Sensitivity and Bias in Decision-Making under Risk: Evaluating the Perception of Reward, Its Probability and Value

Background There are few clinical tools that assess decision-making under risk. Tests that characterize sensitivity and bias in decisions between prospects varying in magnitude and probability of gain may provide insights in conditions with anomalous reward-related behaviour. Objective We designed a simple test of how subjects integrate information about the magnitude and the probability of reward, which can determine discriminative thresholds and choice bias in decisions under risk. Design/Methods Twenty subjects were required to choose between two explicitly described prospects, one with higher probability but lower magnitude of reward than the other, with the difference in expected value between the two prospects varying from 3 to 23%. Results Subjects showed a mean threshold sensitivity of 43% difference in expected value. Regarding choice bias, there was a ‘risk premium’ of 38%, indicating a tendency to choose higher probability over higher reward. An analysis using prospect theory showed that this risk premium is the predicted outcome of hypothesized non-linearities in the subjective perception of reward value and probability. Conclusions This simple test provides a robust measure of discriminative value thresholds and biases in decisions under risk. Prospect theory can also make predictions about decisions when subjective perception of reward or probability is anomalous, as may occur in populations with dopaminergic or striatal dysfunction, such as Parkinson's disease and schizophrenia.


APPENDIX S1
In prospect theory, choices are made not by objective measures of probability and magnitude as in expected value theory, but by subjective perception of these variables, which show important non-linearities that are captured by weighting functions [22].
For the reward at stake, subjects consider the utility of the gain v(x), rather than its absolute magnitude x. The most common weighting function used to model utility is a power function (Image I, left): As the sizes of rewards increase, further increments in reward are generally considered less valuable, and so ρ is usually <1.
For the probability of reward, a single-parameter Prelec function [26] has been shown to capture adequately the tendency of humans to overestimate small probabilities and underestimate large ones (Image I, middle): These perceived probabilities and utilities are combined multiplicatively as in standard utility theory, to give the function for U(L), the perceived value of prospect L: The difference between the perceived value of two prospects L 1 and L 2 is then simply: When choosing between these two prospects, the likelihood that a subject will choose the first prospect (P(L 1 ) can be represented by a logit function: The parameter λ is introduced as a stochastic constant that controls the steepness of the function [24], which in turn reflects the subject's sensitivity to the difference in perceived value between L 1 and L 2 .
To model our experimental parameters in prospect theory, we used constants for Equations 1 and 2 derived from independent studies of other healthy populations, as reviewed comprehensively elsewhere [26]. For the utility function (1), we used ρ = 0.57; for the Prelec function, we used α = 0.77. With these constants, the relation between the V(x,p) function of prospect theory and the EV-ratio of expected value theory is depicted in Image I (right). While there is an approximately linear relation between the two, a key finding is that the point of equivalence between perceived value of the two prospects (when V(x,p) = 0) occurs when the EV-ratio is negative. Hence prospect theory with current best estimates of the parameters in its non-linear functions predicts a choice bias in favour of prospects with higher probability over prospects with larger reward.
This suggests that the choice bias we found when our parameters are expressed as an EV-ratio may be minimized or eliminated if they are expressed as V(x,p). However, because P(L 1 ) is constrained in equation 4 to have an equivalence point of zero, since P(L 1 ) = 0.5 when U(L 1 ) = U(L 2 ), we added a second 'intercept constant' k to the equation: If the best fit to our data is found when k = 0, this would suggest that the choice bias we found can be attributed to the non-linearities of equations 2 and 3. Hence we estimated the stochastic constant λ and the intercept constant k by least-squares fit of curve predictions to the actual data. The best fit to the group data was obtained with λ = 7.2 and k = 0.04 (see Figure 3 in main text): the choice bias seen when plotted against EV-ratio is nearly eliminated when the data are re-plotted in terms of differences in perceived value V(x,p).
We can also consider hypothetical impact of disease states on the parameters of prospect theory. In pathological gambling, one might surmise that subjects are more prone to choose a prospect with larger reward despite its low probability. This could occur because of distorted perceptions of either gain or probability. We can model the first by showing a family of utility functions (1) v(x), with ρ increasing from 0.8 to 1.4, while the α parameter in the perceived probability function (2) w(p) is held constant at 0.77 (Image II). When we plot the predicted choices made at different EV-ratios, we find that the curves gradually shift to the right as ρ increases, indicating greater likelihood of choosing the prospect with larger reward.
We can also model the converse, the anticipated effects of altered perceptions of probability. We can model a family of Prelec functions (2) w(p) for perceived probability, with α declining from 0.7 to 0.25, while ρ in the utility function (1) v(x) is held constant at 0.57. Note that, while there is an increasingly inflated perception for low probabilities as α decreases, there is also a progressive flattening of the slope of the Prelec function in the mid-range of probabilities. Again, we find that for predicted choice as a function of EV-ratios, the curves shift to the right, indicating a shift to choosing the prospect with larger gain.
Thus, a tendency to choose the side with greater reward can occur as the result of either increasing the exponential term in the utility function (1) v(x), or decreasing the exponential term in the Prelec function for perceived probability (2) w(p). However, the difference between the two is that the change in the utility function for reward size also leads to steeper slopes, and therefore increased sensitivity to differences in EV-ratio, whereas the change in the probability function leads to shallower slopes, and therefore reduced sensitivity, likely because of the flattening of the Prelec function in the midrange. Hence these two scenarios in prospect theory predict diametrically opposite effects on discriminative thresholds, which leads to testable predictions in patient studies.
Finally it is also possible to model individual subject performance with prospect theory [26,44]. We used a three--parameter--fit iterative procedure to find optimal estimates of ρ, α, and λ for each subject's data, with the aim of minimizing the summed