The proportional bettor's return on investment

zyxw zyx zyx J. Appl. Rob. 20,563-573 (1983) Printed in Israel @Applied h b a b i l i t y Trust 1983 zyxwvutsrqponm THE PROPORTIONAL BETTOR'S RETURN ON INVESTMENT S. N. ETHIER,. Michigan State University S . TAVA&,** Colorado State University AbstrPet Suppose you repeatedly play a game of chance in which you have the advantage. Your return on investment is your net gain divided by the total amount that you have bet. It is shown that the ratio of your return on investment under optimal proportional betting to your return on investment under constant betting converges to an exponential distribution with mean 1 as your advantage tends to 0. The case of non-optimal proportional betting is also treated. zyxwvu zyxwvutsrqpon zyxwv CONVERGENCE IN DISTRIBUTION; GAMMA DISTRIBUTION, BE'ITING SYSTEMS 1. Introduction Consider a game of chance that is played repeatedly and in which at each trial the bettor either wins or loses the amount of his bet. Suppose that the win probability p satisfies f < p < 1, so that the game is advantageous. Given f E (0,1], one possible strategy is for the bettor to wager a proportion f of his current fortune at each trial. Letting X,, Xz,. be independent and identically distributed (i.i.d.) with -- 1 (1.1) with probability p xl={ -1 with probability 1- p, the proportional bettor's fortune after n trials is Received 30 September 1982. * Postal address: Department of Statistics and Probability, Michigan State University, East Lansing, MI 48824, U.S.A. Supported in part by NSF Grant Ma-8102063. ** Postal address: Department of Statistics, Colorado State University, Fort Collins, CO 80523, U.S.A. 563 564 zyxwvutsr zyxwv zyxw zy zyxwv zyxwvuts zyxwv zy zyx S. N. ETHIER AND S. TAVARk where Fo is his initial fortune (a positive constant). The exponential rate of growth of his fortune is defined by (1.3) Gp (f) = E log(1+ fxi), since limn,,(l/n)log(Fn/Fo) = G p ( f )a.s. by the law of large numbers. In particular, if f is such that Gpcf) > 0, then F, +t~ a s . The choice f * of f that maximizes Gpcf) (namely, f * = 2p - 1) results in an optimal betting system (Kelly (1956), Breiman (1961), Finkelstein and Whitley (1981)), which is often referred to as the Kelly system. Our interest centers on the proportional bettor’s return on investment (i.e., net gain divided by total amount bet), which after n trials is given by - Assume that f is such that Gpcf)> 0. In Section 2 we show that as n + t ~ , R, (1.5) 0 R and where (1.7) denotes convergence in distribution and 1 R= f $ (Fo/Fk) -1 The random variable R, which by (1.5) represents the proportional bettor’s (asymptotic as n +t~) return on investment, can be shown to satisfy Cb< R < 1 almost surely and ER < EX1. (1.8) The latter result says that the expected return on investment under proportional betting is less than it is under constant betting. For fixed p and f, it is difficult to say by how much, but asymptotic results can be obtained by letting p +f + and f+O+. Fix (Y E (0,2). Then G ( l + r y z>( 0~ for ~ ) E positive and sufficiently small, and for such E we let X1(c), X*(E), . be i.i.d. with XI(&)given by (1.1) with p = (1 + ~ ) / 2and , we define R - ( E )by (1.7) and (1.2) with Xi = Xi( E ) and f = ( Y E . This corresponds to the case in which the betting proportion is o! times the optimal betting proportion. In Section 3 we show that as E 4 0 + , - zyxwvuts zyxwvuts zyxwvu zyxwvut -zyxwv zyxw zyxwv zyxwv 565 The proportional bettor's return on investment R"(E)/EX~(E) 0 gamma (i- 1,;) , where gamma(8, A ) denotes the distribution on ( 0 , ~ )with density g(x) = T(B)-'A 'x '-'e-"'. Moreover, we have uniform integrability, so In the case of optimal proportional betting, a = 1 and (1.9) becomes R ' ( E )/Ex~(E)A exponential&), (1.11) where exponential(A) denotes the distribution on (0, w) with density g(x) = Ae-"", while (1.10) reduces to lim ER'(E)/EX~(E) = i. (1.12) c 4 + In a recent paper, Wong (1981) argued that the optimal proportional bettor's return on investment is only about one-half of that of the constant bettor, at least when the actual number of wins is approximately the expected number. By (1.6), the latter condition is unnecessary. Ignoring it and assuming that Wong meant expected return on investment, we see that (1.12) can be viewed as a precise formulation of Wong's assertion. Observe that h ( a )E !$+ P{R"(E)/EXI(E)> 1) (1.13) represents the (asymptotic as E +0 + ) probability that proportional betting outperforms constant betting in terms of return on investment when the betting proportion is a times the optimal betting proportion. Using (1.9), it can be checked that h(O+)=1, h(l)=e-', and h(2-)=0. (We believe that h is monotone decreasing on (0,2) but do not have a proof.) However, it would probably be a mistake to regard this or (1.10) as an indictment of proportional betting. As Wong (1981) put it (referring only to the optimal case), '. . . proportional betting costs you about half of your arithmetic expectation. You can think of this as being the premium you have to pay for the insurance against going broke that you get with proportional betting.' Up until now, we have assumed that XI, the bettor's net gain per unit bet, is { - 1, 1)-valued. This assumption, however, is unnecessary. In particular, (1.5) and (1.8) hold with XI,X,, taken to be i.i.d. non-degenerate [ - 1,m>valued random variables with positive (finite) expectation (assuming of course that Elog(l+fX1)>O). As for (1.9) and (l.lO), we require for some M > O that XI(&),X*(E), * be i.i.d. non-degenerate [ - 1,MI-valued random variables for each E E [0, EO), that X l ( ~ ) a X 1 ( 0 as ) E +O+, and that E X l ( ~ ) > O > - 566 zyx zy zyxwvutsr zyxwv s. N. ETHIER AND s. TAVAF& E[X1(&)/(1+XI(&))]for 0 < E < co and EXl(0) = 0. (The boundedness assumption on XI(&)is probably stronger than necessary but involves no real loss of applicability.) In this case, (1.9) and (1.10) hold with R a ( & ) defined for E sufficiently small by (1.7) and (1.2) with Xi = Xi (E) and f = a f * ( ~ )f,* ( ~ being ) the choice of f that maximizes Elog(1 +fXl(s)), the exponential rate of growth of the bettor's fortune. Clearly, these results generalize the previously stated ones. One of the best-known applications of proportional betting occurs in the game of blackjack (Griffin (1981)). While it seems likely that the above results hold at least qualitatively in this context, it should be recognized that several of our assumptions fail to hold here. First, successive blackjack hands are not independent unless separated by a shuffle. Second, successive hands are not identically distributed from the card-counter's point of view; in particular, his advantage fluctuates. Third, our requirement that X I Z -1 (i.e., the bettor cannot lose more than the amount of his bet) seems to preclude the blackjack options of pair splitting, doubling down, and insurance; however, by suitably rescaling, it it easy to generalize the above results, replacing - 1 by an arbitrary negative constant. Finally, we have assumed implicitly that money is infinitely divisible, which of course is not the case in a gambling casino. zyxw zyxwvu 2. Asymptotic distribution of R. as n +m Our first result concerns the random variable R defined by (1.7), though here we do not assume (1.1). Let XI,X,, be i.i.d. non-degenerate [ - 1,m>valued random variables with 0 < EX, < a. Fix f € (0,1] such that Proposition 2.1. + E log(1 fX1) > 0, (2.1) -- - and let Fo be a positive constant. Define Fl,F,, by (1.2), R1, Rz, * by (1.4), and R by (1.7). (a) 0 < R < ess sup XI a s . (b) R . A R as n+m. (c) There exists a random variable R ' (defined on the same probability space 9 are defined on) such that R ' = R, R ' is independent of XI, and that XI,X,, - - R (2.2) = (1 +fXI)R'/(l+ fR'). (d) ER < EX,. zyxwvutsrqp Proof. (a) By the law of large numbers and (2.1), lim (&/,)I" k- = lim exp{ - (l/k)log(K /Fo)} k- + = exp{ - E log(1 fX,)} < 1 as., zyxwvutsrqpon zyxwv zyxwvut zyxw zyxwvu zyxwv zyxwvu 567 Thc proportional bettor's ntwn on investment so the series in (1.7) converges almost surely by the root test. Hence R > O almost surely. For the second inequality, if esssup Xl = 00, there is nothing to prove. If M = ess sup Xi< 00, then with equality only if Xi = M for each i h 1. Hence the non-degeneracy of XI implies that R < M almost surely. (b) Observe that as n +w, where the equality in distribution follows by reversing the order of XI, X2, * .,Xn; we have also used Fn +m a s . as n +00, valid by (2.1). (c) Define R ' in terms of Xz, X3, exactly in the same way that R is defined Xz, * . Then in terms of X17 - - - which is equivalent to (2.2). (d) By (2.2), R 5 (1 +fXl)/f, and thus R has finite expectation. Now the function *(r)= r/(l +f') is strictly concave on (O,w), so by (c) and Jensen's hequality (using the non-degeneracy of Xland therefore R ), ER = E[l +fXi]E[R/(l+ fR)] < (1 +fEXl)ER/(l +fER), from which the desired result follows. The next result will allow us to prove a generalization of (1.6). --- Xz, * f, Fo,Fl, Fz, , and R be as in Proposition Proposition 2.2. Let Xl, 2.1. For each n h 1, let XI,,, .,X, be [ - 1,w)-valued random variables, and define - a, 568 zyxwvutsr zyxw zyxw zyxw zyxw S. N. ETHIER AND S. TAVARfi 0 (2.6) (X~n,*.*,Xmn)- (X,,*.*,Xrn)as n+m, m 2 1 , 1 5 Itl 5 n, O < u < 1. E(FdFmn)Y5 E(FdFm)Y, (2.7) 0 Then R , - R as n+w. Proof. As in (2.3), we infer from (2.5) that so it will suffice to show that FdF,5 0 and zyxwvu By (2.6), the latter will follow.from Theorem 4.2 of Billingsley (1968), provided we can show that lim sup P (2.8) m- nBm+l [2 (FdFkn)Z 61 = O k-m+l for every 6 >O; the former will also be a consequence of (2.8). Now 4(t)= Eexp( - t log(1 + fXl)) satisfies b(0)= 1 and d’(0)< 0, so there exists u E (0,l) with 4 ( u ) < 1. Thus, the probability in (2.8) is bounded by sa-” 2 E(FdFk)”=6-“ k-m+l 2 4(U)k k-m+l S s - Y ~ ( U ) m + l/ (+l ( u ) ) for each n > m 2 1, where we have used (2.7). This implies (2.8) and completes the proof. Corollary 2.3. Let Xl, -,f, Fo,R1,Rz, * - and R be as in Proposition X2, * 2.1. Suppose in addition that 1 Z 2 and there exist distinct &, * .,6 E [ - 1, w) such that pj = P{Xl = 5 )> 0 for j = 1, * ., 1 and Ei-lpj = 1. For each n 2 1, let mln, * ., mln be non-negative integers summing to n, and put - An = [2 x{x,-e) = mjn, j = 1, * Assume that limn- m,,, /n = pj for j = 1, - a, * *, I 0 1. Then Rn An+R as n +a. Proof. For each n 2 1, let (XI,, .,X,,,,)have the conditional distribution of (XI, - X,,)given A,,, and apply Proposition 2.2. Conditions (2.5) and (2.6) are easily checked, while (2.7) follows from Theorem 4 of Hoeffding (1963). a, zyxwvutsrqponm zyxwvuts zyxwvuts zyx zyxwvu zyxwv 569 ??le proportional benor’s retwn on inwshnent zyxwvuts 3. Asymptotic distribution of R/EXl The main result of this section specifies the asymptotic distribution of R/EXl as the bettor’s advantage tends to 0. Lemma 3.1. Let M and eo be positive constants. Let X1(c) be a nondegenerate, [ - 1,MI-valued random variable for each E E [0, E ~ ) .Suppose that X1(E)ZX1(0)as E + o + , (3.1) EXI(E)>O>E[XI(E)/(~+XI(E))], O< E <EO, where - 1/0= - m, and EXl(0)= 0. For 0 < E < E ~ define , G. (f) = Elog(1 +fXi(~)), O S f 5 1, and ml(&)= EXI(&),m2(&)= EXl(E)2. (a) For each E E (0, E ~ ) , G. (f) is strictly concave in f and has a unique maximum at f = f * ( ~ )say, , in (0,l). l (o(1)) l as E d o + . (b) f * ( ~=) r n l ( ~ ) m 2 ( ~ ) -+ (c) For each a E(0,2), as E + O + , G, (af*(~)) = af*(E)ml(E)(l -+a + o(1)). Proof. (a) Fix E E (0, E ~ ) .Since G, (f) is strictly concave in f, G:(f)is strictly decreasing in f, and G:(O)> 0 > G:(l-) by (3.1). Hence there exists a unique f * ( ~E) (0,l) with G:(f*(e))= 0; clearly, this choice of f maximizes G.(f). (b) First we claim that f*(&)+O as E + O +. Suppose not. Then there exists ~ . - - , 0 +with f * ( ~ . ) + f ~ E ( O , l ] .If f o < l then , by Jensen’s inequality, a contradiction. If fo = 1 the second equality in (3.2) becomes an inequality ( 5 )and limn- is replaced by limsup.,,. Here we are using Theorem 5.3 of Billingsley (1968) and the assumption that X1(&)is bounded above by M. This establishes the claim. Now since G : ( ~ * ( E=)0, ) we have &(E)~ - f * ( ~ ) m 2 (+ ~~ ) * ( E ) ~ E 1+f*(E)XI(E)= O and hence (3.3) 570 zy zyxwvutsr zyxwvutsrqp zyxwvutsrqpon zyx zyxwvu zyxwv zyx s. N. ETHIER AND s. TAVARB for O < E < eo. The desired conclusion now follows from (3.3) and the initial claim. (c) By a Taylor expansion and (b), as E --* 0 + , G. (a!*(&))= Elog(l+ af*(~)Xi(~)) = af*(E)ml(E)-IaZf*(E)Zmz(&)+ O(~*(E)~) = af* (E )m1( E ){ 1- laf* (E )ml(E )-' mz(E ) = af*(E)ml(E)(l -la + o (1)) + o(1)). Zkorem 3.2. Let M, c0, and Fo be positive constants. Let X&), X&), * be i.i.d. random variables for each E E [0, eo) with X1(e),0 5 E < eo, satisfying the conditions of Lemma 3.1. Fix a E (0,2), and, using the notation of Lemma 3.1, define R " ( E )by (1.7) and (1.2) with Xi = X i ( & )and f = a f * ( ~for ) each E E (0, eo) for which G, (af*(E)) > 0. (This is possible by Lemma 3.1 (c).) Then, as E -+ 0 + ,(1.9) holds. Moreover, R"(E)/EXl(&) is uniformly integrable in E , so (1.10) also holds. Proof. Let E > 0 be such that G. (af*(~)) > 0, and denote the n th moment of RP(&)by p , ( ~ for ) each n B 1. Let n B 1. By Proposition 2.l(c), p,(~)= E(1+ Qlf*(E)X,(&))nE[R"(E)n/(l + a f * ( ~ ) R " ( ~ ) ) n ] . Since 1- nx 5 (1 + x)-" 5 1- nx + (";l)xZ for all x > 0, and since P,,+~5 MP,,+~, we have (1 + q ) ( p n - mf*b+l) s /.tm 5 (1 + q ) where the dependence on Rearranging, E [ pn - naf* (1 -+ ~ a f *p n)+ 1 } is implicit and where q = E(l +af*XJ' - 1. .= pzpnIm Y, plpn /my 5 pn+iImn1+ l = (3.4) where p1= q/naf*ml(l + q), Letting E -+ 0 + and noting that q = naf*ml + (3 azf*zmz+ ocf*3), we conclude from Lemma 3.l(b) that both p1and p2converge to 1+ ((n - 1)/2)a. By induction applied to (3.4) (using Proposition 2.l(d) for the initial step), zyxwvutsrq zyxwvutsrqp zyxwvut zyxwvuts zy zyxwvuts 571 Ihc pmportional benor's return on investment sup. p,, /m: < 00, and hence (R"/EXl)"is uniformly integrable in E for each n 2 1 (Billingsley (1968), p. 32). Also from (3.4), if lim,+,,+p,,/m; exists, then lime4+ p,,+l/m;+l exists, and (3.5) zyxwvu lim p , , + ~ m ;+I = (I r+o+ +n-l a) j k p,,/m ;. 2 Since R"/EXI is uniformly integrable in E,it is tight (Billingsley (1968), p. 41), hence relatively compact. Let U be any weak limit as E + O + , so that R"IEXl -% U as E + O + through some subsequence. Let v,, denote the nth moment of U for each n 2 1. By the uniform integrability proved above, p,,/m;+ v,, as E + O + through the subsequence for each n 2 1. By (3.9, vn+l = (1 +n-l a ) v,,, 2 n 21. Define 8 = 2 / a - 1 and A =2/a. Using (3.6), we find that U has Laplace transform Ee-IU=l+ - (-t)"v,,/n! n-1 =1+ = n-1 ( - t)"(e + n - 1)s- .(e [ Zo($(; + l)vl/h"-'n! e~) ] 5 v1+1-- h VI e = ( l + f ) - e $ + l -A- Y]. e As t + m , this tends to l-(A/O)vl, which must therefore be non-negative. Hence U is a mixture of gamma(@A ) and So, the unit mass concentrated at 0. Let 0 < S < min(1, e). We claim that (3.7) zyxw lim sup E(R"(E)/EX](E))-' < 00. #-bo+ Granting this for the moment, it follows that U must be purely gamma(8,A). Since U was an arbitrary weak limit as E +0+ of R"(E)/EX](E), we conclude that (1.9) holds as E +O+. We turn to the proof of (3.7). As E + O + , + E(1+ af*Xl)-* = 1 - &rf*mI fS(S (3.8) + 1)azf*zm2+ O(f*') = l-&rf*ml(l-f(S +l)af*m;'m2+o(l)) = 1 - &rf*ml(l-$(a + 1)a + o(1)) zyxwvutsr - 572 zyxwvuts zyxw zyxw zyx zyxw zyxw S. N. ETHIER AND S. TAVARh by Lemma 3.l(b). Since 0 < S < 8, this is less than 1for all E sufliciently small. In particular, for such E, (2 E(R")-' = (af*)"E (FdFk))' < 00. Now by Proposition 2.l(c) (in particular, (2.4)), E (')6 = E(1+ af*XJ6E (af*ml+')". Using the inequality ( x + c ) 5 ~ x 6 + &x'-', valid for all x > 0 and c > 0 (since 0 < 6 d l), and Proposition 2.l(d) together with Jensen's inequality, we find that and hence (3.9) E(R"/mJ8 5 Gaf*ml{l - E(1+ ~ r f * X ~ ) - ~ } - ' . By (3.8), the right side of (3.9) tends to (1 - f ( G and completing the proof. + l)a)-' as E -*O + ,pro\..ig (3.7) Remark 3.3. We outline an alternative proof of (1.9), valid when XI(&) is given by (1.1) with p = (1 + ~ ) / 2and somewhat more generally. By (3.7), (R"(E)/EX,(E))-~(defined for E positive and sufliciently small) is uniformly integrable in E, hence tight, and therefore relatively compact. Let V be an arbitrary weak limit as E -* 0 + ,and denote its Laplace transform by + ( t ) , t L 0. Using Proposition 2.l(c) (in particular, (2.4)), a Taylor expansion, and Lemma 3.l(b), we find that 4 satisfies the differential equation (3.10) + 4att$"(t) (a- 1)4'(t) - 4(t)= 0, t > 0. Moreover, 4 is monotone decreasing and +(O+) = 1. It follows from Gradshteyn and Ryzhik (1980), Equation 8.494(5), that (3.11) 4 ( t )= q ~ t ) ~ ~ ~ ~ ( 2 V T t ) / r (t e>)0,, zy zyxwvutsrqpon zyxwvuts zyxwvut zyxwvut zyxwvu zyxwvuts Thc proportional 6ettor's return on investment 573 where 8 = 2/a - 1, A = 2/a, and KBis the (decreasing) modified Bessel function of order 8. We conclude from Gradshteyn and Ryzhik (1980), Equation 3.471(9), that 1/V is gamma(8,h). Since V was an arbitrary weak limit of (R (&)/EXl(&))-' as E +0 + , the desired result follows. Acknowledgements We are grateful to Peter A. GrifEn for valuable correspondence and to Thomas G. Kurtz and Stanford Wong for helpful suggestions. References BILLINOSLEY, P. (1968) Convergence of Rubability Measures. Wiley, New York. BREIMAN,L. (1961) Optimal gambling systems for favorable games. h c . 4th Berkeley Symp. Math. Statist. Rob. 1, 65-78. FINKELSTEIN, M. AND WHITLEY,R. (1981) Optimal strategies for repeated games. A d a A M . &ob. 13, 415428. G l u ~ s mI., S . AND RYZHIK,I. M.(1980) Tabk of Integrals, Series, and Products. Academic Press, New York. GRIFFIN,P . A. (1981) The Theory of Blackjack, 2nd edn. GBC Press, Las Vegas. HOEFR)ING, W. (1963) Probability inequalities for sums of bounded random variables. J. Amrr. Statist. Assoc. SS,13-30. KELLY, J. L., JR. (1956) A new interpretation of information rate. &I1 System Tech. J. 35, 917-926. Worn, S. (1981) What proportionalbetting does to your win rate. Blackjack World 3,162-168.