Figure 1
Ambiguous Outcome Magnitude Prospect Theory Model. Prospect theory includes (unambiguous) risk aversion/preference (ρ) and (unambiguous) loss aversion/preference (λ). Our model builds upon prospect theory by additionally parameterizing preference/aversion towards ambiguous outcome magnitudes (α). In this figure, we separately parameterize ambiguous gain aversion (αG) and ambiguous loss aversion (αl) for concision (akin to Model 3; see Methods). For illustrative purposes, we show Objective Utility (depicting rational decision making), risk preference, risk aversion, loss aversion, aversion of ambiguous loss magnitudes, and aversion of ambiguous gain magnitudes. We show additive effects of parameters (e.g., risk aversion vs risk aversion and ambiguous gain aversion). The model shows that risk preference/aversion affects the curvature of the subjective utility function (i.e., via its exponential calculation); (unambiguous) loss aversion, ambiguous loss aversion, and ambiguous gain aversion shift the curve up/down (i.e., via their multiplicative calculation; see Table 2). For unambiguous “True Values,” the monetary value of the gamble is explicitly known; for ambiguous “True Values,” the monetary value represents the most likely or central value of the ambiguous values (e.g., mean). Parameter values were 1 unless otherwise specified here: Risk Preference (ρ = 1.5); Risk Aversion (ρ = .5); Risk and Ambiguous Gain Aversion (ρ = .5, αG = .5); Risk and Loss Aversion (ρ = .5, λ = 2); and Risk, Loss, and Ambiguous Loss Aversion (ρ = .5, λ = 2, αL = 1.5). Please see our repository for an interactive figure of this model, where the user can change inputs (e.g., risk preference) and observe the outputs (file is titled “Zbozinek et al – Prospect Theory Ambiguity Model.html”): https://github.com/tzbozinek/economic-decision-making-ambiguity.
Figure 2
Experimental Conditions. This figure presents a schematic of each condition. Gains are color-coded with green text, losses with red text, and $0 with white text. Green question marks (i.e., “$?”) indicate ambiguous gain magnitude; red question marks (i.e., “–$?”) indicate ambiguous loss magnitude. Starting payments were $4.50 (Study 1) or $24 (Study 2), with total possible payments ranging $3.60 to $5.70 (Study 1) and $6 to $48 (Study 2). Unambiguous gains ranged from $.05 to $1.20 (Study 1) or $1 to $24 (Study 2). Unambiguous losses ranged from –$.05 to –$.90 (Study 1) or –$1 to –$18 (Study 2). Circles without a vertical line indicate a 100% chance of receiving that outcome. Circles with a vertical line indicate a 50%/50% chance of receiving each outcome. Conditions that involve Risk, Loss, or Ambiguity are indicated within each condition’s box.
Figure 3
Gambling Propensity Per Condition in High vs Low Stakes. Bars represent mean gambling percentage for high and low stakes per condition (error bars are standard error). “Gambling” refers to choosing the risky 50%/50% option in Conditions 1–6 or the ambiguous option in Conditions 7–8. Dots indicate individual data points; they are arranged in ascending order within each condition per an empirical cumulative distribution function. Effects of low vs high stakes are significant within each condition. Below X-axis is an example of a trial from each condition; “Risk,” “Loss,” and “Ambiguity” indicate whether each parameter type is present in that condition. All significant differences pass Holm-Bonferroni cutoffs for multiple comparisons.
Table 1
Model predictive accuracy was calculated in two out-of-sample ways. In the within-subjects analysis, model parameters were estimated for each subject using approximately 5/6 of their data and used to test accuracy in the remaining trials. In the between-subjects analysis, group model parameters are estimated in 29/30 participants and used to test accuracy in the remaining subjects. Note that for Null Models 1 and 2, the accuracy is calculated in-sample since the models have no parameters. Model 5 was the best-fitting model in both studies, and model fit improved with higher stakes.
| (A) MODEL COMPARISON (STUDY 1: LOW STAKES $3.60–$5.70) | |||||
|---|---|---|---|---|---|
| MODEL DESCRIPTION | NUMBER OF PARAMETERS | R2 | AIC | MODEL ACCURACY: WITHIN-SUBJECTS OUT-OF-SAMPLE | MODEL ACCURACY: BETWEEN-SUBJECTS OUT-OF-SAMPLE |
| Null Model 1: 50% Probability to Gamble on Each Trial | 0 | .000 | 460.4 | 50.0% | |
| Null Model 2: Average Gambling Rate for Given Participant on Each Trial | 0 | .162 | 385.9 | 69.4% | |
| Model 1: Traditional Prospect Theory | 3 | .334 | 312.6 | 68.9% | 61.9% |
| Model 2: General Ambiguity | 4 | .412 | 278.6 | 72.8% | 63.1% |
| Model 3: Ambiguous Gains and Losses | 5 | .441 | 267.2 | 74.3% | 63.2% |
| Model 4: Ambiguous Loss or No-Loss Contexts | 5 | .445 | 265.3 | 74.5% | 63.3% |
| Model 5: Ambiguous Sure/Risky Gains/Losses | 7 | .473 | 256.4 | 75.6% | 65.1% |
| (B) MODEL COMPARISON (STUDY 2: HIGH STAKES $6–$48) | |||||
| MODEL DESCRIPTION | NUMBER OF PARAMETERS | R2 | AIC | MODEL ACCURACY: WITHIN-SUBJECTS OUT-OF-SAMPLE | MODEL ACCURACY: BETWEEN-SUBJECTS OUT-OF-SAMPLE |
| Null Model 1: 50% Probability to Gamble on Each Trial | 0 | .000 | 460.8 | 50.0% | |
| Null Model 2: Average Gambling Rate for Given Participant on Each Trial | 0 | .145 | 393.9 | 68.3% | |
| Model 1: Traditional Prospect Theory | 3 | .416 | 275.2 | 73.1% | 65.6% |
| Model 2: General Ambiguity | 4 | .496 | 240.2 | 77.1% | 66.7% |
| Model 3: Ambiguous Gains and Losses | 5 | .529 | 227.2 | 78.6% | 67.0% |
| Model 4: Ambiguous Loss or No-Loss Contexts | 5 | .522 | 230.4 | 78.3% | 66.7% |
| Model 5: Ambiguous Sure/Risky Gains/Losses | 7 | .552 | 220.5 | 79.5% | 67.7% |
Figure 4
Model 5 Real Gambling Rate vs Model-Recovered Gambling Rate. Figure shows correlations within each condition for Model 5 between real gambling rate and model-recovered gambling rate. Results show our model was very accurate in its prediction of real gambling rate, suggesting the model’s validity. Panel a is Study 1 (low stakes: $3.60–$5.70), and panel b is Study 2 (high stakes: $6–$48).
Figure 5
Model 5 Parameter Results – High and Low Stakes. Figure shows point estimates of parameters and 95% confidence intervals. Values that are significant are in color (all significant results passed Holm-Bonferroni correction); null results are in gray. Study 1 was low stakes ($3.60–$5.70; N = 367), and Study 2 was high stakes ($6–$48; N = 210). λ = Loss Aversion, ρ = Risk Preference, αRL = Ambiguous Risky Loss Aversion, αSL = Ambiguous Sure Loss Aversion, αRG = Ambiguous Risky Gain Preference, and αSG = Ambiguous Sure Gain Preference. For “Aversion” parameters (i.e., λ, αRL, αSL), greater values indicate greater aversion. For “Preference” parameters (e.g., ρ, αRG, αSG), greater values indicate greater preference.
Figure 6
Low and High Stakes Results – Prospect Theory Model with Ambiguous Outcome Magnitudes. This figure shows the true parameter values derived from the best-fitting model (Model 5) in Study 1 (Low Stakes; $3.60–$5.70) and Study 2 (High Stakes; $6–$48). For concision, we collapsed the ambiguity parameters into two parameters: ambiguous gain preference/aversion and ambiguous loss preference/aversion. This was done within each study by averaging the a) ambiguous risky gain and ambiguous sure gain parameters and b) ambiguous risky loss and ambiguous sure loss parameters using the actual mean parameter values derived in each study. Parameter values were as follows: Low Stakes (ρ = 1.217, αG = 1.447, λ = 2.361, αL = 1.129) and High Stakes (ρ = 1.050, αG = .803, λ = 2.258, αL = .903).
Figure 7
Trait Anxiety and Depression. Figure shows a bar plot of means with standard error and an empirical cumulative distribution function, where individual scores are plotted as dots. Figure shows a wide and consistent distribution of trait anxiety and trait depression in both the low stakes and high stakes studies.
Figure 8
Model 5 Anxiety and Depression Results. Figure shows effects of anxiety and depression on model parameters and 95% confidence intervals. All results are null and therefore in gray. Study 1 was low stakes ($3.60–$5.70), and Study 2 was high stakes ($6–$48). λ = Loss Aversion, ρ = Risk Preference, αRL = Ambiguous Risky Loss Aversion, αSL = Ambiguous Sure Loss Aversion, αRG = Ambiguous Risky Gain Preference, and αSG = Ambiguous Sure Gain Preference. For “Aversion” parameters (i.e., λ, αRL, αSL), greater values indicate greater aversion. For “Preference” parameters (e.g., ρ, αRG, αSG), greater values indicate greater preference.
Table 2
Computational Modeling Calculations.
| CONDITION | U(GAMBLE) = | U(SURE) = |
|---|---|---|
| 1: Mixed gain/loss, unambiguous | 0.5 * gainρ – 0.5 * λ* (loss)ρ | 0 |
| 2: Mixed gain/loss, ambiguous risky loss | 0.5 * gainρ– 0.5 *λ* αRL * (8.15)ρ | 0 |
| 3: Mixed gain/loss, ambiguous risky gain | 0.5 * αRG* 14.15ρ– 0.5 * λ* (loss)ρ | 0 |
| 4: No–loss, unambiguous | 0.5 * gainρ | sureGρ |
| 5: No–loss, ambiguous risky gain | 0.5 * αRG* 15ρ | sureGρ |
| 6: No–loss, ambiguous sure gain | 0.5 * gainρ | αSG * 5ρ |
| 7: No risk, ambiguous sure gain | αSG * 7ρ | sureGρ |
| 8: No risk, ambiguous sure loss | –λ* αSL* 7ρ | λ* sureLρ |
[i] Table shows the mean values for ambiguous gains and ambiguous losses for Study 2 (high stakes; $6-$48). Mean values for condition-specific parameters in Study 1 (low stakes; $3.60 to $5.70) are (in cents): Condition 2 (41), Condition 3 (71), Condition 5 (75), Condition 6 (28), Condition 7 (35), and Condition 8 (35).