Dynamic choice under familiarity-based attention

Abstract

Evidence from consumer research indicates that people tend to focus their attention on options with which they are familiar; for this reason, they are also more likely to choose them. To study this phenomenon axiomatically, we extend the standard choice data to specify the timing of choices. This allows us to consider several (nested) time-evolving, experience-based choice models. Specifically, the agent in our theory first considers only the subset of available options with which he is familiar, and only if none of the options in this subset is satisfying (according to a fixed threshold) does he consider the entire possibility set. Importantly, we allow the set of familiar options to expand as the agent acquires experience. We provide choice-theoretic foundations for maximizing a single preference relation under this dynamic, familiarity-based attention, and show how one can reveal from the observed behavior an agent’s preferences and his threshold level. We also provide a comparative measure of preference for familiar options that is observable from choice and relate it to our main representation.

Appendices

Appendix 1: Dynamic limited consideration

The aim of this appendix is to provide a choice-theoretic foundation to the dynamic limited consideration model, which restricts the G-DEF model using the condition:

$$\begin{aligned} x\in \Gamma (A_t)\backslash X_{t}(A) \text{ implies } x\notin A_{\tau }\backslash \Gamma (A_{\tau }) \quad \text{for all } A_{\tau }\subseteq A_{t}. \end{aligned}$$

(3)

Condition (3) states that, if the new option x grabs the agent’s attention in a menu, then it will do so in any subset of that menu, which includes x,  within the same sequence. In the context of the G-DEF model, different options may grab the agent’s attention in different sequences due to, for example, different unobservable consumption histories.

In the presence of condition (3), when a new option, x,  is chosen from menu $A_{t},$ it, clearly, grabs the agent attention. This means that no subset of $A_{t},$ including x was available before time t;  otherwise, x would have been chosen before time t and it was not new. This strongly suggests the following axiom.

Attention Contraction Axiom (ACA):

If $c^{A}_{t}\notin X_{t}(A),$ then $A_{\tau }\ni c^{A}_{t}$ with $\tau <t$ implies $A_{\tau }\nsubseteq A_{t}.$

Proposition 4

Let c be a G-DEF dynamic choice function. Then, c satisfies condition (3) if and only if CAC holds.

Appendix 2: Between Theorems 1 and 2

The aim of this appendix is to show that Theorem 2, Corollary 2, and Proposition 1 work in much the same way when only some of the options can be identified as new options. Formally, we disentangle a subset $X^{*}\subseteq X$, such that $x\in X^{*}$ means that option x was introduced after time $t=1$. Note that if $X^{*}=\emptyset ,$ then we are back to the framework of Theorem 1, while if $X^{*}=X$, we are in the framework of Theorem 2. We can now reformulate the definition of revealed desirability and SARP-EF to apply to the new framework, as follows.

Definition 4

An alternative x is directly revealed undesirable, if there exists $A\in {\mathcal {A}}$ such that $y=c(A_{t})$, $x\in A_{t}$, $x=c^{A}_{i}$ for some $i<t$, $y\ne c^{A}_{i}$ for all $i<t,$ and $x\in X^{*}$; x is indirectly revealed undesirable (denoted $x\in U^{*}$) if there exists a sequence $x_{i},\ i=0,1,\ldots ,n$ such that $x_{0}$ is directly revealed undesirable, $x_{n}=x,$ and $x_{i-1}Rx_{i}$ for all $i>0.$

Definition 4 is a generalization of Definition 1, which regard option as new, only if it was not previously chosen and it belongs to $X^{*}$.

Intermediate Axiom of Revealed Preference for Experienced-based Focus (IARP-EF):

For any nonempty $A_{t}$, there exists $x^{*}\in A_{t}$ such that for any $B_{\tau }\ni x^{*}$ and $x^{*}\ne c_{\tau }^{B }\in A_{t}$, we have that (i) $x^{*}\ne c_{i}^{B }$ for all $i<\tau$, (ii) $c_{\tau }^{B }=c_{i}^{B }$ for some $i<\tau$, and (iii) $c_{\tau }^{B }\notin U^{*}$.

IARP-EF applies SARP-EF whenever a new option can in fact be verified as new (i.e., belongs to $X^{*}$).

Proposition 5

A dynamic choice function c satisfies IARP-EF if and only if it admits the G-DEF model such that¹³:

$$\begin{aligned} A_{t}\cap X_{t}(A)\subseteq \Gamma (A_{t})\subseteq {A_{t}}\cap (X_{t}(A)\cup (X\backslash X^{*}))\quad \text{for all }t. \end{aligned}$$

The model in Proposition 5 restricts that the G-DEF model such that no option that can be verified as new is in the agent’s focus set. Clearly, when $X^{*}=\emptyset$, the revealed preference coincides with $P_{T}$, but as $X^{*}$ gets larger, the revealed preference approaches $P_{T}^{*}$, which is obtained when $X^{*}=X$. Moreover, an immediate corollary of Proposition 5 is that we can now bound the threshold option also from below as follows: $r\succ r_{**}$ for all $r_{**}\in \max _{R}U^{*}$.¹⁴ Note that all components are observable from choice. In other words, whenever the agent chooses a (verifiable) new option over alternative x, we can be sure that x fails the threshold. Notice that as long as there is one option in $X^{*}$ that is sufficiently close to the real threshold but fails it, $r_{**}$ provides a very good approximation for the real threshold.

Appendix 3: Formal proofs

Proof of Theorem 1

Let P defined as in the main text (i.e., xPy if there exists $A\in {\mathcal {A}}$ such that $x=c^{A}_{t}$ and $y\in A_{t}\cap X_{t}(A)$). As explained in the main text, for c that admits the G-DEF model, xPy implies $x\succeq y$, and by the transitivity of $\succeq ,$ $xP_{T}y$, with $P_{T}$ denoting the transitive closure of P, also implies $x\succeq y$. And because $\succeq$ is antisymmetric, we have that P must be acyclic. Now, assume that WARP-EF fails, then there is a menu $A_{t}$ such that for all $x\in A_{t}$ there is $B\in {\mathcal {A}}$ with $B_{\tau }\ni x$ such that $x\ne c_{\tau }^{B }\in A_{t}$ and $x=c^{B}_{i}$ for some $i<\tau .$ Then, we have $c^{B}_{\tau }Px$ with $c_{\tau }^{B }\in A_{t}$ for all $x\in A_{t}$. This means that P has a cycle, and the “if part” is complete.

For the “only if part,” we first show that WARP-EF implies that P is acyclic. Assume that there exists $z_{i},$ $i=0,1,\ldots ,n$ such that $z_{i-1}Pz_{i}$ for all $i>0$ and $z_{k}Pz_{0}$, and take $A_{t}=\bigcup \{z_{i}\}$ in WARP-EF. Then, no option in $A_{t}$ can serve the role of $x^{*}$ in the axiom, as for any $x\in A_{t}$, we have $y\in A_{t}$ such that yPx; that is, $\exists B$ such that $y=c_{\tau }^{B }$ with $B_{\tau }\ni x$ and $x\in X_{\tau }(B)$. By its acyclicity, standard result implies that P can be extended to a linear order $\succeq$ on X. Next, let R be a binary relation on X defined by xRy if there exists $A_{t}$ such that $x=c^{A}_{t}$ and $y\in A_{t}$, and define $Z:=\{z\in X\mid z\succeq x$ implies that if xRy, then $x\succeq y\}$. Now, let $r=\min _{\succeq }X$ if Z is empty, and $r={ ma}x_{\succeq }Z$ otherwise. Note that because $\succeq$ is a linear order and Z and X are finite, such an r exists. Finally, define $\Gamma$ such that for any $A\in {\mathcal {A}}$, $\Gamma (A_{t})={A_{t}}\cap \bigcup _{i \mathbf {\le }t}\{c^{A}_{i}\}$ for all t.

We now show that the G-DEF model holds for the constructed $\succeq$, $\Gamma (A_{t})$, and r. Fix $A\in {\mathcal {A}}$, and assume $y\succ c^{A}_{t}$ for some $y\in A_{t}$. If, in addition, $y\in X_{t}(A)$, then $\succeq$ is not antisymmetric—a contradiction. If, on the other hand, we have in addition, $x\notin \Gamma (A_{t})$, then by the definition of $\Gamma$, $x\ne c^{A}_{t}$—a contradiction. Finally, if, in addition, $r\succ x$, then Z is nonempty and by the definition of Z, xRy implies $x\succeq y$—a contradiction. This completes the proof. $\square$

Proof of Theorem 2

Let $P^{*}$ defined be as in the main text (i.e., $xP^{*}y$ if there exists $A\in {\mathcal {A}}$ such that $x=c^{A}_{t},\ y\in A_{t},$ and: $(\text{a})\ y\in X_{t}(A)$, or $(\text{b})\ x\notin X_{t}(A)$, or $(\text{c})\ x\in U$. As explained, for c that admits R-DEF, $xP^{*}y$ implies $x\succeq y$, and by the transitivity of $\succeq$, $xP_{T}^{*}y$, with $P_{T}^{*}$ denoting the transitive closure of $P^{*}$, also implies $x\succeq y$. And because $\succeq$ is antisymmetric, we have that $P^{*}$ must be acyclic. Now, assume that SARP-EF fails, then there is a menu $A_{t}$, such that for all $x\in A_{t}$, there is $B\in {\mathcal {A}}$ such that $B\ni x$ with $x\ne c_{\tau }^{B }\in A_{t}$ and at least one of (i) $x\notin B$, (ii) $c_{\tau }^{B }\in X_{\tau }(B)$, and (iii) $c^{A}_{\tau }\notin U$ must fail. Then, we have $c_{\tau }^{B }P^{*}x$ with $c^{A}_{\tau }\in A_{t}$ for all $x\in A_{t}$. This means that $P^{*}$ has a cycle, and the “if part” is complete.

For the “only if part,” we first show that SARP-EF implies that $P^{*}$ is acyclic on X: assume that there exists $z_{i},$ $i=0,1,\ldots ,n,$ such that $z_{i-1}P^{*}z_{i}$ for all $i>0$ and $z_{k}P^{*}z_{0}$, and take $A_{t}=\bigcup \{z_{i}\}$ in SARP-EF. Then, no option in $A_{t}$ can serve the role of $x^{*}$ in the axiom, as for any $y\in A_{t}$, we have $x\in A_{t}$ such that $xP^{*}y$; that is, there is $B\in {\mathcal {A}}$ such that $x=c_{\tau }^{B}$, $y\in B$, and one of the conditions (a)–(c) holds (i.e., one of the conditions (i)–(iii) fails). Thus, $P^{*}$ is acyclic also on $U\subseteq X,$ so there exists a linear order $\succeq _{*}$ on U that extends $P^{*}$. Let $r^{\prime }:=\max _{\succeq _{*}}U$ and define $W:=(X\backslash U)\cup \{r^{\prime }\}$. Clearly, $P^{*}$ is acyclic also on W. Note that $r^{\prime }P^{*}y$ for no $y\in W\backslash \{r^{\prime }\}$ for otherwise $r^{\prime }Ry$ (i.e., $r^{\prime }=c^{A}_{t}$ for some $A_{t}\ni y)$, and because $r^{\prime }\in U$, $y\in U$—a contradiction. Thus, $P^{*}$ can be extended to a linear order $\succeq ^{*}$ on Y,  such that $y\succeq ^{*}r^{\prime }$ for all $y\in W$. Now let $\succeq$ be a binary relation on X such that $\succeq \supseteq (\succeq _{*}\cup \succeq ^{*})$ and for all $y\in W$ and $x\in U$, we have $y\succeq x$. It is readily verified that $\succeq$ is a linear order. Note that because $xP^{*}y$ holds for no x, y such that $x\in U$ and $y\in W,\ \succeq$ is an extension of $P^{*}$.

We now show that R-DEF holds for the constructed $\succeq$ and $r:=\min _{\succeq ^{*}}W$. Assume that for some $A\in {\mathcal {A}}$, $x=c^{A}_{t}$ and $y\succ x$ for some $y\in A_{t}$. If, in addition, $y\in X_{t}(A)$, then we have $yP^{*}x$ and $\succeq$ is not antisymmetric—a contradiction. If, on the other hand, we have, in addition, $x\notin \Gamma (A_{t})$, then $x\notin X_{t}(A)$, so once again $\succeq$ is not asymmetric. Finally, if addition, $r\succ x$, then $r^{\prime }\succeq x$ and $x\in U$, but then $x=c^{A}_{t}$ and $y\in A_{t}$ implies that $xP^{*}y$—a contradiction to $y\succ x$. This completes the proof. $\square$

Proof of Corollaries 1 and 2

The “if part” of Corollary 1(2) follows directly from Theorem 1(2). The “only if part” follows by a standard result (originally due to Dushnik and Miller 1941), stating that an asymmetric and transitive relation coincides with the intersections of its linear extensions. $\square$

Proof of Proposition 1

Part (a): assume that c admits R-DEF and let $x\in U.$ Then, $yP^{*}x$ for some y that is directly revealed undesirable. That is, there exists $A\in {\mathcal {A}}$ such that $y\in A_{t}$, $y=c^{A}_{i}$ for some $i<t$, and $c^{A}_{t}\ne c^{A}_{i}$ for all $i<t$. Then, $c^{A}_{t}\notin \Gamma (A_{t})\ni y,$ by the R-DEF model, we have $r\succ y,$ and by Corollary 2, we find that $r\succ x.$ Now, let $x\in D:=\{x\in X\mid x=c^{A}_{t}$ and $yP_{T}^{*}x$ for some $y\in A_{t}\}$ and for the sake of contradiction, assume that $r\succ x$. Then, $x=c^{A}_{t}$ with $x\ne yP^{*}_{T}x$ for some $y\in A_{t}$ implies that $x\succ y\succ x$—a contradiction. This completes part (a).

Part (b): assume that c is an R-DEF with $\succeq$ and r, and take $A\in {\mathcal {A}}^{*}$ such that $A_{1}=\{min_{\succeq }X\},\ A_{2}=\{c^{A}_{1},\{min_{\succeq }(X\backslash X_{1})\}\},\ A_{3}=\{c^{A}_{2},\{min_{\succeq }(X\backslash X_{2})\}\},\ldots , A_{n}=\{c^{A}_{n-1},\{max_{\succeq }X\}\},\ A_{n+1}=\{max_{\succeq }X\},\ A_{n+2}=\{c^{A}_{n},c_{n+1}\}.$ Then, $r=c^{A}_{n}$ is identified uniquely. To see that the identified r uniquely satisfy $r^{*}\succeq r\succ r_{*}$, first note that since the proof of Theorem 2 establishes that c is rationalized by $r=\sup _{P^{*}}U,$ $r_{*}$ is in fact the highest lower bound for r. For the other direction, note that $r=c^{A}_{n}$, with $max_{\succeq }X\in A_{n}$, but from $max_{\succeq }X=c^{A}_{n+2}$ and $r\in X_{n+2}$, we have $max_{\succeq }XP^{*}r$. Thus, $r\in D,$ and the proof is complete. $\square$

Proof of Proposition 2

Part (a): Assume that $r\succ r^{\prime }$ and consider the sequence $A=(\{r^{\prime }\},\{r,r^{\prime }\})$. Then, we have $c^{A}_{1}=c_{1}^{\prime A} =c_{2}^{\prime A}=r^{\prime }$, $c^{A}_{2}=r$, and c has no weakly more PFO than $c^{\prime }$. For the other direction, assume that $r^{\prime }\succeq r$ and for the sake of contradiction, let A be a sequence such that ${c^{\prime }_{t}}^{A}\in X^{\prime }_{t}$, but $c^{A}_{t}\notin X_{t}(A)$. Then, $c_{t}^{\prime A}\succeq r^{\prime }$ and if $c_{t}^{\prime A}\in X_{t}(A)$, then $r\succ c_{t}^{\prime A}$—a contradiction. If, on the other hand, $c_{t}^{\prime A}\notin X_{t}(A),$ then there exists $\tau <t$ such that $c^{\prime A}_{\tau }=c_{t}^{\prime A}$ and $c^{A}_{\tau }\ne c_{t}^{\prime A}$, and because $\succeq =\succeq ^{\prime }$, it must be the case that $c^{\prime } _{\tau }\in X^{\prime }_{\tau }$, thus, if $c^{\prime A}_{\tau }\in X_{\tau }$ holds, we have again $r\succ c_{t}^{\prime A}$—a contradiction. Otherwise there is $j<$ $\tau$ such that $c_{j}^{\prime A}=c_{t}^{\prime A}$, $c^{A}_{j}\ne c_{t}^{\prime A}$ and without loss of generality $c_{j}^{\prime A}\in X^{\prime }_{j}$, continue in this manner, we obtain (since T is finite) that $c_{t}^{\prime A}\in X_{0}$—a contradiction. This completes part (a) of the proof. For part (b), assume that $\succeq \ne \succeq ^{\prime }$, then we have $x,y\in X$ such that $[x\succ y$ and $y\succ ^{\prime }x]$. Now, let $A=(\{x,y\},\{y\}),$ then $c(A)=(x,y)$, while $c^{\prime }(A)=(y,y)$, hence, c has no weakly more PFO than $c^{\prime }$. The fact that $c^{\prime }$ has no weakly more PFO than c follows in the same way and, hence, the proof is complete. $\square$

Proof of Proposition 3

For the “only if part,” assume that $\left| \{x\mid x\succeq r\}\right| <\left| \{x\mid x\ \succeq ^{\prime } r^{\prime }\}\right|$, then for any bijection $\pi :X\rightarrow X$, we can find an option $x_{\pi }$ such that $x_{\pi }\succeq ^{\prime }r^{\prime }$ but $r\succ \pi (x_{\pi })$. Thus, for the sequence $A=(\{x_{\pi }\},\{x_{\pi },r^{\prime }\} )$, we have $c^{\prime }(A)=(x_{\pi },x_{\pi })$. Now, if $c(\pi (A))=(\pi (x_{\pi }),\pi (r^{\prime }))$, then c has no more PFO than $c^{\prime }$. Thus, assume that $c(\pi (A))=(\pi (x_{\pi }),\pi (x_{\pi })),$ and we have $\pi (x_{\pi })\succeq \pi (r^{\prime })$ and by $r\succ \pi (x_{\pi })$, we find that $r\succ \pi (r^{\prime })$. Now, let $B=(\{r^{\prime }\},$ $\{\pi ^{-1}(r),r^{\prime }\})$ and we have $c( B^{\pi })=(\pi (r^{\prime }),r)$ and $c^{\prime }(B)=(r^{\prime },r^{\prime })$. Thus, c has no more PFO than $c^{\prime }$.

For the other direction, assume that $\left| \{x\mid x\succeq r\}\right| \ge \left| \{x\mid x\ \succeq ^{\prime }\ r^{\prime } \}\right|$ and let $\pi :X\rightarrow X$ be a bijection that satisfies $\pi (x)\succeq \pi (y)$ if and only if $x\ \succeq ^{\prime }\ y$ (i.e., the $\succeq ^{\prime }$–maximal options is mapped to the $\succeq$–maximal options, the $\succeq ^{\prime }$–second best option is mapped to the $\succeq$–second best option, and so on). Now, consider another R-DEF choice function ${\hat{c}}$ with the underlying preference relation $\succeq ^{\prime }$ and a threshold ${\hat{r}}=\pi ^{-1}(r)$. Note that by the definition of $\pi$, $\left| \{\pi ^{-1}(x)\mid \pi ^{-1}(x)\ \succeq ^{\prime }\ \pi ^{-1}(y)\}\right| =\left| \{x\mid x \succeq y\}\right|$, hence, by $\left| \{x\mid x\succeq r\}\right| \ge \left| \{x\mid x\ \succeq ^{\prime }\ r^{\prime } \}\right|$, we have $\left| \{\pi ^{-1}(x)\mid \pi ^{-1}(x)\ \succeq ^{\prime }\ \pi ^{-1}(r)\}\right| \ge \left| \{x\mid x\ \succeq ^{\prime }\ r^{\prime } \}\right|$ and, thus, $r^{\prime }\ \succeq ^{\prime }\ \pi ^{-1}(r)$. We can now apply the “if part” of Proposition 2(a) to show that ${\hat{c}}$ has more PFO than $c^{\prime }$. Thus, to complete the proof, it is enough to show that for any $({\hat{c}}^{A}_{1},\ldots ,{\hat{c}}^{A}_{T})={\hat{c}}(A)$ and $(c^{\pi (A)}_{1},\ldots ,c^{\pi (A)}_{T})=c(\pi (A)),$ we have ${\hat{c}}^{A}_{t}=\pi ^{-1}(c^{\pi (A)}_{t})$ for all t and for all $A\in {\mathcal {A}}.$ However, this follows directly from the facts that $\pi (x)\succeq \pi (y)$ if and only if $x\succeq ^{\prime }\ y;$ and that $x\succeq ^{\prime }{\hat{r}}$ if and only if $\pi (x)\succeq r$ (recall that ${\hat{r}}=\pi ^{-1}(r)$). This completes the proof. $\square$

Proof of Proposition 4

Assume that for a G-DEF dynamic choice function c,  condition (3) holds, but ACA fails. Then, we have $A_{\tau }\subseteq A_{t}$ with $\tau <t$ such that $c^{A}_{t}\notin X_{t}(A)$ and $c^{A}_{t}\in A_{\tau }.$ By (3), $c^{A}_{t}\in \Gamma (A_{\tau });$ however, by $c^{A}_{t}\notin X_{t}(A),\ c^{A}_{t}\ne c^{A}_{\tau }$ must be the case. Thus, we find that $c^{A}_{t}\in \Gamma (A_{\tau })$ and $c^{A}_{\tau }\in \Gamma (A_{t})$—a contradiction to c being a G-DEF choice function. For the other direction, we need to show that if (a) $x\in \Gamma (A_{t})\backslash X_{t}(A)$ and (b) $x\in A_{\tau }\backslash \Gamma (A_{\tau })$, then $A_{\tau }\nsubseteq A_{t}$. Let $\Gamma$ be as defined in the proof of Theorem 1 (i.e., $\Gamma (A_t)=A_t\cap (X_{t}(A)\cup \{c^{A}_{t}\})$), by the definition of $\Gamma$, (a) implies that $x=c_{t}^{A}$; in addition, $x\notin X_{t}(A)$ together with (b) imply that $\tau <t$. Now, (a)–(b) together with $x=c_{t}^{A}$ imply that $c_{t}^{A}\notin X_{t}(A)$ and $c_{t}^{A}\in A_{\tau }$, thus, by ACA, we have $A_{\tau }\nsubseteq A_{t}$. This completes the proof. $\square$