Revealed Attention

Abstract

The standard revealed preference argument relies on an implicit assumption that a decision maker considers all feasible alternatives. However, the marketing and psychology literatures provide well-established evidence that consumers do not consider all brands in a given market before making a purchase (Limited Attention). In this chapter, we illustrate how one can deduce both the decision maker’s preference and the alternatives to which she pays attention and inattention from the observed behavior. We illustrate how seemingly compelling welfare judgements without specifying the underlying choice procedure are misleading. Further, we provide a choice theoretical foundation for maximizing a single preference relation under limited attention.

The original article first appeared in the American Economic Review 102(5):2183–2205, 2012. A newly written addendum has been added to this book chapter.

Appendices

Appendix

1.1 Proofs

Notice that the if-parts of Theorems 1 and 2 have been already shown in the main text. The following proofs use these results.

1.1.1 Proof of Theorem 3

Suppose c is a CLA represented by $(\succ,\varGamma )$. Then Theorem 1(if part) implies that $\succ$ must include P so P must be acyclic. Therefore, by Lemma 1, c must satisfy WARP(LA).

Now suppose that c satisfies WARP(LA). By Lemma 1, P is acyclic so there is a preference $\succ$ that includes P. Pick any such preference arbitrarily and define

$$\displaystyle{ \varGamma (S) =\{ x \in S\:\ c(S) \succ x\} \cup \{ c(S)\}. }$$

(19.2)

Then, it is clear that c(S) is the unique $\succ$-best element in Γ(S) so all we need to show is that Γ is an attention filter. Suppose x ∈ S but $x\not\in \varGamma (S)$ (so x ≠ c(S)). By construction, $x \succ c(S)$ so it cannot be c(S)Px. Hence, it must be $c(S) = c(S\setminus x)$ so we have $\varGamma (S) =\varGamma (S\setminus x)$. $\square$

1.1.2 Proof of Theorem 1 (The Only-If Part)

Suppose xP _Ry does not hold. Then there exists a preference that includes P _R and ranks y better than x. The proof of Theorem 3 shows that c can be represented by such a preference so x is not revealed to be preferred to y. $\square$

1.1.3 Proof of Theorem 2 (The Only-If Parts)

(Revealed Inattention) Suppose x is not revealed to be preferred to c(S). Then pick a preference that includes P _R and puts c(S) above x. The proof of Theorem 3 shows that c can be represented by such a preference and an attention filter Γ with x ∈ Γ(S).

(Revealed Attention) Suppose there exists no T which satisfies the condition. We shall prove that if c is a CLA then it can be represented by some attention filter Γ with $x\not\in \varGamma (S)$. If c(S)P _R x does not hold, we have already shown that c can be represented with $x \succ c(S)$ and $x\not\in \varGamma (S)$ so x is not revealed to attract attention at S, so we focus on the case when c(S)P _R x.

Now construct a binary relation, $\tilde{P}$, where $a\tilde{P}b$ if and only if “aP _R b” or “a = c(S) and not bP _R c(S).” That is, $\tilde{P}$ puts c(S) as high as possible as long as it does not contradict P _R . Since P _R is acyclic and c is represented by an attention filter, one can show that $\tilde{P}$ is also acyclic. Given this, take any preference relation $\succ$ that includes $\tilde{P}$, which includes P _R as well. We have already shown that $\tilde{\varGamma }(S) \equiv \{ z \in S: c(S) \succ z\} \cup \{ c(S)\}$ is an attention filter and $(\tilde{\varGamma },\succ )$ represents c. Now define Γ as follows:

$$\displaystyle{ \varGamma \left (S^{{\prime}}\right ) = \left \{\begin{array}{cc} \tilde{\varGamma }\left (S^{{\prime}}\right ) & \mbox{ for }\ S^{{\prime}}\notin \mathcal{D}, \\ \tilde{\varGamma }\left (S^{{\prime}}\right )\setminus x&\mbox{ for }\ S^{{\prime}}\in \mathcal{D}, \end{array} \right. }$$

where $\mathcal{D}$ is a collections of sets such that

$$\displaystyle{ \mathcal{D} =\left \{S^{{\prime}}\subset X: \begin{array}{cc} c\left (S^{{\prime}}\right ) = c\left (S\right )\mbox{ } &\,\mbox{ and} \\ \mathit{zP}_{R}c\left (S\,\right )\ \mbox{ for all }\ z \in \left (S\setminus S^{{\prime}}\right ) \cup \left (S^{{\prime}}\setminus S\right ).& \end{array} \right \} }$$

That is, Γ is obtained from $\tilde{\varGamma }$ by removing from x any budget set $S^{{\prime}}$ where $c(S) = c(S^{{\prime}})$ and any item that belongs to S or $S^{{\prime}}$ but not to both is revealed to be better than c(S). Notice that x cannot be c(S) because if this true, the condition of the statement is satisfied for T = S. Hence, $\varGamma (S^{{\prime}}) \subset \tilde{\varGamma } (S^{{\prime}})$ always includes $c(S^{{\prime}})$. Furthermore the proof of Theorem 3 shows that $(\tilde{\varGamma },\succ )$ represents c. Therefore, $(\varGamma,\succ )$ also represents c so we only need to show that Γ is an attention filter.

To do that, it is useful to notice that $\tilde{\varGamma }$ is an attention filter and $c(T^{{\prime}}) = c(T^{{\prime\prime}})$ whenever $\tilde{\varGamma }(T^{{\prime}}) =\tilde{\varGamma } (T^{{\prime\prime}})$ because $(\tilde{\varGamma },\succ )$ represents c.

Suppose $y\not\in \varGamma (T)$. We shall prove $\varGamma (T) =\varGamma (T\setminus y)$.

• Case I: y = x

  If $T\not\in \mathcal{D}$, then we have $\varGamma (T) =\tilde{\varGamma } (T) =\tilde{\varGamma } (T\setminus x) =\varGamma (T\setminus x)$. If $T \in \mathcal{D}$, then it must be $c(T) = c(T\setminus x)$ (otherwise, the condition of the statement is satisfied) so by construction of $\tilde{\varGamma }$ and Γ, we have $\varGamma (T) =\tilde{\varGamma } (T)\setminus x =\tilde{\varGamma } (T\setminus x) =\varGamma (T\setminus x).$

• Case II: $T \in \mathcal{D}$ and y ≠ x

  Since $y\not\in \varGamma (T)$ is equivalent to $y\not\in \tilde{\varGamma }(T)$, we have $\tilde{\varGamma }(T) =\tilde{\varGamma } (T\setminus y)$. Therefore, $c(T\setminus y) = c(T) = c(S)$. By construction of Γ and $\tilde{\varGamma }$, it must be $y \succ c(S)$, which implies yP _Rc (S) by construction of $\succ$. Therefore, $T\setminus y \in \mathcal{D}$. Therefore, $\varGamma (T) =\tilde{\varGamma } (T)\setminus x =\tilde{\varGamma } (T\setminus y)\setminus x =\varGamma (T\setminus y)$.

• Case III: $T\not\in \mathcal{D}$ and y ≠ x

  If $T\setminus y \in \mathcal{D}$, analogously to the previous case, we have $c(T) = c(T\setminus y) = c(S)$ and yP _R c(S) so it must be $T \in \mathcal{D}$, which is a contradiction. Hence, $T\setminus y\not\in \mathcal{D}$ so we have $\varGamma (T) =\tilde{\varGamma } (T) =\tilde{\varGamma } (T\setminus y) =\varGamma (T\setminus y)$. $\square$

Addendum

This addendum has been newly written by Daisuke Nakajima for this book chapter.

The article in this chapter, since first presented in 2008, have triggered many interesting studies.²⁶ It is not possible to address all of them so I am presenting two most closely related ones in this addendum.

2.1 Incorporating Auxiliary Data

As we discuss in Sect. 6, it is not always possible to completely elicit DM’s preference from her choice data, which contains only pairs of a feasible set and her choice. In many environments, we can access data beyond feasible sets. For instance, marketing analysts knows the amount of time and money spent on advertisements for each product.

Consider the following situation. The DM chooses x from a certain set of alternatives. Now her choice suddenly shifts to y when a third product z is more advertised. I argue that she prefers y to z. To see this, suppose she does not pay attention to z in the latter environment. Then, it is hard to imagine she does so in the former case where z is less advertised. Thus, z is never considered in neither environment. Extending the idea of attention filters, I conclude that the advertising such an ignored product will not affect her attention span, so should not cause the choice shift. This is a contradiction so she must have considered z when it is more advertised. Thus, we can conclude she prefers y to z.

Iwata (2013) generalizes this idea, which illustrates how to utilize observable salience factors possessed by each alternative in each decision problem. Iwata’s model, called generalized CLA, consists of a stable preference and a consideration set mapping like our original CLA model. The difference is that Iwata’s consideration set mapping now depends not only on a feasible set but also on salience factors of all alternatives. Adapting the idea of our attention filter, Iwata requires DM’s attention filter unaffected when an unconsidered alternative is removed or when an unconsidered alternative’s salience decreases. Iwata defines revealed preference, attention and inattention as we do, and characterizes them like our Theorems 1 and 2.

2.2 Limited Data

We assume all choice data are available. That is, DM’s choices from all subsets of X are all observed. Although this requirement is not uncommon among the theoretical literatures, it sounds too much in many of empirical and experimental settings.

The lack of complete choice data does not undermine the effectiveness of our identification. We can conclude that the DM prefers x to y based only on as few as two decision problems. Nevertheless, de Clippel and Rozen (2014) illustrates Theorems 1 and 3 cannot be literally extended when choice data is limited. Example 2 in their study demonstrates this issue as follows. Suppose there are five potentially available alternatives a, b, d, e, f, but we observe only five choice data:

$$\displaystyle{ c(\mathit{ae}) = e,\ \ c(\mathit{ef }) = f,\ \ c(\mathit{abd}) = d,\ \ c(\mathit{ade}) = a.c(\mathit{bde}) = b. }$$

Suppose c is a CLA. This data contains only one choice reversals (between ade and ad) so $a \succ d$ seems the only revealed preference.

We can, however, also conclude that the DM prefers b to e. Imagine that the DM faced another binary decision problem between b and d. He would choose d. To see this, note that $a\not\in \varGamma (\mathit{ade})$ because $a \succ c(\mathit{ade}) = d$. Thus, removing a would not affect her consideration span nor her choice so it would be c(bd) = d. Notice that we observe c(bde) = b so removing e from bde, although it is not actually observed, would cause a choice reversal. Considering these hypothetical choices, we must conclude she prefers b over e. This example shows that, when the choice data is limited, Theorem 1 overlooks some identifiable parts of the DM’s preference.

Now imagine that we have one more extra choice data c(bef ) = e, which is the original version of Example 2 in Clippel and Rozen’s paper. This extra information, together with c(ef ) = d, makes it possible to identify $e \succ b$. This does not sound a contradiction if we do not imagine that the DM could face the choice between b and d. Thus one may wonder these six data are compatible with the CLA model but we have already shown that the first five data reveals $b \succ e$ by considering the hypothetical binary choice. In sum, this example illustrates that data seemingly satisfying the WALP(LA) may not be explained by the CLA model.

de Clippel and Rozen (2014) emphasize the second issue, and propose the remedy of the behavioral characterization of the CLA model (Proposition 2). Nevertheless, they remark that their results illustrates a weakness of our theory as follows:

Being subject to this pitfall is not a weakness of a theory. Rather, the moral is that one cannot limit the test of consistency to finding a story that explains the observed data, without thinking whether that story extends. This extensibility problem is precisely avoided (for any theory) by employing Definition 1.

In line with their remark, I would like to emphasize the following point: the first issue (the overlooked revealed preference) shows that our model can generate more preference information by the careful investigations rather than mindlessly applying Theorem 1. It does not show a pitfall of but a power of our model, which is more than it appears to have.