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

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

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

Transcription

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 long-term 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 α-to-, β-to-, 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 5-to- which refer to an event with probability of 5+. In Kelly cae, he would quote the odd a 6-to- for an event with a probability of 6. The difference in the two reult from the definition of earning. In the prior example 5-to-, 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 odd-maker 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 m-to- 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 non-negative. 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 Karuh-Kuhn-Tucker Theorem locate local minimum for non-linear 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 Non-Unique 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 non-zero. 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 non-empty 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 non-unique 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 non-unique 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 non-unique 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),

Unit 11 Using Linear Regression to Describe Relationships

Unit 11 Using Linear Regression to Describe Relationships Unit 11 Uing Linear Regreion to Decribe Relationhip Objective: To obtain and interpret the lope and intercept of the leat quare line for predicting a quantitative repone variable from a quantitative explanatory

More information

State-space analysis of control systems: Part I

State-space analysis of control systems: Part I Why a different approach? State-pace analyi of control ytem: Part I Uing a tate-variable approach give u a traightforward way to analyze MIM multiple-input, multiple output ytem. A tate variable model

More information

MSc Financial Economics: International Finance. Bubbles in the Foreign Exchange Market. Anne Sibert. Revised Spring 2013. Contents

MSc Financial Economics: International Finance. Bubbles in the Foreign Exchange Market. Anne Sibert. Revised Spring 2013. Contents MSc Financial Economic: International Finance Bubble in the Foreign Exchange Market Anne Sibert Revied Spring 203 Content Introduction................................................. 2 The Mone Market.............................................

More information

A note on profit maximization and monotonicity for inbound call centers

A note on profit maximization and monotonicity for inbound call centers A note on profit maximization and monotonicity for inbound call center Ger Koole & Aue Pot Department of Mathematic, Vrije Univeriteit Amterdam, The Netherland 23rd December 2005 Abtract We conider an

More information

Decision Theory and the Normal Distribution

Decision Theory and the Normal Distribution MODULE 3 Deciion Theory and the Normal Ditribution LEARNING OBJECTIVE After completing thi module, tudent will be able to: 1. Undertand how the normal curve can be ued in performing break-even analyi.

More information

Chapter 4: Mean-Variance Analysis

Chapter 4: Mean-Variance Analysis Chapter 4: Mean-Variance Analyi Modern portfolio theory identifie two apect of the invetment problem. Firt, an invetor will want to maximize the expected rate of return on the portfolio. Second, an invetor

More information

MA 408 Homework m. f(p ) = f ((x p, mx p + b)) = s, and. f(q) = f (x q, mx q + b) = x q 1 + m2. By our assumption that f(p ) = f(q), we have

MA 408 Homework m. f(p ) = f ((x p, mx p + b)) = s, and. f(q) = f (x q, mx q + b) = x q 1 + m2. By our assumption that f(p ) = f(q), we have MA 408 Homework 4 Remark 0.1. When dealing with coordinate function, I continually ue the expreion ditance preerving throughout. Thi mean that you can calculate the ditance in the geometry P Q or you can

More information

Optical Illusion. Sara Bolouki, Roger Grosse, Honglak Lee, Andrew Ng

Optical Illusion. Sara Bolouki, Roger Grosse, Honglak Lee, Andrew Ng Optical Illuion Sara Bolouki, Roger Groe, Honglak Lee, Andrew Ng. Introduction The goal of thi proect i to explain ome of the illuory phenomena uing pare coding and whitening model. Intead of the pare

More information

On the optimality of List Scheduling for online uniform machines scheduling

On the optimality of List Scheduling for online uniform machines scheduling On the optimality of Lit Scheduling for online uniform machine cheduling Fangqiu HAN Zhiyi TAN Yang YANG Abtract Thi paper conider claical online cheduling problem on uniform machine We how the tight competitive

More information

MBA 570x Homework 1 Due 9/24/2014 Solution

MBA 570x Homework 1 Due 9/24/2014 Solution MA 570x Homework 1 Due 9/24/2014 olution Individual work: 1. Quetion related to Chapter 11, T Why do you think i a fund of fund market for hedge fund, but not for mutual fund? Anwer: Invetor can inexpenively

More information

1 Safe Drivers versus Reckless Drunk Drivers

1 Safe Drivers versus Reckless Drunk Drivers ECON 301: General Equilibrium IV (Externalitie) 1 Intermediate Microeconomic II, ECON 301 General Equilibrium IV: Externalitie In our dicuion thu far, we have implicitly aumed that all good can be traded

More information

A Note on Profit Maximization and Monotonicity for Inbound Call Centers

A Note on Profit Maximization and Monotonicity for Inbound Call Centers OPERATIONS RESEARCH Vol. 59, No. 5, September October 2011, pp. 1304 1308 in 0030-364X ein 1526-5463 11 5905 1304 http://dx.doi.org/10.1287/opre.1110.0990 2011 INFORMS TECHNICAL NOTE INFORMS hold copyright

More information

Queueing systems with scheduled arrivals, i.e., appointment systems, are typical for frontal service systems,

Queueing systems with scheduled arrivals, i.e., appointment systems, are typical for frontal service systems, MANAGEMENT SCIENCE Vol. 54, No. 3, March 28, pp. 565 572 in 25-199 ein 1526-551 8 543 565 inform doi 1.1287/mnc.17.82 28 INFORMS Scheduling Arrival to Queue: A Single-Server Model with No-Show INFORMS

More information

Bidding for Representative Allocations for Display Advertising

Bidding for Representative Allocations for Display Advertising Bidding for Repreentative Allocation for Diplay Advertiing Arpita Ghoh, Preton McAfee, Kihore Papineni, and Sergei Vailvitkii Yahoo! Reearch. {arpita, mcafee, kpapi, ergei}@yahoo-inc.com Abtract. Diplay

More information

Lecture Notes for Laplace Transform

Lecture Notes for Laplace Transform Lecture Note for Laplace Tranform Wen Shen April 9 NB! Thee note are ued by myelf. They are provided to tudent a a upplement to the textbook. They can not ubtitute the textbook. Laplace Tranform i ued

More information

A technical guide to 2014 key stage 2 to key stage 4 value added measures

A technical guide to 2014 key stage 2 to key stage 4 value added measures A technical guide to 2014 key tage 2 to key tage 4 value added meaure CONTENTS Introduction: PAGE NO. What i value added? 2 Change to value added methodology in 2014 4 Interpretation: Interpreting chool

More information

Health Insurance and Social Welfare. Run Liang. China Center for Economic Research, Peking University, Beijing 100871, China,

Health Insurance and Social Welfare. Run Liang. China Center for Economic Research, Peking University, Beijing 100871, China, Health Inurance and Social Welfare Run Liang China Center for Economic Reearch, Peking Univerity, Beijing 100871, China, Email: rliang@ccer.edu.cn and Hao Wang China Center for Economic Reearch, Peking

More information

Partial optimal labeling search for a NP-hard subclass of (max,+) problems

Partial optimal labeling search for a NP-hard subclass of (max,+) problems Partial optimal labeling earch for a NP-hard ubcla of (max,+) problem Ivan Kovtun International Reearch and Training Center of Information Technologie and Sytem, Kiev, Uraine, ovtun@image.iev.ua Dreden

More information

Analyzing Myopic Approaches for Multi-Agent Communication

Analyzing Myopic Approaches for Multi-Agent Communication Analyzing Myopic Approache for Multi-Agent Communication Raphen Becker Victor Leer Univerity of Maachuett, Amhert Department of Computer Science {raphen, leer, hlomo}@c.uma.edu Shlomo Zilbertein Abtract

More information

Growth and Sustainability of Managed Security Services Networks: An Economic Perspective

Growth and Sustainability of Managed Security Services Networks: An Economic Perspective Growth and Sutainability of Managed Security Service etwork: An Economic Perpective Alok Gupta Dmitry Zhdanov Department of Information and Deciion Science Univerity of Minneota Minneapoli, M 55455 (agupta,

More information

A New Interpretation of Information Rate

A New Interpretation of Information Rate A New Interpretation of Information Rate reproduced with permission of AT&T By J. L. Kelly, jr. (Manuscript received March 2, 956) If the input symbols to a communication channel represent the outcomes

More information

MECH 2110 - Statics & Dynamics

MECH 2110 - Statics & Dynamics Chapter D Problem 3 Solution 1/7/8 1:8 PM MECH 11 - Static & Dynamic Chapter D Problem 3 Solution Page 7, Engineering Mechanic - Dynamic, 4th Edition, Meriam and Kraige Given: Particle moving along a traight

More information

HUMAN CAPITAL AND THE FUTURE OF TRANSITION ECONOMIES * Michael Spagat Royal Holloway, University of London, CEPR and Davidson Institute.

HUMAN CAPITAL AND THE FUTURE OF TRANSITION ECONOMIES * Michael Spagat Royal Holloway, University of London, CEPR and Davidson Institute. HUMAN CAPITAL AND THE FUTURE OF TRANSITION ECONOMIES * By Michael Spagat Royal Holloway, Univerity of London, CEPR and Davidon Intitute Abtract Tranition economie have an initial condition of high human

More information

Control of Wireless Networks with Flow Level Dynamics under Constant Time Scheduling

Control of Wireless Networks with Flow Level Dynamics under Constant Time Scheduling Control of Wirele Network with Flow Level Dynamic under Contant Time Scheduling Long Le and Ravi R. Mazumdar Department of Electrical and Computer Engineering Univerity of Waterloo,Waterloo, ON, Canada

More information

v = x t = x 2 x 1 t 2 t 1 The average speed of the particle is absolute value of the average velocity and is given Distance travelled t

v = x t = x 2 x 1 t 2 t 1 The average speed of the particle is absolute value of the average velocity and is given Distance travelled t Chapter 2 Motion in One Dimenion 2.1 The Important Stuff 2.1.1 Poition, Time and Diplacement We begin our tudy of motion by conidering object which are very mall in comparion to the ize of their movement

More information

Chapter 10 Stocks and Their Valuation ANSWERS TO END-OF-CHAPTER QUESTIONS

Chapter 10 Stocks and Their Valuation ANSWERS TO END-OF-CHAPTER QUESTIONS Chapter Stoc and Their Valuation ANSWERS TO EN-OF-CHAPTER QUESTIONS - a. A proxy i a document giving one peron the authority to act for another, typically the power to vote hare of common toc. If earning

More information

Review of Multiple Regression Richard Williams, University of Notre Dame, http://www3.nd.edu/~rwilliam/ Last revised January 13, 2015

Review of Multiple Regression Richard Williams, University of Notre Dame, http://www3.nd.edu/~rwilliam/ Last revised January 13, 2015 Review of Multiple Regreion Richard William, Univerity of Notre Dame, http://www3.nd.edu/~rwilliam/ Lat revied January 13, 015 Aumption about prior nowledge. Thi handout attempt to ummarize and yntheize

More information

Thus far. Inferences When Comparing Two Means. Testing differences between two means or proportions

Thus far. Inferences When Comparing Two Means. Testing differences between two means or proportions Inference When Comparing Two Mean Dr. Tom Ilvento FREC 48 Thu far We have made an inference from a ingle ample mean and proportion to a population, uing The ample mean (or proportion) The ample tandard

More information

Profitability of Loyalty Programs in the Presence of Uncertainty in Customers Valuations

Profitability of Loyalty Programs in the Presence of Uncertainty in Customers Valuations Proceeding of the 0 Indutrial Engineering Reearch Conference T. Doolen and E. Van Aken, ed. Profitability of Loyalty Program in the Preence of Uncertainty in Cutomer Valuation Amir Gandomi and Saeed Zolfaghari

More information

DIHEDRAL GROUPS KEITH CONRAD

DIHEDRAL GROUPS KEITH CONRAD DIHEDRAL GROUPS KEITH CONRAD 1. Introduction For n 3, the dihedral group D n i defined a the rigid motion 1 of the plane preerving a regular n-gon, with the operation being compoition. Thee polygon for

More information

CHARACTERISTICS OF WAITING LINE MODELS THE INDICATORS OF THE CUSTOMER FLOW MANAGEMENT SYSTEMS EFFICIENCY

CHARACTERISTICS OF WAITING LINE MODELS THE INDICATORS OF THE CUSTOMER FLOW MANAGEMENT SYSTEMS EFFICIENCY Annale Univeritati Apuleni Serie Oeconomica, 2(2), 200 CHARACTERISTICS OF WAITING LINE MODELS THE INDICATORS OF THE CUSTOMER FLOW MANAGEMENT SYSTEMS EFFICIENCY Sidonia Otilia Cernea Mihaela Jaradat 2 Mohammad

More information

Online story scheduling in web advertising

Online story scheduling in web advertising Online tory cheduling in web advertiing Anirban Dagupta Arpita Ghoh Hamid Nazerzadeh Prabhakar Raghavan Abtract We tudy an online job cheduling problem motivated by toryboarding in web advertiing, where

More information

Auction Theory. Jonathan Levin. October 2004

Auction Theory. Jonathan Levin. October 2004 Auction Theory Jonathan Levin October 2004 Our next topic i auction. Our objective will be to cover a few of the main idea and highlight. Auction theory can be approached from different angle from the

More information

AMS577. Repeated Measures ANOVA: The Univariate and the Multivariate Analysis Approaches

AMS577. Repeated Measures ANOVA: The Univariate and the Multivariate Analysis Approaches AMS577. Repeated Meaure ANOVA: The Univariate and the Multivariate Analyi Approache 1. One-way Repeated Meaure ANOVA One-way (one-factor) repeated-meaure ANOVA i an extenion of the matched-pair t-tet to

More information

Growth and Sustainability of Managed Security Services Networks: An Economic Perspective

Growth and Sustainability of Managed Security Services Networks: An Economic Perspective Growth and Sutainability of Managed Security Service etwork: An Economic Perpective Alok Gupta Dmitry Zhdanov Department of Information and Deciion Science Univerity of Minneota Minneapoli, M 55455 (agupta,

More information

Redesigning Ratings: Assessing the Discriminatory Power of Credit Scores under Censoring

Redesigning Ratings: Assessing the Discriminatory Power of Credit Scores under Censoring Redeigning Rating: Aeing the Dicriminatory Power of Credit Score under Cenoring Holger Kraft, Gerald Kroiandt, Marlene Müller Fraunhofer Intitut für Techno- und Wirtchaftmathematik (ITWM) Thi verion: June

More information

DISTRIBUTED DATA PARALLEL TECHNIQUES FOR CONTENT-MATCHING INTRUSION DETECTION SYSTEMS. G. Chapman J. Cleese E. Idle

DISTRIBUTED DATA PARALLEL TECHNIQUES FOR CONTENT-MATCHING INTRUSION DETECTION SYSTEMS. G. Chapman J. Cleese E. Idle DISTRIBUTED DATA PARALLEL TECHNIQUES FOR CONTENT-MATCHING INTRUSION DETECTION SYSTEMS G. Chapman J. Cleee E. Idle ABSTRACT Content matching i a neceary component of any ignature-baed network Intruion Detection

More information

Harmonic Oscillations / Complex Numbers

Harmonic Oscillations / Complex Numbers Harmonic Ocillation / Complex Number Overview and Motivation: Probably the ingle mot important problem in all of phyic i the imple harmonic ocillator. It can be tudied claically or uantum mechanically,

More information

DISTRIBUTED DATA PARALLEL TECHNIQUES FOR CONTENT-MATCHING INTRUSION DETECTION SYSTEMS

DISTRIBUTED DATA PARALLEL TECHNIQUES FOR CONTENT-MATCHING INTRUSION DETECTION SYSTEMS DISTRIBUTED DATA PARALLEL TECHNIQUES FOR CONTENT-MATCHING INTRUSION DETECTION SYSTEMS Chritopher V. Kopek Department of Computer Science Wake Foret Univerity Winton-Salem, NC, 2709 Email: kopekcv@gmail.com

More information

Scheduling of Jobs and Maintenance Activities on Parallel Machines

Scheduling of Jobs and Maintenance Activities on Parallel Machines Scheduling of Job and Maintenance Activitie on Parallel Machine Chung-Yee Lee* Department of Indutrial Engineering Texa A&M Univerity College Station, TX 77843-3131 cylee@ac.tamu.edu Zhi-Long Chen** Department

More information

Office of Tax Analysis U.S. Department of the Treasury. A Dynamic Analysis of Permanent Extension of the President s Tax Relief

Office of Tax Analysis U.S. Department of the Treasury. A Dynamic Analysis of Permanent Extension of the President s Tax Relief Office of Tax Analyi U.S. Department of the Treaury A Dynamic Analyi of Permanent Extenion of the Preident Tax Relief July 25, 2006 Executive Summary Thi Report preent a detailed decription of Treaury

More information

On a Generalization of Meyniel s Conjecture on the Cops and Robbers Game

On a Generalization of Meyniel s Conjecture on the Cops and Robbers Game On a Generalization of Meyniel Conjecture on the Cop and Robber Game Noga Alon Tel Aviv Univerity and Intitute for Advanced Study, Princeton nogaa@tau.ac.il Abba Mehrabian Department of Combinatoric and

More information

TRADING rules are widely used in financial market as

TRADING rules are widely used in financial market as Complex Stock Trading Strategy Baed on Particle Swarm Optimization Fei Wang, Philip L.H. Yu and David W. Cheung Abtract Trading rule have been utilized in the tock market to make profit for more than a

More information

Two Dimensional FEM Simulation of Ultrasonic Wave Propagation in Isotropic Solid Media using COMSOL

Two Dimensional FEM Simulation of Ultrasonic Wave Propagation in Isotropic Solid Media using COMSOL Excerpt from the Proceeding of the COMSO Conference 0 India Two Dimenional FEM Simulation of Ultraonic Wave Propagation in Iotropic Solid Media uing COMSO Bikah Ghoe *, Krihnan Balaubramaniam *, C V Krihnamurthy

More information

The Cash Flow Statement: Problems with the Current Rules

The Cash Flow Statement: Problems with the Current Rules A C C O U N T I N G & A U D I T I N G accounting The Cah Flow Statement: Problem with the Current Rule By Neii S. Wei and Jame G.S. Yang In recent year, the tatement of cah flow ha received increaing attention

More information

Name: SID: Instructions

Name: SID: Instructions CS168 Fall 2014 Homework 1 Aigned: Wedneday, 10 September 2014 Due: Monday, 22 September 2014 Name: SID: Dicuion Section (Day/Time): Intruction - Submit thi homework uing Pandagrader/GradeScope(http://www.gradecope.com/

More information

Assessing the Discriminatory Power of Credit Scores

Assessing the Discriminatory Power of Credit Scores Aeing the Dicriminatory Power of Credit Score Holger Kraft 1, Gerald Kroiandt 1, Marlene Müller 1,2 1 Fraunhofer Intitut für Techno- und Wirtchaftmathematik (ITWM) Gottlieb-Daimler-Str. 49, 67663 Kaierlautern,

More information

NETWORK TRAFFIC ENGINEERING WITH VARIED LEVELS OF PROTECTION IN THE NEXT GENERATION INTERNET

NETWORK TRAFFIC ENGINEERING WITH VARIED LEVELS OF PROTECTION IN THE NEXT GENERATION INTERNET Chapter 1 NETWORK TRAFFIC ENGINEERING WITH VARIED LEVELS OF PROTECTION IN THE NEXT GENERATION INTERNET S. Srivatava Univerity of Miouri Kana City, USA hekhar@conrel.ice.umkc.edu S. R. Thirumalaetty now

More information

1 Introduction. Reza Shokri* Privacy Games: Optimal User-Centric Data Obfuscation

1 Introduction. Reza Shokri* Privacy Games: Optimal User-Centric Data Obfuscation Proceeding on Privacy Enhancing Technologie 2015; 2015 (2):1 17 Reza Shokri* Privacy Game: Optimal Uer-Centric Data Obfucation Abtract: Conider uer who hare their data (e.g., location) with an untruted

More information

Global Imbalances or Bad Accounting? The Missing Dark Matter in the Wealth of Nations. Ricardo Hausmann and Federico Sturzenegger

Global Imbalances or Bad Accounting? The Missing Dark Matter in the Wealth of Nations. Ricardo Hausmann and Federico Sturzenegger Global Imbalance or Bad Accounting? The Miing Dark Matter in the Wealth of Nation Ricardo Haumann and Federico Sturzenegger CID Working Paper No. 124 January 2006 Copyright 2006 Ricardo Haumann, Federico

More information

Basic Quantum Mechanics in Coordinate, Momentum and Phase Space

Basic Quantum Mechanics in Coordinate, Momentum and Phase Space Baic Quantum Mechanic in Coorinate, Momentum an Phae Space Frank Rioux Department of Chemitry College of St. Beneict St. Johnʹ Univerity The purpoe of thi paper i to ue calculation on the harmonic ocillator

More information

T-test for dependent Samples. Difference Scores. The t Test for Dependent Samples. The t Test for Dependent Samples. s D

T-test for dependent Samples. Difference Scores. The t Test for Dependent Samples. The t Test for Dependent Samples. s D The t Tet for ependent Sample T-tet for dependent Sample (ak.a., Paired ample t-tet, Correlated Group eign, Within- Subject eign, Repeated Meaure,.. Repeated-Meaure eign When you have two et of core from

More information

Mixed Method of Model Reduction for Uncertain Systems

Mixed Method of Model Reduction for Uncertain Systems SERBIAN JOURNAL OF ELECTRICAL ENGINEERING Vol 4 No June Mixed Method of Model Reduction for Uncertain Sytem N Selvaganean Abtract: A mixed method for reducing a higher order uncertain ytem to a table reduced

More information

A Resolution Approach to a Hierarchical Multiobjective Routing Model for MPLS Networks

A Resolution Approach to a Hierarchical Multiobjective Routing Model for MPLS Networks A Reolution Approach to a Hierarchical Multiobjective Routing Model for MPLS Networ Joé Craveirinha a,c, Rita Girão-Silva a,c, João Clímaco b,c, Lúcia Martin a,c a b c DEEC-FCTUC FEUC INESC-Coimbra International

More information

Project Management Basics

Project Management Basics Project Management Baic A Guide to undertanding the baic component of effective project management and the key to ucce 1 Content 1.0 Who hould read thi Guide... 3 1.1 Overview... 3 1.2 Project Management

More information

TIME SERIES ANALYSIS AND TRENDS BY USING SPSS PROGRAMME

TIME SERIES ANALYSIS AND TRENDS BY USING SPSS PROGRAMME TIME SERIES ANALYSIS AND TRENDS BY USING SPSS PROGRAMME RADMILA KOCURKOVÁ Sileian Univerity in Opava School of Buine Adminitration in Karviná Department of Mathematical Method in Economic Czech Republic

More information

6. Friction, Experiment and Theory

6. Friction, Experiment and Theory 6. Friction, Experiment and Theory The lab thi wee invetigate the rictional orce and the phyical interpretation o the coeicient o riction. We will mae ue o the concept o the orce o gravity, the normal

More information

Proceedings of Power Tech 2007, July 1-5, Lausanne

Proceedings of Power Tech 2007, July 1-5, Lausanne Second Order Stochatic Dominance Portfolio Optimization for an Electric Energy Company M.-P. Cheong, Student Member, IEEE, G. B. Sheble, Fellow, IEEE, D. Berleant, Senior Member, IEEE and C.-C. Teoh, Student

More information

Queueing Models for Multiclass Call Centers with Real-Time Anticipated Delays

Queueing Models for Multiclass Call Centers with Real-Time Anticipated Delays Queueing Model for Multicla Call Center with Real-Time Anticipated Delay Oualid Jouini Yve Dallery Zeynep Akşin Ecole Centrale Pari Koç Univerity Laboratoire Génie Indutriel College of Adminitrative Science

More information

Exercises for Section 6.1

Exercises for Section 6.1 Exercie for Section 6.1 2. The ph of an acid olution ued to etch aluminum varie omewhat from batch to batch. In a ample of 50 batche, the mean ph wa 2.6, with a tandard deviation of 0.3. Let μ repreent

More information

Risk Management for a Global Supply Chain Planning under Uncertainty: Models and Algorithms

Risk Management for a Global Supply Chain Planning under Uncertainty: Models and Algorithms Rik Management for a Global Supply Chain Planning under Uncertainty: Model and Algorithm Fengqi You 1, John M. Waick 2, Ignacio E. Gromann 1* 1 Dept. of Chemical Engineering, Carnegie Mellon Univerity,

More information

Linear energy-preserving integrators for Poisson systems

Linear energy-preserving integrators for Poisson systems BIT manucript No. (will be inerted by the editor Linear energy-preerving integrator for Poion ytem David Cohen Ernt Hairer Received: date / Accepted: date Abtract For Hamiltonian ytem with non-canonical

More information

Buying High and Selling Low: Stock Repurchases and Persistent Asymmetric Information

Buying High and Selling Low: Stock Repurchases and Persistent Asymmetric Information RFS Advance Acce publihed February 9, 06 Buying High and Selling Low: Stock Repurchae and Peritent Aymmetric Information Philip Bond Univerity of Wahington Hongda Zhong London School of Economic Share

More information

Stochastic House Appreciation and Optimal Mortgage Lending

Stochastic House Appreciation and Optimal Mortgage Lending Stochatic Houe Appreciation and Optimal Mortgage Lending Tomaz Pikorki Columbia Buine School tp2252@columbia.edu Alexei Tchityi UC Berkeley Haa tchityi@haa.berkeley.edu December 28 Abtract We characterize

More information

2. METHOD DATA COLLECTION

2. METHOD DATA COLLECTION Key to learning in pecific ubject area of engineering education an example from electrical engineering Anna-Karin Cartenen,, and Jonte Bernhard, School of Engineering, Jönköping Univerity, S- Jönköping,

More information

Quadrilaterals. Learning Objectives. Pre-Activity

Quadrilaterals. Learning Objectives. Pre-Activity Section 3.4 Pre-Activity Preparation Quadrilateral Intereting geometric hape and pattern are all around u when we tart looking for them. Examine a row of fencing or the tiling deign at the wimming pool.

More information

322 CHAPTER 11 Motion and Momentum Telegraph Colour Library/FPG/Getty Images

322 CHAPTER 11 Motion and Momentum Telegraph Colour Library/FPG/Getty Images Standard 7.7.4: Ue ymbolic equation to how how the quantity of omething change over time or in repone to change in other quantitie. Alo cover: 7.2.6, 7.2.7 (Detailed tandard begin on page IN8.) What i

More information

Change Management Plan Blackboard Help Course 24/7

Change Management Plan Blackboard Help Course 24/7 MIT 530 Change Management Plan Help Coure 24/7 Submitted by: Sheri Anderon UNCW 4/20/2008 Introduction The Univerity of North Carolina Wilmington (UNCW) i a public comprehenive univerity, one of the ixteen

More information

Reputation in a model with a limited debt structure

Reputation in a model with a limited debt structure Review of Economic ynamic 8 (2005 600 622 www.elevier.com/locate/red Reputation in a model with a limited debt tructure Begoña omínguez epartment of Economic, The Univerity of Auckland, Private Bag 92019,

More information

A Duality Model of TCP and Queue Management Algorithms

A Duality Model of TCP and Queue Management Algorithms A Duality Model of TCP and Queue Management Algorithm Steven H. Low CS and EE Department California Intitute of Technology Paadena, CA 95 low@caltech.edu May 4, Abtract We propoe a duality model of end-to-end

More information

Bi-Objective Optimization for the Clinical Trial Supply Chain Management

Bi-Objective Optimization for the Clinical Trial Supply Chain Management Ian David Lockhart Bogle and Michael Fairweather (Editor), Proceeding of the 22nd European Sympoium on Computer Aided Proce Engineering, 17-20 June 2012, London. 2012 Elevier B.V. All right reerved. Bi-Objective

More information

A relationship between generalized Davenport-Schinzel sequences and interval chains

A relationship between generalized Davenport-Schinzel sequences and interval chains A relationhip between generalized Davenport-Schinzel equence and interval chain The MIT Faculty ha made thi article openly available. Pleae hare how thi acce benefit you. Your tory matter. Citation A Publihed

More information

January 21, 2015. Abstract

January 21, 2015. Abstract T S U I I E P : T R M -C S J. R January 21, 2015 Abtract Thi paper evaluate the trategic behavior of a monopolit to influence environmental policy, either with taxe or with tandard, comparing two alternative

More information

Auction-Based Resource Allocation for Sharing Cloudlets in Mobile Cloud Computing

Auction-Based Resource Allocation for Sharing Cloudlets in Mobile Cloud Computing 1 Auction-Baed Reource Allocation for Sharing Cloudlet in Mobile Cloud Computing A-Long Jin, Wei Song, Senior Member, IEEE, and Weihua Zhuang, Fellow, IEEE Abtract Driven by pervaive mobile device and

More information

Free Enterprise, the Economy and Monetary Policy

Free Enterprise, the Economy and Monetary Policy Free Enterprie, the Economy and Monetary Policy free (fre) adj. not cont Free enterprie i the freedom of individual and buinee to power of another; at regulation. It enable individual and buinee to create,

More information

Heuristic Approach to Dynamic Data Allocation in Distributed Database Systems

Heuristic Approach to Dynamic Data Allocation in Distributed Database Systems Pakitan Journal of Information and Technology 2 (3): 231-239, 2003 ISSN 1682-6027 2003 Aian Network for Scientific Information Heuritic Approach to Dynamic Data Allocation in Ditributed Databae Sytem 1

More information

Is Mark-to-Market Accounting Destabilizing? Analysis and Implications for Policy

Is Mark-to-Market Accounting Destabilizing? Analysis and Implications for Policy Firt draft: 4/12/2008 I Mark-to-Market Accounting Detabilizing? Analyi and Implication for Policy John Heaton 1, Deborah Luca 2 Robert McDonald 3 Prepared for the Carnegie Rocheter Conference on Public

More information

You may use a scientific calculator (non-graphing, non-programmable) during testing.

You may use a scientific calculator (non-graphing, non-programmable) during testing. TECEP Tet Decription College Algebra MAT--TE Thi TECEP tet algebraic concept, procee, and practical application. Topic include: linear equation and inequalitie; quadratic equation; ytem of equation and

More information

INFORMATION Technology (IT) infrastructure management

INFORMATION Technology (IT) infrastructure management IEEE TRANSACTIONS ON CLOUD COMPUTING, VOL. 2, NO. 1, MAY 214 1 Buine-Driven Long-term Capacity Planning for SaaS Application David Candeia, Ricardo Araújo Santo and Raquel Lope Abtract Capacity Planning

More information

THE IMPACT OF MULTIFACTORIAL GENETIC DISORDERS ON CRITICAL ILLNESS INSURANCE: A SIMULATION STUDY BASED ON UK BIOBANK ABSTRACT KEYWORDS

THE IMPACT OF MULTIFACTORIAL GENETIC DISORDERS ON CRITICAL ILLNESS INSURANCE: A SIMULATION STUDY BASED ON UK BIOBANK ABSTRACT KEYWORDS THE IMPACT OF MULTIFACTORIAL GENETIC DISORDERS ON CRITICAL ILLNESS INSURANCE: A SIMULATION STUDY BASED ON UK BIOBANK BY ANGUS MACDONALD, DELME PRITCHARD AND PRADIP TAPADAR ABSTRACT The UK Biobank project

More information

FEDERATION OF ARAB SCIENTIFIC RESEARCH COUNCILS

FEDERATION OF ARAB SCIENTIFIC RESEARCH COUNCILS Aignment Report RP/98-983/5/0./03 Etablihment of cientific and technological information ervice for economic and ocial development FOR INTERNAL UE NOT FOR GENERAL DITRIBUTION FEDERATION OF ARAB CIENTIFIC

More information

1) Assume that the sample is an SRS. The problem state that the subjects were randomly selected.

1) Assume that the sample is an SRS. The problem state that the subjects were randomly selected. 12.1 Homework for t Hypothei Tet 1) Below are the etimate of the daily intake of calcium in milligram for 38 randomly elected women between the age of 18 and 24 year who agreed to participate in a tudy

More information

OPINION PIECE. It s up to the customer to ensure security of the Cloud

OPINION PIECE. It s up to the customer to ensure security of the Cloud OPINION PIECE It up to the cutomer to enure ecurity of the Cloud Content Don t outource what you don t undertand 2 The check lit 2 Step toward control 4 Due Diligence 4 Contract 4 E-dicovery 4 Standard

More information

Efficient Pricing and Insurance Coverage in Pharmaceutical Industry when the Ability to Pay Matters

Efficient Pricing and Insurance Coverage in Pharmaceutical Industry when the Ability to Pay Matters ömmföäfläafaäflaflafla fffffffffffffffffffffffffffffffffff Dicuion Paper Efficient Pricing and Inurance Coverage in Pharmaceutical Indutry when the Ability to Pay Matter Vea Kanniainen Univerity of Helinki,

More information

SCM- integration: organiational, managerial and technological iue M. Caridi 1 and A. Sianei 2 Dipartimento di Economia e Produzione, Politecnico di Milano, Italy E-mail: maria.caridi@polimi.it Itituto

More information

RISK MANAGEMENT POLICY

RISK MANAGEMENT POLICY RISK MANAGEMENT POLICY The practice of foreign exchange (FX) rik management i an area thrut into the potlight due to the market volatility that ha prevailed for ome time. A a conequence, many corporation

More information

Position: The location of an object; in physics, typically specified with graph coordinates Introduction Position

Position: The location of an object; in physics, typically specified with graph coordinates Introduction Position .0 - Introduction Object move: Ball bounce, car peed, and pacehip accelerate. We are o familiar with the concept of motion that we ue ophiticated phyic term in everyday language. For example, we might

More information

Availability of WDM Multi Ring Networks

Availability of WDM Multi Ring Networks Paper Availability of WDM Multi Ring Network Ivan Rado and Katarina Rado H d.o.o. Motar, Motar, Bonia and Herzegovina Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, Univerity

More information

Independent Samples T- test

Independent Samples T- test Independent Sample T- tet With previou tet, we were intereted in comparing a ingle ample with a population With mot reearch, you do not have knowledge about the population -- you don t know the population

More information

Politicians, Taxes and Debt

Politicians, Taxes and Debt Review of Economic Studie 00) 77, 806 840 0034-657/0/0040$0.00 doi: 0./j.467-937X.009.00584.x Politician, Taxe and Debt PIERRE YARED Columbia Univerity Firt verion received January 008; final verion accepted

More information

A Life Contingency Approach for Physical Assets: Create Volatility to Create Value

A Life Contingency Approach for Physical Assets: Create Volatility to Create Value A Life Contingency Approach for Phyical Aet: Create Volatility to Create Value homa Emil Wendling 2011 Enterprie Rik Management Sympoium Society of Actuarie March 14-16, 2011 Copyright 2011 by the Society

More information

Chapter 8 Transients & RLC

Chapter 8 Transients & RLC Chapter 8 Tranient & C Chapter 8 Tranient & C... 8. Introduction... 8.3 Tranient... 8.3. Firt Order Tranient... 8.3. Circuit... 3 8.3.3 C Circuit... 4 8.9 C Sytem epone... 5 8.9. C Equation... 5 8.9. Sytem

More information

Morningstar Fixed Income Style Box TM Methodology

Morningstar Fixed Income Style Box TM Methodology Morningtar Fixed Income Style Box TM Methodology Morningtar Methodology Paper Augut 3, 00 00 Morningtar, Inc. All right reerved. The information in thi document i the property of Morningtar, Inc. Reproduction

More information

Stochastic House Appreciation and Optimal Subprime Lending

Stochastic House Appreciation and Optimal Subprime Lending Stochatic Houe Appreciation and Optimal Subprime Lending Tomaz Pikorki Columbia Buine School tp5@mail.gb.columbia.edu Alexei Tchityi NYU Stern atchity@tern.nyu.edu February 8 Abtract Thi paper tudie an

More information

Research in Economics

Research in Economics Reearch in Economic 64 (2010) 137 145 Content lit available at ScienceDirect Reearch in Economic journal homepage: www.elevier.com/locate/rie Health inurance: Medical treatment v diability payment Geir

More information

Analysis of Fuzzy Erlang s Loss Queuing Model: Non Linear Programming Approach

Analysis of Fuzzy Erlang s Loss Queuing Model: Non Linear Programming Approach International Journal of Fuzzy Mathematic and Sytem. Volume 1, Number 1 (2011), pp. 1-10 Reearch India Publication http://www.ripublication.com Analyi of Fuzzy Erlang o Queuing Model: Non inear Programming

More information

BLAST and FASTA Heuristics in pairwise sequence alignment

BLAST and FASTA Heuristics in pairwise sequence alignment An Introduction to Bioinformatic Algorithm www.bioalgorithm.info BLAST and FASTA Heuritic in pairwie euence alignment (baed on material of Chritoph Dieterich Department of Evolutionary Biology Max Planck

More information

Performance of Multiple TFRC in Heterogeneous Wireless Networks

Performance of Multiple TFRC in Heterogeneous Wireless Networks Performance of Multiple TFRC in Heterogeneou Wirele Network 1 Hyeon-Jin Jeong, 2 Seong-Sik Choi 1, Firt Author Computer Engineering Department, Incheon National Univerity, oaihjj@incheon.ac.kr *2,Correponding

More information

Socially Optimal Pricing of Cloud Computing Resources

Socially Optimal Pricing of Cloud Computing Resources Socially Optimal Pricing of Cloud Computing Reource Ihai Menache Microoft Reearch New England Cambridge, MA 02142 t-imena@microoft.com Auman Ozdaglar Laboratory for Information and Deciion Sytem Maachuett

More information

Research Article An (s, S) Production Inventory Controlled Self-Service Queuing System

Research Article An (s, S) Production Inventory Controlled Self-Service Queuing System Probability and Statitic Volume 5, Article ID 558, 8 page http://dxdoiorg/55/5/558 Reearch Article An (, S) Production Inventory Controlled Self-Service Queuing Sytem Anoop N Nair and M J Jacob Department

More information