Non-Binary Menus with Ordered Options

This section develops an approach for preference recovery over non-binary menus. There are many interesting possibilities for generalizations, but we focus here on choice situations g ∈ G consisting of a fixed, finite menu of K ordered options X = {1, 2, ..., K} and one of two frames, d ∈ {d^h, dl}. Intuitively, one can think of d^h and dl as a “high” frame and a “low” frame. For example, we might suppose that an individual chooses from a menu of insurance plans, ordered from low-cost, low-benefit plans to high-cost, high-benefit plans, and the frame either emphasizes or de-emphasizes the individual’s risk of developing a serious illness. We will assume that we observe each individual i in exactly one frame, as before.

Recall that in the binary case, the consistency principle and frame monotonicity imply that individuals who choose the “low” option in the “high” frame prefer the low option. We will use this same intuition to develop an identification strategy for the non-binary setting.

The preferences of agent i are represented by choice function y_i^∗ ∈ X. We continue to assume frame separability (A1), so that y_i^∗ does not depend on d.

We strengthen the frame monotonicity assumption as follows:

D1 (Frame monotonicity for many options) For all individuals, yi(dh) ≥ yⁱ(dl).

This frame monotonicity assumption imposes an implicit ordering on the menu and assumes that all individuals are pushed in the same direction by the frames. We also strengthen the consistency principle with the following assumption

D2 (Partition-Consistency principle) For all individuals i and options k ∈ X, y_i(dl) ≥ k =⇒ yi^∗ ≥ k

y_i(dh) ≤ k =⇒ yi^∗ ≤ k. (40)

The name of this assumption comes from the following: suppose that we partition the menu into X⁰ = {J, J + 1, ...K} and X⁰⁰ = X \ X⁰, for some J and K ≥ J. If the individual

consistently chooses within X⁰ across both frames, so yi(dh) ∈ X⁰, and yi(dl) ∈ X⁰, then assumption (40) implies that y_i^∗ ∈ X⁰. Note also that the partition consistency principle implies the consistency principle used in previous sections: if yi(dh) = yi(dl), then assumption (40) implies that yi(d) = y^∗_i.Finally, note that the partition consistency principle and frame monotonicity together imply that ∀i, yi(d^h) ≥ y^∗i ≥ yi(d^l).

For each k = 1, ..., K, we define partition consistency at k, c^k_i, as follows c^k_i ≡ 1{yⁱ(d^h) ≤ k and yⁱ(d^l) ≤ k} + 1{yⁱ(d^h) > k and yi(d^l) > k}

Intuitively, c^k_i captures whether an individual consistently chooses an option above or below k. Note also that frame monotonicity implies that one of the conditions inside each indicator function will be implied by the other condition.

Proposition A2 Let Gj(k) ≡ P(yⁱ(dj) ≤ k|dⁱ = dj) for k = 1, ..., N, j = h, l and let Gj(0) ≡ 0. Let Y^k≡ _Gh(k)+1−G^Gh(k) _l(k) for k = 0, ..., K. Frame separability (A1), frame monotonic-ity (D1), partition consistency (D2), and unconfoundedness (A4) imply that for k = 1, ..., K, (A2.1) The fraction of partition-consistent individuals at k with y^∗_i ≤ k is given by P (y^∗i ≤

k|c^ki = 1) = Yk.

(A2.2) The fraction of partition-consistent individuals at k is given by E[c^k_i] = Gh(k) + 1 − Gl(k).

(A2.3) The fraction of the population who prefer option k is bounded as follows: p(y^∗_i = k) ∈ [Gl(k) − Gh(k − 1), Gh(k) − Gl(k − 1)].

(A2.4) If we additionally assume strong decision-quality independence,∀k, yi^∗ ⊥ c^ki, then the fraction of the population who prefer option k is p(y_i^∗ = k) = Yk− Y^k−1.

Proof

Throughout the proof, we denote the fraction of individuals preferring some option k by

¯φk ≡ p(yi^∗ = k).

Proof of (A2.1) and (A2.2): Fix some k ∈ {1, ..., K − 1}. Let X⁰ = {x1, ...xk}and X⁰⁰ = {xk+1, .., xK} Note that we can write the many-choices problem into a binary menu choice problem between X⁰ and X⁰⁰. Similarly, note that frame separability (A1), frame monotonicity (D1), partition consistency (D2), and partition unconfoundedness (A4) imply the binary analogues to these assumptions A1-A4. As such, (A2.1) and (A2.2) follows directly from the application of Proposition 1 to this problem.

Proof of (A2.3): First suppose that k = 1. Applying Proposition 2 to the binary menu choice problem with X⁰ = {1} and X⁰⁰ = {2, ..., K} implies that

E[φ1] ∈ [Gl(1), Gh(1)] (41)

Note that this confirms the desired result for k = 1 since Gh(0) = Gl(0) = 0 by definition.

Next, applying the same proposition for k = 2, we have φ₁+ φ₂ ∈ [Gl(2), Gh(2)]. Combined with (41), this implies

φ₂ ∈ [Gl(2) − Gh(1), Gh(2) − Gl(1)] (42) Similarly with k = 3, we have that φ₁+ φ₂+ φ₃ ∈ [Gl(3), Gh(3)], and applying (41) and (42) implies that φ₃ ∈ [Gl(3) − Gh(2), Gh(3) − Gl(2)]. Proceeding recursively, suppose that for some k, we know that for any k⁰ < k,

φ_k0 ∈ [G^l(k⁰) − G^h(k⁰− 1), G^h(k⁰) − G^l(k⁰− 1)] (43) Then application of Proposition 2 to the binary menu choice problem with X⁰ = {x1, ..., x_k} yields φ₁+φ₂+...+φ_k ∈ [Gl(k), Gh(k)], so φ_k ∈ [Gl(k)−(φ1+φ₂+...+ ¯φk+1), Gh(k)−(φ1+ φ₂+ ... + ¯φk+1)]. Applying the lower and upper bounds from (43) and simplifying yields the desired result.

Proof of (A2.4): Along with (A2.1), strong decision-quality independence implies that for any k,

P(y^∗_i ≤ k|c^ki = 1) = P (y^∗_i ≤ k) = Yk (44) Applying (44) at k = 1 yields

¯φ1 = Y1 (45)

Applying (44) at k = 2 yields ¯φ1+ ¯φ2 = Y2 and substituting equation (45) yields

¯φ₂ = Y2 − Y1

As in the proof of (A2.3), we proceed recursively to obtain the desired result. Given some k,suppose that for any k⁰ < k we have

¯φk⁰ = Yk⁰ − Yk⁰−1 (46)

Applying (44) at k yields ¯φ1+ ¯φ2+..., +¯φk= Yk.Applying (46) for ¯φ1, ..., ¯φk−1 and simplifying

yields the desired result. 

Discussion of Proposition A2 If we partition the menu of choices into options above and below some option k, then frame monotonicity and the partition-consistency principle

transform the problem to a binary problem, allowing us to use earlier propositions to identify individuals whose preferred choice is above or below k. The first two results, (A2.1) and (A2.2), are therefore the analogue of Proposition 1 in this setting.

Return to the insurance example described above, where the frame either emphasizes or de-emphasizes the risk of serious illness. When some individuals choose a benefit, low-cost plan under the frame that emphasizes the risk of serious illness, our assumptions imply that they prefer an option with costs and benefits at least as low as the ones they choose.

The first two results in Proposition A2 allow us to estimate the fraction of decision-makers who consistently choose an insurance plan that is above or below some specified cost-benefit level, and among those people, how many prefer the low-cost plan.

As in Proposition 2, we can also bound population preferences, reflected in (A2.3). In this case, the many-options problem has a new and interesting structure. Even if individuals are highly susceptible to framing effects when they prefer some option far away from k, our estimate for the fraction of people preferring option k can still be precise, because the partition consistency principle permits us to ignore individuals who consistently choose options above or below k.

Finally, with a stronger version of the decision-quality independence assumption, we can recover the distribution of preferences for the full population. Strong decision-quality independence guarantees that the tendency to be partition-consistent for any partition is un-related to an individuals’ preferences.²² Under strong decision-quality independence, obtain-ing the preferences of partition-consistent individuals will yield the distribution of preferred choices in the population. The equivalence of this problem to the binary problem implies that we could generalize other identification strategies from the binary case. For example, we can identify the preferences of the population using observables via a conditional strong decision-quality independence assumption (the generalization of the matching approach), and in the absence of any decision-quality independence assumptions we can study variation induced by a decision-quality instrument.