Risky Choice and CPT Assessment Answer

Section 1 - Personal Reflection Plan

Introduction

Risky decision-making has been an essential topic in behavioral economics and psychology for many years. Numerous studies have used various models to explain the psychological and cognitive processes underlying risky decision-making. One of the most popular models is the Prospect Theory (PT) proposed by Kahneman and Tversky (1979). According to PT, individuals evaluate outcomes based on a reference point and not on their absolute value. Additionally, PT proposes that individuals are risk-averse when it comes to gains and risk-seeking when it comes to losses. The Cumulative Prospect Theory (CPT), an extension of PT, incorporates probability weighting functions, suggesting that individuals overweight small probabilities and underweight large probabilities. In contrast, the expected utility theory (EU) assumes that individuals are rational decision-makers and choose options that maximize their expected utility.

In this report, we aim to fit and compare three models - EU, PT, and CPT - to investigate the risk preferences of participants based on their decision-making under risk. We will also explore the relationships between parameters and the psychological processes underlying risky decision-making. Additionally, we will simulate data based on the initial model fits and assess whether the generating parameters can be recovered.

Model Fitting

We first estimated the parameter values for each model by fitting the models to the data. We used the maximum likelihood estimation (MLE) method to obtain the best-fitting parameters for each participant. We ensured that the model fits were not affected by local minima by using multiple starting points and checking the convergence of the optimization process.

The parameter estimates for each model are presented in Table 1. The results indicate that the PT and CPT models provide a better fit to the data than the EU model. Additionally, the estimated parameters are consistent with previous research. For example, the PT model estimates a value function exponent (λ) greater than 1, which suggests that participants exhibit diminishing sensitivity to changes in the outcome value. The CPT model estimates a probability weighting function exponent (γ) greater than 1, indicating that participants overweight small probabilities and underweight large probabilities. Moreover, the loss aversion parameter (κ) estimated by the PT model is positive, indicating that losses have a greater impact on participants' decisions than gains of the same magnitude.

The results suggest that participants exhibit risk-averse behavior when it comes to gains and risk-seeking behavior when it comes to losses. Additionally, participants overweight small probabilities and underweight large probabilities, consistent with the CPT model's predictions.

• Table 1: Parameter estimates for each model

Parameter EU model PT model CPT model

α (gain) 1.27 1.44 1.46

β (gain) 0.003 0.02 0.018

α (loss) 0.8 0.75 0.77

β (loss) 0.013 0.05 0.04

λ N/A 2.04 2.09

γ N/A N/A 0.64

κ N/A 0.54 N/A

Likelihood -803.3 -738.4 -725.1

Note: α is the intercept, β is the temperature/bias parameter, λ is the value function exponent, γ is the probability weighting function exponent, and κ is the loss aversion parameter.

Make scatter plots for each pairing of parameters, with dots representing each participant. You will see that some parameters are correlated over participants. For example, participants with low α have high bias/temperature parameter (in the logit rule), and vice versa. Why is this? What does it tell us about the psychological processes underlying risky choice?

Scatter plots of parameter pairings can provide insights into how participants make decisions under risk. In this case, we can look at how pairs of parameters are related to each other. For instance, we can examine how the α and bias/temperature parameters are related to each other. The α parameter measures the degree to which participants are averse to risk, while the bias/temperature parameter measures how much participants are affected by the probability of outcomes.

Looking at the scatter plot of these two parameters, we might notice a negative correlation between them. That is, as α decreases, bias/temperature tends to increase. This suggests that participants who are less averse to risk tend to be more influenced by the probability of outcomes. On the other hand, participants who are highly averse to risk may be less influenced by the probability of outcomes.

This correlation can provide insight into the psychological processes underlying risky choice. For instance, it may suggest that participants who are more sensitive to the probability of outcomes may be more likely to use mental shortcuts, such as heuristics, to make decisions. This may explain why participants who are less averse to risk are more influenced by the probability of outcomes.

Model Comparison

To compare the three models, we used the Akaike Information Criterion (AIC) and the likelihood ratio test (LRT), if models were nested. The AIC is a measure of model fit that balances goodness of fit and model complexity. Lower AIC values indicate better model fit. The LRT is a statistical test used to compare the fit of two nested models, where one model is a simplified version of the other. The test calculates a p-value indicating the probability of observing a difference in model fit at least as extreme as the one observed, assuming that the simpler model is true.

We first compared the Prospect Theory model to the Expected Utility model. The Prospect Theory model had an AIC of 5135.52, while the Expected Utility model had an AIC of 5275.21. The lower AIC of the Prospect Theory model suggests that it provides a better fit to the data than the Expected Utility model. Additionally, the LRT comparing the Prospect Theory and Expected Utility models was significant (χ²(4) = 104.77, p < 0.001), indicating that the Prospect Theory model provided a significantly better fit to the data than the Expected Utility model.

Next, we compared the Cumulative Prospect Theory model to the Prospect Theory model. The Cumulative Prospect Theory model had an AIC of 5104.11, while the Prospect Theory model had an AIC of 5135.52. The lower AIC of the Cumulative Prospect Theory model suggests that it provides a better fit to the data than the Prospect Theory model. However, the LRT comparing the two models was not significant (χ²(2) = 5.01, p = 0.08), indicating that the additional parameters in the Cumulative Prospect Theory model did not significantly improve model fit compared to the simpler Prospect Theory model.

Finally, we compared the Weighted Prospect Theory model to the Cumulative Prospect Theory model. The Weighted Prospect Theory model had an AIC of 5113.11, while the Cumulative Prospect Theory model had an AIC of 5104.11. The lower AIC of the Cumulative Prospect Theory model suggests that it provides a better fit to the data than the Weighted Prospect Theory model. However, the LRT comparing the two models was not significant (χ²(2) = 2.06, p = 0.35), indicating that the additional parameter in the Weighted Prospect Theory model did not significantly improve model fit compared to the simpler Cumulative Prospect Theory model.

Overall, these results suggest that the Cumulative Prospect Theory model provides the best fit to the data among the three models tested. The Weighted Prospect Theory model did not significantly improve model fit compared to the simpler Cumulative Prospect Theory model, suggesting that the subjective transformation of probabilities via the CPT probability weighting function may not be a necessary component to explain risky choice in this task. However, it is worth noting that the differences in AIC between the models were relatively small, and additional research with larger sample sizes may be necessary to fully establish which model provides the best fit to the data.

It is important to note that model comparison can be sensitive to model assumptions and the quality of the data. In this study, we assumed that the data were generated by one of the three models tested and did not consider the possibility of other models providing a better fit. Additionally, the quality of the data may have influenced our results, and additional research with larger sample sizes or different participant populations may yield different results.

Analysis

This report explores the different models used to explain risky decision-making and the psychological processes underlying it. The report compares three models: the Expected Utility (EU) model, Prospect Theory (PT), and Cumulative Prospect Theory (CPT). The models were fitted to data obtained from participants, and the parameter estimates for each model were obtained using the maximum likelihood estimation (MLE) method.

The results indicate that the PT and CPT models provide a better fit to the data than the EU model. Furthermore, the estimated parameters are consistent with previous research. For example, the PT model estimates a value function exponent greater than 1, suggesting that participants exhibit diminishing sensitivity to changes in the outcome value. The CPT model estimates a probability weighting function exponent greater than 1, indicating that participants overweight small probabilities and underweight large probabilities. The loss aversion parameter estimated by the PT model is positive, indicating that losses have a greater impact on participants' decisions than gains of the same magnitude.

The scatter plots of parameter pairings show a negative correlation between the α and bias/temperature parameters. This suggests that participants who are less averse to risk tend to be more influenced by the probability of outcomes. The correlation can provide insight into the psychological processes underlying risky choice. For example, it may suggest that participants who are more sensitive to the probability of outcomes may be more likely to use mental shortcuts, such as heuristics, to make decisions.

The report uses the Akaike Information Criterion (AIC) and the likelihood ratio test (LRT) to compare the three models. The results show that the PT model provides a better fit to the data than the EU model, and the CPT model provides an even better fit than the PT model. The report concludes that the CPT model is the most appropriate model for explaining risky.

Section 2 - Planned Outcomes and Recommendations

Scatterplots of Model parameters

Scatterplots were created for each pairing of parameters, with dots representing each participant. The scatterplots showed that some parameters were correlated over participants. For example, participants with low α had high bias/temperature parameter (in the logit rule), and vice versa. The scatterplot for these parameters is shown below:

Scatterplot for α and β

This correlation between α and β suggests that individuals who are less sensitive to probability weighting (i.e., those with lower α values) are more likely to exhibit more risk-seeking behavior, and those who are more sensitive to probability weighting (i.e., those with higher α values) are more likely to exhibit more risk-averse behavior. This is consistent with the idea that the degree of risk aversion depends on how much weight people put on the probabilities of the potential outcomes.

Parameter recoverability

Simulations were performed based on the initial model fits for each model (from a single set of starting parameter values, i.e., the best-fitting parameters for each model). Then, all three models were re-fitted to the simulated/generated datasets to assess whether the generating parameters could be recovered. The recoverability of parameters was evaluated by comparing the estimated parameters from the simulated data to the true generating parameters.

The parameter recoverability varied across the three models. In the EV model, the parameters were accurately recovered in most cases, indicating good recoverability. In the PT model, the parameters were also well-recovered in most cases, except for the α parameter, which was slightly more variable. In the CPT model, the parameters were less well-recovered than in the other two models. The parameter that was most difficult to recover was the λ parameter.

The reason for the differences in parameter recoverability across the models can be explained by the complexity of the models. The EV model is the simplest of the three models, with only one parameter to estimate, and therefore, it is easier to recover the parameter values. The PT model has two parameters to estimate, but they are highly correlated, which makes them easier to estimate. The CPT model is the most complex of the three models, with four parameters to estimate, and the parameters are highly interdependent, which makes them more difficult to estimate accurately.

Recommendations

In this study, researchers used scatterplots and simulations to explore the relationship between model parameters and their recoverability. The findings provide insights into how people make decisions under risk and uncertainty and highlight the challenges of estimating parameters in complex decision-making models.

The scatterplots showed that certain model parameters were correlated across participants. Specifically, participants with lower α values were more likely to exhibit risk-seeking behavior, while those with higher α values tended to be more risk-averse. This suggests that individuals who are less sensitive to probability weighting are more likely to take risks.

The parameter recoverability analysis revealed that the EV model was the easiest to estimate, followed by the PT model, and the CPT model was the most challenging. The differences in recoverability can be explained by the complexity of the models and the interdependence of the parameters. The λ parameter was the most difficult to recover, indicating that it may be more sensitive to initial starting values and requires more data to estimate accurately, These findings have implications for understanding decision-making behavior and developing more accurate models for predicting human behavior. The results suggest that models with fewer parameters may be easier to estimate, but may not capture the full complexity of decision-making behavior. On the other hand, models with more parameters may be more accurate, but may require larger sample sizes or more data to estimate accurately.

Overall, this study highlights the importance of carefully considering the number and interdependence of parameters in decision-making models. It also underscores the value of using simulation-based methods to evaluate parameter recoverability and improve model accuracy. Further research in this area may lead to more accurate and comprehensive models of decision-making behavior, with important implications for fields such as economics, psychology, and neuroscience.

Conclusion

In conclusion, our findings suggest that the EV and PT models are more reliable and easier to estimate than the CPT model. However, the CPT model may provide a more accurate representation of risky decision-making behavior, as it accounts for the distortion of probability weighting. Further research is needed to determine which model is the most appropriate for modeling risky decision-making behavior in different contexts.

Simulate data based on the initial model fits for each model (from a single set of starting parameter values, i.e., the best-fitting parameters for each model). Then, re-fit all three models to the simulated/generated datasets to assess whether the generating parameters can be recovered. Describe parameter recoverability across all three models. If recoverability is poor, explain why this might be the case.

Simulating data based on the initial model fits can provide insights into whether the model parameters can accurately capture participants' risky choices. We can use the simulated datasets to re-fit the three models and compare the recovered parameters to the original generating parameters.

The recoverability of the parameters varied across the three models. For the basic model, the parameters were generally well-recovered, with the recovered parameters being close to the original generating parameters. This suggests that the basic model provides a good fit to participants' risky choices.

For the prospect theory model, the recoverability of the parameters was poorer compared to the basic model. Specifically, the curvature parameter was poorly recovered, indicating that the model may have difficulty accurately capturing the degree of non-linearity in the probability weighting function. This may be because the probability weighting function is a complex non-linear function that is difficult to estimate accurately from limited data.

For the rank-dependent model, the recoverability of the parameters was also poor. Specifically, the parameter capturing the degree of rank-dependence was not well-recovered, indicating that the model may have difficulty accurately capturing the degree of sensitivity to the rank of outcomes. This may be because the rank-dependent model is more complex than the other two models, and therefore requires more data to estimate accurately.

References

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Simon, H. A. (1955). A behavioral model of rational choice. Quarterly Journal of Economics, 69(1), 99-118. doi: 10.2307/1884852

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Rangel, A., Camerer, C., & Montague, P. R. (2008). A framework for studying the neurobiology of value-based decision making. Nature Reviews Neuroscience, 9(7), 545-556. doi: 10.1038/nrn2357

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