Senior Thesis. Horse Play. Optimal Wagers and the Kelly Criterion. Author: Courtney Kempton. Supervisor: Professor Jim Morrow


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1 Senior Thei Hore Play Optimal Wager and the Kelly Criterion Author: Courtney Kempton Supervior: Profeor Jim Morrow June 7, 20
2 Introduction The fundamental problem in gambling i to find betting opportunitie that generate poitive return. After identifying thee attractive opportunitie, the gambler mut decide how much of her capital he intend to bet on each opportunity. Thi problem i of great interet in probability theory, dating back to at leat the 8th c. when The Saint Peterburg Paradox paper by Daniel Bernoulli aroued dicuion over the topic []. For a rational peron, it would eem that they would chooe the gambling opportunitie that maximized expected return. However the reulting paradox, illutrated by Daniel Bernoulli, wa that a gambler hould bet no matter the cot. Intead of maximizing return, economit and probablit (like Bernoulli) have argued that it i ome utility function that the gambler hould maximize. Economit ue utility function a a way to meaure relative atifaction. Utility function are often modeled uing variable uch a conumption of variou good and ervice, poeion of wealth, and pending of leiure time. All utility function rely on a diminihing return. The difference between utility function and return arie becaue people attach a weight of importance to certain monetary value. For example, a bet which involve the lo of one life aving of $00,000 a contrated with the poibility of doubling the life aving would never be conidered even with a high probability. A rational peron attache more utility to the initial $00,000 then the poibility of the additional value. A uggeted by Daniel Bernoulli, a gambler hould not maximize expected return but rather the utility function log x where x i the return on the invetment. John Larry Kelly Jr., a cientit at Bell Lab, extended Bernoulli idea and uncovered ome remarkable propertie of the utility function log x [2]. Although hi interet lie in tranmiion of information over a communication channel, hi reult can be extended to a gambler determining which bet he hould make and how much. In particular, he determined the fixed fraction of one capital that hould be bet in order to maximize the function and provide the gambler with the highet atifaction given certain condition. Thi i known a the Kelly Criterion. Currently, the Kelly Criterion ue extend into invetment theory. It i of importance to invetor to determine the amount they hould invet in the tock market. The Kelly Criterion provide a olution although it ha it drawback namely it longterm orientation. 2 Simple Example 2. Maximizing Expected Return Suppoe you are confronted with an infinitely wealthy opponent who will wager even bet on repeated independent trial of a biaed coin. Let the probability of u winning on each trial be p > 2 and converly loing q = p. Aume we tart with an initial capital of X 0 and ubequently X i will be our capital after i coin toe. How much hould we bet (B i ) on each to? Let T i = be a win on the ith to and T i = a lo. A a reult, n X n = X 0 + B i T i, our initial capital plu or minu the amount we win. The expected value then i n n E[X n ] = X 0 + E[B i T i ] = X 0 + pe[b i ] qe[b i ] i= i= = X 0 + i= n (p q)e[b i ]
3 Hence, we ee that in order to maximize our expected return we hould maximize the amount we bet on each trial E[B i ]. The trategy ugget that on each to of the coin we hould wager our entire capital (i.e. B = X 0 and o forth). However, we ee that with one lo our capital i reduced to zero and ruin occur. Ultimately, the trategy would lead to the claic gambler ruin. From the example above, we expect that the optimal wager hould be a fraction of our total capital. Auming we bet a fixed fraction, B i = fx n where 0 f, then our capital after n trial i given by X n = X 0 ( + f) S ( f) L, where S + L = n and S i the number of win and L i the number of loe. In thi cenario, gambler ruin i impoible for P r(x n = 0) = 0. Finding the optimal fraction to bet i the bai of the Kelly Criterion. 2.2 Introduction to Kelly Criterion In order to determine the fixed fraction to bet, Kelly employ the ue of a quantity called G. The derivation of G follow from above: X n X 0 = ( + f) S ( f) L log X n = S log( + f) + L log( f) X ( ) 0 n log Xn = S X 0 n log( + f) + L log( f) n [ ( )] G = E n log Xn S = lim n n log( + f) + L log( f) n In the long run, S n = p and imilarly L n = p = q. Therefore, X 0 G = p log( + f) + q log( f) where f < G meaure the expected exponential rate of increae per trial. Originally, Kelly paper ued log bae 2 to reflect if G = you expect to double your capital, which in turn fit the tandard of uing log 2 in information theory. However given the relationhip between log of different bae, hi reult hold if we ue a log bae e and are eaier to compute. For the ret of the paper, log will refer to log bae e. Kelly chooe to maximize G with repect to f, the fixed fraction one hould bet at each trial. Uing calculu, we can derive the value of f that optimize G: G (f) = p + f q f = p q f(p + q) ( + f)( f). Setting thi to zero, yield that the optimal value f = p q. Now, G (f) = p ( + f) 2 q ( f) 2. Since p, q are probabilitie, p, q 0. Therefore, G (f) < 0, o that G(f) i a trictly concave function for f [0, ). Given that G(f) i concave over [0, ) and f atifie G (f) = 0, then G(f) ha a global maximum at f and by Propoition G max = p log( + p q) + q log( p + q) = log(2) + p log(p) + q log(q) 2
4 2.2. Example of the Kelly Criterion Elliot play againt an infinitely rich enemy. Elliot win even money on ucceive independent flip of a biaed coin with probability of winning p =.75. Applying f = p q, Elliot hould bet =.5 or 50% of hi current capital on each bet to grow at the fatet rate poible without going broke. Let that Elliot doen t want to bet 50% of hi capital (f.5). We can ee exactly how Elliot growth rate differ with variou f. Recall that the growth rate i given by G = p log( + f) + q log( f) G =.75 log( + f) +.25 log( f). Applying thi formula numerically with varying f yield: Fraction bet (f) Growth Rate (G) % % % % % We ee that Elliot deviation from f =.5 reult in a lower rate of growth. The further the deviation from the optimal, the greater the difference in growth rate. Kelly aume that the gambler will alway bet to maximize G ince if expected return are optmized then the gambler will eventually reach ruin. In thi imple example, we only bet on one outcome and the odd of winning are even (i.e. 2:). Kelly examine different cenario that exit for gambler with alway the ame reult of maximizing G. 3 Odd and Track Take Before continuing a brief dicuion on the term odd and track take i required in order to proceed. Aume there i a betting pool and that you can bet on 4 different team. You are allowed to bet a much a you want on each team. Let uppoe A dollar are bet on the firt team, B on the econd, C on the third, and D on the fourth. If team one win the people who bet on that team hare the pot A + B + C + D in proportion to what they each bet. Let A + B + C + D = α A where for every one dollar bet you get α dollar back if A won. Label β = A+B+C+D B, γ, δ imilarly. Therefore, let the odd be quoted a αto, βto, etc. Notice that in thi cenario all the money would be returned to the gambler. that receive a cut of the money. Moreover, α + β + γ + δ = A + B + C + D A + B + C + D =. There i no third party Aume now that there exit a bookie who keep E dollar and return A + B + C + D E to the winner. The new odd would be α = A + B + C + D E A and imilarly for β, γ, δ. Alo, notice that We ee that the cae where α + β + γ + δ = A + B + C + D A + B + C + D E >. α + β + γ + δ < 3
5 i meaningle. Thi cenario would ugget that more money i returned that what wa betted on. In gambling, thi i an improbable if not impoible cenario. A a reult, Kelly doe not conider thi option in hi analyi. When Kelly talk about odd, he i referring to the type of odd dicued here. Thi i in contrat to quoted odd uch a 5to which refer to an event with probability of 5+. In Kelly cae, he would quote the odd a 6to for an event with a probability of 6. The difference in the two reult from the definition of earning. In the prior example 5to, the 5 refer to the additional earning, o you receive $6 but you bet $ and thu your earning i $5. On the other hand, Kelly look at the total earning or the pure amount you receive and doe not ubtract off the dollar that you lot in making the bet. Note an important obervation regarding odd: Odd do not directly tranlate into probabilitie. An odd i imply a calculation baed on the amount of money in the pool and can/will vary from the probability. Hence, he aume the oddmaker do not know the underlying true probabilitie. 4 Generalizing Kelly Criterion Kelly explore the cae when an information channel ha multiple channel tranmitting different ignal. In tead of uing Kelly notation and definition, we retated hi definition with gambling terminology. p the probability that an event (i.e. a hore) will win α a() b r α σ the odd paid on the occurrance that the hore win. α repreent the number of dollar returned (including the one dollar bet) for a one dollar bet (e.g. α = 2 ay that for you win $2 for every $ bet). Thi i often tated at mto where m i an value greater than or equal to one. the fraction of the gambler capital that he decide to bet on the hore. the fraction of the gambler capital that i not bet the interet rate earned on the fraction not bet. α +r. Slight modification on the odd that incorporate the rate earned when you don t bet. α = σ The um of the reciprocal of the odd over all outcome (all ). A pecial remark: if σ > or α > +r then their exit a track take; if σ = or α = +r, no track take; and if σ < or α < +r then more money returned than betted on. We can conider Kelly information channel a imilar to betting on the outcome of a ingle game. An extenion of the reult given by Kelly i to ue tatitically independent identically ditributed game being played at the ame time (i.e. the odd are the ame for each game). 5 Kelly Criterion The Kelly Criterion focue on anwering:. a() the fraction the gambler bet on each 2. b the fraction the gambler doe not bet 3. Γ where Γ i the et of indice,, for which we place a bet (i.e. a() > 0). Let Γ be the et of indice,, in which we don t place a bet. 4
6 5. Setting up the Maximizing Function Kelly focu wa on information theory and the tranfer of information acro channel. For him, the nuiance urrounding money did not play a ubtantial part of hi analyi. For intance, he ignored time value of money becaue information channel do not earn interet when not in ue. When conidering hi analyi under the concept of money, one logical quetion to ak i: if I don t bet part of my capital, I could earn interet, o how doe that effect the fraction I bet and which hore do I bet on? One could ay that we are comparing the rate of return on a rik free aet (i.e. Treaury bond) with the expected rate earned by betting. We expand on Kelly analyi to include earning a rate on the money not bet. Let V N be the total capital after N iteration of the game and V 0 be the gambler initial capital. Moreover the fraction we don t bet, b, earn an interet rate of r. Let aume on one iteration we bet on and 2, then V = (a( )α +(+r)b)v 0 or (a( 2 )α 2 +(+r)b)v 0 depending upon whether or 2 hore won repectively. Occaionally, a()α will be repeated. Therefore let W be the number of time in the equence that win. Under Kelly aumption, omething alway win. It never occur that no hore win the race. Moreover, the bettor ue up her entire capital either by betting on the hore or with holding ome capital not bet. Thu, the gambler loe money on every bet that he made except for the hore that actually won. Hence, n V N = (( + r)b + a()α ) W V 0. From thi, Kelly deduce G by ( ) VN log = log (( + r)b + α a()) W V 0 ( ) G = lim N N log VN = W V N log(( + r)b + α a()) 0 n G = p() log(( + r)b + α a()) where in the long run lim N W N We will now lightly modify G by changing the odd to α = ( + r)α. Replacing G with n G = p() log(( + r)b + ( + r)α a()) G = G = n p() log(b + α a()) + n Therefore, the function i given by n p() log( + r) p() log(b + α a()) + log( + r) for p() =. max n p() log(b + α a()) + log( + r) = p(). over the variable b and a(). The known parameter are α and p() for each. Note that log( + r) i a contant and doe not effect the critical value. In Kelly paper, he imply et r = 0 and proceeded with maximizing the function. We are jut generalizing hi concluion to include the interet earned when you don t bet. Contraint Two general contraint emerge: 5
7 . The total capital i the um of the fraction bet plu the fraction not bet. b + Γ a() = 2. Due to the log, G i defined only when Thi implie that if b = 0 then for all a() > 0. b + α a() > 0 for all. Moreover, 0 b and 0 a() a the fraction bet and aved mut be nonnegative. One important obervation to note i that p() =. The Optimization Problem Generally for conitency in optimization problem, optimization problem are written in the form min f(x).t. g i 0 h j = 0 for i =,..., l for j =,..., k However any optimization problem can be converted to the form decribed above (i.e. max f(x) = min f(x). Converting Kelly contraint, the optimization problem primarily decribed in Kelly paper i min n p() log(b + α a()) + log( + r) n.t. b + a() = () = (b + ( + r)α a()) < 0 for =,..., n (2) a() 0 for =,..., n (3) a() 0 for =,..., n (4) b 0 (5) b 0 (6) for the variable b and a(). 5.2 KKT Criterion The KaruhKuhnTucker Theorem locate local minimum for nonlinear optimization function ubject to contraint. Kelly did not utilize thi theorem in contructing hi paper. He examine three different optimization problem, ditinguihing them baed on the value of σ and p()α. However, the ame reult can be concluded uing the KKT approach with a ingle optimization problem. 6
8 Applying the KKT condition, the following et of equation hold n p() b + α a() = µ + µ 2 + λ where µ (b) = 0, µ 2 (b ) = 0, µ 0, µ 2 0 (7) for all p()α b + α a() = µ + µ 2 + λ where µ (a()) = 0, µ 2 (a() ) = 0, µ 0, µ 2 0 (8) If a b and et of a() exit uch that the KKT condition hold, then thi i a neceary condition for a local minimum. However, G i a concave function. Let y = b + α a() and g(x) = log(x). Becaue log(x) i an increaing function and both y and g(x) are concave function, the compoition g(y) i alo concave. In addition, the um of concave function i alo concave, and adding a contant, log( + r) preerve it concavene. Hence, n p() log(b + α a()) + log( + r) i concave. The ignificance of thi tatement i that if there i a x that atifie KKT then x i global maximum of G. The goal i to find critical point and optimal value under certain condition facing a gambler. In particular, when the gambler confront different max(p()α ) and σ. In order to determine whether point are critical, we need to find appropriate b, λ, µ k, µ j, and a() that atify (7) and (8) a well a the feaibility condition ()(6). Although in Kelly paper optimization i a central feature, he doe not employ the ue of KKT. In fact, we are lightly deviating from Kelly by manipulating hi three ditinct cae and converting it to one optimization problem with three olution depending on the ituation facing the gambler a well a generalizing the reult to include the interet rate earned. Let examine the cae: σ and σ. 6 α A( gambler encountering thi ituation ha two option: either all the money i returned to the winner ) α = or the bookie ha injected money into the ytem o more money i returned than betted ( ) on α <. In both cae, there exit critical point that atify KKT. However, the optimal value depend on the relationhip between p() and α. 6. If b = 0 Claim. If b = 0, then a() = p() for all with an optimal value of G(0, p()) = p() log(α p()) + log( + r) if and only if α or equivalently α +r. Proof. Since b = 0 from (2), a() > 0 for all. Moreover from () and b = 0, a() =. Thee tatement provide a bound on a(), 0 < a() < =,..., n. Given thi bound, µ j = 0 for all j and becaue b = 0, the KKT condition become p()α a()α = λ =,..., n (9) p() α a() = λ µ (0) 7
9 Hence (9), and umming up over all p() = λa() =,..., n p() = λ a() = λ. Thu, a() = p() Becaue λ = and a() = p(), (0), α = µ, mut hold at the critical value. So critical point exit if and only if µ = α > 0. Hence α or α +r. Replacing the critical point into the objective function yield: G max = G max = p() log(α p()) + log( + r) p() log ( ) p()α + log( + r) + r A gambler facing a α or α +r (no track take) will optimize G by betting on everything in proportion to the probability, a() = p(), with b = 0. The gambler hould ignore the poted odd and bet with the probability and earn G max = p() log(α p()) + log( + r), or equivalently, G max = p() log ( ) α p() + log( + r). + r Thi agree with Kelly olution regarding no track take cae. We don t need to proceed further with the breakdown of max{p()α } becaue the reult only depend on σ. Let take a cloer look at the optimal value G given that a() = p() and = c. We want to examine how the bookie could limit the growth rate. Since the only control the bookie have i the odd, we want to minimize G with repect to a() ubject to α = c. The critical point would be the odd the bookie would like to quote to minimize the return to the gambler. α 8
10 Lemma. Suppoe α = cp() and Furthermore when, α min G = = c where c i a poitive number. Then, the minimum of G occur when ( p() log p() cp() ) + log( + r) = log(c) + log( + r). c < = G min > log( + r) c = = G min = log( + r) c > = G min < log( + r) Proof. The optimization problem i min p() log(p()α ) + log( + r) with repect to α..t. α = However intead of optimizing thi ytem, let β = α. Hence, the new problem i min ) p() log (p() β + log( + r) = min p() log(p()) p() log (β ) + log( + r) with repect to β..t. β = c Given thi new ytem, we ee that log(β ) i a convex function. Since the um of convex function i convex and contant do not change the function, G i convex. Due to convexity, any point that atifie KKT i a global minimum. Applying Lagrange multiple give, Summing over, we get Hence, a global minimum value i When β = cp(), the global minimum i p() log(p() p() = λβ for all. = cλ. β = cp() or α = cp(). p() log(cp()) + log( + r) = log(c) p() + log( + r) = log(c) + log( + r) The lat tatement follow from ubtituting in different value for c. 9
11 Corollary. Let c =. If α p(), G = p() log(p()α ) + log( + r) > log( + r). Proof. In Lemma, we demontrated that α = any light modification, where α p() but earned by placing money in bank. p() α produce a global minimum value of log( + r). Hence =, will reult in a growth rate greater than the rate In Kelly paper, he make a pecial note of thi tatement. In hi cae, he looked at when r = 0 and tated that when the odd are unfair (i.e. α differ from p() ) then the gambler ha an advantage and can expect a poitive growth rate. When Kelly tate unfair odd, he implie that the gambler ha inider information. In eence, the gambler mut know that the real probability of the hore winning i different then what the odd ay. Hence, the gambler mut have inider information; hence if it were public the odd would reflect that information. Therefore, a gambler with inider information can and will gain a poitive growth rate. 6.2 If 0 < b α Above we howed that if then an optimal value exit at b = 0 and a() = p(). Recall that there may be multiple critical point but only one optimal value. Thi i the ituation with thi cae. A gambler only want to optimize G. Hence from above, any point that atifie KKT i a global optimal, o if that point fulfill KKT condition then it i one way of achieving the optimal value G. Hence, we found that the global maximum when α would be achieved by b = 0 and a() = p(). There may be additional way to have the ame global optimal value, with 0 < b, but it would produce the ame optimal value NonUnique Critical Point Already hown i that a critical point exit at b = 0 with. I it poible that there are more? Moreover, doe a critical point exit when b i nonzero. The anwer i ye. Note that the ame optimal value reult when a() = p() b α (i.e. G max = p() log(p()α ) + log( + r)). However, the hitch i that a() 0; thu, a() = p()α b α α 0 for all. Becaue α > 0, then for a() > 0, 0 < b min(p()α ). Alo, we know that a() = b. Summing over all, a() = b = b p() b α α Thi tatement will only be true if α =. 0
12 If α = and the gambler choe any b with 0 < b min(p()α ) = min(p()α) +r, then the bettor ave b dollar, bet a() = p() b α = p() b(+r) α dollar, and get the ame expected growth rate G max = = p() log(p()α ) + log( + r) ( p() log p() α ) + log( + r) + r Summary α A gambler with = c where c (no track take) will optimize G by betting on everything in proportion to the probability, a() = p() and b = 0. If c < = G = c = = G = p() log(α p()) + log( + r) > log( + r) p() log(α p()) + log( + r) = log( + r) Moreover if α p() for at leat ome and α = then G > log( + r). α For =, multiple critical point exit. The bettor can ave b uch that 0 < b min(p()α ) and bet a() = p() b α and receive the ame optimal value of G max = p() log(p()α ) + log( + r). 7 α > α Now we will look at another cae. When >, thi i indicative of a track take. Hence, there exit a bookie that take part of the pool off the top. In thi cae, how hould the gambler bet? To Kelly, thi wa the mot intereting cae. In effect, he derive an algorithm to determine which to bet and how much. Diviion into additonal cae i neceary. 7. b = Auming b =, () lead to a() = 0 for all. Thi alo atifie the feaibility condition (2). Hence, the optimal value i G(, 0) = log( + r) if a() = 0 for all and b = i a critical point.
13 A a reult, the KKT condition, (7) and (8), are p() b Letting b = and umming over all p(), () become From (2), = µ 2 + λ where µ = 0 () and p()α = λ µ for all (2) = µ 2 + λ. (3) λ = p()α + µ for all. (4) Given that p()α > 0 for all and feaibility of KKT, µ 0, then (4) implie λ > 0. Moreover ince at feaibility µ 2 0 and λ > 0, from (3) 7.. max{p()α } 0 < λ. Claim 2. There i a critical point at a() = 0 for all and the correponding optimal value i if and only if max{p()α }. G(, 0) = log( + r) Proof. Now we jut need to how there exit λ, µ, µ 2 uch that µ value, then the above i a critical point. 0 and µ 2 0. If there exit thee. Suppoe b =, a() = 0 i a critical point, then there i an λ, 0 < λ and µ 2 = λ 0. Moreover, µ 0 uch that λ = p()α +µ for all. Hence if 0 < λ and µ 0, implie by (4) p()α for all. 2. Suppoe max{p()α } implying that p()α for all. Now let λ =. By (3), µ 2 = 0, which atifie the KKT condition. Moreover letting λ = and p()α for all, (4) i p()α = µ 0 for all. Thu, all KKT condition are fullfilled and a() = 0 and b = i a critical point with optimal value G(, 0) = log( + r). From the gambler perpective, if the max{p()α }, then the gambler bet growth rate i log( + r). Hence one way of achieving the optimal growth rate with max{p()α } i by betting 0 on all max{p()α } > A we howed in Claim 2, there can not exit an optimal value when max{p()α } > if b =. 2
14 7.2 If 0 < b < Recall that Γ i the et of all indice in which a() > 0 and Γ i the et of all indice in which a() = 0. If 0 < b < and () hold, then there exit at leat one where a() > 0 (Γ i a nonempty et). In addition, a() < by (). With thee aumption, the KKT equation (7) and (8) are and breaking up (7) p()α b + α a() = λ for a() >, Γ (5) p()α = λ µ for Γ, µ 0 (6) b p() b + α a() = λ (7) Γ p() b We will how in Propoition 2 that λ = and + Γ p() b + α a() = λ. (8) a() = p() b α for a() > 0. Now that we know a(), it i till dependent upon knowing the value of b. Computing b require an introduction of new notation. Denote σ Γ = Γ α p Γ = Γ p() Note that σ Γ and σ differ. We can think of σ Γ a an intermediate value of σ. Put another way, σ Γ i the um of the reciprocal odd for all hore we bet on. Contrating, σ = α i the um of the reciprocal odd of all the hore. Hence, σ Γ σ. With thi notation, a value of b can be computed. Uing (8) with p() =, ( ) p() + p() b b + α Γ Γ a() = λ Propoition (2) and (23) conclude that b ( Γ p() ) + Γ α =. From the definition, p Γ b = σ Γ, (9) 3
15 and hence ince 0 p Γ and b > 0 then σ Γ. At thi tage, two poibilitie emerge from (9): σ Γ < or σ Γ =. Kelly conider only one of thee cae when σ Γ <. He regard that a poitive b only reult when a track take occur ( α > ). By Propoition 3, σ Γ < only reult if σ > or σ <. If σ, we have handled the cae in Section 6. However, we are only conidering in thi ection when σ >. If σ >, we can ay that at optimal σ Γ < (Propoition 3), o there exit at leat one in which we don t bet on. Thi implie p Γ <. The reult directly come from Becaue b > 0 and > p Γ > 0, then and hence σ Γ < at optimal. Moreover, we can olve for b and with p Γ b = σ Γ. σ Γ > 0 b = p Γ σ Γ, a() = p() b α for Γ we know the critical point. Becaue a() > 0 and from (2), the following mut be true at optimum But there i one quetion: p()α > b for Γ (20) p()α b for Γ (2) σ Γ < (22) What i Γ, Γ? To olve thi problem, let introduce ome new notation. Notation and aumption:. Order the product p()α o that p()α p( + )α + When the product are equal the ordering i arbitrary. Let t be the index deignating the ordering where t i an integer greater than or equal to 0. t = i the correponding to the max{p()α }. 2. p() > 0, n t p() =, p t = p() = = 3. Hence if σ t < then t < n. α > 0, n α = t, σ t = α = 4
16 4. Let F t = p t σ t, F 0 = Conider how F t = p t σ t varie with t. Let t be the critical index. It i the break point that determine which hore we bet or don t bet on (Γ and Γ et). When we are at optimal, F t = b. Kelly provide an algorithm for finding the t that optimize G. He compute t by t = min{min{t : p(t + )α t+ < F t }, max{t : σ t < }}, a given by Theorem 2. The proce tart by computing the firt condition with t = 0 and increaing t by one until p(t + )α t+ < F t. Then repeat but ue the econd condition, σ t <. If t = 0 then we do not bet on anything max{p()α } Now if max{p()α } = p()α (i.e. p()α F 0 ), F t increae with t until σ t by Theorem. Becaue p()α F 0, then by Theorem F 0 F o p()α F. Therefore, it fail (20). In thi cae by Theorem 2, t = 0, Γ =, and b =. The gambler i better off not betting on anything and hence G max = log( + r). Thi agree with the previou cae when b = and max{p()α }. Thi demontrate that there doe not exit any critical value point if 0 < b <. By not betting on anything, we reach a contradiction for the only way 0 < b < i if there exit at leat one a() > 0. In thi cae, the bet option i to reort to the previou cae where b =. If max{p()α } and Section 7.. and bet b = with α > then there are no critical value point. The gambler would ue G max = log( + r) max{p()α } > If max{p()α } > or equivalently p()α >, F t decreae with t until p(t + )α t < F t or σ t by Theorem and 2. In eence if p()α >, then by Theorem, F 0 > F, and hence p()α > F. Thi atifie (20). A long a p(t)α t > F t, then, F t > F t, and it will fulfill (20). At optimum, (20) mut hold true, o we need to find all that meet (20) and (2). By uing thi procedure, we will find t, which indicate Γ = {t : 0 < t t }. Now that we know t, the optimization problem i olved and critcal value exit. If σ > and max{p()α } >, G = Γ p() log(p()α ) + Γ p() log(b) + log( + r) where a() = p() b α, b = p t σ where Γ = {t : 0 < t t t } and t = min{min{t : p(t + )α t+ < F t }, max{t : σ t < }}. 5
17 Kelly noted that if σ > and p()α < then no bet are placed, but if max{p()α } > ome bet might be made for which p()α < (i.e. the odd are wore than the probabilitie and hence the return i le than expected). Thi contrat with the claic gambler intuition which i to never bet on uch an event. Summary. If σ > ( α > +r ) and max{p()α } (max{p()α } ( + r)) then and b =, a() = 0 for all. G max = log( + r). 2. If σ > ( α > +r ) and max{p()α } > (max{p()α } > + r) then G max = Γ p() log(p()α ) + Γ p() log(b) + log( + r) with a() = p() b α = p() b(+r) α and b = p t σ t and Γ = {t : 0 < t t } 8 σ and max{p()α } Combination that Don t Exit It i important to remark that there exit a few combination of σ and max{p()α } which don t exit. 8. σ < and max{p()α } Let conider the ituation where a gambler not only ha no track take but money i injected into the pool (i.e. < ). Thi cenario will never occur if max{p()α }. In fact, α Proof. Aume max{p()α } then for all α < implie max{p()α } >. p()α = p() α. Summing over all,, α but thi i a contradiction for α < and hence max{p()α } >. 8.2 max{p()α } < and σ =. A imilar concluion i reached for a game where max{p()α } < and =. In thi cenario, the winning gambler receive all of the money in the pool. Intuitively, the tatement i untrue becaue if one p()α <, then another p()α > to counter it in the ummation of the reciprocal of α. In general, α = implie max{p()α }. α 6
18 Proof. Aume max{p()α } <. Then, for all p()α < = p() < α. Summing over all, <, α but we reach a contradiction a α =. Hence, max{p()α }. 9 Concluion When gambler enter a bet, they have acce to only three piece of information:. The p()α for all 2. α 3. The rate earned on money not bet, r Kelly uncovered under the permutation of the above information the fixed fraction to bet, a(), and b, the fraction not bet to optimize the gambler growth rate. If the odd are conitent with the probabilitie, the gambler can expect a growth rate equal to log(+r). Growth rate larger than the rik free rate only occur if the gambler ha inider information (i.e. know that the true probabilitie differ from the odd). Moreover when gambler experience track take (σ > ) and at leat one of the probabilitie i inconitent with the odd, the gambler will gain a growth rate greater than the rik free rate, r. Interetingly a pointed out by Kelly, gambler will make bet when the odd (i.e. gambler return) are wore then the probability (p() < α ). Thi i counterintuitive, why would a gambler bet on omething that i expecting a negative return? Kelly wa baffled by thi reult. Kelly original paper doe not include the rate earned on money not bet. Hi reearch wa on information theory and the tranfer of ignal acro channel. Hence, earning an additional rate on the money not bet wa not conidered in hi analyi. We reach the ame concluion a he doe by etting r = 0. Moreover, Kelly ignored the poibility of multiple critical value point. A we howed in Section 6.2., there are nonunique critical point when α = +r. The chart below ummarize our finding: 7
19 Gambler Situation No Track Take α +r Track Take > α +r max{p()α } < + r max{p()α } = + r max{p()α } > + r max{p()α } + r max{p()α } > + r Gambler Should Gambler Should Gambler Should Gambler Should 8 Doe not happen Bet b = 0 and a() = p() Growth( Rate: ) p() log p()α + +r log( + r) If α = + r, there are nonunique critical point (View Section 6.2.) Bet b = 0 and a() = p() Growth( Rate: ) p() log p()α + +r log( + r) If α = + r, there are nonunique critical point (View Section 6.2.) Bet b = and a() = 0 for all Growth Rate: log( + r) Bet b = p Γ σ Γ and a() = p() b(+r) α for Γ Growth Rate: Γ p() log ( p() α +r Γ p() log(b) + log( + r) ) +
20 A tated above, Kelly olved the optimization problem under the aumption that r = 0. To conclude, the following exhibit Kelly original finding. Gambler Situation: Kelly Original Finding No Track Take α Track Take α > max{p()α } < max{p()α } = max{p()α } > max{p()α } max{p()α } > 9 Gambler Should Gambler Should Gambler Should Gambler Should Doe not happen Bet b = 0 and a() = p() Growth Rate: p() log(p()α) Bet b = 0 and a() = p() Growth Rate: p() log(p()α) Bet b = and a() = 0 for all Growth Rate: 0 Bet b = p Γ σ Γ and a() = p() b α for Γ Growth Rate: Γ p() log (p()α) + Γ p() log(b)
21 0 Propoition and Theorem Propoition. G max = log(2) + p log(p) + q log(q) Proof. From the tatement, the optimal value for G(f) occur when f = p q. Subtituting into G yield: G max = p log( + p q) + q log( p + q) Since p + q =, which can be reduced further to Hence, G max = p log(2p) + q log(2q), (p + q) log(2) + p log(p) + q log(q). G max = log(2) + p log(p) + q log(q). Propoition 2. If 0 < b <, then Proof. From (5), and ubtituting into (8) Recall that, p() = and hence λ = and a() = p() b α. p() = λ(b + α a()) α Γ Γ p() b + Γ λ α for Γ (23) = λ. p() = Γ p(). Replacing into the above equation and uing (23) yield λ b = Γ λ α Γ Note that from (), rearranging produce b = Γ o λ =. λa() b a() b λ b = λ ( b + λ Γ α which ubtituted in reult ), Due to λ = and (5), a() = p() b α for a() > 0. Propoition 3. σ Γ = if and only if σ = σ Γ < if and only if σ > or σ < 20
22 Proof. By (9), If σ Γ = and b > 0 then p b = σ Γ. p = 0 Hence, p =. A a reult becaue p() =, p = implie that at critical point we bet on all. Thu, σ Γ = σ. Thi will only be true if and only if σ = to tart with. For any other σ value, σ Γ. Theorem. Let T = {t : σ t < } and aume t + T and σ <. Then F > 0 and for t p(t + )α t+ = F t F t+ = F t (24) p(t + )α t+ > F t F t+ < F t (25) p(t + )α t+ < F t F t+ > F t (26) Proof. The aumption that σ < directly reult from the baic of odd. By definition, σ = α and σ > only if α <. In term of odd, α < indicate that the gambler receive le money than they bet. Thi cenario i improbable; odd will never be le than. Hence, the fact that F t > 0 under the tated aumption i clear and probable. I will prove (25) and the two other proof for (26) and (24) are imilar. Since 0 < σ t < σ t+ <, then σ t > 0 and σ t+ < 0. Alo p t > 0 ince t < n. Thu, F t+ < F t i equivalent to the following equalitie p t+ = p t p(t + ) σ t+ σ t < p t σ α t t+ σ t p t + p t σ t p(t + ) + p(t + )σ t < σ t α t+ + p t σ t + p t α t+ p t Cancelling out, σ t, p t σ t, p t, thi become p(t + ) + p(t + )σ t < α t+ + p t α t+ Hence becaue ( σ t ) >, p(t + )( σ t ) < α t+ ( p t )) p(t + )α t+ > p t σ t = F t Theorem 2. There exit a unique index t uch that p()a() > b for t (27) p()a() b for > t (28) where b = F t. Thi t repreent the critical index that determine Γ and Γ, the et you bet on and don t bet on repectively. Hence, Γ = {t : t t } and Γ = {t : t > t } where the et maximize G. 2
23 Ue the following procedure to find t : t = min{ min(t : p(t + )α (t+) < F t), max(t : σ t < ) } In word, tart with t = 0 and tet the firt condition. Increae t until p(t + )α (t+) < F t. Then repeat for σ t until σ t. Compare the two t value and chooe the minimum to be t. Aume σ t < and. If p()α then t = If p()α > then t where t i choen by the method above. Proof. From Propoition (2), Subtituting into G, a() = p() b α for Γ. ( ( p() log b + α p() b )) α + p() log(b) Γ Γ p() log (α p()) + p() log(b). Γ Γ Hence, we want to find all uch that p()α > b for thee will maximize G. By reordering the value a decribed above,. If p()α < = F 0, then by Theorem () Equation (26), F > F 0 o p()α < F. Moreover p(2)α 2 < p()α ; thu by the ame logic F 0 < F < F 2 and again p(2)α 2 < F 2. By induction, we can continue for all t and conclude p(t)α t < F t. Becaue at maximum F t = b and auming σ t <, none of thee t value are candidate for no t value atifie the condition that p()α > b Hence, t = 0 which implie b = by (2). However, thi i a contradiction for we aumed that 0 < b < and therefore thi doe not occur. 2. If p()α >, then F < F 0. Thi atifie the condition at optimal (i.e. p()α > b). Therefore, we bet on at leat on, =. From Theorem if p(t + )α t+ > F t then F t+ < F t. Let aume t = 2 atifie thi condition then F 2 < F < F 0. Becaue p(2)α 2 > F then p(2)α 2 > F 2 and p()α > F 2. A a reult, thi atified the condition at optimality. By induction, we can ee that a long a p(t + )α t+ > F t then for all t le than or equal to t + will atify the optimality condition p()α > F t+. Thu, we hould bet on thee outcome. If p(t + )α t+ F t then by Theorem F t F t+. Once thi occur, F t+ p(t + )α t+. However at optimality, we need F t+ < p(t + )α t+ and therefore we do not include t + a part of Γ. With thee two tatement, it allow u to find t. We note from Propoition 3 that at optimality σ Γ <. Hence t will be computed by min{min(t : p(t + )α (t+) < F t, max(t : σ t < ). A hown above, computing t with thi method guarantee that the condition at optimality hold. 22
24 Reference [] Edward Thorp, The Kelly Criterion in Blackjack Sport Betting, and the Stock Market, 0th International Conference on Gambling and Rik Taking (997), 45. [2] J.L. Kelly, A New Interpretation of Information Rate, The Bell Sytem Technical Journal (March 2, 956),
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