# Preregistration: Fourfold Pattern of Risk Attitudes (Prospect Theory Replication) **Complete implementation — items, hypotheses, and analysis plan** **Date:** [To be filled at registration] **OSF project:** [Link to be added] **Design reference:** Kahneman & Tversky (1979); Tversky & Kahneman (1992); Ruggeri et al. (2020), *Nature Human Behaviour*. --- ## 1. Objectives We aim to replicate the **fourfold pattern** of risk attitudes from prospect theory. The pattern predicts that risk preferences vary with (1) **outcome domain** (gains vs. losses) and (2) **probability level** (high vs. low): | Domain | High probability | Low probability | |----------|-------------------|------------------| | **Gains** | Risk-averse | Risk-seeking | | **Losses**| Risk-seeking | Risk-averse | Success is defined as: (a) modal choices in the predicted direction in all four cells (H1–H4), and (b) a reflection effect (H6): preferences in gains reverse in losses at matched probability levels. **Monetary convention:** *M* denotes a monetary amount in local currency (e.g., 3000). Replace *M* with the chosen amount in the actual materials. All prospects are “outcome *x* with probability *p*, otherwise 0” (or the loss analogue). --- ## 2. Hypotheses Let *p* denote the proportion of participants choosing the **predicted** option in a given item. All primary tests are **one-sided** (predicted direction). ### 2.1 Cell-specific (primary) | ID | Cell | Prediction | Formal hypothesis | |-----|---------------------------|-----------------------------|------------------------| | H1 | High-probability gains | Risk-averse (choose safe) | *p*safe > 0.5 | | H2 | Low-probability gains | Risk-seeking (choose risky) | *p*risky > 0.5 | | H3 | High-probability losses | Risk-seeking (choose risky) | *p*risky > 0.5 | | H4 | Low-probability losses | Risk-averse (choose safe) | *p*safe > 0.5 | ### 2.2 Pattern-level - **H5 (Fourfold pattern):** All four cell-specific hypotheses H1–H4 hold simultaneously (proportion choosing predicted option > 0.5 in each of the four primary items). - **H6 (Reflection effect):** The preference in the gains domain is reversed in the loss domain at matched probability levels. Tested via domain × probability interaction in a logistic regression (not a single item). --- ## 3. Complete item list (implementation) Each item is a **binary choice**. Option A and Option B are as below. **Predicted option** is the option prospect theory predicts the majority will choose; the hypothesis is that the proportion choosing it exceeds 0.5. Left–right position of A and B should be counterbalanced in the actual survey (record which option was left/right for analysis). --- ### 3.1 Primary items (1–4) — one per cell of the fourfold pattern **Item 1 — High-probability gains (H1: risk-averse)** - **Option A:** *M* for sure. - **Option B:** 1.33*M* with probability 0.80, otherwise 0. - **Predicted option:** A. - **Source:** Kahneman & Tversky (1979), Table I / reflection pair. **Item 2 — Low-probability gains (H2: risk-seeking)** - **Option A:** 1.33*M* with probability 0.20, otherwise 0. - **Option B:** *M* with probability 0.25, otherwise 0. - **Predicted option:** A. - **Source:** Kahneman & Tversky (1979), Table I / reflection pair. **Item 3 — High-probability losses (H3: risk-seeking)** - **Option A:** −1.33*M* with probability 0.80, otherwise 0. - **Option B:** −*M* for sure. - **Predicted option:** A. - **Source:** Kahneman & Tversky (1979), Table I / reflection pair. **Item 4 — Low-probability losses (H4: risk-averse)** - **Option A:** −*M* with probability 0.25, otherwise 0. - **Option B:** −1.33*M* with probability 0.20, otherwise 0. - **Predicted option:** A. - **Source:** Kahneman & Tversky (1979), Table I / reflection pair. --- ### 3.2 Robustness items (5–8) — parallel parameterization (optional) **Item 5 — High-probability gains (H1: risk-averse)** - **Option A:** *M* with probability 0.90, otherwise 0. - **Option B:** 2*M* with probability 0.45, otherwise 0. - **Predicted option:** A. - **Source:** K&T Problem 7 (0.90/0.45). **Item 6 — Low-probability gains (H2: risk-seeking)** - **Option A:** 2*M* with probability 0.001, otherwise 0. - **Option B:** *M* with probability 0.002, otherwise 0. - **Predicted option:** A. - **Source:** K&T Problem 8 (0.002/0.001). **Item 7 — High-probability losses (H3: risk-seeking)** - **Option A:** −2*M* with probability 0.45, otherwise 0. - **Option B:** −*M* with probability 0.90, otherwise 0. - **Predicted option:** A. - **Source:** K&T Problem 7′ (loss). **Item 8 — Low-probability losses (H4: risk-averse)** - **Option A:** −*M* with probability 0.002, otherwise 0. - **Option B:** −2*M* with probability 0.001, otherwise 0. - **Predicted option:** A. - **Source:** K&T Problem 8′ (loss). --- ### 3.3 Item–hypothesis mapping (summary) | Hypothesis | Cell | Primary item | Robustness item | |------------|--------------------------|--------------|-----------------| | H1 | High-probability gains | 1 | 5 | | H2 | Low-probability gains | 2 | 6 | | H3 | High-probability losses | 3 | 7 | | H4 | Low-probability losses | 4 | 8 | | H5 | All four cells | Items 1–4 | — | | H6 | Reflection (regression) | All items | — | --- ## 4. Design - **Design:** Within-subjects. Each participant completes at least the four primary items (1–4). Optionally, all eight items (1–8). - **Stimuli:** As in Section 3. *M* in local currency (e.g., 1–2% of typical monthly income). - **Counterbalancing:** For each item, randomize whether the predicted option appears as left or right; record option position for analysis. - **Order:** Item order randomized (or gains block then losses block; state choice at registration). - **Instructions:** “Choose the option you prefer. There is no correct answer. One choice per question.” No time limit (or generous limit, e.g., 60 s per item). - **Setting:** Hypothetical choices (to match original) or one randomly selected decision for payment. --- ## 5. Participants - **Target N:** At least **400** complete responses (all primary items 1–4). If robustness items 5–8 are used, same participants complete all eight. - **Recruitment:** [Specify: e.g., convenience sample, online panel, student sample.] - **Inclusion:** [Specify: e.g., age 18+, fluent in (language), no prior exposure to these exact problems.] - **Exclusions (preregistered):** See Section 8. --- ## 6. Primary analysis plan ### 6.1 Data preparation - Merge choices with item definitions so that each row has: participant_id, item_id, choice (A or B), predicted_option (A or B), and a binary **chosen_predicted** = 1 if choice equals predicted_option, 0 otherwise. Account for counterbalancing so that “predicted” is the substantive option, not left/right. - Primary analyses use **items 1–4 only**. Robustness uses items 5–8 if collected. ### 6.2 Cell-level tests (H1–H4) - For each primary item *i* ∈ {1, 2, 3, 4}: **one-sided binomial test** that the proportion choosing the predicted option is greater than 0.5. - Report for each item: *n*, *k* (number choosing predicted option), proportion, 95% CI for proportion, one-sided *p*-value. - **Alpha:** 0.05 per test (one-sided). [Multiplicity: no correction / Bonferroni over 4 tests — state choice.] ### 6.3 H5 (Fourfold pattern) - **Replication criterion:** The fourfold pattern is declared **replicated** if and only if **all four** primary item-level tests (items 1–4) are significant in the predicted direction at α = 0.05 (one-sided). - No partial credit: all four must go through. ### 6.4 H6 (Reflection effect) - **Model:** Logistic regression with **chosen_predicted** (0/1) as outcome. Predictors: **domain** (0 = gains, 1 = losses), **probability** (0 = high, 1 = low), **domain × probability** interaction. Use all available item-level data (primary; or primary + robustness if both collected). - **Primary test:** Significance of the **domain × probability** interaction (two-sided *p*-value for the interaction term). - **Interpretation:** Reflection effect supported if the interaction is significant and in the direction where high-p gains (safe preferred) vs high-p losses (risky preferred) and low-p gains (risky preferred) vs low-p losses (safe preferred) are opposite. - Option: cluster standard errors by participant if appropriate for the software used. --- ## 7. Robustness analysis (items 5–8) - If items 5–8 were collected: run the **same** one-sided binomial test per item (5, 6, 7, 8) for proportion choosing predicted option > 0.5. - Report: “Robustness fourfold: all four go through” if and only if all four robustness items are significant in the predicted direction at α = 0.05 (one-sided). - Report proportions and *p*-values for items 5–8 in a separate table or section. This is **exploratory/robustness**; the primary replication claim rests on items 1–4 and H5. --- ## 8. Exclusion criteria - Exclude participants who do not complete all primary items (1–4). If robustness is analyzed, exclude those who do not complete items 5–8 when reporting robustness. - Exclude duplicate IPs / duplicate participant IDs (one response per participant). - [ ] Exclude participants who fail an attention check or comprehension item (specify if used). - [ ] Optional: Exclude responses with RT < X s per item (speeders) or > Y s (dropouts); specify thresholds if used. - **Primary analysis:** Apply the above exclusions; report *N* after exclusions. - **Sensitivity:** Report results with no exclusions (or minimal exclusions) as a sensitivity analysis. --- ## 9. Sensitivity analyses (optional) - Re-run primary analyses (H1–H5) and H6 without exclusions (or with only completion and duplicate exclusions). - If both primary and robustness items were collected: report H5 for items 1–4 only; report “robustness fourfold” for items 5–8; optionally run H6 on primary-only vs all items. --- ## 10. Deliverables - De-identified dataset: participant_id, item_id, choice (A/B), option position (if counterbalanced), demographics (if collected). - Analysis script (e.g., R or Python) that reproduces primary and robustness analyses from the raw data. - Short report: descriptive statistics, results for H1–H6 (and robustness table if applicable), conclusion on replication success (H5 and H6). --- ## 11. References - Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. *Econometrica*, *47*(2), 263–291. - Tversky, A., & Kahneman, D. (1992). Advances in prospect theory: Cumulative representation of uncertainty. *Journal of Risk and Uncertainty*, *5*(4), 297–323. - Ruggeri, K., et al. (2020). Replicating patterns of prospect theory for decision under risk. *Nature Human Behaviour*, *4*(6), 622–633.