The Larger the . . . It Is Desirable to Know . . . How the Amount of . . . Formulas Are Needed, . . . How the Proportion of People Who This Formula Is Purely . . . Agree to Perform a Task Depends on the Let Us Reformulate . . . Stimulus: A Theoretical Explanation of What Do We Know . . . the Empirical Formula Resulting Formulation . . . Proof: Let Us . . . Laxman Bokati 1 , Vladik Krenovich 1 , Home Page and Doan Thanh Ha 2 Title Page 1 Computational Science Program ◭◭ ◮◮ University of Texas at El Paso ◭ ◮ El Paso, Texas 79968, USA laxman@miners.utep.edu, vladik@utep.edu Page 1 of 20 2 Banking University of Ho Chi Minh City 36 Ton That Dam, District 1 Go Back Ho Chi Minh City, Vietnam, hadt@buh.edu.vn Full Screen Close Quit
The Larger the . . . It Is Desirable to Know . . . 1. The Larger the Stimulus, the More People Agree How the Amount of . . . to Do the Task Formulas Are Needed, . . . • In economics, we need to entice people to perform cer- This Formula Is Purely . . . tain tasks – whether it is Let Us Reformulate . . . What Do We Know . . . – planting crops Resulting Formulation . . . – or working on a factory Proof: Let Us . . . – or writing a software package. Home Page • When the corresponding stimulus is too small, no one Title Page will agree to perform the task. ◭◭ ◮◮ • When the stimulus is very high, everyone will agree. ◭ ◮ • The proportion p of people who agree to perform a task Page 2 of 20 will increase with the increase in the stimulus s . Go Back Full Screen Close Quit
The Larger the . . . It Is Desirable to Know . . . 2. It Is Desirable to Know the Exact Amount of How the Amount of . . . Stimulus Formulas Are Needed, . . . • A company wants certain tasks to be performed, so it This Formula Is Purely . . . has to use some stimulus. Let Us Reformulate . . . What Do We Know . . . • It is therefore desirable to find the exact amount of Resulting Formulation . . . stimulus needed: Proof: Let Us . . . – if the stimulus is too low, no one will volunteer, Home Page – if it is very high, the tasks will be performed, but Title Page the company will lose too much money. ◭◭ ◮◮ ◭ ◮ Page 3 of 20 Go Back Full Screen Close Quit
The Larger the . . . It Is Desirable to Know . . . 3. How the Amount of Stimulus Is Usually Deter- How the Amount of . . . mined Now Formulas Are Needed, . . . • In many cases, the selection of the right stimulus is This Formula Is Purely . . . done mostly by trial and error. Let Us Reformulate . . . What Do We Know . . . • This is, e.g., how airline companies, in an overbooked Resulting Formulation . . . situation, ask for volunteers to give up their seats. Proof: Let Us . . . • They increase the award offered to potential volunteers Home Page until they get enough volunteers. Title Page ◭◭ ◮◮ ◭ ◮ Page 4 of 20 Go Back Full Screen Close Quit
The Larger the . . . It Is Desirable to Know . . . 4. Formulas Are Needed, And There Are Such How the Amount of . . . Formulas Formulas Are Needed, . . . • Trial-and-error is a lengthy process, difficult to predict. This Formula Is Purely . . . Let Us Reformulate . . . • It is therefore desirable to have an expressions helping What Do We Know . . . us select the right amount of stimulus. Resulting Formulation . . . • Such expressions exist. Proof: Let Us . . . s q Home Page • The most empirically adequate expression is p = s q + c Title Page for some constants q and c . ◭◭ ◮◮ ◭ ◮ Page 5 of 20 Go Back Full Screen Close Quit
The Larger the . . . It Is Desirable to Know . . . 5. This Formula Is Purely Empirical How the Amount of . . . • One of the main limitation of this formula is that: Formulas Are Needed, . . . This Formula Is Purely . . . – it is purely empirical, Let Us Reformulate . . . – it does not have a convincing theoretical explana- What Do We Know . . . tion. Resulting Formulation . . . • Practitioners are usually very suspicious of best-fit purely Proof: Let Us . . . empirical formulas. Home Page • They are reluctant so use these formulas. Title Page • They prefer formulas for which some theoretical expla- ◭◭ ◮◮ nation exists. ◭ ◮ • Indeed, empirical formulas often turn out to be wrong. Page 6 of 20 • And in economics and related areas, such later-wrong Go Back empirical formulas are ubiquitous. Full Screen • When a country has a boom, empirical formulas pre- dict exponential growth forever. Close Quit
The Larger the . . . It Is Desirable to Know . . . 6. This Formula Is Purely Empirical (cont-d) How the Amount of . . . • When, in the 1920s, the number of telephone operators Formulas Are Needed, . . . started growing exponentially: This Formula Is Purely . . . Let Us Reformulate . . . – empirical formulas predicted that in a few decades, What Do We Know . . . – half of the population will be telephone operators. Resulting Formulation . . . • There are many examples like that. Proof: Let Us . . . Home Page • It is thus desirable to come up with a theoretical ex- planation for empirical formulas. Title Page ◭◭ ◮◮ • In this talk, we provide a theoretical explanation for the above formula. ◭ ◮ Page 7 of 20 Go Back Full Screen Close Quit
The Larger the . . . It Is Desirable to Know . . . 7. Let Us Reformulate the Problem in Terms of How the Amount of . . . Probabilities Formulas Are Needed, . . . • In the above text, we talked about proportion of people This Formula Is Purely . . . who take on the task. Let Us Reformulate . . . What Do We Know . . . • From the mathematical viewpoint, a proportion is not Resulting Formulation . . . something about which we know much. Proof: Let Us . . . • But what is proportion? Home Page • It is simply the probability that a randomly selected Title Page person will take on the task. ◭◭ ◮◮ • So, whatever we said about proportions can be refor- ◭ ◮ mulated in terms of probabilities. Page 8 of 20 • And about probabilities, we know a lot! Go Back Full Screen Close Quit
The Larger the . . . It Is Desirable to Know . . . 8. What Do We Know About Probabilities? How the Amount of . . . • One of the most widely used facts about probabilities Formulas Are Needed, . . . is that: This Formula Is Purely . . . Let Us Reformulate . . . – if we add new evidence E , What Do We Know . . . – the probability of each hypothesis H i changes ac- Resulting Formulation . . . cording to the Bayes formula. Proof: Let Us . . . • Namely, it changes from the original value p 0 ( H i ) to Home Page the new value Title Page p 0 ( H i ) · p ( E | H i ) p ( H i | E ) = p 0 ( H j ) · p ( E | H j ) . ◭◭ ◮◮ � j ◭ ◮ • In our case, we have two hypotheses: Page 9 of 20 – the hypothesis H 0 that the person will take on the Go Back task whose probability is p ( H 0 ), and Full Screen – the hypothesis H 1 that the person will not take on the task; its probability is p ( H 1 ) = 1 − p ( H 0 ). Close Quit
The Larger the . . . It Is Desirable to Know . . . 9. What We Know About Probabilities (cont-d) How the Amount of . . . • In this case, the Bayes formula takes the form Formulas Are Needed, . . . This Formula Is Purely . . . p 0 ( H 0 ) · p ( E | H 0 ) p ( H 0 | E ) = p 0 ( H 0 ) · p ( E | H 0 ) + (1 − p 0 ( H 0 )) · p ( E | H 1 ) = Let Us Reformulate . . . What Do We Know . . . p ( H 0 ) · p ( E | H 0 ) p ( H 0 ) · ( p ( E | H 0 ) − p ( E | H 1 )) + p ( E | H 1 ) , i.e. Resulting Formulation . . . Proof: Let Us . . . p · p ( E | H 0 ) p ′ = Home Page p · ( p ( E | H 0 ) − p ( E | H 1 )) + p ( E | H 1 ) . Title Page • Here, we denoted p = p 0 ( H 0 ) and p ′ = p ( H 0 | E ). ◭◭ ◮◮ • If we divide both the numerator and the denominator ◭ ◮ of this formula by p ( E | H 1 ), we get: Page 10 of 20 p · p ( E | H 0 ) Go Back a · p p ( E | H 1 ) p ′ = � , = 1 + (1 − a ) · p. � p ( E | H 0 ) Full Screen 1 + p · p ( E | H 1 ) − 1 Close Quit
The Larger the . . . It Is Desirable to Know . . . 10. What We Know About Probabilities (cont-d) How the Amount of . . . = p ( E | H 0 ) Formulas Are Needed, . . . def • Here, we denoted a p ( E | H 1 ) . This Formula Is Purely . . . • In other words: Let Us Reformulate . . . What Do We Know . . . – the change of the probability from the previous Resulting Formulation . . . value p to the new value p ′ Proof: Let Us . . . – is described by a fractional-linear formula. Home Page • Our idea is that: Title Page – when we increase the stimulus, ◭◭ ◮◮ – the resulting change of the probability should follow ◭ ◮ such a formula. Page 11 of 20 Go Back Full Screen Close Quit
The Larger the . . . It Is Desirable to Know . . . 11. How Can We Formalize This Idea How the Amount of . . . • What does it mean “increase the stimulus”? Formulas Are Needed, . . . This Formula Is Purely . . . • Intuitively, it means that we increase all the previous Let Us Reformulate . . . stimuli the same way. What Do We Know . . . • What does that mean? Resulting Formulation . . . • If we add $10 to all the stimulus values, this does not Proof: Let Us . . . Home Page mean that we increases all the stimuli the same way. Title Page • For example: ◭◭ ◮◮ – if the previous stimulus was $5, this is a drastic 3-times increase, but ◭ ◮ – if the previous stimulus was $1000, this is a barely Page 12 of 20 noticeable 1% increase. Go Back Full Screen Close Quit
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