand, if he has flat bets of value equal to the initial Kelly bet, namely $ = $10, then, with 101 wins and 99 losses his net gain is $20,almost twice that of the Kelly bettor. We shall prove this result and show further that approximately 68% of the times, the gambler would be better off making flat bets rather than Kelly bets. Considering that the expected value of the final capital for the Kelly gambler is greater than that for the flat bettor, this shows that there are situations where the expected value does not quite tell the whole story. The second disadvantage that Wong considered, possibly of some interest to gamblers, is that the win to action ratio, (action being defined as the total amount bet) for Kelly betting with the optimal fraction f * is about one-half of that for flat betting. We shall consider this in some detail in Section 4. Finally, we consider, for the Kelly bettor, situations analogous to the ruin problem for the flat bettor, i.e. as in the classical gambler s ruin problem. Since, for values of f < f c, ruin is not possible and the capital increases to infinity almost surely, we consider, for the Kelly bettor, the problem of reduction of the capital to a certain 3

4 factor a, before its increase to a multiple b. Somewhat arbitrarily we take, following Chen and Ingenoso (2007) a = 1/2, and b = 2. The main result derived here, namely equation (15), has previously been derived in different forms and with different methods. Gottlieb (1985) derived it from a blackjack-specific model, Taylor and Karlin from a version using results from geometric Brownian motion, while Chen and Ingenoso (2007) derived it using stochastic calculus. We shall show that with the optimal Kelly fraction f * the probability of the reduction of the gambler s capital to half before its eventual doubling is 1 /3, whatever the gambler s edge! We shall show that this can be alleviated by having the proportionality factor a fraction k of the Kelly fraction f *, say, k = 1/2 or even k = 1/6. For k = 1/3 the probability of the reduction of the capital to half its initial value is only 1/32, but the expected number of games required to halve or double the capital is considerably increased. Finally, in section 6 we compare Kelly and flat betting with respect to the probability of ruin for flat betting and the reduced return for Kelly betting for a scenario with the most expected number of wins. 2. Kelly betting : Basics We will consider the simplest form of gambling games where the probability of winning a unit amount is p (0 < p <1) and the probability of losing the unit amount bet is q (where p + q = 1). Let X 0 be the initial capital and let us assume that we are wagering a fraction f of the capital at each stage. Then, if X n is the capital at the end of n games, we have, X n = X 0 (1 + f ) S (1 f ) F (1) where S and F are respectively the number of successes and failures in n trials (i.e. S + F = n). The quantity, 4

5 ln (X n / X 0 )/n = S ln (1 + f)/n +F ln (1 f)/n measures the exponential rate of growth per trial. The Kelly criterion maximizes the expected value of this growth, namely G(f) = p ln (1 + f) + q ln (1 f). The value of f that maximizes G(f) is called the Kelly criterion fraction and is denoted by f *. For the simple game considered here, f * is equal to (p q ). It is easy to work out the mean and variance of X n. For any value of f, we have X n { X X n 1 n 1 (1 (1 f ) f ) with with probability probability p q Thus, E (X n X n-1 ) = X n-1 (1 + f (p q)), and, hence, E(X n ) = X 0 K n, where K = 1 + f(p q). Similarly, one can obtain, 2 2 ( X n ) = X o (L n K 2n ), where L = 1 + f 2 + 2f(p q). For the particular case f = kf * = k(p q) where k is a fraction of f *, the expectation and variance of X n are E(X n ) = X 0 [ 1 + k(f * ) 2 ] n, (2) σ 2 ( X n ) = X 2 0 [ (1 + (kf * ) 2 + 2k(f * ) 2 ) n (1 + k (f * ) 2 ) 2n ]. (3) For large values of n, we have, from (2) and (3), E (X n ) ~ X 0 exp (nk (f * ) 2 ), (4) σ 2 ( X n ) ~ X 2 0 [ exp[ n(kf * ) 2 + 2nk(f * ) 2 ] exp (2nk(f * ) 2 ) ]. (5) 5

6 3. Comparison of net gain We consider here a series of the simple games outlined in section 2. We show that, if the number of games won is equal to the expected number, the net gain of the Kelly bettor (with optimal Kelly fraction f * ) is about half of that of the corresponding flat bettor, when n(f * ) 2 is small. Let the original capital be 1 unit. Then the capital at the end of n games, when the number of games won equals the expected number np, and the proportionality factor is f is, X n = ( 1 + f ) np (1 f ) nq. (6) Expanding the right-hand side of (6) we have, X n ~ [1 + npf + np(np 1)f 2 / 2 + np(np 1)(np 2)f 3 /6 + ] [1 nqf + nq (nq 1)f 2 /2 nq(nq 1)(nq 2)f 3 /6 + ], which simplifies to X n ~ 1+ nf(p q) + f 2 [ n 2 (p q) 2 n ]/2 +. Taking, f = kf * = k (p q), we have X n 1 + nk (f * ) 2 + (kf * ) 2 [ (nf * ) 2 n]/2 + = 1 + nk (f * ) 2 nk 2 (f * ) 2 / 2 + k 2 (nf * ) 4 /2n 2 For small values of n(f * ) 2 we may ignore the last term above and we have, X n ~ 1 + nk(f * ) 2 nk 2 (f*) 2 /2 (7) so that the net win is nk(f * ) 2 nk 2 (f * ) 2 /2, thus showing the approximation is valid when (nf * ) 2 is small. For flat betting with bets of size f, the net gain is nf(p q) = nk(f * ) 2 when the number of wins is equal to the expected number np. We therefore see that for all values of k the net gain for the Kelly bettor is less than that for the flat bettor. For the particular case of k = 1, the net gain for the Kelly bettor is n(f * ) 2 /2, as against n(f * ) 2 6

7 for the flat bettor, showing that the net gain for the Kelly bettor with optimal Kelly fraction is about half of that for the flat bettor. Table 1 shows the results for the case p = 0.505, q = 0.495, f * = 0.01, and n = 200 for the number of wins around the expected value, namely 101. No. of wins Net Gain for Kelly Bettor Net Gain for Flat Bettor Table 1. Comparison of the Kelly and flat bettor s net gain for n = 200, p = 0.505, around the expected number of wins It is seen that for the number of wins in the range , the net gain for the Kelly bettor is less than that for the flat bettor, with the gain for the most likely case, r = 101, being about half of that of the flat bettor. We get similar results for other cases 7

8 For example, for the case p = 0.505, n = 1000, the net gain when the number of wins is 505 is , which is about half of that with flat betting (0.10) and that in the range ( ), the net gain for the Kelly bettor is less than that for the flat bettor. The question arises as to whether there is a range of number of wins where the Kelly bettor does not do as well as the flat bettor. For r wins the net gain for a Kelly bettor is K (r) = (1+ f * ) r ( 1 f * ) n-r 1,while for a flat bettor for bets of size f * it is F (r) = (2r n )f *. Expanding K (r) and ignoring powers of n(f * ) 2 we have, K (r ) ~ (2r n)f * + (4r 2 4nr +n 2 n )(f * ) 2 /2, so that, the relation K ( r ) F (r) reduces to (2r n ) 2 n or r n/2 n / 2. Since p ½, n/2 is the approximate expected number of wins and n/2 =, the standard deviation of the number of wins. So the range of values of the number of wins for which the net gain for the Kelly bettor is less than that for the flat bettor is: Number of wins Expected number of wins. This means that although, overall, the expected value for the Kelly bettor is greater than that for the flat bettor, in the range displayed above, if n is large, then approximately 68 % of the time, the Kelly bettor is worse off than the flat bettor. The increase in the expected value of the overall gain occurs for approximately 32% of the time. 4. The Gain to Action ratio. We now consider a result similar to, but somewhat different from that in the previous section. Wong (1981) stated that for the Kelly bettor with optimal 8

9 fraction f *, the rate of return on action is about one-half of that for the flat bettor, the latter quantity being f *, action being defined as the total amount wagered. There is some ambiguity in Wong s treatment. He gives an informal proof of the above statement which is supposed to apply to the expected rate of return on action, i.e. for expectation taken over all possible results of a series of trials. This is in contrast to the situation in the previous section where the equivalent statement applies to a particular result, namely for the case when the number of wins is equal to the expected value. Also, here, action depends not only on the initial capital, the proportionality factor, and the number of trials, but also on the order of the results of the trials. Wong also gives an example of 5 trials (where the action can be explicitly determined) and taking p = 0.6, demonstrates the approximate validity of the statement for the expected result, namely 3 wins and 2 losses. Ethier and Tavare (1983) show that Wong s main statement is true in the limit when n and p 0.5. We shall prove it for the limiting case. Let us first consider the relationship between return on capital and the expected value of the exponential rate of growth, G (f). Let the rate of return on capital for the Kelly bettor with optimal fraction f * be r *. Then, if r * is taken to be the limit of the ratio of return when n tends to infinity, we have r * = lim n (X n /X 0 ) 1/n 1. Now, G(f * ) = lim n ln (X n / X 0 )/n, so that exp (G (f * )) = lim n (X n / X 0 ) 1/n. Hence, r * = exp (G (f * )) 1. (8) Now, since only a fraction f * of the capital is bet at each stage, the rate of return on action is r * / f *. We shall show that r * /f * is approximately f * /2. 9

10 We have, from (8), r * = exp ( G ( f * )) 1 = (1 + f * ) p (1 f * ) q 1. (9) Expanding the right hand side of (9) as in section 3, we have, ignoring higher powers of f *, r * ~ (f * ) 2 /2 + (f * ) 4 /2 ~ (f * ) 2 /2. (10) Thus, r * / f * ~ f * /2. We note that only when r * is evaluated as the limit of (X n /X 0 ) 1/n 1 as n that it is related to G(f * ) and (10) follows. For small values of n, r * / f * is considerably greater than f * /2, although when the number of wins is equal to or around the expected value, the ratio win/action is around half of f *. For Wong s case of n = 5 and p = 0.6, the ratio win/ action ranges from 0.08 to 0.13 for the case of 3 wins and 2 losses, but varies from 0.44 to 1.00 for 4 wins and above, and varies from 0.23 to 1.00 for 2 wins or less. The average over all 32 values is , somewhat greater than f * /2. 5. Halving and Doubling Capital We now derive the probability that the Kelly bettor s capital is reduced to a small fraction a (0 < a <1 ) of the original before increasing to a multiple b (b > 1) of the original. To derive this and other associated results, we need some preliminaries. 1. A random variable X has the lognormal distribution with parameters and if Y = ln (X) has the normal distribution with mean and variance 2. The mean and variance of X are given by: X = exp ( + 2 / 2 ), X 2 = exp (2 + 2 ) ( exp ( 2 ) 1 ) (11) 2. Consider a sequence { Z i }, ( i = 1,2,3 ) of independently and identically distributed random variables and let S n = 1 n Z i. The probability that the 10

11 random walk { S n } (n = 1,2,3, ), starting from zero, and with absorbing barriers at A (< 0), and B (> 0), is absorbed at A ( using, Wald s Identity, or the Optional Stopping Theorem) and ignoring the overshoot over the barriers A and B, is given by P (A ) = [ 1 exp(b 0 ) ] / [ exp (A 0 ) exp(b 0 ) ], (12) where 0 ( 0) is the solution of M( ) =1, M( ) being the moment generating function of Z i. If the random variables { Z i } have the normal distribution with mean and variance 2, we have 0 = 2 / 2, and (12) becomes P (A) = [ 1 exp ( 2 B/ 2 )]/ [ exp ( 2 A/ 2 ) exp ( 2 B/ 2 ) ] (13) Now, if the quantities B and A are large compared to the mean and variance of the increments { Z i }, it follows that n is large, and hence for random variables { Z i } (i = 1, 2, 3, ) which are not necessarily normal, S n is asymptotically normal with mean n and variance n 2 by the Central Limit Theorem so that the result (13) holds asymptotically. We shall now use the above two preliminaries to derive the probability mentioned at the beginning of the section. Consider now the process of the Kelly bettor s capital X n at stage n, where X n is given by (1). From (4) and (5), we have for X 0 = 1, E (X n ) exp [ nk( f * ) 2 ], Var ( X n ) exp ( nk 2 ( f * ) 2 + 2nk(f * ) 2 ) exp (2nk (f * ) 2 ) (14) Taking logarithms of (1), we have for X 0 = 1, and f = kf * ln ( X n ) = S ln ( 1 + k f * ) + ( n S )ln ( 1 kf * ) = n ln ( 1 k f * ) + S ln [ ( 1+ k f * )/ ( 1 k f * )]. 11

12 Since ln (X n ) is asymptotically normal, X n is asymptotically lognormal. Equating the E ( X n ) and Var ( X n ) from (11) and (14) we have, + 2 /2 nk (f * ) 2, n [ k 2 (f * ) 2 + k (f * ) 2 ], so that 2 nk 2 (f * ) 2, = n (f * ) 2 (k k 2 /2 ), or 2 / 2 = 1 2/k. Now, the probability of the random walk X n being absorbed at a is the same as the probability of Y = ln (X n ) being absorbed at ln (a) = A, and is given by P(A) in (13) above. Hence the probability that the Kelly bettor s capital is reduced to a fraction a before reaching b is, P ( A ) = [ 1 b 1-2/k ]/ [ a 1-2/k b 1-2/k ], (15) where ln (b) = B. It is significant that this probability does not depend on the bettor s edge f *, but f * does enter into calculations of the expected time to reach a or b. For the random walk {S n } the expected time E(T) to absorption at a or b, is given by E (T) = [ P ( A ) ln (a) + ( 1 P ( A ) )ln (b) ] /E (Z ), and, so for the random walk {X n }, we have E ( T ) = [P(A)ln (a) + ( 1 P ( A ) ) ln (b) ] / [(f * ) 2 ( k k 2 / 2)] (16) It is interesting to consider some particular cases of (15) and (16). Literature on gambling (see, for example, Wong (1981), and Chen and Ingenoso (2007)) tells us that the focus of a gambler s interest lies mainly on the values a=1/2 and b=2, ( i.e. with his capital being halved or doubled), perhaps, because he is aware of the fact that he cannot be ruined and that in the long run he will be infinitely rich We shall, 12

13 somewhat arbitrarily focus on the probability that his capital reduces to half and the expected time for the capital to double. For k = 1, P(A) = a(b 1)/(b a), (17) which, for the case b = 1/a, reduces to P(A) = a/(1+ a). That is, with the optimal Kelly betting, the probability of reducing the capital to half its value before doubling it is 1/3, whatever the edge. The edge comes into calculation only for the expected time for the capital to reach half its or to double. For example, for f * = 0.06, the expected time to reach these levels is E (T) = [ ln(1/2)/3 + 2 ln(2)/3 ] / ( /2) = Once again, consider the case k=1. The probability of ever reducing the capital to the fraction a can be calculated from (17) by letting b. We get the probability as a. On the other hand, an optimistic gambler, knowing that P (B) = 1, is interested only in the expected time it takes to double his original capital, i.e. reaching b = 2 (irrespective of the level a=1/2). Taking, once again, the case f * = 0.06, this expected time is equal to 2ln(2)/ = If the gambler considers that betting with the full Kelly factor is risky, he may want to know how a fractional Kelly betting affects the above probabilities and expected values. From (15) as b the probability of the capital reaching a is a 2/k-1, which for k = 1 is a, and for k = 1/2 is a 3. Table 2 gives the probability P * of the Kelly bettor reducing his capital to half, and the expected value, E(T) of the time for him to double his initial capital for various values of k and f *. From Table 2 we see that a gambler who is playing a game with an edge of 0.06, say, has the choice of the proportionality factor k as a fraction of the Kelly fraction f *. He 13

14 may want to play it safe and choose k=1/3, so that the probability of reducing his capital to half its initial value is very small, equal to 1/32, but is prepared to wait an average of 693 trials to double his capital. On the other hand, he may choose k=1, with the probability ½ of reducing his capital by half, but having to wait only for an average of 385 trials to double it. f * / k 1 1/2 1/ P * 1/2 1/8 1/32 E(T) P * 1/2 1/8 1/32 E(T) P * 1/2 1/8 1/32 E(T) Table 2. Values of P * and E(T) for various values of k and f * 6. Comparison between Kelly and Flat Betting Suppose a gambler has the option of Kelly betting (with the choice of a value of k) or the corresponding flat betting with the same value of k, i.e. with bets of fixed size X 0 kf * at all the stages. The disadvantage of this flat betting is that the gambler can be ruined, which, as we know, is not possible in the Kelly form of gambling. On the other hand, as we saw in section 3, the net gain for Kelly betting for outcomes around the most probable case is less than that for the equivalent flat betting. We shall work out for various values of k and f *, the probability of ruin for flat betting and the ratio R 14

15 of the net gain for Kelly betting to that of flat betting for the most probable outcome. Both these quantities are independent of the value of f *. First consider the flat betting case with bets of size X 0 kf *. Since the edge per unit is f *, we have that the expectation for the bet is X 0 k (f * ) 2. The variance is Var = X 2 0 k 2 ( f * ) 2 X 2 0 k 2 (f * ) 4 ~ X 2 0 k 2 (f * ) 2, ignoring the higher powers of f *. Putting this in the setting of a random walk with normal increments with mean μ and variance σ 2 as in section 5, we have, 0 = 2μ/σ 2 = 2/(kX 0 ). Now, for a random walk with only one absorbing barrier at a (<0) the probability of absorption is asymptotically given by exp ( aθ 0 ). So, with a = X 0, we have that the probability of ruin is P (Ruin ) = exp ( 2/k ) for all values of f *. As we saw before, the ratio R is also independent of the value of f *. Table 3 gives the values of R and of the probabilities of ruin for the equivalent flat betting. This means that for any value of f *, the gambler has the option of, (i) Kelly betting with k = 1, avoiding the 13.5% chance of being ruined, but having a net gain of about a half that with flat betting or, (ii) flat betting with k = 1/3, thereby incurring a chance of ruin of only 0.25%, but increasing the net gain over Kelly betting by a factor of 6:5. k 1 1/2 1/3 R 1/2 3/4 5/6 Prob (Ruin ) Table 3. Values of Probability of ruin and R 15

16 Acknowledgement The author expresses his thanks to an anonymous referee and to Professor J. Gani for suggestions and corrections on an earlier draft of the paper. References BREIMAN, L.(1961). Optimal gambling systems for favourable games. In Proc. 4 th Berkeley Symp. Math. Statist. Pub., vol. I, University of California Press, Berkeley, CHEN, W. AND INGENOSO, M (2007). Risk formulae for proportional Betting. In Optimal play: Mathematical studies of games and gambling. Eds. Ethier,S. and Eadington, W. Publisher, Institute for the study of gambling and commercial gaming. Reno, ETHIER, S. AND TAVARE, S. (1983) The proportional bettor s return on investment. J. Appl. Prob. 20, GOTTLIEB, G (1985). An Analytic Derivation of Blackjack Win Rates. Operations Research, 33, No.5. p GRIFFIN, P.A. (1981). The theory of Blackjack, 2 nd edn. GBC Press, Las Vegas. KELLY, J.L. JR. (1956). A new interpretation of information rate. Bell System Tech. J. 35, PHATARFOD, R. (2007). Some aspects of gambling with the Kelly Criterion. Math. Scientist. 32, ROTANDO, L.M., AND THORP, E.O. (1992). The Kelly criterion and the stock market. Amer. Math. Monthly. 99, TAYLOR, S. AND KARLIN, S. (1998) An Introduction to Stochastic Modeling, Academic Press. 16

17 THORP, E.O. (1966) Beat the Dealer, New York, Random House. WONG, S. (1981). What proportional betting does to your win rate. Blackjack World 3,