The Technion Prediction Tournament

Organized by: Ido Erev, Eyal Ert, and Alvin E. Roth

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7.1. The Baseline Models - One shot decisions under risk (Condition Descprition)

7.1.1 Original (5- parameter) Cumulative prospect theory (CPT)

Download SAS example of the CPT model here

Download MATLAB example of the CPT model here

According to cumulative prospect theory (Tversky & Kahneman, 1992), decision-makers are assumed to select the prospect with the highest weighted value.  The weighted value of Prospect X that pays x1 with probability p1 and x2 otherwise is: 

(1)                                                                  

where V(xi) is the subjective value of outcome xi, and π (pi) is the subjective weight of outcome xi

The subjective values are given by a value function that can be described as follows:

 (2)                                                                                                             

The parameters 0 < α < 1 and 0 < β < 1 capture the assumption of diminishing sensitivity in the gain and the loss domain respectively.  The parameter λ >1 captures loss aversion assertion.

            The subjective weights are assumed to depend on the outcomes' rank, sign, and on a cumulative weighting function.  When the two outcomes are of different sign, the weight of outcome i is:

(3)                                                                        

 

The parameters 0 < γ < 1 and 0 < δ < 1 capture the tendency to overweight low-probability outcomes. 

When the outcomes are of the same sign, the weight of the most extreme outcome (largest absolute value) is computed with equation (3) (as if it is the sole outcome of that sign), and the weight of the less extreme outcome is the difference between that value and 1.

The prediction of CPT is estimated with three sets of parameters: Tversky and Kahneman’s parameters (for the median decision maker), the parameters estimated in Ert and Erev (2007), and the parameters that best fit the current data.  Table 2 presents the different predictions.  It show that all three set of parameters capture the median choice relatively well: Proportion of agreement between the median choice and the prediction is above 90% in all cases.  However, the mean square deviation (MSD) score is relatively high.

 

7.1.2 Stochastic cumulative prospect theory (SCPT). 

The second model considered here is the stochastic variant of cumulative prospect theory proposed by Erev, Roth, Slonim and Barron (2002, and see a similar idea in Busemeyer, 1985).

The model assume that the probability of selecting the risky prospect (R) over the safe prospect (S) is

 

(4)                                                                                           

 

The parameter μ captures payoff sensitivity, and D is the absolute distance between the cumulative functions implied by the two prospects.  The computation of D requires a normalization of the weights of the different outcomes.  The normalized weight of outcome x1 is 

(5)                                                                                        

Assuming x1> x2, the cumulative normal value of gamble X at point z (0 < z < 1) is

(6)                                                                            

D is the absolute distance between the cumulative normal payoffs of the two prospects. 

            Following Ert and Erev (2007) we focused on a three parameters simplification of SCPT.  The simplification involves the assumption of gain loss symmetry that implies: λ =1, β = α,  δ = γ.  Table 2 presents the predictions of this model with the parameters estimated by Ert and Erev (2007), and with the parameters that best fit the current data.  The results show that SCPT matches the proportion of agreement of the original model and reduces the MSD score.

 

7.1.3  The priority heuristic

(a minor bug was corrected on August 14, 2008)

According to the priority heuristic (Brandstätter, Gigerenzer, & Hertwig, 2006), decision-makers are assumed to follow a lexicographic rule that evaluate alternatives by sequential comparison of their minimum value, their maximum value, and their respective probabilities.

               The priority rule for nonnegative prospects asserts that the first comparison is done between the two minimum gains, the stopping rule is determined by a free cutoff parameter s: if the minimum gains differ by s (or more) of the maximum gain then the examination is stopped and the prospect with the better minimum gain is selected. If the difference does not pass this cutoff then the two probabilities of the minimum gains are matched. The examination is stopped if these probabilities differ by s (or more) of the probability scale. In the case that this cutoff rule is also not satisfied, the decision maker selects whichever prospect with the higher maximum gain. The same procedure applies for mixed problems and for nonpositive prospects (in such a case the word "gain" is simply replaced with the word "loss"). 

               The prediction of the Priority Heuristic is estimated with two sets of parameters: the stopping rule parameter used by Brandstätter et al. (2006) and the stopping rule parameter that best fit the current data. The results show that this model capture the median choice as well as the CPT model: Proportion of agreement between the median choice and the prediction is above 90%.  However, the mean square deviation (MSD) score is relatively high.

 

Table 2: Condition Description: The upper panel presents the aggregated proprotion of risky choices (Prisk) and the predictions of the baseline models.  The lower panel presents summary statistic.

One shot decisions under risk (Condition Description)

 

Risk

Safe

observed

CPT

SCPT

Priority Rule

#

High

P(High)

Low

Med

Prisk

Org

α=.88

β=.88

λ=2.25

γ=.61

δ=.69

Ert & Erev

α=β=.86

λ=1

γ= δ=.5

 

Fit

α=.70

β=.70

λ=1

γ=.65

δ=.65

Org

α=β=.77

λ=1

γ=δ=.71

μ=2.04

Fit

α=.89

β=.98

λ=1.5

γ=.7

δ=.7

μ=2.15

Org*

s=.1

1

-0.3

0.96

-2.1

-0.3

0.20

0

0

0

0.12

0.11

0

2

-0.9

0.95

-4.2

-1.0

0.20

0

0

0

0.26

0.22

0

3

-6.3

0.30

-15.2

-12.2

0.60

1

1

1

0.59

0.58

1

4

-10.

0.20

-29.2

-25.6

0.85

1

1

1

0.69

0.68

1

5

-1.7

0.90

-3.9

-1.9

0.30

0

0

0

0.36

0.33

0

6

-6.3

0.99

-15.7

-6.4

0.35

0

0

0

0.26

0.23

0

7

-5.6

0.70

-20.2

-11.7

0.50

1

1

1

0.63

0.6

1

8

-0.7

0.10

-6.5

-6.0

0.75

1

1

1

0.75

0.75

1

9

-5.7

0.95

-16.3

-6.1

0.30

0

0

0

0.29

0.25

0

10

-1.5

0.92

-6.4

-1.8

0.15

0

0

0

0.32

0.28

0

11

-1.2

0.02

-12.3

-12.1

0.90

1

1

1

0.8

0.8

1

12

-5.4

0.94

-16.8

-6.4

0.10

0

0

0

0.45

0.41

0

13

-2.0

0.05

-10.4

-9.4

0.50

1

1

1

0.58

0.56

1

14

-8.8

0.60

-19.5

-15.5

0.70

1

1

1

0.72

0.72

1

15

-8.9

0.08

-26.3

-25.4

0.60

1

1

1

0.79

0.79

1

16

-7.1

0.07

-19.6

-18.7

0.55

1

1

1

0.73

0.73

1

17

-9.7

0.10

-24.7

-23.8

0.90

1

1

1

0.79

0.79

1

18

-4.0

0.20

-9.3

-8.1

0.65

1

1

1

0.63

0.63

1

19

-6.5

0.90

-17.5

-8.4

0.55

1

0

1

0.54

0.51

1

20

-4.3

0.60

-16.1

-4.5

0.05

0

0

0

0.13

0.12

0

21

2.0

0.10

-5.7

-4.6

0.65

1

1

1

0.68

0.6

0

22

9.6

0.91

-6.4

8.7

0.05

0

0

0

0.19

0.17

0

23

7.3

0.80

-3.6

5.6

0.15

0

0

0

0.25

0.23

0

24

9.2

0.05

-9.5

-7.5

0.50

1

1

1

0.61

0.49

0

25

7.4

0.02

-6.6

-6.4

0.90

1

1

1

0.84

0.82

1

26

6.4

0.05

-5.3

-4.9

0.65

1

1

1

0.81

0.78

1

27

1.6

0.93

-8.3

1.2

0.15

0

0

0

0.24

0.18

0

28

5.9

0.80

-0.8

4.6

0.35

0

0

0

0.27

0.28

0

29

7.9

0.92

-2.3

7.0

0.40

0

0

0

0.24

0.24

0

30

3.0

0.91

-7.7

1.4

0.40

0

0

0

0.45

0.34

0

31

6.7

0.95

-1.8

6.4

0.10

0

0

0

0.16

0.15

0

32

6.7

0.93

-5.0

5.6

0.25

0

0

0

0.29

0.26

0

33

7.3

0.96

-8.5

6.8

0.15

0

0

0

0.19

0.17

0

34

1.3

0.05

-4.3

-4.1

0.75

1

1

1

0.83

0.82

1

35

3.0

0.93

-7.2

2.2

0.25

0

0

0

0.31

0.24

0

36

5.0

0.08

-9.1

-7.9

0.40

1

1

1

0.75

0.69

0

37

2.1

0.80

-8.4

1.3

0.10

0

0

0

0.22

0.17

0

38

6.7

0.07

-6.2

-5.1

0.65

1

1

1

0.71

0.63

0

39

7.4

0.30

-8.2

-6.9

0.85

1

1

1

0.84

0.82

0

40

6.0

0.98

-1.3

5.9

0.10

0

0

0

0.13

0.12

0

41

18.8

0.80

7.6

15.5

0.35

0

0

0

0.45

0.45

0

42

17.9

0.92

7.2

17.1

0.15

0

0

0

0.26

0.25

0

43

22.9

0.06

9.6

9.2

0.75

1

1

1

0.88

0.89

1

44

10.0

0.96

1.7

9.9

0.20

0

0

0

0.15

0.14

0

45

2.8

0.80

1.0

2.2

0.55

0

0

0

0.49

0.5

0

46

17.1

0.10

6.9

8.0

0.45

1

1

1

0.59

0.62

1

47

24.3

0.04

9.7

10.6

0.65

1

1

1

0.58

0.61

1

48

18.2

0.98

6.9

18.1

0.10

0

0

0

0.16

0.15

0

49

13.4

0.50

3.8

9.9

0.05

0

0

0

0.31

0.31

0

50

5.8

0.04

2.7

2.8

0.70

1

1

1

0.72

0.74

1

51

13.1

0.94

3.8

12.8

0.15

0

0

0

0.19

0.18

0

52

3.5

0.09

0.1

0.5

0.35

1

1

0

0.47

0.54

0

53

25.7

0.10

8.1

11.5

0.40

0

0

0

0.42

0.44

0

54

16.5

0.01

6.9

7.0

0.85

1

1

1

0.74

0.77

1

55

11.4

0.97

1.9

11.0

0.15

0

0

0

0.27

0.26

0

56

26.5

0.94

8.3

25.2

0.20

0

0

0

0.28

0.28

0

57

11.5

0.60

3.7

7.9

0.35

0

0

0

0.46

0.47

0

58

20.8

0.99

8.9

20.7

0.25

0

0

0

0.18

0.17

0

59

10.1

0.30

4.2

6.0

0.45

1

0

0

0.5

0.52

0

60

8.0

0.92

0.8

7.7

0.20

0

0

0

0.19

0.18

0

Mean

0.40

0.45

0.42

0.42

0.46

0.45

0.33

MSD (X100)

9.99

9.99

9.32

1.34

1.16

11.16

Correlation with observed data

0.84

0.83

0.85

0.91

0.92

0.76

Proportion of agreement

0.91

0.91

0.95

0.93

0.91

0.91

* The fit of the priority heuristic yielded the same parameter value as the original model.