Discounting and . . . A Simple Theoretical . . . Practical Problem . . . Discounting: . . . Do It Today Or Do It Analysis of the Problem Tomorrow: Empirical Which Re-Scalings Are . . . Main Result Non-Exponential Discussion Proof Discounting Explained by Home Page Symmetry Ideas Title Page ◭◭ ◮◮ Francisco Zapata 1 , Olga Kosheleva 1 ◭ ◮ Vladik Kreinovich 1 , Thongchai Dumrongpokaphan 2 Page 1 of 17 1 University of Texas at El Paso, El Paso, Texas 79968, USA fazg74@gmail.com, olgak@utep.edu, vladik@utep.edu Go Back 2 Department of Mathematics, Chiang Mai University Full Screen Chiang Mai 50200 Thailand, tcd43@hotmial.com Close Quit
Discounting and . . . A Simple Theoretical . . . 1. Discounting and Procrastination Practical Problem . . . • Future awards are less valuable than the same size Discounting: . . . awards given now. Analysis of the Problem Which Re-Scalings Are . . . • This phenomenon is known as discounting . Main Result • Suppose that we have a task due by a deadline: e.g., Discussion submitting a grant proposal, or a paper to a conference. Proof • The reward is the same no matter when we finish this Home Page task – as long as it is before the deadline. Title Page • Similarly, the negative effect caused by the need to do ◭◭ ◮◮ some boring stuff is the same no matter when we do it. ◭ ◮ • The further in the future is this negative effect, the Page 2 of 17 smaller is its influence on our today’s happiness. Go Back • Thus, a natural way to maximize today’s happiness is to postpone this task as much as possible. Full Screen • This is exactly what people do. Close Quit
Discounting and . . . A Simple Theoretical . . . 2. A Simple Theoretical Model of Discounting Practical Problem . . . • Suppose that 1 dollar tomorrow is equivalent to D < 1 Discounting: . . . dollars today. Analysis of the Problem Which Re-Scalings Are . . . • Thus, 1 dollar at the day t +1 is equivalent to D dollars Main Result in day t . Discussion • In particular, 1 dollar in day t 0 + 2 is equivalent to D Proof dollars at time t 0 + 1. Home Page • But D dollars on day t 0 + 1 are equivalent to D · D Title Page dollars on day t 0 . ◭◭ ◮◮ • Thus, 1 dollar at day t 0 + 2 is equivalent to D 2 dollars ◭ ◮ at moment t 0 . Page 3 of 17 • In general, 1 dollar at moment t 0 + t is equivalent to = D t dollars at the current moment t 0 . def D ( t ) Go Back • Here, D ( t ) = D t = exp( − a · t ), where a def Full Screen = − ln( D ); this discounting is thus known as exponential . Close Quit
Discounting and . . . A Simple Theoretical . . . 3. Practical Problem with Exp Discounting Practical Problem . . . • For large t , exp( − a · t ) is indistinguishable from 0. Discounting: . . . Analysis of the Problem • So, a person looks for an immediate reward even if Which Re-Scalings Are . . . there is a negative downside in the distant future. Main Result • Such behavior indeed happens: Discussion – a young man takes many loans without taking into Proof Home Page account that in the future, he will have to pay; – a young person ruins his health by using drugs, Title Page – a person commits a crime ignoring that eventually, ◭◭ ◮◮ he will be caught and punished. ◭ ◮ • Such behavior does happen, but such behavior is ab- Page 4 of 17 normal. Go Back • This means that for most people, discounting decreases Full Screen much slower than the exponential function. Close Quit
Discounting and . . . A Simple Theoretical . . . 4. Discounting: Empirical Data. Practical Problem . . . • Empirical data shows: discounting indeed decreases Discounting: . . . much slower than predicted by the exp function. Analysis of the Problem Which Re-Scalings Are . . . • Namely, 1 dollar at moment t 0 + t is equivalent to 1 Main Result D ( t ) = 1 + k · t dollars at moment t 0 . Discussion • This formula is known as hyperbolic discounting . Proof Home Page • In principle, there exist many functions that decrease Title Page slower than the exponential function exp( − a · t ). ◭◭ ◮◮ • So why, out of all these functions, we observe the hy- perbolic one? ◭ ◮ Page 5 of 17 • In this talk, we use symmetries to provide a theoretical explanation for the empirical discounting formula. Go Back Full Screen Close Quit
Discounting and . . . A Simple Theoretical . . . 5. Analysis of the Problem Practical Problem . . . • Let D ( t ) denote the discounting of a reward which is t Discounting: . . . moments into the future. Analysis of the Problem Which Re-Scalings Are . . . • In other words, getting D ( t ) dollars now is equivalent Main Result to getting 1 dollar after time t . Discussion • By definition, D (0) means getting 1 dollar with no de- Proof lay, so D (0) = 1. Home Page • It is also reasonable to require that as the time period Title Page time t increases, the value of the reward goes to 0: ◭◭ ◮◮ t → + ∞ D ( t ) = 0 . lim ◭ ◮ Page 6 of 17 • It is also reasonable to require that a small change in t should lead to small changes in D ( t ). Go Back Full Screen • So, D ( t ) should be differentiable (smooth). Close Quit
Discounting and . . . A Simple Theoretical . . . 6. Analysis of the Problem (cont-d) Practical Problem . . . • If we delay all the rewards by some time s , then each Discounting: . . . value D ( t ) will be replaced by a smaller value D ( t + s ). Analysis of the Problem Which Re-Scalings Are . . . • We can describe this replacement as D ( t + s ) = Main Result F s ( D ( t )), where the function F s ( x ): Discussion – re-scales the original discount value D ( t ) Proof – into the new discount value D ( t + s ). Home Page • For the exponential discounting, the re-scaling F s ( x ) is Title Page def linear: D ( t + s ) = C · D ( t ), where C = exp( − a · s ). ◭◭ ◮◮ • So, we have F s ( x ) = C · x . ◭ ◮ Page 7 of 17 • For the hyperbolic discounting, the corresponding re- scaling F s ( x ) is not linear. Go Back • Which re-scaling should we select? Full Screen Close Quit
Discounting and . . . A Simple Theoretical . . . 7. Which Re-Scalings Are Reasonable? Practical Problem . . . • Of course, linear re-scalings should be reasonable. Discounting: . . . Analysis of the Problem • Also, intuitively, if a re-scaling is reasonable, then its Which Re-Scalings Are . . . inverse should also be reasonable. Main Result • Similarly: Discussion – if two re-scalings are reasonable, Proof – then applying them one after another should also Home Page lead to a reasonable re-scaling. Title Page • So, the class of all reasonable re-scalings should be ◭◭ ◮◮ closed under inversion and composition: be a group . ◭ ◮ • We want to be able to determine the transformation Page 8 of 17 from this group based on finitely many experiments. Go Back • In each experiment, we gain a finite number of values. Full Screen • So, after a finite number of experiments, we can only determine a finite number of parameters. Close Quit
Discounting and . . . A Simple Theoretical . . . 8. Which Re-Scalings Are Reasonable (cont-d) Practical Problem . . . • Thus, we should be able: Discounting: . . . Analysis of the Problem – to select an element of the desired transformation Which Re-Scalings Are . . . group Main Result – based on the values of finitely many parameters. Discussion • So, the corresponding transformation group should be Proof finite-dimensional . Home Page • Summarizing: we want all the transformations F s ( x ) Title Page to belong to: ◭◭ ◮◮ – a finite-dimensional transformation group of func- ◭ ◮ tions of one variable Page 9 of 17 – that contains all linear transformations. Go Back Full Screen Close Quit
Discounting and . . . A Simple Theoretical . . . 9. Which Re-Scalings Are Reasonable: Answer Practical Problem . . . • Known: the only finite-dimensional transformation Discounting: . . . groups that contain all linear transformations are: Analysis of the Problem Which Re-Scalings Are . . . – the group of all linear transformations and Main Result – the group of all fractional-linear transformations Discussion a + b · x Proof 1 + c · x. Home Page • Thus, each reasonable re-scaling is fractionally linear: Title Page D ( t + s ) = a ( s ) + b ( s ) · D ( t ) ◭◭ ◮◮ 1 + c ( s ) · D ( t ) . ◭ ◮ Page 10 of 17 Go Back Full Screen Close Quit
Discounting and . . . A Simple Theoretical . . . 10. Main Result Practical Problem . . . • We say that a smooth decreasing function D ( t ) is a Discounting: . . . reasonable discounting function if: Analysis of the Problem Which Re-Scalings Are . . . – we have D (0) = 1 and lim t →∞ D ( t ) = 0, and Main Result – for every s , there exist values a ( s ), b ( s ), and c ( s ) Discussion for which Proof D ( t + s ) = a ( s ) + b ( s ) · D ( t ) Home Page 1 + c ( s ) · D ( t ) . Title Page Proposition. D ( t ) is a reasonable discounting func- ◭◭ ◮◮ tion if and only if it has one of the following forms: ◭ ◮ 1 D ( t ) = exp( − a · t ) , D ( t ) = 1 + k · t, Page 11 of 17 Go Back 1 + a a D ( t ) = 1 + a · exp( k · t ) , D ( t ) = ( a + 1) · exp( k · t ) − 1 . Full Screen Close Quit
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