When expectation-maximization-based theories work or do not work: An eye-tracking study of the discrepancy between everyone and every one

doi:10.3724/SP.J.1041.2022.01517

Abstract

Abstract:

Mainstream theorists in risky decision-making have developed various expectation theories with the ambitious goal of capturing everyone’s choices. However, ample evidence has revealed that these expectation theories could not capture every individual’s (“every one’s”) actual risky choice as descriptive theories. With doubts about the default compatibility between everyone (full set) and every one (subset), we used an eye-tracking technique to explore whether a theory for everyone would work well for every one. We found that expectation theories could capture the choice of an individual when making decisions for everyone and for self in a multiple-play condition, but could not capture the choice of an individual when making decisions for self in a single-play condition. Our findings contribute to a better understanding of the boundaries of expectation theories and those of heuristic/non-expectation models, and may shed light on the general issue of the classification of risky decision-making theories.

Key words: risky choice, decision for everyone, expectation-maximization, discrepancy between everyone and every one, eye-tracking

LIU Hong-Zhi, LI Xingshan, LI Shu, RAO Li-Lin. (2022). When expectation-maximization-based theories work or do not work: An eye-tracking study of the discrepancy between everyone and every one. Acta Psychologica Sinica, 54(12), 1517-1531.

Figures/Tables 8

References 62

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Model	Decision rule
Expected value (EV) theory	Calculate the sum of all weighted possible outcomes using the formula Σ p_i · x_i. Choose the gamble with the highest weighted sum.
Expected utility (EU) theory	Calculate the sum of all weighted outcomes using the following formula: Σ p_i · u(x_i). Choose the gamble with the highest weighted sum. We assumed the utility function u(x_i) = log(x_i) in this study (Su et al., 2013).
Cumulative prospect theory (CPT)	Calculate the sum of all weighted outcomes using the following formula: Σ π(p_i) · v(x_i), π(p_i) = p_i^γ/[ p_i^γ + (1 - p_i)^γ]^1/^γ, v(x_i) = x_i^α. Choose the gamble with the highest weighted sum. We estimated the values of parameters in individual level.
Equate-to-differentiate (ETD) model	Choose the gamble with more attractive gain on the dimension (best or worst payoff) with the greatest intradimensional utility difference. We assumed the utility function u(x_i) = log(x_i) in this study (Su et al., 2013).
Maximax heuristic (MH)	Choose the gamble with the highest monetary payoff.
Tallying heuristic (TH)	Give a tally mark to the gamble with (a) the higher minimum gain, (b) the higher maximum gain, (c) the lower probability of the minimum gain, and (d) the higher probability of the maximum gain. Select the gamble with the higher number of tally marks.

Model	Decision rule
Expected value (EV) theory	Calculate the sum of all weighted possible outcomes using the formula Σ p_i · x_i. Choose the gamble with the highest weighted sum.
Expected utility (EU) theory	Calculate the sum of all weighted outcomes using the following formula: Σ p_i · u(x_i). Choose the gamble with the highest weighted sum. We assumed the utility function u(x_i) = log(x_i) in this study (Su et al., 2013).
Cumulative prospect theory (CPT)	Calculate the sum of all weighted outcomes using the following formula: Σ π(p_i) · v(x_i), π(p_i) = p_i^γ/[ p_i^γ + (1 - p_i)^γ]^1/^γ, v(x_i) = x_i^α. Choose the gamble with the highest weighted sum. We estimated the values of parameters in individual level.
Equate-to-differentiate (ETD) model	Choose the gamble with more attractive gain on the dimension (best or worst payoff) with the greatest intradimensional utility difference. We assumed the utility function u(x_i) = log(x_i) in this study (Su et al., 2013).
Maximax heuristic (MH)	Choose the gamble with the highest monetary payoff.
Tallying heuristic (TH)	Give a tally mark to the gamble with (a) the higher minimum gain, (b) the higher maximum gain, (c) the lower probability of the minimum gain, and (d) the higher probability of the maximum gain. Select the gamble with the higher number of tally marks.

Variable	df	F	p	η²_p	Variable	df	F	p	η²_p
PTIS					SMI
Task	2, 94	5.02	0.008	0.10	Task	2, 94	8.59	< 0.001	0.16
ED	2, 94	0.19	0.827	0.00	ED	2, 94	0.72	0.487	0.02
OD	2, 94	3.68	0.029	0.07	OD	2, 94	0.32	0.725	0.01
Task × ED	4, 188	1.03	0.396	0.02	Task × ED	4, 188	1.60	0.177	0.03
Task × OD	4, 188	2.96	0.021	0.06	Task × OD	4, 188	0.16	0.961	0.00
ED × OD	4, 188	0.57	0.686	0.01	ED × OD	4, 188	6.84	< 0.001	0.13
Task × ED × OD	8, 376	0.93	0.488	0.02	Task × ED × OD	8, 376	1.20	0.298	0.03
MFD					PSTB
Task	2, 94	10.94	< 0.001	0.19	Task	2, 94	9.03	< 0.001	0.16
ED	2, 94	22.75	< 0.001	0.33	ED	2, 94	1.49	0.230	0.03
OD	2, 94	2.93	0.058	0.06	OD	2, 94	5.14	0.008	0.10
Task × ED	4, 188	5.78	< 0.001	0.11	Task × ED	4, 188	0.96	0.433	0.02
Task × OD	4, 188	1.22	0.304	0.03	Task × OD	4, 188	0.86	0.491	0.02
ED × OD	4, 188	3.65	0.007	0.07	ED × OD	4, 188	0.33	0.860	0.01
Task × ED × OD	8, 376	1.79	0.079	0.04	Task × ED × OD	8, 376	1.30	0.244	0.03

Variable	df	F	p	η²_p	Variable	df	F	p	η²_p
PTIS					SMI
Task	2, 94	5.02	0.008	0.10	Task	2, 94	8.59	< 0.001	0.16
ED	2, 94	0.19	0.827	0.00	ED	2, 94	0.72	0.487	0.02
OD	2, 94	3.68	0.029	0.07	OD	2, 94	0.32	0.725	0.01
Task × ED	4, 188	1.03	0.396	0.02	Task × ED	4, 188	1.60	0.177	0.03
Task × OD	4, 188	2.96	0.021	0.06	Task × OD	4, 188	0.16	0.961	0.00
ED × OD	4, 188	0.57	0.686	0.01	ED × OD	4, 188	6.84	< 0.001	0.13
Task × ED × OD	8, 376	0.93	0.488	0.02	Task × ED × OD	8, 376	1.20	0.298	0.03
MFD					PSTB
Task	2, 94	10.94	< 0.001	0.19	Task	2, 94	9.03	< 0.001	0.16
ED	2, 94	22.75	< 0.001	0.33	ED	2, 94	1.49	0.230	0.03
OD	2, 94	2.93	0.058	0.06	OD	2, 94	5.14	0.008	0.10
Task × ED	4, 188	5.78	< 0.001	0.11	Task × ED	4, 188	0.96	0.433	0.02
Task × OD	4, 188	1.22	0.304	0.03	Task × OD	4, 188	0.86	0.491	0.02
ED × OD	4, 188	3.65	0.007	0.07	ED × OD	4, 188	0.33	0.860	0.01
Task × ED × OD	8, 376	1.79	0.079	0.04	Task × ED × OD	8, 376	1.30	0.244	0.03