Models of Choice. Agenda Administrivia –Readings –Programming –Auditing –Late HW...
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Transcript of Models of Choice. Agenda Administrivia –Readings –Programming –Auditing –Late HW...
Agenda
• Administrivia– Readings– Programming– Auditing– Late HW– Saturated– HW 1
• Models of Choice– Thurstonian scaling– Luce choice theory– Restle choice theory
• Quantitative vs. qualitative tests of models.• Rumelhart & Greeno (1971)• Conditioning…• Next assignment
Choice
• The same choice is not always made in the “same” situation.
• Main assumption: Choice alternatives have choice probabilities.
Overview of 3 Models
• Thurstone & Luce– Responses have an associated ‘strength’.– Choice probability results from the
strengths of the choice alternatives.
• Restle– The factors in the probability of a choice
cannot be combined into a simple strength, but must be assessed individually.
Thurstone Scaling
• Assumptions– The strongest of a set of alternatives will
be selected.– All alternatives gives rise to a probabilistic
distribution (discriminal dispersions) of strengths.
Thurstone Scaling
• Let xj denote the discriminal process produced by stimulus j.
• The probability that Object k is preferred to Stimulus j is given by – P(xk > xj) = P(xk - xj > 0)
Thurstone Scaling
• Assume xj & xk are normally distributed with means j & k, variances j & k, and correlation rjk.
• Then the distribution of xk - xj is normal with – mean k - j
– variance j2 + k
2 - 2 rjkjk = jk2
Thurstone Scaling
€
P(xk > x j )
= P(xk − x j > 0)
= N(μ k −μ j ,σ jk2 )
0
∞
∫
= N(0,1)−zkj
∞
∫ €
zkj =μ k −μ jσ jk
μ k −μ j = zkj ⋅σ jk
Thurstone Scaling
• Special cases:– Case III: r = 0
• If n stimuli, n means, n variances, 2n parameters.
– Case V: r = 0, j2 = k
2
• If n stimuli, n means, n parameters.
Luce’s Choice Theory
• Classical strength theory explains variability in choices by assuming that response strengths oscillate.
• Luce assumed that response strengths are constant, but that there is variability in the process of choosing.– The probability of each response is
proportional to the strength of that response.
Luce’s Choice Theory
• For Thurstone with 3 or more alternatives, it can be difficult to predict how often B will be selected over A. The probabilities of choice may depend on what other alternatives are available.
• Luce is based on the assumption that the relative frequency of choices of B over C should not change with the mere availability of other choices.
Luce’s Choice Axiom
• Mathematical probability theory cannot extend from one set of alternatives to another. For example, it might be possible for:– T1 = {ice cream, sausages}
• P(ice cream) > P(sausage)
– T2 = {ice cream, sausages, sauerkraut}• P(sausage) > P(ice cream)
• Need a psychological theory.
Luce’s Choice Axiom
• Assumption: The relative probabilities of any two alternatives would remain unchanged as other alternatives are introduced.– Menu: 20% choose beef, 30% choose
chicken.– New menu with only beef & chicken: 40%
choose beef, 60% choose chicken.
Luce’s Choice Axiom
• PT(S) is the probability of choosing any element of S given a choice from T.– P{chicken, beef, pork, veggies}(chicken, pork)
Luce’s Choice Axiom
• Let T be a finite subset of U such that, for every S T, Ps is defined, Then:– (i) If P(x, y) 0, 1 for all x, y T, then for
R S T, PT(R) = PS(R) PT(S)
– (ii) If P(x, y) = 0 for some x, y in T, then for every S T, PT(S) = PT-{x}(S-{x})
Luce’s Choice Axiom•(ii) If P(x, y) = 0 for some x, y in T, then for every S T, PT(S) = PT-{x}(S-{x})
•Why? If x is dominated by any element in T, it is dominated by all elements. Causes division problems.
S
T
X
Luce’s Choice Theorem
• Theorem: There exists a positive real-valued function v on T, which is unique up to multiplication by a positive constant, such that for every S T,
€
PS (x) =v(x)
v(y)y∈S
∑
Luce’s Choice Theorem
• Proof: Define v(x) = kPT(x), for k > 0. Then, by the choice axiom (proof of uniqueness left to reader),
€
PS (x) =PT (x)
PT (S)
=kPT (x)
kPT (x)y∈s
∑
=v(x)
v(y)y∈s
∑
Thurstone & Luce
• Thurstone's Case V model becomes equivalent to the Choice Axiom if its discriminal processes are assumed to be independent double exponential random variables– This is true for 2 and 3 choice situations.– For 2 choice situations, other discriminal
processes will work.
Restle
• A choice between 2 complex and overlapping choices depends not on their common elements, but on their differential elements.– $10 + an apple– $10
XXX X
XXX
P($10+A, $10) = (4 - 3)/(4 - 3 + 3 - 3) = 1
Quantitative vs. Qualitative Tests
Dimensions
Stimulus Legs Eye Head Body
A1 1 1 1 0
A2 1 0 1 0
A3 1 0 1 1
A4 1 1 0 1
A5 0 1 1 1
B1 1 1 0 0
B2 0 1 1 0
B3 0 0 0 1
B4 0 0 0 0
Quantitative vs. Qualitative Tests
Dimensions
Stimulus Legs Eye Head Body
A1 1 1 1 0
A2 1 0 1 0
A3 1 0 1 1
A4 1 1 0 1
A5 0 1 1 1
B1 1 1 0 0
B2 0 1 1 0
B3 0 0 0 1
B4 0 0 0 0
Prototype vs.ExemplarTheories
Quantitative Test
P(Correct)
Stimulus Data Prototype Exemplar
A1 .58 .65 .60
A2 .66 .60 65
A3 .58 .61 .61
A4 .71 .74 .78
A5 .45 .45 .40
B1 .41 .42 .40
B2 .47 .46 .45
B3 .59 .60 .60
B4 .65 .61 .63
GOF .0119 .0103 Made-up #s
Qualitative Test
Dimensions
Stimulus Legs Eye Head Body
A1 1 1 1 0
A2 1 0 1 0
A3 1 0 1 1
A4 1 1 0 1
A5 0 1 1 1
B1 1 1 0 0
B2 0 1 1 0
B3 0 0 0 1
B4 0 0 0 0
<- More ‘protypical’<- Less ‘prototypcial’
Qualitative Test
Dimensions
Stimulus Legs Eye Head Body
A1 1 1 1 0
A2 1 0 1 0
A3 1 0 1 1
A4 1 1 0 1
A5 0 1 1 1
B1 1 1 0 0
B2 0 1 1 0
B3 0 0 0 1
B4 0 0 0 0
<- Similar to A1, A3<- Similar to A2, B6, B7
Prototype: A1>A2Exemplar: A2>A1
Quantitative Test
P(Correct)
Stimulus Data Prototype Exemplar
A1 .58 .65 .60
A2 .66 .60 65
A3 .58 .61 .61
A4 .71 .74 .78
A5 .45 .45 .40
B1 .41 .42 .40
B2 .47 .46 .45
B3 .59 .60 .60
B4 .65 .61 .63
GOF .0119 .0103 Made-up #s
Quantitative vs. Qualitative Tests
• You ALWAYS have to figure out how to split up your data.– Batchelder & Riefer, 1980 used E1, E2, etc
instead of raw outputs.– Rumelhart & Greeno, 1971 looked at
particular triples.