Formulation of the . . . Formulation of the . . . Comment Empirical Power Law for Comment (cont-d) Our Explanation Company Losses: Our Explanation (cont-d) Our Explanation (cont-d) A Probability-Based References Explanation Home Page Title Page Ricardo Alvarez 1 , Laxman Bokati 2 , Panfeng Liang 1 , ◭◭ ◮◮ Adrian Lopez 1 ,, Carlos Salda˜ na 1 , Ricardo Sanchez 1 , ◭ ◮ Angel Villapando 1 , and Vladik Kreinovich 1 , 2 1 Department of Computer Science and 2 Computational Science Program Page 1 of 9 University of Texas at El Paso, El Paso, TX 79968, USA, ralvarezlo@miners.utep.edu, lbokati@miners.utep.edu, Go Back pliang@miners.utep.edu, ajlopez18@miners.utep.edu, Full Screen casaldanamatamoros@miners.utep.edu, rasanchez7@miners.utep.edu, avillalpando4@miners.utep.edu, vladik@utep.edu Close Quit
Formulation of the . . . 1. Formulation of the Problem Formulation of the . . . Comment • Companies compete in the market. Comment (cont-d) • Both a given company and its competitors constantly Our Explanation develop new products. Our Explanation (cont-d) Our Explanation (cont-d) • One of these products becomes a winner. References • The efforts of all other companies do not lead to success Home Page and thus, qualify as losses: Title Page – the more money and efforts the company invests in ◭◭ ◮◮ the development of the new products, ◭ ◮ – the higher the probability that this company will succeed. Page 2 of 9 • Vice versa, the fewer money is invested, the higher the Go Back probability of failure. Full Screen Close Quit
Formulation of the . . . 2. Formulation of the Problem (cont-d) Formulation of the . . . Comment • Analysis of different companies shows that, on average: Comment (cont-d) – the probability of failure p is approximately in- Our Explanation versely proportional to Our Explanation (cont-d) – the overall investment I in development of new Our Explanation (cont-d) products. References c Home Page • To be more precise, p ≈ for some constants c I + I 0 Title Page and I 0 . ◭◭ ◮◮ • The problem is that there is no convincing explanation ◭ ◮ for the above formula. Page 3 of 9 • In this talk, we provide such an explanation. Go Back Full Screen Close Quit
Formulation of the . . . 3. Comment Formulation of the . . . Comment • Similar dependencies can be found in many application Comment (cont-d) areas. Our Explanation • Historically the first such dependence was Zipf’s law – Our Explanation (cont-d) first formulated in linguistics. Our Explanation (cont-d) References • This law states that if the text is large enough, then: Home Page – when we order words in the decreasing order of fre- Title Page quency, ◭◭ ◮◮ – the frequency f n of n -th word is approximately equal c to f n ≈ for some constants c and n 0 . ◭ ◮ n + n 0 Page 4 of 9 Go Back Full Screen Close Quit
Formulation of the . . . 4. Comment (cont-d) Formulation of the . . . Comment • Similar formulas work well: Comment (cont-d) – when we sort cities in the decreasing order of their Our Explanation population, Our Explanation (cont-d) – when we sort companies in the decreasing order of Our Explanation (cont-d) their sizes, References Home Page – when we sort papers by number of citations, Title Page – when we sort earthquakes by magnitude, etc. ◭◭ ◮◮ ◭ ◮ Page 5 of 9 Go Back Full Screen Close Quit
Formulation of the . . . 5. Our Explanation Formulation of the . . . Comment • Let k denote the number of new products developed Comment (cont-d) by a given company. Our Explanation • Let a be the average investment needed to develop a Our Explanation (cont-d) new product. Our Explanation (cont-d) References • Then, the overall company’s investment is equal to Home Page I = a · k. Title Page • So, in terms of the investment I , the value k has the ◭◭ ◮◮ form k = I/a . ◭ ◮ • Let C denote the average number of new products pro- Page 6 of 9 posed by the competition. Go Back • Then, the overall number of competing products is Full Screen k + C. Close Quit
Formulation of the . . . 6. Our Explanation (cont-d) Formulation of the . . . Comment • It is reasonable to assume that all these products are Comment (cont-d) equally reasonable. Our Explanation • Thus, each of these products has the same probability Our Explanation (cont-d) of becoming a commercial success. Our Explanation (cont-d) References • This probability is equal to 1 / ( k + C ). Home Page • The probability that the given company loses: Title Page – can thus be estimated as the probability that one ◭◭ ◮◮ of C competitors’ products will succeed, ◭ ◮ C – and is, therefore, equal to p = k + C . Page 7 of 9 • Substituting k = I/a into this formula, we get Go Back C Full Screen p = . I a + C Close Quit
Formulation of the . . . 7. Our Explanation (cont-d) Formulation of the . . . Comment C • Reminder: p = . Comment (cont-d) I a + C Our Explanation Our Explanation (cont-d) • Multiplying both the numerator and the denominator a · C Our Explanation (cont-d) by a , we conclude that p = I + a · C . References c Home Page • So, we indeed get the desired expression p ≈ , I + I 0 Title Page with c = I 0 = a · C. ◭◭ ◮◮ ◭ ◮ Page 8 of 9 Go Back Full Screen Close Quit
8. References Formulation of the . . . Formulation of the . . . • J. Hall, Risk Management and Financial Institutions , Comment Comment (cont-d) Prentice Hall, Upper Saddle River, New Jersey, 2006. Our Explanation Our Explanation (cont-d) Our Explanation (cont-d) References Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 9 of 9 Go Back Full Screen Close Quit
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