Compromise Effect Why This Is Irrational? This is Not Just an . . . Wiener’s Symmetry . . . Interval and Symmetry What Do We Know . . . Approaches to Uncertainty – Natural . . . What Can We . . . Pioneered by Wiener – Summarizing Conclusion Help Explain Seemingly Home Page Irrational Human Behavior: Title Page A Case Study ◭◭ ◮◮ ◭ ◮ Joe Lorkowski and Vladik Kreinovich Page 1 of 13 Department of Computer Science Go Back University of Texas at El Paso Full Screen El Paso, TX 79968, USA lorkowski@computer.org, vladik@utep.edu Close Quit
Compromise Effect Why This Is Irrational? 1. Compromise Effect This is Not Just an . . . • A customer shopping for an item has choices: some Wiener’s Symmetry . . . cheaper, some more expensive but of higher quality. What Do We Know . . . Natural . . . • Examples: shopping for a camera, for a hotel room. What Can We . . . • Researchers asked the customers to select one of the Summarizing three randomly selected alternatives. Conclusion Home Page • They expected all three to be selected with equal prob- ability. Title Page • Instead, in the overwhelming majority of cases, cus- ◭◭ ◮◮ tomers selected the intermediate alternative. ◭ ◮ • The intermediate alternative provides a compromise Page 2 of 13 between the quality and cost. Go Back • So, this phenomenon was named compromise effect . Full Screen Close Quit
Compromise Effect Why This Is Irrational? 2. Why This Is Irrational? This is Not Just an . . . • Selecting the middle alternative seems reasonable. Wiener’s Symmetry . . . What Do We Know . . . • But let’s consider alternatives a 1 < a 2 < a 3 < a 4 sorted Natural . . . by price (and quality). What Can We . . . • If we present the user with three choices a 1 < a 2 < a 3 , Summarizing the user will select the middle choice a 2 . Conclusion Home Page • This means that, to the user, a 2 is better than a 3 . Title Page • But if we present the user with three other choices a 2 < a 3 < a 4 , the same user will select a 3 . ◭◭ ◮◮ • So, to the user, the alternative a 3 is better than a 2 . ◭ ◮ Page 3 of 13 • If in a pair-wise comparison, a 3 is better, then the first choice is wrong, else the second choice is wrong. Go Back • In both cases, one of the two choices is irrational. Full Screen Close Quit
Compromise Effect Why This Is Irrational? 3. This is Not Just an Experimental Curiosity, This is Not Just an . . . Customers’ Have Been Manipulated This Way Wiener’s Symmetry . . . • At first glance, this seems like an optical illusion or a What Do We Know . . . logical paradox: interesting but not very important. Natural . . . What Can We . . . • Actually, it is important: customers have been manip- Summarizing ulated into buying a more expensive product. Conclusion • If there are two types of a product, a company adds an Home Page even more expensive third option. Title Page • Recent research shows the compromise effect only hap- ◭◭ ◮◮ pens when a customer has no additional information. ◭ ◮ • In situations when customers were given access to ad- ditional information, their selections were consistent. Page 4 of 13 • However, in situation when decisions need to be made Go Back under major uncertainty, this effect is clearly present. Full Screen • How to explain such a seemingly irrational behavior? Close Quit
Compromise Effect Why This Is Irrational? 4. Wiener’s Symmetry Approach: Main Idea This is Not Just an . . . • Main idea: Wiener’s Symmetry . . . What Do We Know . . . – if the situation is invariant with respect to some Natural . . . natural symmetries, What Can We . . . – then it is reasonable to select an action which is Summarizing also invariant with respect to all these symmetries. Conclusion • This approach has indeed been helpful in dealing with Home Page uncertainty. In particular, it explains: Title Page – the use of a sigmoid activation function s ( z ) = ◭◭ ◮◮ 1 1 + exp( − z ) in neural networks, ◭ ◮ – the use of the most efficient t-norms and t-conorms Page 5 of 13 in fuzzy logic, Go Back – etc. Full Screen Close Quit
Compromise Effect Why This Is Irrational? 5. What Do We Know About the Utility of Each This is Not Just an . . . Alternative? Wiener’s Symmetry . . . • The utility of each alternatives comes from two factors: What Do We Know . . . Natural . . . – the first factor u 1 comes from the quality: the higher What Can We . . . the quality, the better – i.e., the larger u 1 ; Summarizing – the second factor u 2 comes from price: the lower Conclusion the price, the better – i.e., the larger u 2 . Home Page • We have alternatives a < a ′ < a ′′ characterized by pairs Title Page u ( a ) = ( u 1 , u 2 ), u ( a ′ ) = ( u ′ 1 , u ′ 2 ), and u ( a ′′ ) = ( u ′′ 1 , u ′′ 2 ). ◭◭ ◮◮ • We do not know the utility values, we only know that ◭ ◮ u 1 < u ′ 1 < u ′′ 1 and u ′′ 2 < u ′ 2 < u 2 . Page 6 of 13 • Since we only know the order, we can mark the values Go Back u i as L (Low), M (Medium), and H (High). Full Screen • Then u ( a ) = ( L, H ), u ( a ′ ) = ( M, M ), u ( a ′′ ) = ( H, L ). Close Quit
Compromise Effect Why This Is Irrational? 6. Natural Transformations and Symmetries This is Not Just an . . . • We do not know a priori which of the utility compo- Wiener’s Symmetry . . . nents is more important. What Do We Know . . . Natural . . . • It is thus reasonable to treat both components equally. What Can We . . . • So, swapping the two components is a reasonable trans- Summarizing formation: Conclusion Home Page – if we are selecting an alternative based on the pairs Title Page u ( a ) = ( L, H ) , u ( a ′ ) = ( M, M ) , and u ( a ′′ ) = ( H, L ) , ◭◭ ◮◮ – then we should select the exact same alternative ◭ ◮ based on the “swapped” pairs Page 7 of 13 u ( a ) = ( H, L ) , u ( a ′ ) = ( M, M ) , and u ( a ′′ ) = ( L, H ) . Go Back Full Screen Close Quit
Compromise Effect Why This Is Irrational? 7. Transformations and Symmetries (cont-d) This is Not Just an . . . • Similarly, there is no reason to a priori prefer one al- Wiener’s Symmetry . . . ternative versus the other. What Do We Know . . . Natural . . . • So, any permutation of the three alternatives is a rea- What Can We . . . sonable transformation. Summarizing • We start with Conclusion u ( a ′ ) = ( M, M ) , u ( a ′′ ) = ( H, L ) . u ( a ) = ( L, H ) , Home Page • If we rename a and a ′′ , we get Title Page u ( a ) = ( H, L ) , u ( a ′ ) = ( M, M ) , u ( a ′′ ) = ( L, H ) . ◭◭ ◮◮ ◭ ◮ • For example: Page 8 of 13 – if we originally select an alternative a with Go Back u ( a ) = ( L, H ) , Full Screen – then, after the swap, we should select the same al- ternative – which is now denoted by a ′′ . Close Quit
Compromise Effect Why This Is Irrational? 8. What Can We Conclude From These Symme- This is Not Just an . . . tries Wiener’s Symmetry . . . • We start with What Do We Know . . . Natural . . . u ( a ′ ) = ( M, M ) , u ( a ′′ ) = ( H, L ) . u ( a ) = ( L, H ) , What Can We . . . • If we swap u 1 and u 2 , we get Summarizing u ( a ′ ) = ( M, M ) , u ( a ′′ ) = ( L, H ) . u ( a ) = ( H, L ) , Conclusion Home Page • Now, if we also rename a and a ′′ , we get Title Page u ( a ′ ) = ( M, M ) , u ( a ′′ ) = ( H, L ) . u ( a ) = ( L, H ) , ◭◭ ◮◮ • These are the same utility values with which we started. ◭ ◮ • So, if originally, we select a with u ( a ) = ( L, H ) , in the Page 9 of 13 new arrangements we should also select a . Go Back • But the new a is the old a ′′ . • So, if we selected a , we should select a ′′ – a contradic- Full Screen tion. Close Quit
Compromise Effect Why This Is Irrational? 9. What Can We Conclude (cont-d) This is Not Just an . . . • We start with Wiener’s Symmetry . . . What Do We Know . . . u ( a ′ ) = ( M, M ) , u ( a ′′ ) = ( H, L ) . u ( a ) = ( L, H ) , Natural . . . • If we swap u 1 and u 2 , we get What Can We . . . Summarizing u ( a ′ ) = ( M, M ) , u ( a ′′ ) = ( L, H ) . u ( a ) = ( H, L ) , Conclusion • Now, if we also rename a and a ′′ , we get Home Page u ( a ′ ) = ( M, M ) , u ( a ′′ ) = ( H, L ) . u ( a ) = ( L, H ) , Title Page ◭◭ ◮◮ • These are the same utility values with which we started. ◭ ◮ • So, if originally, we select a ′′ with u ( a ′′ ) = ( H, L ) , in Page 10 of 13 the new arrangements we should also select a . • But the new a ′′ is the old a . Go Back Full Screen • So, if we selected a ′′ , we should select a – a contradic- tion. Close Quit
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