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UCLA STATS 596 - Kelly

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The Kelly Betting System for Favorable Games.Thomas Ferguson, Statistics Department, UCLAA Simple Example. Suppose that each day you are offered a gamble with probability2/3 of winning and probability 1/3 of losing. You may bet any positive amount you like,provided you have it. The amount you bet is either doubled or lost, independently eachday. This gamble is only offered for 20 days. The question is: how much should you beton each day?Let X0denote your initial fortune, and let Xkdenote your fortune after the kthgamble. On day k,youmaybetbkprovided0 ≤ bk≤ Xk−1(1)If you win on day k your fortune increases to Xk−1+bk; if you lose it decreases to Xk−1−bk.So,Xk=Xk−1+ bkwith probability 2/3Xk−1− bkwith probability 1/3(2)It is an easy problem to find, by backward induction, the betting system that maximizesE(X20|X0), your expected fortune after the 20 days have passed. It is to bet all you haveeach day. On the last day, given you have fortune X19, to maximize E(X20|X19)youbetitall,b20= X19, and your expected final fortune is (2/3)2X19+(1/3)0 = (4/3)X19.Since E(X20|X18) = E(E(X20|X19)|X18)=(4/3)E(X19|X18)=(4/3)2X18, the analysis bybackward induction shows that E(X20|X0)=(4/3)20X0= 315.34X0.However, there is something disturbing about this solution. The distribution of X20using the strategy of betting everything each time isX20=220X0with probability (2/3)20= .000300 with probability .99970(3)In other words, you will end up destitute with high probability. This is silly. You areplaying a favorable game. You can control your destiny. You should want your fortune totend to infinity if you play infinitely long. Instead you fortune tends to zero almost surely.Proportional Betting Systems. To avoid this fate, consider using a proportionalbetting system. Such a system dictates that you bet a fixed proportion, call it π,ofyourfortune at each stage, 0 <π<1. This requires that we consider money infinitely divisible.Under such a system, your fortune will tend to infinity almost surely. There remains theproblem of which value of π to use.Let us search for a π that will make our fortune tend to infinity at the fastest rate. Ifyou bet the same proportion π at each stage, then after n stages if you have won Zntimesand lost n − Zntimes, you fortune will beXn=(1+π)Zn(1 − π)n−ZnX0, (4)II – 1where Znhas the binomial distribution, B(n, 2/3). The fortune has increased by the factor(1 + π)Zn(1 − π)n−Zn=exp{Znlog(1 + π)+(n − Zn)log(1− π)}=exp{n[(Zn/n) log(1 + π)+(1− (Zn/n)) log(1 − π)]} exp{n[(2/3) log(1 + π)+(1/3) log(1 − π)]}(5)We can see that (1 + π)Zn(1 − π)n−Zntends to infinity almost surely exponentially fast.The rate of convergence is defined as the limit of (1/n) log((1 + π)Zn(1 − π)n−Zn), whichhere is(2/3) log(1 + π)+(1/3) log(1 − π). (6)The value of π that maximizes the rate of convergence is found by setting the derivative of(6) to zero and solving for π.Itisπ =1/3. Therefore the optimal rate of convergence ofXnto infinity is achieved by wagering one-third of your fortune at each stage. The optimalrate is(2/3) log(4/3) + (1/3) log(2/3) = .0566 ··· (7)This means that if two bettors compete, one using π =1/3 and the other using π =1/3,the fortune of the former will eventually be greater than that of the latter, and stay greaterfrom some stage on.The Kelly Betting System and Log Utility. Consider the same example now butwith 2/3 replaced by an arbitrary probability of winning, p, and let us find the proportionalbetting system that maximizes the rate at which Xntends to infinity. Of course if p<1/2,you cannot achieve having your fortune tend to infinity. You might as well bet proportion0 at each stage.But suppose p ≥ 1/2. The same analysis as above yields the rate of convergence tobe (in place of (6))p log(1 + π)+(1− p)log(1− π). (8)This is easily seen to be maximized at π =2p−1. This gives us what is known as the Kellybetting system; see Kelly (1956). If, for an even money bet, the probability of success isp, bet proportion π(p) of your fortune, whereπ(p)=0ifp ≤ 1/22p − 1ifp>1/2.(9)The Kelly Betting Proportion.0.5 101II – 2This rule also has an interpretation as the optimal rule of an investor who has thelogarithm of the fortune as his utility function and wishes to maximize the expectation ofthe utility of his fortune at each stage. Given the fortune at stage k − 1andbk= πkXk−1with 0 ≤ πk≤ 1, the expectation of log(Xk)givenXk−1isE(log(Xk)|Xk−1)=p log(Xk−1+ bk)+(1− p)log(Xk−1− bk)= p log(Xk−1(1 + πk)) + (1 − p)log(Xk−1(1 − πk))= log(Xk−1)+[p log(1 + πk)+(1− p)log(1− πk)](10)This is maximized by the same value of π that maximizes (8). This gives a small samplejustification of the use of the Kelly betting system.This betting system may also be used if the win probabilities change from stage tostage. Thus, if there are n stages and the win probability at stage i is pifor i =1,...,n,the Kelly betting system at each stage uses the myopic rule of maximizing the expectedlog, one stage ahead. Thus at stage k, you bet proportion π(pk) of your fortune. Theasymptotic justification of the Kelly Betting System described above has a generalizationthat holds in this situation also. See Breiman (1961).A General Investment Model with Log Utility. A striking fact is that thismyopic rule is globally optimal for maximizing E(log(Xn)) under quite general conditions.One might think that if some future piare close to 1, one should pass up marginal piandwait for those that really make a difference. You do not have to look ahead; you don’t evenneed to know the future pito act at stage k. Here is a quite general model that shows this.(For example, see Ferguson and Gilstein (1985).) It allows information to be gathered ateach stage that may alter the bettor’s view of his future winning chances.Play takes place in stages. The bettor starts with initial fortune X0. At the beginningof the kth stage, the bettor chooses an amount bkto bet and his fortune goes up or downby that amount depending on whether he wins or loses. We assume0 ≤ bk≤ Xk−1and Xk= Xk−1+ bkYkfor k =1,...,n (11)where Ykrepresents the random variable which is plus one if he wins the kth bet andminus one if he loses it. Choice of bkmay depend on the information gathered throughthe previous k − 1 stages. Let Z0denote the information known to the decision makerbefore he makes the

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