What Is Anchoring Effect? A Formula Describing . . . What Are the Values . . . What We Want How to Explain the Three Natural Properties Anchoring Formula in Fourth Natural . . . Definitions and the . . . Behavioral Economics Proof Explaining the . . . Laxman Bokati 1 , Vladik Krenovich 1 , Home Page and Chon Van Le 2 Title Page 1 Computational Science Program ◭◭ ◮◮ University of Texas at El Paso El Paso, Texas 79968, USA ◭ ◮ laxman@miners.utep.edu, vladik@utep.edu 2 International University Page 1 of 28 Ho Chi Minh City, Vietnam Go Back lvchon@hcmiu.edu.vn Full Screen Close Quit
What Is Anchoring Effect? A Formula Describing . . . 1. What Is Anchoring Effect? What Are the Values . . . • Traditional economics assumes: What We Want Three Natural Properties – that people know the exact value of each possible Fourth Natural . . . item, and Definitions and the . . . – that this value determines the price that they are Proof willing to pay for this item. Explaining the . . . • The reality is more complicated. Home Page • In many practical situations: Title Page ◭◭ ◮◮ – people are uncertain about the value of an item, and thus, ◭ ◮ – uncertain about the price they are willing to pay Page 2 of 28 for this item. Go Back • This happens, e.g., when hunting for a house. Full Screen Close Quit
What Is Anchoring Effect? A Formula Describing . . . 2. What Is Anchoring Effect (cont-d) What Are the Values . . . • In many such situations, the price that the customer What We Want is willing to pay is affected by the asking price: Three Natural Properties Fourth Natural . . . – if the asking price is higher, the customer is willing Definitions and the . . . to pay a higher price, but Proof – if the asking price is lower, the price that the cus- Explaining the . . . tomer is willing to pay is also lower. Home Page • This phenomenon is known as the anchoring effect : Title Page – just like a stationary ship may move a little bit, but ◭◭ ◮◮ cannot move too far away from its anchor, ◭ ◮ – similarly, a customer stays closer to the asking price Page 3 of 28 – which thus acts as a kind of an anchor. Go Back • The anchoring effect may sound somewhat irrational. Full Screen • However, it makes some sense. Close Quit
What Is Anchoring Effect? A Formula Describing . . . 3. What Is Anchoring Effect (cont-d) What Are the Values . . . • If the owner lists his/her house at an unexpectedly high What We Want price, then maybe: Three Natural Properties Fourth Natural . . . – there are some positive features of the house Definitions and the . . . – of which the customer is not aware. Proof • After all, the owner does want to sell his/her house. Explaining the . . . Home Page • So he/she would not just list an outrageously high price without any reason. Title Page ◭◭ ◮◮ ◭ ◮ Page 4 of 28 Go Back Full Screen Close Quit
What Is Anchoring Effect? A Formula Describing . . . 4. What Is Anchoring Effect (cont-d) What Are the Values . . . • Similarly, if the owner lists his/her house at an unex- What We Want pectedly low price, then maybe: Three Natural Properties Fourth Natural . . . – there are some drawbacks of the house or of its Definitions and the . . . location Proof – of which the customer is not aware. Explaining the . . . • After all, the owner does want to get his/her money Home Page back when selling his/her house. Title Page • So he/she would not just list an outrageously low price ◭◭ ◮◮ without any reason. ◭ ◮ Page 5 of 28 Go Back Full Screen Close Quit
What Is Anchoring Effect? A Formula Describing . . . 5. A Formula Describing the Anchoring Effect What Are the Values . . . • Let p 0 be the price that the customer would suggest in What We Want the absence of an anchor. Three Natural Properties Fourth Natural . . . • Of course: Definitions and the . . . – if the asking price a is the same value a = p 0 , Proof – there is no reason for the customer to change the Explaining the . . . price p that he/she is willing to pay for this item, Home Page – i.e., this price should still be equal to p 0 . Title Page • It turns out that each anchoring situation can be de- ◭◭ ◮◮ scribed by a coefficient α ∈ [0 , 1]. ◭ ◮ • This coefficient is called an anchoring index . Page 6 of 28 • The idea is that: Go Back – if we consider two different asking prices a ′ � = a ′′ , – then the difference p ′ − p ′′ between the resulting Full Screen customer’s prices should be equal to α · ( a ′ − a ′′ ). Close Quit
What Is Anchoring Effect? A Formula Describing . . . 6. Anchoring Effect Formula (cont-d) What Are the Values . . . • We know that p = p 0 when a = p 0 . What We Want Three Natural Properties • Thus, the above idea enables us to come up with the Fourth Natural . . . formula describing the anchoring effect. Definitions and the . . . • Indeed, for anchor a , the difference p − p 0 is equal to Proof Explaining the . . . α · ( a − p 0 ) . Home Page • Since p − p 0 = α · ( a − p 0 ), we thus have Title Page p = p 0 + α · ( a − p 0 ) = (1 − α ) · p 0 + α · a. ◭◭ ◮◮ ◭ ◮ • How can we explain this empirical formula? Page 7 of 28 Go Back Full Screen Close Quit
What Is Anchoring Effect? A Formula Describing . . . 7. What Are the Values of the Anchoring Index What Are the Values . . . • In different situations, we observe different values of What We Want the anchoring index. Three Natural Properties Fourth Natural . . . • When people are not sure about their original opinion, Definitions and the . . . the anchoring index is usually close to 0.5. Proof • For a regular person buying a house, this index is equal Explaining the . . . to 0 . 48 ≈ 0 . 5. Home Page • For people living in a polluted city, Title Page – when asked what living costs they would accept to ◭◭ ◮◮ move to an environmentally clean area, ◭ ◮ – the anchoring index was also close to 0.5. Page 8 of 28 Go Back Full Screen Close Quit
What Is Anchoring Effect? A Formula Describing . . . 8. Values of the Anchoring Index (cont-d) What Are the Values . . . • For other situations: What We Want Three Natural Properties – when a decision maker in more confident in his/her Fourth Natural . . . original opinion, Definitions and the . . . – we can get indices between 0.25 and 0.5. Proof • For a real estate agent buying a house, this index is Explaining the . . . equal to 0.41. Home Page • For a somewhat similar situation of charity donations, Title Page this index is equal to 0.30. ◭◭ ◮◮ • How can we explain these values? ◭ ◮ Page 9 of 28 Go Back Full Screen Close Quit
What Is Anchoring Effect? A Formula Describing . . . 9. What We Do in This Talk What Are the Values . . . • In this talk, we try our best to answer both questions. What We Want Three Natural Properties • We provide a formal explanation for the general for- Fourth Natural . . . mula. Definitions and the . . . • We provide a somewhat less formal explanation for the Proof empirically observed values of the anchoring index. Explaining the . . . Home Page • To make our explanations more convincing, we have tried to make the mathematics as simple as possible. Title Page ◭◭ ◮◮ ◭ ◮ Page 10 of 28 Go Back Full Screen Close Quit
What Is Anchoring Effect? A Formula Describing . . . 10. What We Want What Are the Values . . . • We are given two numbers: What We Want Three Natural Properties – the price p 0 that the customer is willing to pay be- Fourth Natural . . . fore getting the asking price, and Definitions and the . . . – the actual asking price a , Proof • We want a function that, given p 0 and a , produces the Explaining the . . . price p ( p 0 , a ) that the customer is willing to pay. Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 11 of 28 Go Back Full Screen Close Quit
What Is Anchoring Effect? A Formula Describing . . . 11. Three Natural Properties What Are the Values . . . • As we have mentioned, if a = p 0 , then we should have What We Want Three Natural Properties p ( p 0 , a ) = p ( p 0 , p 0 ) = p 0 . Fourth Natural . . . • Small changes in p 0 and a should not lead to drastic Definitions and the . . . changes in the resulting price. Proof • In mathematica terms, this means that the function Explaining the . . . p ( p 0 , a ) should be continuous. Home Page • Intuitively, the change from p 0 to p should be in the Title Page direction to the anchor, i.e.: ◭◭ ◮◮ – if a < p 0 , we should have p ( p 0 , a ) ≤ p 0 , and ◭ ◮ – if p 0 < a , we should have p 0 ≤ p ( p 0 , a ) . Page 12 of 28 • When the changed value p ( p 0 , a ) moves in the direction Go Back of the asking price a , it should not exceed a : Full Screen – if a < p 0 , we should have a ≤ p ( p 0 , a ), and – if p 0 < a , we should have p ( p 0 , a ) ≤ a. Close Quit
What Is Anchoring Effect? A Formula Describing . . . 12. Three Natural Properties (cont-d) What Are the Values . . . • These three property can be summarized by saying What We Want that: Three Natural Properties Fourth Natural . . . – for all p 0 and a , Definitions and the . . . – the price p ( p 0 , a ) should always be in between the Proof original price p 0 and the asking price a . Explaining the . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 13 of 28 Go Back Full Screen Close Quit
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