|Other Abstract||In single-outcome risk choices, the common patio effect, which violates the independence axiom, and intransitivity, which violates the transitivity axiom, have been reported in the literature. Prospect theory can account for the common ratio effect but not for intransitivity, whereas the proportional difference model can account for intransitivity but not for the common ratio effect. The equate-to-differentiate theory (ETD) is proposed so that weak-dominance principle can be made applicable in binary choice. In single-outcome risk choices, ETD suggests that people tend to compare the subjective difference in outcomes (△outcome) and the subjective difference in probabilities (△probability) between the two options, and if △outcome/△probability is greater, then people will treat the smaller △probability/△outcome as if there is no difference (equate).
In other words, the two options are treated as if they have a weak-dominance relationship. Following the weak-dominance principle, people are likely to choose the option with greater outcome or probability (differentiate) than other options.
Although both common ratio effect and intransitivity can be explained by ETD, the theory is, thus far, a qualitative theory of decision making. Therefore, the aim of the current study is two-fold: 1)to develop a quantitative mathematical model for ETD and 2) to carry out model comparison among the newly modeled ETD and two quantitative mathematical models, namely, prospect theory and proportional difference model. Three studies were designed to achieve this goal.
In Study 1，we use the parameter-free elicitation and visual analogue scale to determine the shape of each subject's utility function and subjective probability function of ETD in the gain domain using a standard outcome sequence and a standard probability sequence. A within-subj ects, two-condition (increasing and decreasing sequences) experimental design was developed in this study. According to the results: 1)the proportion of subjects with concave utility functions was significantly greater than 50%, either when evaluated simultaneously in both conditions or separately in each condition, thus supporting the hypothesis of diminishing marginal utility, and 2) the proportion of subjects with the inverse S-shaped subjective probability function was significantly greater than 50% in decreasing sequences, whereas this proportion did not differ from 50% in decreasing sequences or in both conditions, thus indicating a large individual variation in subjective probability functions, rather strongly supporting the场pothesis of S-shaped subjective probability function, with the largest number of subjects exhibiting inverse S-shaped functions, but the subjective probability functions of 32% of the subjects and 25% of the subjects, respectively, were uncertain or concave in at least one condition.
In Study 2, we proposed a quantitative mathematical representation of ETD. First based on our finding, which suggested a diminishing marginal utility, we chose the logarithmic function proposed by Kirby (2011)as the utility function of ETD to describe people's subjective perception of money. Second, also based on our finding, which suggested that the subjective function is S-shaped or concave, we chose the two-parameter exponential function proposed by Prelec (1998) as the subjective probability function of ETD to describe people's subjective perception of objective probability. Third, taking account of previous research, we proposed an attribute difference function of ETD to describe people's perceptions of the differences between attributes (outcome vs. probability).
In Study 3, we collected data of individuals' decisions on 20 pairs of single-outcome risk choices, and randomly selected 16 pairs as the training sets for fitting ETD, the proportional difference model, and prospect theory. Then, we used the fitting results of the three models to predict individuals' decision outcomes in each of the remaining four pairs (test set). Results showed that, of the three competing models, ETD provided the largest goodness-of-fit and the highest out-of-sample predictability of the data from training set. Furthermore, ETD provided the best account of the data from the test set.
In summary, this study developed a refined version of ETD, proposing a mathematical representation of ETD for single-outcome risk choices in the gain domain. The findings of this work could help lay a foundation for further modeling for single-outcome risk choices in the loss domain and, hopefully, for modelling for intertemporal choices and spatial choices in the future.|