copyright robert j. marks ii ece 5345 multiple random variables: a gambling sequence

copyright Robert J. Marks II

ECE 5345Multiple Random Variables: A Gambling Sequence


Optimal % Betting

• You play a sequence of Bernoulli games with the chance of you winning = p >1/2. (Why must his be?)

• You start with a stash of $D. • If you bet and loose, you loose what you

bet. If you win, your bet is matched.• What is the optimal % bet for steady state

winning?


Optimal % Betting

• Example Sequence– Win D[1]= D + %D =(1+%)D– Loose D[2]=D[1]- %D[1]= (1+%) (1-%)D– Loose D[3]=D[2]- % D[2]= (1+%) (1-%)2D – Win D[4]= (1+%) 2 (1-%)2D

• n trials and k wins leaves a stash ofD[n]=(1+%) k (1-%)n-k D


Optimal % Bettingn trials and k wins leaves a stash of

D[n]=(1+%) k (1-%)n-k D

What value of % maximizes D[n] for large n?

Note: Maximizing D[n] for a fixed n is the same as maximizing

)Dlog(%1log%1log

][Dlog1

][

n

kn

n

k

nn

nB


Optimal % Betting

The law of large numbers says

and

Thus

)Dlog(%1log)1(%1log][ ppnB

pn

k

k

lim p

n

kn

k

1lim

%1

1

%10

%

][

pp

d

ndB


Optimal % Betting

Solving

The optimal return is then

D[n] = (1+%) k (1-%)n-k D = 2n pk (1-p)n-k D

Since

12% p

1/n1/1 D)1(2][D ppn ppn

Sanity Check:

100% for p=1

0% for p=1/2


Optimal % Betting

The number of turns to double your stash is obtained by setting this to two and solving for n. The result is

npp ppn )1()1(2

D

][D

122double )1(log)1(log1 ppppn


0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

5

10

15

20

25

30

35

Optimal % Betting 1


p

Notes:

At p=1/2, ndouble =

At p=1, ndouble = 1


0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 110

0

101

102

103

104

Optimal % Betting

p



Roulette

p = even or green = 20/36 = 0.5557

% = 2p-1 = 11.13%

ndouble = 111.655


What if there are odds?

Extend the problem to where you loose % of your bet when you loose and get w% of your bet when you win. Either or w can exceed one.

What is the relation among w, ,and p to assure you are in a position to win.

Compute the optimal % of your bet.

Optimal % Betting

copyright robert j. marks ii ece 5345 multiple random variables: a gambling sequence

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