Revealing Choice Bracketing

The paper addresses how individuals bracket interdependent choices—whether they consider interactions across parts of a decision (broad bracketing) or optimize each part in isolation (narrow bracketing). Many economic and behavioral models implicitly assume a bracketing mode, which affects measured risk attitudes, fairness judgments, and welfare. Prior experimental evidence mainly rejects broad bracketing but does not test narrow bracketing directly. The authors propose a theoretical framework with testable revealed-preference implications that discriminate among narrow, broad, and intermediate (partial-narrow) bracketing. They implement this through experiments in risky portfolio choice, social allocation, and induced-value consumer choice to measure individual-level bracketing and its welfare consequences.

Section 2 surveys evidence on choice bracketing. Classic designs (Tversky & Kahneman, 1981; Kahneman & Tversky, 1979) document violations of broad bracketing but cannot falsify narrow bracketing because any choice pattern is consistent with it. Other studies show that endowments or background risks are often not integrated into choices, again contradicting broad bracketing. Work on fungibility and mental accounting (Thaler, 1985; 1999; Hastings & Shapiro, 2013, 2018) shows that categories constrain spending, but mental accounting can coexist with either broad or narrow bracketing across parts. Dynamic settings reveal additional non-broad phenomena (e.g., myopic loss aversion, disposition and house-money effects), while some recent work suggests less narrow bracketing of future risks. The authors note the need for designs that provide direct, individual-level tests of both narrow and broad bracketing and allow intermediate forms.

The framework considers T ≥ 1 decisions, each with K_t ≥ 1 parts, where each part (t,k) offers a compact, non-empty feasible set B^{t,k} ⊂ R^n. A subject chooses x^{t,k} ∈ B^{t,k} for each part, and the payoff of decision t depends on the final alternative x^t = Σ_k x^{t,k}.

• Broad bracketing (BB): there exists an increasing u such that x^t maximizes u over the aggregate feasible set B^t = {Σ_k y^{t,k} : y^{t,k} ∈ B^{t,k}} for each decision.
• Narrow bracketing (NB): there exists an increasing u such that each x^{t,k} maximizes u over B^{t,k} (optimization part-by-part).
• Partial-narrow bracketing (PNB): for α ∈ [0,1], choices maximize α Σ_k u(y^{t,k}) + (1−α) u(Σ_k y^{t,k}). PNB-PE is a related intrapersonal equilibrium version optimizing part-by-part with the same aggregator. Revealed-preference tests:
• Necessary predictions include NB-WARP and BB-WARP (WARP applied at part vs. decision level) and BB-Mon (monotonicity at the decision-level frontier).
• Full characterizations (Theorem 1): NB-SARP is necessary and sufficient for NB rationalization when applied to the ancillary dataset D^{NB} that treats each part as a separate observation; BB-SARP is necessary and sufficient for BB rationalization when applied to D^{BB} that aggregates parts to decision-level choices.
• PNB test (Theorem 2): an algorithm reduces α-PNB rationalizability to a linear program by mapping each decision to a menu of lotteries whose support weights correspond to α on part-level outcomes and 1−α on their sum; feasibility corresponds to an expected-utility rationalization via LARP conditions. Experimental design:
• Three domains: Risk (portfolio choice among state-contingent assets), Social (allocations between two anonymous recipients), and Shopping (induced-value consumer bundles of apples and oranges with payment pay = (2/5)(√apples + √oranges)).
• Five rounds per experiment; some decisions have two parts. Goods in part 1 are perfect substitutes for corresponding goods in part 2, sharpening differences between BB and NB. Final payment is determined by summing choices across parts in one randomly selected round (modified payment draw for Social to pay other anonymous pairs).
• Budgets: integer lattice bundles from linear (or piecewise linear) budgets; symmetry across states/recipients/goods is exploited to strengthen tests.
• Implementation: paper-based lab sessions (Toronto and SFU labs, June 2019–Feb 2020); comprehension quizzes; randomized orderings of decisions/parts; no calculators allowed on paper. Sample sizes: Risk n=99, Social n=102, Shopping n=101. Online robustness studies varied presentation (Examine prompt; Tabs vs. Side-by-Side) and collected process data (tab clicks; calculator entries).

Direct tests (Risk and Social):

• BB ruled out for most: within one error, only about 20% satisfy BB-WARP, and 8–12% satisfy BB-Mon; with full BB-SARP, 0% (Risk) and 10% (Social) pass with one error.
• NB supported for many: pairwise NB-WARP comparisons show 69–81% (Social) and 75–77% (Risk) are within one error for each pair; 44% (Risk) and 53% (Social) pass all NB-WARP with one error. With full NB-SARP (one error), 34% (Risk) and 35% (Social) pass.
• PNB intermediate: with one error, about 15% (Risk) and 12% (Social) pass PNB but neither NB- nor BB-SARP. Shopping (induced values):
• Model point predictions are testable per decision. With two errors: 40% are consistent with NB across all decisions; only 16% with BB. Few additional are explained by PNB once precision is accounted for.
• α estimation by decision-level predictions yields 64% best described by ranges that include α=1 (narrow), 25% by ranges including α=0 (broad); no subjects best fit α in [0.25,0.71], indicating most are close to a pole. Classification adjusting for predictive precision (Selten score):
• Narrow bracketing dominates: 77.8% (Risk), 75.5% (Social), 67.3% (Shopping).
• Broad bracketing minority: 2.0% (Risk), 9.8% (Social), 26.7% (Shopping).
• Partial-narrow accounts for few: 7.1%, 2.0%, and 4.0% respectively; unclassified small remainders. Process and architecture:
• Online Risk: architecture nudges (Examine, Tabs vs. Side-by-Side) had limited impact on broad bracketing; only 1% passed BB-SARP with two errors; 84.5% classified as narrow.
• Consideration data show over a quarter of narrow bracketers still examined both parts or computed decision-level bundles, indicating that narrow bracketing is not solely due to ignoring other parts. Welfare and interpretation:
• Shopping: narrow bracketers earned on average about $1.29 less than broad bracketers in the two-part decisions (>10% of variable payment), quantifying losses from failing to integrate across parts.
• Social: accounting for bracketing reverses naive inferences about equity: broad bracketers achieve equalized final allocations via specialization across parts; narrow bracketers’ part-level inequality masks decision-level equity once bracketing is considered.

The findings provide individual-level, non-parametric evidence that a majority of subjects optimize part-by-part rather than over aggregate feasible sets, with a notable minority broadly bracketing. By designing two-part decisions with perfect substitutability across parts, the authors generate distinct, testable implications for BB, NB, and PNB, overcoming limitations of prior designs that only rejected broad bracketing. The heterogeneity in bracketing has important implications: misclassifying bracketing can misstate underlying risk or inequity aversion (e.g., small-stakes gamble rejections imply very different risk attitudes under NB vs. BB). Welfare losses from narrow bracketing arise through missed gains from specialization: broad bracketers allocate to the cheaper source first, analogous to comparative advantage in trade, while narrow bracketers forgo those gains. Process data suggest that some subjects consider both parts yet still implement narrow choices, explaining why presentation nudges had limited effect. The approach enables separating preferences from bracketing, improving inference about heterogeneity in risk and social preferences and reconciling mixed evidence in the literature (e.g., contexts that necessitate broader vs. narrower frames).

The paper introduces and validates revealed-preference tests that identify how individuals bracket interdependent choices, distinguishing narrow, broad, and partial-narrow bracketing without parametric assumptions beyond monotonicity (and symmetry in some implementations). Across risk, social allocation, and induced-value consumer choice, most subjects are best described as narrow bracketers, with a smaller share broadly bracketing and few best captured by intermediate models after adjusting for predictive power. The method quantifies welfare losses from non-broad bracketing and demonstrates how bracketing affects inference about preferences. Future research should investigate why individuals adopt particular bracketing modes, explore stronger or more targeted decision aids, extend the framework to dynamic settings with potential dynamic inconsistency, and examine domain and population differences in bracketing prevalence.

• Dynamic settings are not directly tested; extending the approach would require decision trees and assumptions about dynamic consistency.
• Experimental choices are on discrete grids with few decisions to minimize learning; discretization can create measurement coarseness and error thresholds that affect pass rates.
• Broad bracketing often implies corner solutions; extremeness aversion could reduce observed BB. A relaxed extremeness-averse BB-Mon adds few additional BB-consistent subjects, but cannot be fully ruled out as an influence.
• Assumed monotonicity (and symmetry in some tests) constrains interpretations; violations or heterogeneity in these assumptions may affect classifications.
• Broad bracketers are relatively rare, limiting power to analyze their determinants or responsiveness to nudges.
• Subject pools (university labs; Prolific online) and paper vs. online interfaces may influence behavior; differences across domains (Risk, Social, Shopping) suggest contextual sensitivity.
• Partial-narrow models nest NB and BB; while adjusted via predictive success, model comparison remains sensitive to error allowances and predictive areas.
• Welfare comparisons presume no direct utility cost to broad bracketing and rely on induced values (Shopping) or expected/aggregate payoffs (Risk/Social) for interpretation.