Salience and limited attention

Abstract

In this study, we investigate how the salience of alternatives influences the attention of a decision maker in a natural extension of the model of choice with limited attention by Masatlioglu et al. (Am Econ Rev 102(5):2183–2205, 2012). We propose positive salience effects on the attention of a decision maker. For example, when an alternative to which a consumer pays attention becomes more salient in terms of the extent of product advertisement, she will still pay attention to it in a market. We manipulate the salience of alternatives to find choice reversals from observable choice data, which play an important role in our analysis. The advantages of our model over the Masatlioglu et al. (Am Econ Rev 102(5):2183–2205, 2012) model are that it broadens the application of the model of choice with limited attention, and improves the power of inferring the decision maker’s preference and the alternatives to which she pays attention and does not pay attention.

Appendix

We have already proved the if-parts of Theorems 1 and 2 in the body of the paper, which will be useful in proving the remaining results.

Proof of Lemma 1

We first show that if c satisfies WARP(SLA), then P is acyclic. Suppose P has a cycle: $x^{1}Px^{2}P\cdots Px^{t}Px^{1}$. Without loss of generality, it is possible to rearrange the order of alternatives such as $x_{1}Px_{2}P\cdots Px_{t}Px_{1}$. Then, for any $j\in \{1,\ldots ,t-1\}$, there exist $B^{j}\in \mathscr {A}$, $i\in Q$, and $b'_{i(j+1)}\in I_{i}$, with $b^{j}_{i(j+1)}$ $\ge b'_{i(j+1)}$, such that $c(B^{j})=x_{j}\ne c(B^{j}\backslash b'_{i(j+1)})$, and there exist $B^{t}\in \mathscr {A}$, $i\in Q$, and $b'_{i1}\in I_{i}$ with $b^{t}_{i1}\ge b'_{i1}$ such that $c(B^{t})=x_{t}\ne c(B^{t}\backslash b'_{i1})$. Consider a profile A with $\{x_{1},\ldots ,x_{t}\}=F(A)$. Then, for any $x_{k}\in F(A)$, there exists B with $x_{k}\in F(B)$, such that $c(B)\in F(A)$, and $c(B)\ne c(B\backslash b'_{ik})$, for some $i\in Q$ and some $b'_{ik}\in I_{i}$, with $b_{ik}\ge b'_{ik}$ but $c(B)\ne x_{k}$, which implies that c violates WARP(SLA).

Next, we show that if P is acyclic, then c satisfies WARP(SLA). Suppose P is acyclic. Then, for any $A\in \mathscr {A}$, there exists at least one alternative $x_{j}\in F(A)$ such that there is no $x_{k}\in F(A)$ with $x_{k}Px_{j}$, which implies that there is no $x_{k}\in F(A)$ such that $c(B)=x_{k}\ne c(B\backslash b'_{ij})$ for some $B\in \mathscr {A}$, some $i\in Q$, and some $b'_{ij}\in I_{i}$, with $b_{ij}\ge b'_{ij}$. Therefore, for any $B\in \mathscr {A}$ with $x_{j}\in F(B)$, whenever $c(B)\in F(A)$ and $c(B)\ne c(B\backslash b'_{ij})$, for some $i\in Q$ and some $b'_{ij}\in I_{i}$, with $b_{ij}\ge b'_{ij}$, we have $c(B)=x_{j}$, which implies that c satisfies WARP(SLA). $\square$

Proof of Theorem 3

To show that c satisfies WARP(SLA), suppose c is a CSLA represented by $(\succ ,\Gamma )$. Then, the if-part of Theorem 1 implies that $\succ$ includes P. Hence, P must be acyclic. It follows from Lemma 1 that c satisfies WARP(SLA).

We now suppose c satisfies WARP(SLA). Since P is acyclic by Lemma 1, we can find a preference $\succ$ that includes P. Consider a preference $\succ$ that includes P, and define the following consideration set mapping: for any $A,A'\in \mathscr {A}$, $F(A)=F(A')$ implies $\Gamma (A)=\Gamma (A')$ and, for any $A\in \mathscr {A}$,

$$\begin{aligned} \Gamma (A)=\{x_{j}\in F(A):c(A)\succ x_{j}\}\cup \{c(A)\}. \end{aligned}$$

Then, it is clear that c(A) is the unique $\succ$-best alternative in $\Gamma (A)$. We now show that $\Gamma$ is a salience-affected attention filter. We shall show that $\Gamma (A)=\Gamma (A\backslash a'_{ij})$ for any $A\in \mathscr {A}$, any $i\in Q$, any $x_{j}\in X$, and any $a'_{ij}\in I_{i}$, with $a_{ij}\ge a'_{ij}$ whenever $x_{j}\notin \Gamma (A)$. Suppose $x_{j}\in F(A)$, but $x_{j}\notin \Gamma (A)$. Then, we have $c(A)\ne x_{j}$. By construction of $\Gamma$, we have $x_{j}\succ c(A)$ and, therefore, $c(A)Px_{j}$ does not hold, which implies that we have $c(A)=c(A\backslash a'_{ij})$. By construction of $\Gamma$, we have $\Gamma (A)=\Gamma (A\backslash a'_{ij})$. $\square$

Proof of Theorem 1

We will prove the only-if part of Theorem 1. That is, we show that if $x_{j}$ is revealed to be preferred to $x_{k}$, then we have $x_{j}P_{R}x_{k}$. Suppose $x_{j}P_{R}x_{k}$ does not hold. Then, there exists a preference $\succ$ that includes $P_{R}$ and ranks $x_{k}$ higher than $x_{j}$. The proof of Theorem 3 shows that c can be represented by the preference $\succ$. Therefore, we conclude that $x_{j}$ is not revealed to be preferred to $x_{k}$. $\square$

Proof of Theorem 2

We will prove the only-if part of Theorem 2.

(Revealed Inattention): We first show that if $x_{j}$ is revealed not to attract attention at A, then we have $x_{j}P_{R}c(A)$. Suppose $x_{j}P_{R}c(A)$ does not hold. Then, consider a preference $\succ$ that includes $P_{R}$ and ranks c(A) higher than $x_{j}$. By the proof of Theorem 3, c can be represented by the preference and a salience-affected attention filter, with $x_{j}\in \Gamma (A)$.

(Revealed Attention): Next, we show that if $x_{j}$ is revealed to attract attention at A, there exists $B\in \mathscr {A}$ that satisfies the condition of statement 2 in Theorem 2. Suppose there exists no B that satisfies the condition. We prove that if c is a CSLA, then it can be represented by some salience-affected attention filter $\Gamma$, with $x_{j}\notin \Gamma (A)$. If $c(A)P_{R}x_{j}$ does not hold, we have already shown that c can be represented by $(\succ ,\Gamma )$ with $x_{j}\succ c(A)$ and $x_{j}\notin \Gamma (A)$. Thus, $x_{j}$ is not revealed to attract attention at A and, therefore, we only consider the case where $c(A)P_{R}x_{j}$ holds.

Now construct a binary relation $\tilde{P}$ as follows: $a\tilde{P}b$ if and only if “$aP_{R}b$” or “$c(A)=a$ and not $bP_{R}c(A)$”. That is, $\tilde{P}$ ranks c(A) as high as possible unless it contradicts $P_{R}$. Since $P_{R}$ is acyclic and c is represented by a salience-affected attention filter, we can show that $\tilde{P}$ is also acyclic. Given this observation, consider a preference $\succ$ that includes $\tilde{P}$ as well as $P_{R}$. We have already shown that the following consideration set mapping is a salience-affected attention filter: for any $A',A''\in \mathscr {A}$, $F(A')=F(A'')$ implies $\tilde{\Gamma }(A')=\tilde{\Gamma }(A'')$ and, for any $A'\in \mathscr {A}$, $\tilde{\Gamma }(A')\equiv \{x_{j}$ $\in F(A'):c(A')\succ x_{j}\}\cup \{c(A')\}$. In addition, we can show that c is represented by $(\succ ,\tilde{\Gamma })$. We now define $\Gamma$ as follows:

$$\begin{aligned} \Gamma (A')=\left\{ \begin{array}{ll} \tilde{\Gamma }(A') &{}\quad \text {for }A'\notin \Omega \\ \tilde{\Gamma }(A')\backslash x_{j} &{}\quad \text {for }A'\in \Omega \end{array}\right. , \end{aligned}$$

where $\Omega$ is a collection of profiles, such that

$$\begin{aligned} \Omega = \begin{Bmatrix}&(\mathrm{a}) ~c(A')=c(A), \\&(\mathrm{b}) ~ x_{k}P_{R}c(A) ~ \text {for any }x_{k}\in (F(A)\backslash F(A'))\cup (F(A')\backslash F(A)), \\ A'\in \mathscr {A}:&\text {and (c) for any} ~i\in Q\backslash \{1\} \text {and any} x_{k}\in F(A)\cup F(A'), \\&a_{ik}=a'_{ik} \text {unless }x_{k}P_{R}c(A) \end{Bmatrix}. \end{aligned}$$

Thus, $\Gamma$ is obtained from $\tilde{\Gamma }$ by removing $x_{j}$ from $\tilde{\Gamma }(A')$ of any profile $A'\in \mathscr {A}$ where (a) $c(A')=c(A)$, (b) any alternative that belongs to F(A) and $F(A')$, but not to both, is revealed to be preferred to c(A), and (c) for any $i\in Q\backslash \{1\}$ and any $x_{k}\in F(A)\cup F(A')$, $a_{ik}=a'_{ik}$, unless $x_{k}$ is revealed to be preferred to c(A). Note that it follows from $c(A)P_{R}x_{j}$ that $c(A)\ne x_{j}$. Therefore, $\Gamma (A')\subset \tilde{\Gamma }(A')$ always contains $c(A')$. Furthermore, since $(\succ ,\tilde{\Gamma })$ represents c, $(\succ ,\Gamma )$ also represents c. Thus, we only need to show that $\Gamma$ is a salience-affected attention filter.

Note that $\tilde{\Gamma }$ is a salience-affected attention filter and $c(A')=c(A'')$ whenever $\tilde{\Gamma }(A')=\tilde{\Gamma }(A'')$ because $(\succ ,\tilde{\Gamma })$ represents c.

Suppose $x_{k}\notin \Gamma (B)$ for any $B\in \mathscr {A}$. We prove $\Gamma (B)=\Gamma (B\backslash b'_{ik})$ for any $i\in Q$ and any $b'_{ik}\in I_{i}$, with $b_{ik}\ge b'_{ik}$. We now show that $\Gamma (B)=\Gamma (B\backslash b'_{ik})$, for any $i\in Q$ and any $b'_{ik}\in I_{i}$ with $b_{ik}\ge b'_{ik}$. We distinguish three possible cases: (i) $x_{k}=x_{j}$, (ii) $B\in \Omega$ and $x_{k}\ne x_{j}$, and (iii) $B\notin \Omega$ and $x_{k}\ne x_{j}$.

Case (i): If $B\notin \Omega$, then we have $\Gamma (B)=\tilde{\Gamma }(B)=\tilde{\Gamma }(B\backslash b'_{ik})=\Gamma (B\backslash b'_{ik})$. If $B\in \Omega$, then $c(B)=c(B\backslash b'_{ik})$ must hold (otherwise, the condition of the statement is satisfied). It follows from the construction of $\tilde{\Gamma }$ and $\Gamma$ that $\Gamma (B)=\tilde{\Gamma }(B)\backslash x_{j}=\tilde{\Gamma }(B\backslash b'_{ik})=\Gamma (B\backslash b'_{ik})$.

Case (ii): Since $x_{k}\notin \Gamma (B)$ is equivalent to $x_{k}\notin \tilde{\Gamma }(B)$, we have $\tilde{\Gamma }(B)=\tilde{\Gamma }(B\backslash b'_{ik})$, by construction of $\tilde{\Gamma }$. Therefore, we have $c(B\backslash b'_{ik})=c(B)=c(A)$. Since we have $x_{k}\notin \Gamma (B)$, $x_{k}\succ c(A)$ must hold, by construction of $\Gamma$ and $\tilde{\Gamma }$, which implies $x_{k}P_{R}c(A)$ by construction of $\succ$. Then, we obtain $B\backslash b'_{ik}\in \Omega$. Therefore, we have $\Gamma (B)=\tilde{\Gamma }(B)\backslash x_{j}=\tilde{\Gamma }(B\backslash b'_{ik})\backslash x_{j}=\Gamma (B\backslash b'_{ik})$.

Case (iii): If $B\backslash b'_{ik}\in \Omega$, we have $c(B)=c(B\backslash b'_{ik})=c(A)$ and $x_{k}P_{R}c(A)$ by a similar argument to that above. Therefore, $B\in \Omega$ must hold, which is a contradiction. Thus, we have $B\backslash b'_{ik}\notin \Omega$, which implies that $\Gamma (B)=\tilde{\Gamma }(B)=\tilde{\Gamma }(B\backslash b'_{ik})=\Gamma (B\backslash b'_{ik})$. $\square$