Chapter 12 Judgment and Choice This chapter covers

Chapter 12 Judgment and Choice This chapter covers the mathematical models behind the way that consumer decide and choose. We will discuss q The detection of sensory information q The detection of differences between two things q Judgments where consumers compare two things q A model for the recognition of advertisements q How multiple judgments are combined to make a single decision As usual, estimation of the parameters in these models will serve as an important theme for this chapter Mathematical Marketing Slide 12. 1 Judgment and Choice

There Are Two Different Types of Judgments q Absolute Judgment • • Do I see anything? How much do I like that? q Comparative Judgment • • Does this bagel taste better than that one? Do I like Country Time Lemonade better than Minute Maid? Psychologists began investigating how people answer these sorts of questions in the 19 th Century Mathematical Marketing Slide 12. 2 Judgment and Choice

The Early Concept of a “Threshold” Absolute Detection 1. 0 Pr(Detect). 5 0 n Physical measurement Difference Detection 1. 0 Pr(n Perceived > n 2). 5 0 n 1 Mathematical Marketing n 2 n 3 Slide 12. 3 Judgment and Choice

But the Data Never Looked Like That 1. 0 Pr(Detect) . 5 0 Mathematical Marketing n Slide 12. 4 Judgment and Choice

A Simple Model for Detection si is the psychological impact of stimulus i If si exceeds the threshold, you see/hear/feel it We make this assumption Pr[Detect stimulus i] = Pr[si s 0]. ei ~ N(0, 2) so that which then implies We also assume Mathematical Marketing Slide 12. 5 Judgment and Choice

Our Assumptions Imply That the Probability of Detection Is… (Note missing left bracket in Equation 12. 6 in book. ) Converting to a z-score we get (Note missing subscript i on the z in book) Mathematical Marketing Slide 12. 6 Judgment and Choice

Making the Equation Simpler But since the normal distribution is symmetric about 0 we can say: Mathematical Marketing Slide 12. 7 Judgment and Choice

Graphical Picture of What We Just Did 0 Pr(Detection) Mathematical Marketing 0 Slide 12. 8 Judgment and Choice

A General Rule for Pr(a > 0) Where a Is Normally Distributed For a ~ N[E(a), V(a)] we have Pr [a 0] = [E(a) / V(a)] Mathematical Marketing Slide 12. 9 Judgment and Choice

So Why Do Detection Probabilities Not Look Like a Step Function? Mathematical Marketing Slide 12. 10 Judgment and Choice

Paired Comparison Data: Pr(Row Brand > Column Brand) Mathematical Marketing A B C A - . 6 . 7 B . 4 - . 2 C . 3 . 8 - Slide 12. 11 Judgment and Choice

Assumptions of the Thurstone Model ei ~ N(0, ) Cov(ei, ej) = ij = r i j Draw si Draw sj Is si > sj? Mathematical Marketing Slide 12. 12 Judgment and Choice

Deriving the E(si - sj) and V (si - sj) pij = Pr(si > sj ) = Pr(si - sj > 0) Mathematical Marketing Slide 12. 13 Judgment and Choice

Predicting Choice Probabilities For a ~ N[E(a), V(a)] we have Pr [a 0] = [E(a) / V(a)] Below si - sj plays the role of "a" Mathematical Marketing Slide 12. 14 Judgment and Choice

Thurstone Case III =0 =1 How many unknowns are there? How many data points are there? Mathematical Marketing Slide 12. 15 Judgment and Choice

Unweighted Least Squares Estimation Mathematical Marketing Slide 12. 16 Judgment and Choice

Conditions Needed for Minimizing f Mathematical Marketing Slide 12. 17 Judgment and Choice

Minimum Pearson 2 Same model: Different objective function Mathematical Marketing Slide 12. 18 Judgment and Choice

Matrix Setup for Minimum Pearson 2 V(p) = V Mathematical Marketing Slide 12. 19 Judgment and Choice

Minimum Pearson 2 Modified Minimum Pearson 2 Simplifies the derivatives, and reduces the computational time required Mathematical Marketing Slide 12. 20 Judgment and Choice

Definitions and Background for ML Estimation Assume that we have two possible events A and B. The probability of A is Pr(A), and the probability of B is Pr(B). What are the odds of two A's on two independent trials? Pr(A) • Pr(A) = Pr(A)2 In general the Probability of p A's and q B's would be Note these definitions and identities: fij = npij Mathematical Marketing Slide 12. 21 Judgment and Choice

ML Estimation of the Thurstone Model According to the Model Mathematical Marketing According to the general alternative Slide 12. 22 Judgment and Choice

Categorical or Absolute Judgment Love [ ] Like [ ] Dislike [ ] s 1 1 Love Brand 1 Brand 2 Brand 3 Mathematical Marketing . 20. 10. 05 2 Like. 30. 10 s 2 3 Hate [ ] s 3 4 Dislike. 20. 60. 15 Hate. 30. 20. 70 Slide 12. 23 Judgment and Choice

Cumulated Category Probabilities Love Brand 1. 20 Raw Probabilities Cumulated Probabilities Mathematical Marketing Like. 30 Dislike. 20 Hate. 30 Brand 2 . 10 . 60 . 20 Brand 3 . 05 . 10 . 15 . 70 Brand 1 . 20 . 50 . 70 1. 00 Brand 2 . 10 . 20 . 80 1. 00 Brand 3 . 05 . 15 . 30 1. 00 Slide 12. 24 Judgment and Choice

The Thresholds or Cutoffs c 0 = - Mathematical Marketing c 1 c 2 c 3 (c. J-1) c 4 = + Slide 12. 25 Judgment and Choice

A Model for Categorical Data ei ~ N(0, 2) Probability that item i is placed in category j or less Mathematical Marketing Probability that the discriminal response to item i is less than the upper boundary for category j Slide 12. 26 Judgment and Choice

The Probability of Using a Specific Category (or Less) Pr [a 0] = [E(a) / V(a)] Below ci - sj is plays the role of "a" Mathematical Marketing Slide 12. 27 Judgment and Choice

The Theory of Signal Detectability Response Reality Mathematical Marketing S N S Hit Miss N False Alarm Correct Rejection Slide 12. 28 Judgment and Choice