Direct influences vs contextuality in human choices Víctor H. Cervantes 1 , Ehtibar N. Dzhafarov 2 Purdue University 1 cervantv@purdue.edu 2 ehtibar@purdue.edu Purdue Winer Memorial Lectures - 2018 Purdue University November 11, 2018
Introduction Contextuality-by-Default Introduction Contextuality-by-Default Principles Within a context, random variables are jointly distributed. Otherwise, they are stochastically unrelated .
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: R c q
Introduction Contextuality-by-Default Introduction Contextuality-by-Default R 1 R 1 · · c 1 1 2 R 2 R 2 · · c 2 2 3 R 3 R 3 · · c 3 3 4 R 4 R 4 · · c 4 1 4 R q 1 q 2 q 3 q 4
Introduction Couplings Introduction Couplings Coupling A coupling of a set of random variables { X, Y, Z, . . . } � � X, � � Y, � is a random variable Z, . . . (with jointly distributed components), such that X d Y d Z d � � � = X, = Y, = Z, . . . , where d = stands for “has the same distribution as.” A coupling always exists, generally non-uniquely.
Introduction Couplings Introduction Couplings Maximal coupling A maximal coupling of a set of random variables { X, Y, Z, . . . } � � X, � � Y, � is a coupling Z, . . . � � X = � � Y = � with the maximal possible value of Pr Z = . . . . A maximal coupling always exists, generally non-uniquely. The maximal probability equals 1 if and only if X d = Y d = Z d = . . .
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 0 2 / 3 2 / 3 1 / 8 7 / 8 is the maximal coupling of the variables X and Y .
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 .
Contextuality Contextuality in Contextuality-by-Default R 1 R 1 · · c 1 1 2 R 2 R 2 · · c 2 2 3 R 3 R 3 · · c 3 3 4 R 4 R 4 · · c 4 1 4 q 1 q 2 q 3 q 4 R
Contextuality Contextuality in Contextuality-by-Default � � R 1 R 1 S 1 R 1 R 1 · · · · 1 2 1 2 � � R 2 R 2 S 2 R 2 R 2 · · · · 2 3 2 3 ⇒ = � � R 3 R 3 S 3 R 3 R 3 · · · · 3 4 3 4 � � R 4 R 4 S 4 R 4 R 4 · · · · 1 4 1 4
Contextuality Contextuality in Contextuality-by-Default ? ? ? ? = C 1 = C 2 = C 3 = C 4 S 1 S 2 S 3 S 4 � � R 1 R 1 · · 1 2 � � R 2 R 2 · · 2 3 � � R 3 R 3 · · 3 4 � � R 4 R 4 · · 1 4 where C j is the maximal coupling of the two content-sharing random variables in the column.
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 s odd − ( n − 2 ) − ∆ � 0 where � � � n R i i R i s odd = max odd # of − ′ s i = 1 ± i ⊕ 1 � �� � � � � � = � n R i ⊖ 1 R i ∆ � − � i = 1 i i and � X � denotes the expected value of X .
Experiments 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 s odd . Four experiments of rank 3 and two of rank 4. Three or four randomly assigned conditions, respectively.
Experiments Snow Queen Snow Queen Hans Christian Andersen’s “The Snow Queen” story involves the following characters with the following characteristics: The Snow Queen by Elena Ringo. Licensed under the CC Attribution 3.0 Unported license. 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.
Experiments Snow Queen Snow Queen Choices: R 1 R 1 · · c 1 = ( q 1 , q 2 ) 1 2 q 1 Gerda / Troll R 2 R 2 · · c 2 = ( q 2 , q 3 ) q 2 Beautiful / 2 3 Unattractive R 3 R 3 · · c 3 = ( q 3 , q 4 ) 3 4 q 3 Snow Queen / Old R 4 R 4 · · c 4 = ( q 1 , q 4 ) Finn woman 1 4 q 1 q 2 q 3 q 4 R q 4 Kind / Evil
Experiments Snow Queen Snow Queen experiment
Experiments Snow Queen Snow Queen Character choice Characteristic choice N total (correct) ⋆ Gerda ⋆ Beautiful Context 1 447 (425) ⋆ Troll ⋆ Unattractive ⋆ Snow Queen ⋆ Beautiful Context 2 446 (410) ⋆ Old Finn Woman ⋆ Unattractive ⋆ Snow Queen ⋆ Kind Context 3 453 (388) ⋆ Old Finn Woman ⋆ Evil ⋆ Gerda ⋆ Kind Context 4 453 (429) ⋆ Troll ⋆ Evil
Experiments Snow Queen Snow Queen Context 1 Beautiful Ugly Mar. Character Context 4 Kind Evil Mar. Character Gerda 0.887 0 0.887 Gerda 0.841 0 0.841 Troll 0 0.113 0.113 Troll 0 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 0.837 Snow Queen 0 0.626 0.626 Old Finn woman 0 0.163 0.163 Old Finn woman 0.374 0 0.374 Mar. Characteristic 0.837 0.163 1 (equality) Mar. Characteristic 0.374 0.627 0 (equality) s odd − ( n − 2 ) − ∆ = 0.452 > 0
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) s odd − ( n − 2 ) − ∆ = 0.280 > 0
Experiments Snow Queen Snow Queen
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) option. She does not want both courses to be H nor does she want both of them to be L.
Experiments PR rank 3 PR rank 3
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.
Experiments PR rank 3 PR rank 3 Table: Dichotomous choices in experiments 1 to 4 q 1 q 2 q 3 Starters: Main course: Dessert: 1. Meals Soup (H)* or Salad (L) Burger (H)* or Beans (L) Cake (H)* or Coffee (L) Skirt: Blouse: Jacket: 2. Clothes Plain* or Fancy Plain* or Fancy Plain* or Fancy Book: Soft toy (bear): Construction set: 3. Presents Big expensive book (E)* or Smaller book(C) (E)* or (C) (E)* or (C) Arms: Back: Legs: 4. Exercises Hard* or Easy Hard* or Easy Hard* or Easy * Denotes the response encoded with + 1
Experiments PR rank 3 PR rank 3 R 1 R 1 · c 1 = ( q 1 , q 2 ) 1 2 R 2 R 2 · c 2 = ( q 2 , q 3 ) 2 3 R 3 R 3 · c 3 = ( q 1 , q 3 ) 1 3 q 1 q 2 q 3 R 3
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.
Experiments PR rank 3 PR rank 4 Table: Dichotomous choices in experiments 5 and 6 q 1 q 2 q 3 q 4 West-East fork NorthWest-SouthEast fork North-South fork NorthEast-SouthWest fork 5. Directions ← or → տ or ց ↑ or ↓ ր or ւ one of one of one of one of 6. Colored figures For each choice q i , the response encoded by + 1 is the one on the left: e.g., for q 1 in Experiment 5, the response ← was encoded by + 1 .
Recommend
More recommend
