Limited Attention#

Model choice under consideration set constraints. Based on Chambers & Echenique (2016) Chapter 14.

Note

Limited attention models explain apparent irrationality by assuming the decision-maker doesn’t see all available options. A choice is “attention-rational” if it’s optimal among the items actually considered.

Attention Filter Framework#

Function: test_attention_rationality(log)

A decision-maker has limited attention if they maximize utility over a consideration set:

\[c(A) = \arg\max_{x \in \Gamma(A)} u(x)\]

where \(\Gamma(A) \subseteq A\) is the consideration set (items actually considered).

An attention filter \(\Gamma\) maps menus to consideration sets with a key property: removing items outside the consideration set doesn’t change what’s considered.

Definition (Attention Filter)

A map \(\Gamma\) is an attention filter if for all menus \(A\) and items \(x\):

\[x \notin \Gamma(A) \implies \Gamma(A \setminus \{x\}) = \Gamma(A)\]

This captures the intuition that unnoticed items can be removed without affecting consideration.

WARP(LA): WARP with Limited Attention#

Function: test_warp_la(log)

The WARP(LA) axiom (Masatlioglu, Nakajima & Ozbay, 2012) characterizes Choice with Limited Attention (CLA). It defines a revealed preference relation \(P\) as:

\[x \, P \, y \iff \exists \text{ menu } T \text{ such that } c(T) = x \text{ and } c(T \setminus \{y\}) \neq x\]

In words: \(x\) is revealed preferred to \(y\) if removing \(y\) from some menu changes the choice away from \(x\).

Theorem (Masatlioglu et al., 2012)

A choice function \(c\) is rationalizable by an attention filter and preference ordering if and only if the revealed preference relation \(P\) is acyclic.

Interpretation: WARP(LA) is weaker than standard WARP. Data that violates WARP may still satisfy WARP(LA) if the violations can be explained by attention effects.

Recovering Attention Filters#

Function: recover_preference_with_attention(log)

When WARP(LA) is satisfied, we can construct an attention filter that rationalizes the data:

\[\Gamma(S) = \{c(S)\} \cup \{x \in S : c(S) \succ x \text{ in revealed preference}\}\]

This is the minimal consideration set needed to rationalize each choice.

Consideration Set Estimation#

Function: estimate_consideration_sets(log)

Estimate the consideration sets \(\Gamma(A)\) that rationalize observed choices:

\[\Gamma^*(A) = \min \{ \Gamma : c(A) = \arg\max_{x \in \Gamma} u(x) \text{ for some } u \}\]

Salience Weights#

Function: compute_salience_weights(log)

Estimate how likely each item is to enter the consideration set:

\[\sigma_i = \Pr[i \in \Gamma(A) | i \in A]\]

Higher salience items are more likely to be considered.

Random Attention Model (RAM)#

Function: fit_random_attention_model(log)

The Random Attention Model (Cattaneo et al., 2020) extends attention theory to stochastic choice. Instead of deterministic consideration, each item has an attention probability \(\mu_i\).

Model:

Each item \(i\) is considered with probability \(\mu_i\) (independently)
Consumer chooses the most preferred item among those considered
Choice probability depends on both preference rank and attention

\[P(\text{choose } x | S) = \sum_{\Gamma \subseteq S : x \in \Gamma} P(x \text{ is maximal in } \Gamma) \cdot P(\Gamma \text{ is considered})\]

RAM Constraints

Under RAM with preference \(\succ\), choice probabilities must satisfy:

Regularity bounds: \(P(x|S) \leq P(x|T)\) when \(T \subseteq S\) and \(x \in T\)
Monotonicity (optional): \(\mu_i \geq \mu_j\) if \(i \succ j\)

RAM Assumptions#

Function: fit_random_attention_model(log, assumption="...")

Different assumptions on attention probabilities:

Assumption	Description
`"monotonic"`	Higher-ranked items have higher attention probability
`"independent"`	Attention probabilities are item-specific (no ranking constraint)
`"general"`	Minimal restrictions on attention

Attention Bounds#

Function: compute_attention_bounds(log, preference, item, menu)

RAM provides identified bounds on attention probabilities from choice data:

\[\underline{\mu}_i \leq \mu_i \leq \overline{\mu}_i\]

These bounds can be computed using linear programming.

Reference: Masatlioglu, Nakajima & Ozbay (2012), Cattaneo et al. (2020), Chambers & Echenique (2016) Ch. 14