Math Help - Help with Optimization Problem found in research paper

1. Help with Optimization Problem found in research paper

Hi all,

I'm an accounting student who has taken one university math course (required) so my math skills are not that high. However I do have some knowledge regarding calculus and optimization.

I was looking for a way to select my bet sizes at a horse race assuming a logarithmic utility curve (thus to optimize asset growth). Imagine my surprise when I found an article outlining a step-by-step algorithm to do just that.

Optimal Bets In Pari-Mutuel Systems by Nissan Levin
Efficiency of Racetrack Betting Markets - Google Books

The sad thing is I cannot decipher the logarithmic algorithm. I can solve the example for the linear using the given algorithm but I believe the author did not show the complete work for the logarithmic example.

The paper includes an example with answers which is used for the linear and logarithmic utility sections (the other aspects of the paper-fair game,deserted,stochastic, etc are not relevant for me) as well as the algorithm to solve both.

The log algorithm starts as
Step 0: Let t=0
Step 1: Increase t by 1 and find a and G(t) using (15)-(18)

This is where I don't know what to do. First I don't know why a "t" variable is needed but that is beside the point. Formulas (15) to (18) have variables which refer to each other and I cannot solve for G(t). I think the alternate equations for a(i) (11) and "a" (12) would be useful to substitute in but I'm not sure.

Basically I want to know how to arrive at A(4)=1056.1 and G(4)=0.00072 and the chart on pg 119 (table 2) with the parameters given (w,p(i),b(i),c,Q)

Any help would be greatly appreciated.

P.S. Answering this question is very important to me. My hope is that a qualified individual could read the relevant sections (about six pages) and easily show how to answer the question. If this cannot be done (it is a long process to derive a simpler algorithm) a rough estimate of how long a mathematician would require to solve it and where I could possibly hire someone would be appreciated.

2. Re: Help with Optimization Problem found in research paper

I guess nobody wants to read the paper, the relevant section is only on pg 118

Can a (which is the sum of all (a)(i) and G(t) be found using only lemma 9)?
(and the given parameters of course)

I could try and post all the relevant info directly in the thread if that would encourage someone to look at the question but I might miss something

3. Re: Help with Optimization Problem found in research paper

OK so I re-wrote the relevant sections from the paper (hope I didn’t miss anything important) again the relevant sections include: the intro, linear utility and logarithmic utility only and it is written in a straight forward manner. I am a novice regarding calculus and algebra but finding a method to solve these types of questions is important to me. Any help would be greatly appreciated.

Efficiency of Racetrack Betting Markets - Google Books

The goal of the algorithm is to structure Mr. G’s bets on a horse race optimizing his expected logarithmic utility.

n- number of horses

b(i)- public money placed on horse (i)

b-total public money placed on all horses

p(i)-probability of horse (i) winning

Q-percentage left in pool after commission by management

c-carryover- the amount money left in pool from previous games

m(i)-Mr. G’s wager on horse (i)

m-total money placed by Mr. G on all horses

M=[m(1),…,m(n)]

a(i)=b(i)+m(i), 1=1,..,n
total bets placed on horse (i)-public and Mr. G. (decision variable used)

a=b+m
total bets placed on all horses-public and Mr. G

s(i)=(Q*a+c)/a(i), i=1,…,n
share per ticket ($1) when outcome (i) is realized the integrality constraint is relaxed in the example i.e. Mr. G can bet in amounts less than a$1 or between dollars

w- wealth of Mr. G before any bets

w(i)=m(i)*s(i)+w-m i=1,…,n
wealth of Mr. G after realization of horse (i) winning race

U(-)- Mr. G’s utility function defined on his wealth

Assumptions
a. The outcomes are mutually exclusive and totally exhaustive
b. b(i), p(i), Q, c, and w are known parameters
c. Each winning ticket entitles the bettor to one share as defined by (1)
d. Mr. G cannot borrow money

Known Parameters
(i) b(i) p(i) p(i)/b(i)
1 100 0.2 .00200
2 80 0.1 .00130
3 320 0.3 .00094
4 350 0.3 .00086
5 150 0.1 .00067

b=1000
w=300
Q=0.9
c=100

The goal is to maximize expected logarithmic utility (maximize R=E(log(w)) and the algorithm shows how to do so (I’m stuck at step 1 however)

Lemma 1 shows that for all concave utility functions if for some (i) and (j), p(i)/b(i)>=p(j)/b(j) then if it is optimal to bet on outcome j, it is also optimal to bet on outcome i.
Thus order the outcomes according to the “merit order” as done in the table

Lemma 8 states
If r<n then a<w+b
r is the high number of horses bet when Mr. G bets in an optimal fashion in the linear case. The algorithm for the log case states to start at t=1calculate the values in lemma 9 and then increase by 1 (i.e. calculate R when Mr.G bets optimally only on one horse and then add horses to compare which R value is highest)

Lemma 9 (can't solve-need help)

Algorithm
Step 0: Let t=0
Step 1: Increase t by 1 and find (a) and G(t) using (15)-(18)

I do not know how to use these equations to find (a) (total amount bet) or G(t). If t=1 then Mr. G is only betting on horse i=1 and A(t=1)=(a).

I can define A(1)=a(1)-b(1)+b and could substitute this value in for all the A(t=1) in the equations.Then solve for a(1) but then how would I solve for A(2) because A(2)=a(1)+a(2)-b(1)-b(2)+b and I would have two unknowns- a(1) and a(2) (a(1) would be different in when two horses are bet on). The author states it is very similar to the linear utility example (where I can follow the algorithm but am unsure where the similarities lie). There are some alternative equations for (a) and a(i) in the linear case but I do not think they apply to this case.

Here are the answers provided in the example for all (t) values

My second question should be easier to solve (I hope). Trying to get a better grasp of the equations (15)-(18) I took the answers and plugged them into the equations as an error check. I found a(i),  and w(i) but can’t decipher G(t). There is an extra bracket in the series term and I don’t know where the sigma should go. I tried all different combinations but can’t get close to the answer provided. Here are my calculations needed for G(2). I think they are all right.

Data and Calculations for G(t=2)
i a(i) b(i) p(i) w(i) calculated
1 119.1 100 0.2 438.21
2 86 80 0.1 345.49
3 320 320 0.3 274.1
4 350 350 0.3 274.1
5 150 150 0.1 274.1

A(2)=1025.9
(2)=2.535712541

Thank you,
JD