Directional Revision under Two-Horizon Deliberation: A Revealed-Preference Analysis

Technical Summary: Directional Revision under Two-Horizon Deliberation

Problem Statement
The paper investigates deterministic choice behavior when a single decision-maker is observed under two distinct deliberative modes: an "ordinary" mode ($O$) and a "full-horizon" mode ($F$). The setting assumes that the set of available alternatives, factual information, and feasibility constraints remain constant. However, a deliberative intervention alters the salience of one of two fixed ordinal consequence dimensions. These dimensions are labeled "Now" ($N$) and "Hereafter" ($H$). The central question is: if an intervention makes the $H$-dimension more salient without changing the underlying consequence rankings, what restrictions does this directional shift impose on observed choices? The analysis proceeds without specifying a cardinal utility aggregator, a target utility function, or a parametric rule for resolving cross-dimensional tradeoffs.

Methodology and Axioms
The framework utilizes revealed preference theory on a finite set of alternatives $X$ and a set of non-empty menus $\mathcal{M}$. Two single-valued choice functions, $C_O$ and $C_F$, represent the ordinary and full-horizon choices, respectively. The analysis relies on three primary axioms:

1. Path Independence (Axiom 3.1): Both choice functions satisfy Plott's (1973) path independence condition. This ensures that each mode is rationalized by a unique strict linear order ($\succ_O$ and $\succ_F$).
2. Two-Horizon Dominance (Axiom 3.2): Ordinary choice respects a dominance relation defined across both dimensions. If an alternative $x$ is weakly better than $y$ on both $N$ and $H$ dimensions and strictly better on at least one, $C_O$ must select $x$. Formally, if $x P_{N,H} y$, then $C_O(\{x, y\}) = x$.
3. Hereafter-Directed Revision (Axiom 3.3): This is the core directional restriction. If a binary choice reverses between modes (i.e., $C_F(\{x, y\}) \neq C_O(\{x, y\})$), the full-horizon choice must favor the alternative that is strictly better under the $H$-ranking. Specifically, if $C_F(\{x, y\}) = x$ and $C_O(\{x, y\}) = y$, then $x \succ_H y$. The axiom is one-sided; it does not require that every $H$-superior alternative be chosen in the full-horizon mode, only that reversals must move toward the $H$-better option.

Key Characterization Results
The paper establishes a precise characterization of the admissible rankings:

• Protected Partial Order: The author defines a "protected" strict partial order $P_{O,H}$ generated by the intersection of the ordinary ranking and the Hereafter ranking: $x P_{O,H} y \iff x \succ_O y \text{ and } x \succeq_H y$.
• Linear Extensions: The main theorem (Theorem 4.2) states that a full-horizon ranking $\succ_F$ is admissible if and only if it is a linear extension of $P_{O,H}$. Furthermore, the ordinary ranking $\succ_O$ must be a linear extension of the dominance relation $P_{N,H}$.
• Identification:
  • Pairwise: The intersection of all admissible full-horizon rankings recovers exactly the protected relation $P_{O,H}$.
  • Menu-Level: The set of admissible choices for a menu $S$ under the full-horizon mode corresponds to the set of $P_{O,H}$-maximal elements in $S$. Point identification occurs if and only if $S$ has a unique maximal element under $P_{O,H}$.
• Directional Choice Wedge (Theorem 5.4): If the ordinary and full-horizon choices differ on a menu ($b \neq f$), the change is strictly signed: the full-horizon winner is strictly better under $H$ ($f \succ_H b$), and the ordinary winner is strictly better under $N$ ($b \succ_N f$). This implies that observed switches only occur on genuine cross-horizon tradeoffs.

Tie-Free Geometry and Ordered Deliberation
When the Hereafter ranking is strict (no ties), the model admits a metric interpretation using the Kemeny–Kendall distance ($d_K$):

• Kemeny-Betweenness: The full-horizon ranking $\succ_F$ lies "between" the ordinary ranking $\succ_O$ and the Hereafter ranking $\succ_H$. Specifically, $d_K(\succ_O, \succ_H) = d_K(\succ_O, \succ_F) + d_K(\succ_F, \succ_H)$.
• Inversion Sets: Full-horizon deliberation removes "Hereafter inversions" (pairs where the $H$-worse alternative is ranked above the $H$-better one) but never creates new ones. The set of inversions under $\succ_F$ is a subset of those under $\succ_O$.
• Monotone Trajectories: In a sequence of deliberative modes where $H$-salience increases stepwise, the sequence of choices forms a monotone chain in the tradeoff order. An alternative abandoned at an earlier stage cannot reappear later, and the number of switches is bounded by the height of the induced tradeoff order.

Rationalizability and Incomplete Data
The paper provides tests for rationalizability when data is incomplete. A dataset of full-horizon choices is rationalizable if and only if the union of the protected relation $P_{O,H}$ and the directly revealed pairwise comparisons from the data is acyclic. If acyclic, the set of rationalizing rankings corresponds to the linear extensions of the transitive closure of this union.

Significance and Claims
The paper claims to provide an ordinal framework for analyzing preference revision driven by deliberative emphasis rather than new information or preference construction. Its significance lies in:

1. Directional Constraints: It offers a testable restriction on choice revisions that does not assume a lexicographic priority or a specific cardinal tradeoff rule.
2. Sharp Identification: It yields precise bounds on what can be inferred about the full-horizon ranking from observed ordinary choices and the fixed consequence dimensions.
3. Behavioral Geometry: It links the behavioral restriction to standard geometric concepts (Kemeny distance and betweenness) in the space of linear orders, providing a clear structure for how deliberation moves choice.
4. Generality: The framework applies to any setting where two fixed ordinal consequence dimensions exist and an intervention shifts attention directionally, without requiring the model to specify the threshold for tradeoffs or the nature of the "self" making the decision.

The author explicitly notes that the formal results do not depend on the specific Islamic-economic illustration used for motivation, nor do they model decision-tree dynamics or target utility functions. The contribution is the structural link between directional deliberation and the resulting protected partial order.