[PPT] - Direct influences vs contextuality in human choices Vctor H. PowerPoint Presentation

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Direct influences vs contextuality in human choices

Víctor H. Cervantes1, Ehtibar N. Dzhafarov2

Purdue University

1cervantv@purdue.edu 2ehtibar@purdue.edu

Purdue Winer Memorial Lectures - 2018

Purdue University November 11, 2018

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Introduction Contextuality-by-Default

Introduction

Contextuality-by-Default

Principles Within a context, random variables are jointly distributed. Otherwise, they are stochastically unrelated.

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Introduction Contextuality-by-Default

Introduction

Contextuality-by-Default

Principles The identity of a random variable is not completely given by its content. Random variables in different contexts are necessarily different. Context needs to be included in their description. Accordingly, random variables should be labeled both by their content and the contexts: Rc

q

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Introduction Contextuality-by-Default

Introduction

Contextuality-by-Default

R1

1

R1

2

· · c1 · R2

2

R2

3

· c2 · · R3

3

R3

4

c3 R4

1

· · R4

4

c4 q1 q2 q3 q4 R

SLIDE 5

Introduction Couplings

Introduction

Couplings

Coupling A coupling of a set of random variables { X, Y, Z, . . . } is a random variable

X,

Y, Z, . . .

(with jointly distributed components), such that
X d

= X,

Y d

= Y,

Z d

= Z, . . ., where d = stands for “has the same distribution as.” A coupling always exists, generally non-uniquely.

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Introduction Couplings

Introduction

Couplings

Maximal coupling A maximal coupling of a set of random variables { X, Y, Z, . . . } is a coupling

X,

Y, Z, . . .

with the maximal possible value of Pr
X =

Y = Z = . . .

.

A maximal coupling always exists, generally non-uniquely. The maximal probability equals 1 if and only if X d = Y d = Z d = . . .

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Introduction Couplings

Introduction

Couplings

Maximal coupling - Example Let X and Y be two binary random variables that take values 1 or −1. Suppose that Pr(X = 1) = 1/3 and Pr(Y = 1) = 1/8 The pair ( X, Y) with

Y = 1
Y = −1
X =

1

1/8 5/24 1/3

X = −1

2/3 2/3 1/8 7/8

is the maximal coupling of the variables X and Y.

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Contextuality

Contextuality in Contextuality-by-Default

Contextuality (Binary random variables) Given the set of maximal couplings for all pairs of content-sharing variables in a system R, the system is said to be noncontextual if R has a coupling which contains as a marginal of each pair the corresponding maximal coupling. Otherwise R is said to be contextual.

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Contextuality

Contextuality in Contextuality-by-Default

R1

1

R1

2

· · c1 · R2

2

R2

3

· c2 · · R3

3

R3

4

c3 R4

1

· · R4

4

c4 q1 q2 q3 q4 R

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Contextuality

Contextuality in Contextuality-by-Default

R1

1

R1

2

· · · R2

2

R2

3

· · · R3

3

R3

4

R4

1

· · R4

4

= ⇒ S1

R1

1

R1

2

· · S2 ·

R2

2

R2

3

· S3 · ·

R3

3

R3

4

S4

R4

1

· ·

R4

4

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Contextuality

Contextuality in Contextuality-by-Default

S1

?

= C1 S2

?

= C2 S3

?

= C3 S4

?

= C4

R1

1

R1

2

· · ·

R2

2

R2

3

· · ·

R3

3

R3

4

R4

1

· ·

R4

4

where Cj is the maximal coupling of the two content-sharing random variables in the column.

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Contextuality

Contextuality in Contextuality-by-Default

Criterion for cyclic systems (Kujala & Dzhafarov, 2016) A cyclic system of binary random variables taking values ±1 is noncontextual if and only if sodd − (n − 2) − ∆ 0 where sodd = maxodd # of −′s n

i=1 ±

Ri

iRi i⊕1

∆

= n

i=1

Ri⊖1

i

−
Ri

i

and X denotes the expected value of X.

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Experiments

Snow Queen (Cervantes & Dzhafarov, 2018) Four randomly assigned conditions. Two choices:

One of a character from a given pair of characters, and of a suitable characteristic of this character from a given pair of characteristics.

PR-like rank 3 and 4, (Basieva, Cervantes, Dzhafarov, & Khrennikov, 2018) Maximum value of sodd. Four experiments of rank 3 and two of rank 4. Three or four randomly assigned conditions, respectively.

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Experiments Snow Queen

Snow Queen

The Snow Queen by Elena Ringo. Licensed under the CC Attribution 3.0 Unported license.

Hans Christian Andersen’s “The Snow Queen” story involves the following characters with the following characteristics: Snow Queen is Beautiful and Evil. Gerda is Beautiful and Kind. The Troll is Unattractive and Evil. The Old Finn Woman is Unattractive and Kind.

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Experiments Snow Queen

Snow Queen

R1

1

R1

2

· · c1 = (q1, q2) · R2

2

R2

3

· c2 = (q2, q3) · · R3

3

R3

4

c3 = (q3, q4) R4

1

· · R4

4

c4 = (q1, q4) q1 q2 q3 q4 R

Choices: q1 Gerda / Troll q2 Beautiful / Unattractive q3 Snow Queen / Old Finn woman q4 Kind / Evil

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Experiments Snow Queen

Snow Queen experiment

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Experiments Snow Queen

Snow Queen

Character choice Characteristic choice N total (correct) Context 1 ⋆ Gerda ⋆ Beautiful 447 (425) ⋆ Troll ⋆ Unattractive Context 2 ⋆ Snow Queen ⋆ Beautiful 446 (410) ⋆ Old Finn Woman ⋆ Unattractive Context 3 ⋆ Snow Queen ⋆ Kind 453 (388) ⋆ Old Finn Woman ⋆ Evil Context 4 ⋆ Gerda ⋆ Kind 453 (429) ⋆ Troll ⋆ Evil

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Experiments Snow Queen

Snow Queen

Context 1 Beautiful Ugly

Mar. Character

Context 4 Kind Evil

Mar. Character

Gerda 0.887 0.887 Gerda 0.841 0.841 Troll 0.113 0.113 Troll 0.159 0.159

Mar. Characteristic

0.887 0.113 1 (equality)

Mar. Characteristic

0.841 0.159 1 (equality) Context 2 Beautiful Ugly

Mar. Character

Context 3 Kind Evil

Mar. Character

Snow Queen 0.837 0.837 Snow Queen 0.626 0.626 Old Finn woman 0.163 0.163 Old Finn woman 0.374 0.374

Mar. Characteristic

0.837 0.163 1 (equality)

Mar. Characteristic

0.374 0.627 0 (equality)

sodd − (n − 2) − ∆ = 0.452 > 0

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Experiments Snow Queen

Snow Queen

Context 1 Beautiful Ugly

Mar. Character

Context 4 Kind Evil

Mar. Character

Gerda 0.843 0.020 0.864 Gerda 0.797 0.035 0.832 Troll 0.029 0.107 0.136 Troll 0.018 0.150 0.168

Mar. Characteristic

0.872 0.128 0.951 (equality)

Mar. Characteristic

0.815 0.185 0.947 (equality) Context 2 Beautiful Ugly

Mar. Character

Context 3 Kind Evil

Mar. Character

Snow Queen 0.769 0.011 0.780 Snow Queen 0.135 0.537 0.672 Old Finn woman 0.070 0.150 0.220 Old Finn woman 0.320 0.008 0.328

Mar. Characteristic

0.839 0.161 0.919 (equality)

Mar. Characteristic

0.455 0.545 0.143 (equality)

sodd − (n − 2) − ∆ = 0.280 > 0

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Experiments Snow Queen

Snow Queen

SLIDE 21

Experiments PR rank 3

PR rank 3

A fictional Alice is faced with making two choices; each choice was between two alternatives. Experiments 1–4

1. Meals Alice wishes to order a two-course meal. For each course

she can choose a high-calorie (H) or a low-calorie (L)

ption. She does not want both courses to be H nor does

she want both of them to be L.

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Experiments PR rank 3

PR rank 3

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Experiments PR rank 3

PR rank 3

A fictional Alice is faced with making two choices; each choice was between two alternatives. Experiments 1–4

2. Clothes Alice is dressing for work, and chooses two pieces of

clothing.

3. Presents Alice wishes to buy two presents for her nephew’s birthday.
4. Exercises Alice is doing two physical exercises.

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Experiments PR rank 3

PR rank 3

Table: Dichotomous choices in experiments 1 to 4

q1 q2 q3

1. Meals

Starters: Main course: Dessert: Soup (H)* or Salad (L) Burger (H)* or Beans (L) Cake (H)* or Coffee (L)

2. Clothes

Skirt: Blouse: Jacket: Plain* or Fancy Plain* or Fancy Plain* or Fancy

3. Presents

Book: Soft toy (bear): Construction set: Big expensive book (E)* or Smaller book(C) (E)* or (C) (E)* or (C)

4. Exercises

Arms: Back: Legs: Hard* or Easy Hard* or Easy Hard* or Easy

* Denotes the response encoded with +1

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Experiments PR rank 3

PR rank 3

R1

1

R1

2

· c1 = (q1, q2) · R2

2

R2

3

c2 = (q2, q3) R3

1

· R3

3

c3 = (q1, q3) q1 q2 q3 R3

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Experiments PR rank 3

PR rank 4

A fictional Alice is faced with making two choices; each choice was between two alternatives. Experiments 5 and 6

5. Directions Alice goes for a walk, and has to choose path directions at
forks. Alice wants the two directions to be as similar as

possible (i.e., the angle between them to be as small as possible).

6. Colored figures Alice is taking a drawing lesson, and is presented

with two pairs consisting of a square and a circle. Alice needs to choose one figure from each and she wants them to be of similar color.

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Experiments PR rank 3

PR rank 4

Table: Dichotomous choices in experiments 5 and 6

q1 q2 q3 q4

5. Directions

West-East fork NorthWest-SouthEast fork North-South fork NorthEast-SouthWest fork ←

r →

տ

r ց

↑

r ↓

ր

r ւ
6. Colored figures
ne of
ne of
ne of
ne of

For each choice qi, the response encoded by +1 is the one on the left: e.g., for q1 in Experiment 5, the response ← was encoded by +1.

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Experiments PR rank 3

PR rank 4

R1

1

R1

2

· · c1 = (q1, q2) · R2

2

R2

3

· c2 = (q2, q3) · · R3

3

R3

4

c3 = (q3, q4) R4

1

· · R4

4

c4 = (q1, q4) q1 q2 q3 q4 R4

SLIDE 29

Experiments PR rank 3

PR rank 3

c1 R1

2 = 1

R1

PR rank 3

Table: Probability estimates p1, p2, p3 that determine the outcomes of Experiments 1–4, and the cell sample sizes n1, n2, n3.

Experiment c1 c2 c3 p1 n1 p2 n2 p3 n3

1. Meals

0.349 2090 0.658 2052 0.653 2050

2. Clothes

0.639 1996 0.566 2086 0.435 2110

3. Presents

0.547 2081 0.387 2052 0.515 2059

4. Exercises

0.590 2058 0.306 2024 0.580 2110

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Experiments PR rank 3

PR rank 4

Table: Probability estimates p1, p2, p3, p4 that determine the outcomes of Experiments 5 and 6 and the cell sample sizes n1, n2, n3, n4. Experiment c1 c2 c3 c4 p1 n1 p2 n2 p3 n3 p4 n4

5. Directions

0.471 1549 0.706 1504 0.645 1537 0.750 1602

6. Colored figures

0.419 1603 0.819 1589 0.360 1482 0.154 1517∗

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Experiments PR rank 3

PR results

sodd ∆ sodd − (n − 2) − ∆

1. Meals

3 0.639 1.361

2. Clothes

3 0.560 1.440

3. Presents

3 0.452 1.548

4. Exercises

3 0.777 1.223

5. Directions

4 1.242 0.758

6. Colored figures

4 2.984 −0.984

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Experiments PR rank 3

PR results

1.361 1.077 1.456 20000 40000 60000 80000 1.0 1.1 1.2 1.3 1.4 1.5 1.6

sodd − 1 − ∆ Count

1. Meals

1.440 1.154 1.605 20000 40000 60000 80000 1.1 1.2 1.3 1.4 1.5 1.6 1.7

sodd − 1 − ∆ Count

2. Clothes

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Experiments PR rank 3

PR results

1.548 1.382 1.719 20000 40000 60000 80000 1.2 1.3 1.4 1.5 1.6 1.7 1.8

sodd − 1 − ∆ Count

3. Presents

1.223 1.067 1.385 20000 40000 60000 80000 1.0 1.1 1.2 1.3 1.4 1.5 1.6

sodd − 1 − ∆ Count

4. Exercises

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Experiments PR rank 3

PR results

0.758 0.448 1.069 10000 20000 30000 40000 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

sodd − 2 − ∆ Count

5. Directions

−0.984 −1.266 −0.695 10000 20000 30000 40000 −1.4 −1.3 −1.2 −1.1 −1.0 −0.9 −0.8 −0.7 −0.6

sodd − 2 − ∆ Count

6. Colored figures

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Closing remarks

With these experiments, we firmly establish that “quantum-like” contextuality can be observed outside quantum physics. The absence or presence of contextuality in a system is a fundamental aspect of its probabilistic structure. Contextuality or noncontextuality of a system of random variables describing human decisions could become a significant aspect of their comprehensive psychological theory.

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Closing remarks

While the design of all experiments drives the correlations to be maximal in absolute value, noncontextuality is not an unexpected result.

1

For PR rank 3, one third of systems with randomly selected marginals are noncontextual.

2

For PR rank 4, two thirds of systems with randomly selected marginals are noncontextual.

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Closing remarks

Bibliography I

Basieva, I., Cervantes, V. H., Dzhafarov, E. N., & Khrennikov, A. Y. (2018). True contextuality beats direct influences in human decision making. Retrieved from https://arxiv.org/abs/1807.05684 Cervantes, V. H., & Dzhafarov, E. N. (2018). Snow queen is evil and beautiful: Experimental evidence for probabilistic contextuality in human choices. Decision, 5, 193–204. doi: doi: 10.1037/dec0000095 Kujala, J. V., & Dzhafarov, E. N. (2016). Proof of a Conjecture on Contextuality in Cyclic Systems with Binary Variables. Foundations of Physics, 46, 282–299.