Peak-End Rule: . . . Towards an Explanation Need for a Utility- . . . Natural Properties of . . . Peak-End Rule: Main Result A Utility-Based Discussion First Open Problem Explanation Second Open Problem Proof Olga Kosheleva, Martine Ceberio, and Home Page Vladik Kreinovich Title Page University of Texas at El Paso ◭◭ ◮◮ El Paso, Texas 79968, USA ◭ ◮ olgak@utep.edu, mceberio@utep.edu vladik@utep.edu Page 1 of 16 Go Back Full Screen Close Quit
Peak-End Rule: . . . Towards an Explanation 1. Peak-End Rule: Description and Need for an Need for a Utility- . . . Explanation Natural Properties of . . . • Often, people judge their overall experience by the Main Result peak and end pleasantness or unpleasantness. Discussion First Open Problem • In other words, they use only the maximum (minimum) Second Open Problem and the last value. Proof • This is how we judge pleasantness of a medical proce- Home Page dure, quality of the cell phone perception, etc. Title Page • There is a lot of empirical evidence supporting the ◭◭ ◮◮ peak-end rule, but not much of an understanding. ◭ ◮ • At first glance, the rule appears counter-intuitive: why Page 2 of 16 only peak and last value? why not average? Go Back • In this talk, we provide such an explanation based on the traditional decision making theory. Full Screen Close Quit
Peak-End Rule: . . . Towards an Explanation 2. Towards an Explanation Need for a Utility- . . . • Our objective is to describe the peak-end rule in terms Natural Properties of . . . of the traditional decision making theory. Main Result Discussion • According to decision theory, preferences of rational First Open Problem agents can be described in terms of utility . Second Open Problem • A rational agent selects an action with the largest value Proof of expected utility. Home Page • Utility is usually defined modulo a linear transforma- Title Page tion. ◭◭ ◮◮ • In the above experiments, we usually have a fixed sta- ◭ ◮ tus quo level which can be taken as 0. Page 3 of 16 • Once we fix this value at 0, the only remaining non- uniqueness in describing utility is scaling u → k · u . Go Back Full Screen • We want to describe the “average” utility correspond- ing to a sequence of different experiences. Close Quit
Peak-End Rule: . . . Towards an Explanation 3. Need for a Utility-Averaging Operation Need for a Utility- . . . • We assume that we know the utility corresponding to Natural Properties of . . . each moment of time. Main Result Discussion • To get an overall utility value, we need to combine First Open Problem these momentous utilities into a single average. Hence: Second Open Problem – if we have already found the average utility corre- Proof sponding to two consequent sub-intervals of time, Home Page – we then need to combine these two averages into a Title Page single average corresponding to the whole interval. ◭◭ ◮◮ • In other words, we need an operation a ∗ b that: ◭ ◮ – given the average utilities a and b corresponding to Page 4 of 16 two consequent time intervals, Go Back – generates the average utility of the combined two- stage experience. Full Screen Close Quit
Peak-End Rule: . . . Towards an Explanation 4. Natural Properties of the Utility-Averaging Op- Need for a Utility- . . . eration Natural Properties of . . . • If two stages have the same average utility a = b , then Main Result two-stage average should be the same: a ∗ a = a . Discussion First Open Problem • In mathematical terms, this means that the utility- Second Open Problem averaging operation ∗ should be idempotent . Proof • If we make one of the stages better, then the result- Home Page ing average utility should increase (or at least not de- Title Page crease). ◭◭ ◮◮ • In other words, the utility-averaging operation ∗ should be monotonic : if a ≤ a ′ and b ≤ b ′ then a ∗ b ≤ a ′ ∗ b ′ . ◭ ◮ Page 5 of 16 • Small changes in one of the stages should lead to small changes in the overall average utility. Go Back • In precise terms, this means that the function a ∗ b must Full Screen be continuous . Close Quit
Peak-End Rule: . . . Towards an Explanation 5. Properties of Utility Averaging (cont-d) Need for a Utility- . . . • For a three-stage situation, with average utilities a , b , Natural Properties of . . . and c : Main Result Discussion – we can first combine a and b into a ∗ b , and then First Open Problem combine this with c , resulting in ( a ∗ b ) ∗ c ; Second Open Problem – we can also combine b and c , and then combine Proof with a , resulting in a ∗ ( b ∗ c ). Home Page • The resulting three-stage average should not depend Title Page on the order: ( a ∗ b ) ∗ c = a ∗ ( b ∗ c ). ◭◭ ◮◮ • In mathematical terms, the operation a ∗ b must be ◭ ◮ associative . Page 6 of 16 • Finally, since utility is defined modulo scaling u → k · u , Go Back the utility-averaging does not change with scaling: Full Screen ( k · a ) ∗ ( k · b ) = k · ( a ∗ b ) . Close Quit
Peak-End Rule: . . . Towards an Explanation 6. Main Result Need for a Utility- . . . Let a ∗ b be a binary operation on the set of all non-negative Natural Properties of . . . numbers which satisfies the following properties: Main Result Discussion 1) it is idempotent, i.e., a ∗ a = a for all a ; First Open Problem 2) it is monotonic: a ≤ a ′ and b ≤ b ′ imply a ∗ b ≤ a ′ ∗ b ′ ; Second Open Problem 3) it is continuous as a function of a and b ; Proof Home Page 4) it is associative, i.e., ( a ∗ b ) ∗ c = a ∗ ( b ∗ c ) ; Title Page 5) it is scale-invariant, i.e., ( k · a ) ∗ ( k · b ) = k · ( a ∗ b ) for all k , a and b . ◭◭ ◮◮ Then, ∗ coincides with one of the following four operations: ◭ ◮ • a 1 ∗ . . . ∗ a n = min( a 1 , . . . , a n ) ; Page 7 of 16 • a 1 ∗ . . . ∗ a n = max( a 1 , . . . , a n ) ; Go Back • a 1 ∗ . . . ∗ a n = a 1 ; Full Screen • a 1 ∗ . . . ∗ a n = a n . Close Quit
Peak-End Rule: . . . Towards an Explanation 7. Discussion Need for a Utility- . . . • Every utility-averaging operation which satisfies the Natural Properties of . . . above reasonable properties means that we select: Main Result Discussion – either the worst First Open Problem – or the best Second Open Problem – or the first Proof – or the last utility. Home Page • This (almost) justifies the peak-end phenomenon. Title Page ◭◭ ◮◮ • The only exception that in addition to peak and end, we also have the start as one of the options: ◭ ◮ a 1 ∗ . . . ∗ a n = a 1 . Page 8 of 16 Go Back • A similar result can be proven if we take negative a i . Full Screen Close Quit
Peak-End Rule: . . . Towards an Explanation 8. First Open Problem Need for a Utility- . . . • Following the psychological experiments, we only con- Natural Properties of . . . sidered: Main Result Discussion – the case when all experiences are positive and First Open Problem – the case when all experiences are negative. Second Open Problem • What happens in the general case? Proof Home Page • If we impose an additional requirement of shift-invariance, then we can get a result similar to the above: Title Page ◭◭ ◮◮ ( a + u 0 ) ∗ ( b + u 0 ) = a ∗ b + u 0 . ◭ ◮ • But what if we do not impose this additional require- Page 9 of 16 ment? Go Back Full Screen Close Quit
Peak-End Rule: . . . Towards an Explanation 9. Second Open Problem Need for a Utility- . . . • Are all five conditions necessary? Some are necessary: Natural Properties of . . . Main Result 1) a ∗ b = a + b satisfies all the conditions except for Discussion idempotence; First Open Problem 4) a ∗ b = a + b satisfies all the conditions except for Second Open Problem 2 associativity; Proof Home Page 5) the closest-to-1 value from [min( a, b ) , max( a, b )] sat- isfies all the conditions except for scale invariance. Title Page • However, it is not clear whether monotonicity and con- ◭◭ ◮◮ tinuity are needed to prove our results. ◭ ◮ Page 10 of 16 Go Back Full Screen Close Quit
Peak-End Rule: . . . Towards an Explanation 10. Acknowledgments Need for a Utility- . . . This work was supported in part: Natural Properties of . . . Main Result • by the National Science Foundation grants HRD-0734825 Discussion and HRD-1242122 (Cyber-ShARE Center of Excellence) First Open Problem and DUE-0926721, Second Open Problem • by Grants 1 T36 GM078000-01 and 1R43TR000173-01 Proof from the National Institutes of Health, and Home Page • by a grant N62909-12-1-7039 from the Office of Naval Title Page Research. ◭◭ ◮◮ ◭ ◮ Page 11 of 16 Go Back Full Screen Close Quit
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