Choice and Opportunity Costs
What this paper finds — and why it matters
Layer 1 — Overview
This paper develops a unified choice-theoretic framework in which agents evaluate alternatives not in isolation but relative to their opportunity costs — the alternatives they forgo. The central departure from classical theory is the relaxation of additive separability between benefits and costs. In the standard additive model, accounting for opportunity costs is behaviourally equivalent to simple utility maximisation: a decision maker who correctly perceives the feasible set and maximises an additively separable utility will make identical choices whether or not opportunity costs are explicitly considered (the paper calls this the irrelevance of opportunity costs under additivity, formally establishing it as a general result). Once additive separability is relaxed, however, opportunity costs become non-trivial and generate a genuinely distinct theory of choice.
The primitive of the model is a net preference — an asymmetric binary relation on pairs (x, y) of distinct alternatives, where (x, y) ≻ (w, z) means the agent strictly prefers obtaining x while forgoing y over obtaining w while forgoing z. Because the opportunity cost of a chosen alternative depends on what else the agent would choose, and vice versa, choice emerges from an intrapersonal equilibrium rather than from direct maximisation.
The paper defines and axiomatically characterises two nested models. The Recursive Opportunity Model (ROM) adopts a behavioural definition of opportunity costs: the cost of the chosen alternative x in menu A is c(A \ x), the alternative that would actually be chosen were x unavailable; the cost of every unchosen alternative is x itself. This recursive structure is completely characterised by a single observable condition — Weak Path Independence (WPI): if x is chosen when added to a menu A, then x must also be chosen in a pairwise comparison against c(A). WPI is shown to imply Always Chosen (AC) — that a Condorcet winner is always selected — but it permits pairwise cycles of choice (failures of No Binary Cycles). Rationality within the ROM requires additionally that the net preference be a strict order satisfying Congruence, an acyclicity condition on the gross preference induced by the net preference. Even then, the utility function being maximised need not coincide with the gross preference naturally implied by the underlying psychological net preference, raising a welfare identification problem.
The Opportunity Model (OM) generalises the ROM by allowing the opportunity cost of the chosen alternative to be any unchosen alternative rather than the recursively determined one. This relaxation permits both pairwise cycles and menu effects (Condorcet violations). The OM is completely characterised by Never Chosen (NC): an alternative that loses every pairwise comparison within a menu (a Condorcet loser) cannot be chosen. Imposing a strict order and Congruence on the net preference of an OM rules out only pairwise cycles, leaving menu effects intact. Full rationality within the OM is restored only with the additional assumption that opportunity costs are non-decreasing in the induced gross preference as the feasible set expands (the Increasing Opportunity Model).
Extensions characterise multivalued versions of both models (M-ROM and M-OM) via adapted axioms on choice correspondences, and show that several known behavioural models in the literature — including list-rationalizable choice and game-tree rationalizable choice — satisfy WPI and thus are instances of ROM. Applications demonstrate that OMs can represent the attraction effect and the multiple decoy effect, providing a preference-maximisation account without appealing to bounded cognition, and that ROMs can represent intransitive pairwise choices via smooth parametric net preferences, avoiding the discontinuities of lexicographic semiorder models.
Q: What is the paper’s foundational definition of opportunity cost, and how does it differ from the standard textbook definition? A: The paper defines the opportunity cost of the chosen alternative x in menu A as the alternative that would actually be chosen from A \ {x} — that is, c(A \ {x}). The opportunity cost of any unchosen alternative y is the actual choice x. The standard textbook definition — “the next-best feasible alternative” — presupposes context-independent, additively separable preferences, precisely the assumption the paper relaxes. The behavioural definition is grounded directly in the agent’s own choice function, making it consistent with non-separable evaluations.
Q: Under what conditions do opportunity costs become irrelevant, and why? A: If preferences admit an additively separable utility representation u, then for any finite menu A and any two alternatives x and y, u(x) ≥ u(y) if and only if u(x) − max_{a ∈ A{x}} u(a) ≥ u(y) − max_{a ∈ A{y}} u(a). Net utility maximisation and gross utility maximisation rank alternatives identically. Opportunity costs become non-trivial only when additive separability is relaxed — at that point, the agent’s comparative evaluation of (alternative, cost) pairs can produce choices that no gross utility function rationalises.
Q: What is the Recursive Opportunity Model (ROM) and what single axiom characterises it? A: A choice function c is a ROM if there exists a net preference ≻ such that for every menu A and every unchosen alternative x, the chosen alternative evaluated at its opportunity cost is preferred to x evaluated at c(A). This is equivalent to the choice function satisfying Weak Path Independence (WPI): if x ∉ A and x = c(A ∪ {x}), then x = c({x, c(A)}). WPI is necessary and sufficient for a ROM (Theorem 1). It is not sufficient for full rationality, as it permits pairwise cycles while ruling out menu effects.
Q: What kinds of irrationality can a ROM exhibit, and what kinds does it preclude? A: The paper establishes (Corollary 1) that WPI implies Always Chosen — a ROM always selects the Condorcet winner when one exists. Therefore, the only admissible form of irrational behaviour in a ROM is pairwise cycles (failures of No Binary Cycles). Condorcet violations (menu effects) are precluded. A ROM becomes fully rational if and only if it additionally satisfies No Binary Cycles.
Q: What additional condition on the net preference guarantees that a ROM is rational? A: Theorem 2 establishes that a choice function is rational if and only if it is a ROM generated by a net preference that is a strict order (complete, asymmetric, transitive) satisfying Congruence. Congruence requires that the induced binary relation P≻ on alternatives — defined by xP≻y whenever there exists z such that (x, z) ≻ (y, z) or (z, y) ≻ (z, x) — is acyclic. For a (u, v)-additive net preference, Congruence holds if and only if u and v are ordinally equivalent.
Q: Can rational behaviour generated by a ROM be welfare-analysed using revealed preference in the standard sense? A: No — and this is a key warning in the paper. Even when a ROM with a strict order and Congruence produces fully rational behaviour, the utility function being maximised need not coincide with the gross preference P≻ naturally induced by the underlying net preference. The paper provides an explicit example (Remark 1, equation 10) in which the choice-rationalising order P is xPyPz while the induced preference is xP≻zP≻y. The utility “revealed” by choice may diverge from the psychological primitive driving that choice, undermining the normative authority of standard revealed preference welfare analysis.
Q: What is the Opportunity Model (OM) and how does it extend the ROM? A: The OM relaxes the recursive assumption by allowing the opportunity cost of the chosen alternative to be any unchosen element of the menu rather than specifically c(A \ c(A)). This breaks the recursive structure while preserving the intrapersonal equilibrium character (the choice still affects the net value of alternatives). The OM is completely characterised by Never Chosen (NC): no Condorcet loser can be chosen (Theorem 3). Unlike the ROM, an OM may fail to select the Condorcet winner, permitting both pairwise cycles and Condorcet violations.
Q: What is the Increasing Opportunity Model and when does it restore full rationality? A: An IOM is an OM in which the opportunity function o is monotone in the sense that if A ⊃ B and o(A) ≠ o(B), then o(A) is ranked higher than o(B) in the induced gross preference P≻. Intuitively, opportunity costs do not decrease as the feasible set expands. Theorem 5 establishes that a choice function is rational if and only if it is an IOM generated by a net preference that is a strict order satisfying Congruence. Full rationality within the OM thus requires both the internal consistency of the net preference (strict order, Congruence) and this monotonicity of opportunity costs.
Q: How does the paper explain the attraction effect using the OM? A: In the canonical formulation, c({x,y}) = x, c({y,d}) = y, c({x,d}) = x, and c({x,y,d}) = y, where d is a decoy. This pattern is incompatible with gross preference maximisation. The paper represents it as an OM with opportunity function o({x,y,d}) = d and a strict net preference order yd ≻ xy ≻ yx ≻ xd ≻ dx ≻ dy. The psychological interpretation is that the introduction of the decoy shifts the comparator for y from x to d; y looks more favourably comparable to d than x does, so the equilibrium where y is chosen is selected. No bounded cognition or imperfect attention is assumed.
Q: How does the framework account for multiple decoys? A: With decoys dx and dy specific to x and y respectively, the observed pattern c({x,y}) = x and c({x,y,dy}) = y and c({x,y,dx,dy}) = y can be represented as an OM with a transitive net preference satisfying xdx ≻ ydy ≻ xy ≻ yx ≻ dyy ≻ dxx and opportunity function o({x,y,dx,dy}) = dx, o({x,y,dy}) = dy. The paper notes this net preference can be extended to a strict order while preserving the choice pattern. This accommodates a phenomenon that poses a challenge to standard theoretical choice literature (per Masatlioglu, Nakajima and Ozbay [25]).
Q: How does the ROM explain intransitive choices more smoothly than lexicographic semiorder models? A: The paper shows that the Tversky (1969) cyclical pattern c({x,y}) = x, c({y,z}) = y, c({x,z}) = z with x=(115,7), y=(117,3), z=(120,0) can be generated by net preferences that admit smooth parametric representations. Specifically, for any two alternatives w=(a,b) and z=(c,d), the paper proposes (w,z) ≻ (z,w) iff (max{a−c, b−d})² > k(min{a−c, b−d})², where k is a relative sensitivity parameter. For k=1/2 this yields the required cycle. Lexicographic models require sharp discontinuities in preference and systematic avoidance of trade-offs, which are often viewed as implausible within the standard economic paradigm; the smooth parametric form avoids these features.
Q: What is the relationship between ROMs and previously studied choice models in the literature? A: Several known models satisfy WPI and are therefore, by Theorem 1, instances of ROMs: specifically, Rationalizability by Game Trees (Xu and Zhou) and List-Rationalizable Choice (Yildiz) are shown to satisfy WPI. The two-stage choice model of Bajraj and Ulku satisfies NC but not WPI, making it an OM but not a ROM. The net preference being maximised in each case can in principle be recovered using the explicit construction in the proof of Theorem 1.
Q: How does the ROM relate to Koszegi-Rabin personal equilibrium? A: Both models involve preferences that depend on a variable determined endogenously by choice, requiring an intrapersonal equilibrium concept in which the agent’s conjectures about their own behaviour must be internally consistent. The key difference is that in Koszegi-Rabin the psychological primitive is a set of reference-dependent preferences ≻r on alternatives in X (where r is the reference point), and equilibrium requires c(A) ≻{c(A)} y for all y ∈ A \ c(A). In the ROM, the primitive is a preference on pairs of distinct alternatives, and the opportunity cost differs for each alternative being compared (the chosen alternative has one opportunity cost, each unchosen alternative has a different one, namely c(A) itself).
Net preference: An asymmetric binary relation on pairs (x, y) of distinct alternatives, where (x, y) ≻ (w, z) means the agent strictly prefers to be in a situation where they choose x while forgoing y over a situation where they choose w while forgoing z. The primitive is defined on X = {(x, y) ∈ X × X : x ≠ y}, without imposing additive separability.
Recursive Opportunity Model (ROM): A choice function c is a ROM if there exists a net preference ≻ such that for every menu A and every unchosen x, the pair (c(A), c(A \ c(A))) ≻ (x, c(A)). The opportunity cost of the chosen alternative is defined recursively as c(A \ c(A)); choice results from intrapersonal equilibrium rather than simple maximisation.
Opportunity Model (OM): A generalisation of the ROM in which the opportunity cost of the chosen alternative can be any unchosen alternative in the menu (not necessarily the recursively determined one). Characterised by Never Chosen: no Condorcet loser can be chosen. Permits both pairwise cycles and Condorcet violations.
Weak Path Independence (WPI): The axiom characterising ROMs: if x ∉ A and x = c(A ∪ {x}), then x = c({x, c(A)}). Equivalently, if an alternative is chosen upon being added to a menu, it must also win in a pairwise comparison with what was previously chosen from the original menu.
Congruence: A consistency condition on net preferences requiring that the induced binary relation P≻ — defined by xP≻y whenever there exists z such that (x,z) ≻ (y,z) or (z,y) ≻ (z,x) — is acyclic. For a (u,v)-additive net preference, Congruence holds if and only if u and v are ordinally equivalent. Together with a strict net preference order, Congruence in a ROM is equivalent to rational choice.
Intrapersonal equilibrium: The concept underlying both models: an agent is in equilibrium when selecting x from A if they correctly anticipate their own contingent behaviour across hypothetical scenarios (i.e., they use the actual choice function c to evaluate what they would choose from A \ {x}), and the chosen alternative is net-preference-maximal given those consistent conjectures.
Never Chosen (NC): The axiom characterising OMs: an alternative that is a Condorcet loser — losing in every pairwise comparison within a menu — cannot be chosen from that menu. NC is weaker than WPI (which implies both Always Chosen and Never Chosen) and is the precise behavioural content of the OM.