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Chapter 12 Judgment and Choice This chapter covers the mathematical models behind the way that consumer decide and choose. We will discuss The detection of sensory information The detection of differences between two things Judgments where consumers compare two things A model for the recognition of advertisements 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 Absolute Judgment • • Do I see anything? How much do I like that? 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 19th 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 > n2) .5 0 n1 Mathematical Marketing n2 n3 Slide 12.3 Judgment and Choice But the Data Never Looked Like That 1.0 Pr(Detect) .5 0 n Mathematical Marketing 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 Pr[Detect stimulus i] = Pr[si s0] . We make this assumption ei ~ N(0, 2) so that which then implies si ~ N(si, 2 ) We also assume Mathematical Marketing si si ei s0 0 Slide 12.5 Judgment and Choice Our Assumptions Imply That the Probability of Detection Is… 1 2 2 p̂i exp[ ( s s ) / 2 ] ds i i i 2 0 Converting to a z-score we get z i2 1 p̂i exp dz i 2 0 s 2 i (Note missing subscript i on the z in book) Mathematical Marketing Slide 12.6 Judgment and Choice Making the Equation Simpler z i2 1 p̂i exp dz i 2 0 s 2 i But since the normal distribution is symmetric about 0 we can say: si z i2 1 p̂ i exp dz 2 2 [ si / ] Mathematical Marketing Slide 12.7 Judgment and Choice Graphical Picture of What We Just Did Pr( s i ) 2 0 Pr( z) si si 1 Pr(Detection) z si 0 Pr( z) Mathematical Marketing 1 0 si z 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? s1 dim s2 medium s3 bright 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 s i si ei 2 ei ~ N(0, i ) Cov(ei, ej) = ij = rij si Draw si sj 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) E(s i s j ) E ( si e i ) ( s j e j ) si s j 2 i ij 1 V(s i s j ) 1 1 2 ij j 1 i2 2j 2ij i2 2j 2ri j 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" p̂ij Pr(s i s j ) (si s j ) E(si s j ) Mathematical Marketing i2 2j 2ri j V(si s j ) Slide 12.14 Judgment and Choice Thurstone Case III p̂ij Pr(s i s j ) (si s j ) s1 = 0 i2 2j 12 = 1 s2 , s3 , , st , 22 , 32 , , 2t How many unknowns are there? How many data points are there? Mathematical Marketing Slide 12.15 Judgment and Choice Unweighted Least Squares Estimation 1[Pr(s i s j )] 1 (si s j ) i2 2j ẑ12 ( s1 s2 ) 12 22 ẑ13 ( s1 s3 ) 12 32 ẑ ( t 1) t ( st 1 st ) t 1 2t 1 2t t f (z ij ẑ ij ) 2 i 1 ji 1 Mathematical Marketing Slide 12.16 Judgment and Choice Conditions Needed for Minimizing f f /s1 0 f / s 0 2 f / s 0 t f /12 0 2 f / 2 0 2 f / 0 t Mathematical Marketing Slide 12.17 Judgment and Choice Minimum Pearson 2 Same model: p̂ ij (si s j ) i2 2j . Different objective function 2 ( np n p̂ ) ij ij ˆ 2 i j i np̂ ij t Mathematical Marketing t Slide 12.18 Judgment and Choice Matrix Setup for Minimum Pearson 2 p p12 p13 p( t 1) t pˆ p̂12 p̂13 p̂( t 1) t V(pij ) V[pij p̂ij ] p̂ij (1 p̂ij ) n V(p) = V ˆ 2 (p pˆ ) V 1 (p pˆ ) Mathematical Marketing Slide 12.19 Judgment and Choice Minimum Pearson 2 (np ij np̂ ij ) 2 ˆ i j i np̂ ij 2 t t Modified Minimum Pearson 2 (np ij np̂ ij ) 2 ˆ i j i np ij 2 t t 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 Pr( A) p Pr( B) q Note these definitions and identities: fij = npij Mathematical Marketing p ij f ij n p ji 1 pij n f ij n Slide 12.21 Judgment and Choice ML Estimation of the Thurstone Model According to the Model According to the general alternative t 1 t f nf l 0 p̂ ij ij (1 p̂ ij ij ) t 1 t f nf l A p ij ij (1 p ij ij ) i 1 t 1 ji 1 i 1 t ln(l 0 ) L0 f ij ln p̂ij (n f ij ) ln(1 p̂ij ) i 1 ji 1 ˆ 2 2 ln Mathematical Marketing t 1 ji 1 t ln(l A ) L A f ij ln pij (n f ij ) ln(1 pij ) i 1 ji 1 l0 2[ L A L 0 ] lA Slide 12.22 Judgment and Choice Categorical or Absolute Judgment Love [ ] Like [ ] Dislike [ ] s1 1 Love Brand 1 Brand 2 Brand 3 Mathematical Marketing .20 .10 .05 2 Like .30 .10 .10 s2 3 Hate [ ] s3 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 .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 c0 = - Mathematical Marketing c1 c2 c3 (cJ-1) c4 = + Slide 12.25 Judgment and Choice A Model for Categorical Data s i si ei ei ~ N(0, 2) p̂ij Pr[si c j ] Pr[c j si 0] 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) p̂ij c j si i 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 S N S Hit Miss N False Alarm Correct Rejection Reality Mathematical Marketing Slide 12.28 Judgment and Choice Psychophysical, Psychological and Psychomotor Functions n1 s1 V(n1 ) n2 s 2 V( n 2 ) r I(s1 , s 2 ,, s J ) Mathematical Marketing nJ s J V( n J ) y M(r) Slide 12.29 Judgment and Choice