Reward . . . What People Want Rewarding Good . . . Reward for Good Punishing for Mistakes . . . Convex and Concave . . . Performance Works Better Resulting Explanation Than Punishment for Discussion Home Page Mistakes: Economic Title Page Explanation ◭◭ ◮◮ ◭ ◮ Olga Kosheleva 1 , Julio Urenda 2 , 3 , and Vladik Kreinovich 3 1 Department of Teacher Education Page 1 of 19 2 Department of Mathematical Sciences Go Back 3 Department of Computer Science University of Texas at El Paso Full Screen 500 W. University El Paso, TX 79968, USA Close olgak@utep.edu, jcurenda@utep.edu, vladik@utep.edu Quit
Reward . . . 1. Reward vs. Punishment: An Important Eco- What People Want nomic Problem Rewarding Good . . . • One of the most important issues in economics is how Punishing for Mistakes . . . to best stimulate people’s productivity. Convex and Concave . . . Resulting Explanation • What is the best combination of reward and punish- Discussion ment that makes people perform better. Home Page • This problem rises not only in economics, it appears Title Page everywhere. ◭◭ ◮◮ • How do we stimulate students to study better? ◭ ◮ • How do we stimulate our own kids to behave better? Page 2 of 19 Go Back Full Screen Close Quit
Reward . . . 2. Empirical Fact What People Want • A lot of empirical studies were done on this topic. Rewarding Good . . . Punishing for Mistakes . . . • Some of these studies were made by Nobelist Daniel Convex and Concave . . . Kahneman – one of the fathers of behavioral economics. Resulting Explanation • Most confirm that reward for good performance, in Discussion general, works better than punishment for mistakes. Home Page • But why? Title Page • Like many facts from behavioral economics, this fact ◭◭ ◮◮ does not have a convincing theoretical explanation. ◭ ◮ • In this talk, we provide a theoretical explanation for Page 3 of 19 this empirical phenomenon. Go Back Full Screen Close Quit
Reward . . . 3. What People Want What People Want • People spend some efforts e . Rewarding Good . . . Punishing for Mistakes . . . • Based on results of these efforts, they get a reward r ( e ). Convex and Concave . . . • In the first approximation, we can say that the overall Resulting Explanation gain is the reward minus the efforts: r ( e ) − e. Discussion • A natural economic idea is that every person wants to Home Page maximize his/her gain, i.e., maximize r ( e ) − e ; so: Title Page – to explain why rewards work better than punish- ◭◭ ◮◮ ments, ◭ ◮ – we need to analyze what are the reward functions Page 4 of 19 r ( e ) corr. to the two reward strategies. Go Back • We will use simplified “first approximation” models, Full Screen providing qualitative understanding of the situation. Close Quit
Reward . . . 4. What Reward Function Corresponds to Reward- What People Want ing Good Performance Rewarding Good . . . • What does rewarding good performance mean? Punishing for Mistakes . . . Convex and Concave . . . • On the one hand: Resulting Explanation – if the performance is not good, i.e., if the effort e is Discussion smaller than the smallest needed effort e 0 , Home Page – there is practically no reward: r ( e ) = r + for some Title Page r + ≈ 0 . ◭◭ ◮◮ • On the other hand: ◭ ◮ Page 5 of 19 – the more effort the person uses, the larger the re- ward; Go Back – so, every effort beyond e 0 is proportionally rewarded, Full Screen i.e., r ( e ) = r + + c + · ( e − e 0 ) , for some c + . Close Quit
Reward . . . 5. Rewarding Good Performance (cont-d) What People Want Rewarding Good . . . • The constant c + depends on the units used for measur- ing effort and reward: Punishing for Mistakes . . . Convex and Concave . . . – one unit of effort corresponds Resulting Explanation – to c + units of reward. Discussion • These two formulas can be combined into a single for- Home Page mula Title Page r ( e ) = r + +max(0 , c + · ( e − e 0 )) = r + + c + · max(0 , e − e 0 ) . ◭◭ ◮◮ ◭ ◮ Page 6 of 19 Go Back Full Screen Close Quit
Reward . . . 6. Rewarding Good Performance (cont-d) What People Want Rewarding Good . . . • This dependence has the following form: Punishing for Mistakes . . . r ( e ) ✻ � Convex and Concave . . . � � Resulting Explanation � � Discussion � � Home Page � � Title Page � � ◭◭ ◮◮ � � � ◭ ◮ � e ✲ Page 7 of 19 Go Back Full Screen Close Quit
Reward . . . 7. What Can We Say About This Function What People Want • It is easy to see that our function is convex . Rewarding Good . . . Punishing for Mistakes . . . • This means that for all e ′ < e ′′ and for each α ∈ [0 , 1], Convex and Concave . . . we have Resulting Explanation r ( α · e ′ + (1 − α ) · e ′′ ) ≤ α · r ( e ′ ) + (1 − α ) · r ( e ′′ ) . Discussion Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 8 of 19 Go Back Full Screen Close Quit
Reward . . . 8. What Reward Function Corresponds to Pun- What People Want ishing for Mistakes Rewarding Good . . . • What does punishing for mistakes means? Punishing for Mistakes . . . Convex and Concave . . . • On the one hand: Resulting Explanation – if the performance is good, i.e., if the effort e is ≥ Discussion the smallest needed effort e 0 , Home Page – then there is no punishment, i.e., the reward re- Title Page mains the same: r ( e ) = r − for some constant r − ; ◭◭ ◮◮ • On the other hand: ◭ ◮ – the fewer effort the person uses, the most mistakes Page 9 of 19 he/she makes, Go Back – so the larger the punishment and the smaller the resulting reward; Full Screen – so, every effort below e 0 is proportionally penalized, Close i.e., r ( e ) = r − − c − · ( e 0 − e ) , for some c − . Quit
Reward . . . 9. Punishing for Mistakes (cont-d) What People Want Rewarding Good . . . • The constant c − depends on the units used for measur- ing effort and reward: Punishing for Mistakes . . . Convex and Concave . . . – one unit of effort corresponds Resulting Explanation – to c − units of reward. Discussion • These two formulas can be combined into a single for- Home Page mula Title Page r ( e ) = r − − c − · max(0 , e 0 − e ) = r − + c − · min(0 , e − e 0 ) . ◭◭ ◮◮ ◭ ◮ Page 10 of 19 Go Back Full Screen Close Quit
Reward . . . 10. Punishing for Mistakes (cont-d) What People Want Rewarding Good . . . • This dependence has the following form: Punishing for Mistakes . . . r ( e ) ✻ � Convex and Concave . . . � � Resulting Explanation � � Discussion � � Home Page � � Title Page � e ✲ ◭◭ ◮◮ ◭ ◮ Page 11 of 19 Go Back Full Screen Close Quit
Reward . . . 11. What Can We Say About This Function What People Want • It is easy to see that this function is concave . Rewarding Good . . . Punishing for Mistakes . . . • This means that for all E ′ < E ′′ and for each α ∈ [0 , 1], Convex and Concave . . . we have Resulting Explanation r ( α · e ′ + (1 − α ) · e ′′ ) ≥ α · r ( e ′ ) + (1 − α ) · r ( e ′′ ) . Discussion Home Page • Now, we are ready to present the desired explanation. Title Page ◭◭ ◮◮ ◭ ◮ Page 12 of 19 Go Back Full Screen Close Quit
Reward . . . 12. Known Properties of Convex and Concave Func- What People Want tions: Reminder Rewarding Good . . . • It is known that: Punishing for Mistakes . . . Convex and Concave . . . – every linear function is both convex and concave; Resulting Explanation – the sum of two convex functions is convex, and Discussion – the sum of two concave functions is concave. Home Page • In particular, the linear function f ( e ) = − e is both Title Page convex and concave, thus: ◭◭ ◮◮ – when the function r ( e ) is convex, the sum ◭ ◮ r ( e ) + ( − e ) = r ( e ) − e is also convex; and Page 13 of 19 – when the function r ( e ) is concave, the sum r ( e ) + ( − e ) = r ( e ) − e is also concave. Go Back Full Screen Close Quit
Reward . . . 13. Convex and Concave Functions (cont-d) What People Want Rewarding Good . . . • It is also known that: Punishing for Mistakes . . . – for a convex function, the maximum on an interval Convex and Concave . . . is always attained at one of the endpoints; Resulting Explanation – for a concave function, its maximum on an interval Discussion is always attained at some point inside the interval. Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 14 of 19 Go Back Full Screen Close Quit
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