An ownership rule assigns each menu to one preference
The paper studies choice procedures that assign every menu to one of two binary relations. The relation that receives the menu "owns" the decision for that menu, and the owner then selects the option that maximizes its own preference. The authors call this a single ownership rule, formalized as a mapping from menus to a pair of preferences, with the resulting choice equal to the maximum of the menu under the assigned preference.
The owner need not be a person. The paper treats the two preferences abstractly, so they can represent two individuals, such as a parent and a child, or two selves, such as fast and slow thinking, or two rationales competing to justify a decision. What matters for the model is that each menu is imputed to exactly one preference, and that the owner is accountable because it picks its own best option.
This differs from aggregating binary relations across all menus. The paper positions ownership as menu-dependent dictatorship: the identity of the deciding preference can change from one menu to another, which is the feature that generates observable departures from classical rationality.
Three nested environments vary rationality and observability
The characterizations are organized by three nested choice environments that differ along two dimensions: how rational the underlying binary relations are, and whether choice is observed as a function or a correspondence. In the minimal environment, choice is a single-valued function and the two relations are preferences, meaning asymmetric binary relations. In the standard environment, choice remains a function but the relations are orders, meaning connected and transitive preferences. In the rich environment, choice is a correspondence and the relations are orders.
These environments are nested in the sense that each adds structure relative to the previous one. The minimal environment imposes the weakest requirements on the relations and on observability, while the rich environment allows multi-valued choice and requires the most structure. The paper provides a revealed preference characterization for each environment rather than a single unified result.
Monotone ownership is equivalent to weak-α
In the minimal environment, the paper focuses on monotone ownership rules. An ownership rule is monotone if, for any three menus ordered by inclusion, a switch in ownership between the first two menus forces the third menu to keep the new owner. This captures situations where one agent handles small menus and the other handles large menus, and once ownership switches to the large-menu agent it cannot switch back.
The characterizing axiom is weak-α, a relaxation of contraction consistency. It requires that if contracting a larger menu to a smaller one violates consistency, then further contractions of that smaller menu do not violate consistency. In other words, a consistency violation can occur at most once along a chain of menus ordered by inclusion, because a violation reveals an ownership change and monotonicity forbids further flips along the same chain.
Theorem 1 states that a choice function respects monotone ownership in a minimal environment if and only if it satisfies weak-α. The proof intuition is that weak-α partitions menus into two blocks, one closed under unions and one closed under intersections, and standard condition α holds separately on each block. The preference over the union-closed block is acyclic and can be extended to an order, which the authors connect to the parent-child paradigm: the parent deciding over larger menus is rationalized by an order, while the child deciding over smaller menus is rationalized by a preference.
Simple ownership is equivalent to PC plus AP
In the standard environment, the paper studies simple ownership rules. A rule is simple if all binary menus are assigned to the same agent, while ownership over larger menus is unrestricted. This captures cases where choice over two-item menus is handled by one mode of thinking, such as fast thinking, and larger menus may be handled differently.
Two axioms characterize this case. Pairwise consistency requires that if x is chosen over y and y is chosen over z in binary menus, then x is chosen over z. Asymmetric prevalence relaxes the strong axiom of revealed preference: an alternative x prevails over y if there is a sequence of alternatives and menus where each menu's choice contradicts the order used for binary menus, and the axiom requires that prevalence cannot hold in both directions.
Theorem 2 states that a choice function respects simple ownership in a standard environment if and only if it satisfies pairwise consistency and asymmetric prevalence. The paper also shows that the two axioms are independent, providing examples where one holds without the other.
The rich environment characterizes multi-valued choice
The third environment allows choice to be a correspondence rather than a function, so a menu can yield multiple chosen alternatives, and the underlying relations are orders. The paper characterizes choice behavior in this environment through a set of axioms, including pairwise consistency, an addition condition, efficiency, and several expansion conditions labeled EI, NBI, SI, EXP.i, and EXP.ii.
The appendix provides independence examples for each axiom in Theorem 3, showing that dropping any single axiom admits a choice correspondence that violates it while satisfying the others. This establishes that the characterization is tight in the sense that each axiom carries independent force.
The rich environment also accommodates joint ownership, where a menu can be assigned to both preferences and the choice consists of the union of their maximizers. The paper defines joint ownership and general ownership, the latter allowing the set of owners to depend on both the menu and the two binary relations themselves.
The characterization is limited to two binary relations
The paper explicitly restricts the range of an ownership rule to two binary relations. The authors conjecture that the general case of n binary relations is much harder, citing results that rationalizing choice behavior is NP-complete in general and that polynomial computability for a specific sequential rationalization model requires at most two binary relations.
This limitation qualifies the scope of the three characterizations. The equivalences between ownership rules and axioms hold within the two-relation setting and within the specific nested environments studied. The paper does not claim that the same axioms characterize ownership rules with more than two relations, nor does it extend the results to stochastic choice or to structurally richer settings such as consumer choice or choice under uncertainty.
The authors list several research directions: studying ownership in structured settings where ownership may depend on other contextual cues, developing notions of random ownership and extending the analysis to stochastic choice, and applying choice via ownership rules to more than two binary relations. These are presented as open avenues rather than established results.
Ownership rules link menu structure to accountability
The paper's central contribution is to make the menu itself a source of bounded rationality. Because the context, such as menu size, domain, or complexity, can interfere with maximizing a single preference over all choice problems, an ownership rule models how the structure of a choice problem determines which binary relation is used to decide. The owner is accountable because it selects the best option according to its own preference.
The three characterizations show that different restrictions on the ownership rule correspond to different observable axioms. Monotone ownership, where ownership switches at most once along chains of menus, corresponds to weak-α. Simple ownership, where all binary menus share one owner, corresponds to pairwise consistency plus asymmetric prevalence. The rich environment, allowing multi-valued choice, corresponds to a broader axiom set including efficiency and expansion conditions.
For readers comparing choice procedures, the practical distinction is the level at which ownership is restricted. If the relevant question is whether ownership can switch only once as menus grow, weak-α is the testable condition. If the question is whether binary menus are handled by a single agent while larger menus may not be, pairwise consistency and asymmetric prevalence are the relevant conditions. The paper's results apply to two-relation settings, and the authors treat extensions to more relations, stochastic choice, and richer environments as open problems.
Sources used in this answer
Choice via Ownership Rules
The paper characterizes three ownership rules over two binary relations in three nested choice environments: monotone ownership in a minimal environment is equivalent to weak-α, simple ownership in a standard environment is equivalent to pairwise consistency plus asymmetric prevalence, and a rich environment yields a correspondence-based characterization with additional axioms.