Proportional Bettor's Return on Investment

8/9/2019 Proportional Bettor's Return on Investment

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J. Appl. Ro b. 20,563-573 (1983)Printed in Israel@ A p p l ie d h b a b i l i t y Trust 1983

THE PROPORTIONAL BETTOR'S RETURN ON INVESTMENT

S. N. ETHIER,. Michigan State UniversityS . TAVA&,** Colorado State University

AbstrPetSuppose you repeatedly play a game of chance in which you have theadvantage. Your return on investment is your net gain divided by the totalamount that you have bet. It is shown that the ratio of your return on

investment under optimal proportional betting to your return on investmentunder constant betting converges to an exponential distribution with mean 1 asyour advantage tends to 0. The case of non-optimal proportional betting is alsotreated.CONVERGENCE IN DISTRIBUTION; GAMMA DISTRIBUTION, BE'ITING SYSTEMS

1. IntroductionConsider 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 winprobability p satisfies f


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564 S. N.ETHIER AND S.TAVARkwhere Fo is his initial fortune (a positive constant). The exponential rate ofgrowth of his fortune is defined by(1.3) Gp(f) = E log(1+ fxi),since limn,,(l/n)log(Fn/Fo) = Gp(f )a.s. by the law of large numbers. Inparticular, if f is such that Gp f )>0 , then F, + ~ as . The choice f * of f thatmaximizes Gp f ) (namely, f * = 2p - 1) results in an optimal betting system(Kelly (1956), Breiman (1961), Finkelstein and Whitley (1981)), which is oftenreferred to as the Kelly system.

Our interest centers on the proportional bettors 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 ~ ,(1.5) R ,-and

0

where denotes convergence in distribution and1

f $ (Fo/Fk)(1.7) R = -1The random variable R , which by (1.5) represents the proportional bettors(asymptotic as n+ ~) eturn on investment, can be shown to satisfy Cb< R < 1almost surely and(1.8) ER


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The proportional bettor's return on investment 5650R"(E)/EX~(E)- amma (i 1,;) ,

where gamma(8, A ) denotes the distribution on ( 0 , ~ ) ith 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(1.11) R ' ( E )/Ex~(E)A xponential&),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.In a recent paper, Wong (1981) argued that the optimal proportional bettor'sreturn on investment is only about one-half of that of the constant bettor, at leastwhen the actual number of wins is approximately the expected number. By (1.6),the latter condition is unnecessary. Ignoring it and assuming that Wong meantexpected return on investment, we see that (1.12) can be viewed as a preciseformulation of Wong's assertion.

c 4 +(1.12)

Observe that(1.13) h ( a )E !$+ {R"(E)/EXI(E)> 1)represents the (asymptotic as E + + ) probability that proportional bettingoutperforms constant betting in terms of return on investment when the bettingproportion is a times the optimal betting proportion. Using (1.9), it can bechecked that h(O+)=1, h(l )=e-', and h( 2- )=0. (We believe that h ismonotone decreasing on (0,2) but do not have a proof.) However, it wouldprobably be a mistake to regard this or (1.10) as an indictment of proportionalbetting. 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 insuranceagainst 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>valuedrandom variables with positive (finite) expectation (assuming of course thatElog(l+fX1)>O). As for (1.9) and (l.lO), we require for some M>O thatXI(&),X*(E), * be i.i.d. non-degenerate [- 1,MI-valued random variables foreach E E [0,EO) , t h a t X l ( ~ ) a X 1 ( 0 )s E + O + , and that EXl(~)>O>


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566 s.N.ETHIER AND s.TAVAF&E[X1(&)/(1 XI(&))] or 0< E < co and EXl(0)= 0. (The boundedness assump-tion on XI(&) s probably stronger than necessary but involves no real loss ofapplicability.) In this case, (1.9) and (1.10) hold with Ra(&)defined for Esufficiently small by (1.7) and (1.2) with Xi = Xi(E)and f = a f * ( ~ ) ,* ( ~ )eingthe choice of f that maximizes Elog(1 +fXl(s)), the exponential rate of growthof the bettor's fortune. Clearly, these results generalize the previously statedones.

One of the best-known applications of proportional betting occurs in the gameof blackjack (Griffin (1981)). While it seems likely that the above results hold atleast qualitatively in this context, it should be recognized that several of ourassumptions fail to hold here. First, successive blackjack hands are not indepen-dent unless separated by a shuffle. Second, successive hands are not identicallydistributed from the card-counter's point of view; in particular, his advantagefluctuates. Third, our requirement that X I Z -1 (i.e., the bettor cannot losemore than the amount of his bet) seems to preclude the blackjack options of pairsplitting, doubling down, and insurance; however, by suitably rescaling, it it easyto generalize the above results, replacing -1by an arbitrary negative constant.Finally, we have assumed implicitly that money is infinitely divisible, which ofcourse is not the case in a gambling casino.2. Asymptotic distribution of R . as n +m

Our first result concerns the random variable R defined by (1.7), though herewe do not assume (1.1).Proposition 2.1. Let XI,X,, be i.i.d. non-degenerate [-1,m>valued ran-

dom variables with 0


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Thc proportional bettor's ntwn on investment 567so the series in (1.7) converges almost surely by the root test. Hence R > Oalmost surely. For the second inequality, if esssup Xl = 00, there is nothing toprove. If M = ess supXi 0, then

with equality only if Xi=M for each i h 1. Hence the non-degeneracy of XIimplies that R


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568 S. N. ETHIER AND S. TAVARfi0(2.6) (X~n,*.*,Xmn)- (X,,*.*,Xrn)as n+m , m 2 1 ,

(2.7) E(FdFmn)Y5 E(FdFm)Y, 1 5 tl 5 n, O < u < 1.0Then R , - R as n + w .

Proof. As in (2.3), we infer from (2.5) that

so it will suffice to show that F d F , 5 0 nd

By (2.6), the latter will follow.from Theorem 4.2 of Billingsley (1968), providedwe can show that

lim sup P [ 2 (FdFkn)Z 61= Ofor every 6 O ; the former will also be a consequence of (2.8). Now 4 ( t ) =Eexp(- log(1+fXl))atisfies b(0)= 1 and d(0)m 2 , where we have used (2.7). This implies (2.8) and completesthe proof.

Corollary 2.3. Let Xl,2, -,,Fo,R 1,R z , * - and R be as in Proposition2.1. Suppose in addition that 1 Z 2 and there exist distinct &, * .,6 E - ,w)such that pj =P{Xl= 5 )>0 for j = 1, - * .,1 and Ei-lpj= 1. For each n 2 , letmln, * .,mln be non-negative integers summing to n, and put

0

An = [2 {x , -e) = mjn, j = 1, * * *,Assume that limn- m,,,n = p j for j = 1,

Proof. For each n 2 , let (XI,, .,X,,,,) ave the conditional distribution of(XI,a , X,,)given A,,,and apply Proposition 2.2. Conditions (2.5) and (2.6) areeasily checked, while (2.7) follows from Theorem 4 of Hoeffding (1963).

a , 1. Then Rn IAn+ as n +a.


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??leproportional benors retwn on nwshnent 5693. Asymptotic distribution of R/EXl

The main result of this section specifies the asymptotic distribution of R/EXlas the bettors advantage tends to 0.

Lemma 3.1. Let M and eo be positive constants. Let X1(c) be a non-degenerate, [- 1,MI-valued random variable for each E E 0,E ~ ) .Suppose thatX1(E)ZX1(0)s E + o + ,(3.1) EXI(E)>O>E[XI(E)/(~+XI(E))],< E < E O ,where - 1/0= -m, and EXl(0)= 0. For 0< E < E ~ , efine

G. (f) = Elog(1+fXi(~)), O S 5 ,and m l ( & )= EXI(&),m2(&)= EXl(E)2.maximum at f = f * ( ~ ) ,ay, in (0,l).

(a) For each E E 0, E ~ ) ,G. (f) is strictly concave in f and has a unique(b) f * ( ~ )rnl(~)m2(~)- l( lo(1)) as E d o + .(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 ) s strictly decreasing in f, and G:(O)>0>G:(l-) by (3.1). Hence there exists a unique f * ( ~ ) 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]. f f o < l ,hen

by Jensens inequality, a contradiction. If fo = 1 the second equality in (3.2)becomes an inequality ( 5 )nd limn- is replaced by limsup.,,. Here we areusing Theorem 5.3 of Billingsley (1968) and the assumption that X1(&)sbounded above by M. This establishes the claim. Now since G : ( ~ * ( E ) )0, wehave

& ( E ) ~1+f*(E)XI(E)= O* ( ~ ) m 2 ( ~ )~ * ( E ) ~ Eand hence

(3.3)


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570 s. N. ETHIER AND s. TAVARBfor O < E < eo . The desired conclusion now follows from (3.3) and the initialclaim.

(c) By a Taylor expansion and (b), as E --* 0+ ,G.(a!*(&)) Elog(l+ af*(~)Xi(~))

= af*(E)ml(E)-IaZf*(E)Zmz(&)+( ~ * ( E ) ~ )= af (E ) m ( E ) 1- f (E )ml(E )-'mz(E )+ o (1))= af*(E)ml(E)(l - la +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),05 E < eo, satisfyingthe conditions of Lemma 3.1. Fix a E 0,2), and, using the notation of Lemma3.1, define R " ( E ) y (1.7) and (1.2) with X i = X i ( & ) nd f = af*(~)or eachE 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 ) / E X l ( & )s uniformly integrable in E , so(1.10) also holds.

Proof. Let E >0 be such that G.(af*(~))0, and denote the n h moment ofRP(&)y p , ( ~ )or each n B 1. Let n B 1. By Proposition 2.l(c),p , ( ~ ) = (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,,+~ M P , , + ~ ,we have

(1+ q ) ( p n -mf*b+ l ) m 5 (1+ q ) n - naf* 1-+ a f * )n + 1 }where the dependence on E is implicit and where q = E(l + a f * X J ' - 1.Rearranging,

n+ l .=(3.4) plpn / m y5 n+iIm = pzpnm Y,where

p1= q/naf*ml(l+ q),

Letting E -+0+ and noting thatq = naf *m l+(3 z f * zmz+ ocf*3),

we conclude from Lemma 3.l(b) that both p1 nd p2converge to 1+ ((n-1)/2)a.By induction applied to (3.4) (using Proposition 2.l(d) for the initial step),


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Ihcpmportional benor's returnon investment 571sup. p,, /m : < 0, and hence (R"/EXl)" s uniformly integrable in E for each n 2(Billingsley (1968), p. 32). Also from (3.4), if lim,+,,+p,,/m; exists, thenlime4+p,,+l/m;+lexists, and(3.5) lim p , , + ~ m+I = (I +n-l ) jk p,, m ;r+o+ 2

SinceR"/EXI s uniformly integrable in E, t is tight (Billingsley(1968), p. 41),hence relatively compact. Let U be any weak limit as E + O + , so thatR"IEXl-% U as E + O + through some subsequence. Let v,, denote the nthmoment 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 . By (3 .9 ,

vn+l= (1 +n-l) v,,, n 2 1 .2Define 8 = 2 / a - and A = 2 / a . Using (3.6), we find that U has Laplacetransform

E e - I U = l + ( - t ) " v , , / n !n-1-= 1+ ( - t)"(e + n - ) s - . ( e + l )v l /h"-'n!n-1= [Zo$(; e~ ] 5 v 1 +1 - - h VI= ( l + f ) - e $ + l - - Y].

ee

As t + m , this tends to l-(A/O)vl, which must therefore be non-negative.Hence U s a mixture of gamma(@A ) and So, the unit mass concentrated at 0.

Let 0< S


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572 S. N. ETHIER AND S. T A V A R hby Lemma 3.l(b). Since0< S < 8, this is less than 1for all E sufliciently small. Inparticular, for such E,

E(R")-' = (af*)"E 2 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 ) ~ 6 +&x'-', valid for all x > 0 and c >0 (since0< 6 d ), 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 + l)a)-' as E -*O + ,pro\..ig (3.7)and completing the proof.

Remark 3.3. We outline an alternative proof of (1.9), valid when XI(&)sgiven by (1.1) with p = (1+ ~ ) / 2 nd somewhat more generally. By (3.7),(R"(E)/EX,(E))-~ defined for E positive and sufliciently small) is uniformlyintegrable in E, hence tight, and therefore relatively compact. Let V be anarbitrary weak limit as E -*0+ , nd denote its Laplace transform by +(t) , t L 0.Using Proposition 2.l(c) (in particular, (2.4)), a Taylor expansion, and Lemma3.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 Grad-shteyn and Ryzhik (1980), Equation 8.494(5), that(3.11) 4 ( t )= q ~ t ) ~ ~ ~ ~ ( 2 V T t ) / r ( e ) ,>0,


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Thc roportional 6ettor's return on investment 573where 8 = 2/a - , A = 2/a,and KB s the (decreasing) modified Bessel functionof 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(&))-'s E + + , he desired result follows.AcknowledgementsThomas G. Kurtz and Stanford Wong for helpful suggestions.

We are grateful to Peter A. GrifEn for valuable correspondence and to

ReferencesBILLINOSLEY,. (1968) Convergenceof Rubability Measures. Wiley, New York.BREIMAN, . (1961) Optimal gam bling systems for favorable games. h c . th Berkeley Symp.FINKELSTEIN,. AND WHITLEY, . (1981) Optimal strategies for repeated games. A d a A M .G l u ~ s m ,. S . AND RYZHIK,. M. 1980) Tabkof Integrals, Series, and Products. AcademicGRIFFIN,P . A. (1981) The Theory of Blackjack, 2nd edn. GBC Press, Las Vegas.HOEFR)ING, . (1963) Probability inequalities for sums of bounded random variables. J. Amrr.KELLY, J. L., JR. (1956) A new interpretation of information rate. &I1 System Tech. J. 35,Worn, S. (1981) What proportional betting does to your win rate. Blackjack World 3,162-168.

Math. Statist. Rob. 1, 65-78.&ob. 13, 415428.Press, New York.

Statist. Assoc. SS,13-30.917-926.

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Proportional Bettor's Return on Investment