Models & Heuristics

Note

The list of models and heuristics that are currently implemented in Prest and presented here is far from being exhaustive.

Some have been studied in the literature extensively. Others represent more recent developments in the field.

The inclusion/implementation of the latter has been guided by computational feasibility considerations,

and by the developers’ own research interests and expertise.

With the exception of Utility Maximization that appears first, the presentation order does not reflect any preference or priority.

Note

Unless otherwise stated, \(\mathcal{D}\) here is a general dataset, defined on an underlying set \(X\), and without default/status quo options.

Utility Maximization [1]

There is a weak order \(\succsim\) on \(X\) such that for every menu \(A\) in \(\mathcal{D}\)

\[C(A) = \{x\in A: x\succsim y\;\; \text{for all $y\in A$}\}\]

Prest allows for testing either/both special cases of this model:

• Utility Maximization (Strict): there are no distinct alternatives \(x\) and \(y\) such that \(x\sim y\).
• Utility Maximization (Non-Strict): there are some distinct alternatives \(x\) and \(y\) such that \(x\sim y\).

Tip

When analysing other models that generalize Utility Maximization, Prest only considers instances of the more general models that do not overlap with those covered by the above two variants of Utility Maximization. It is therefore recommended that Utility Maximization always be included in all model-estimation tasks.

Tip

When “Utility Maximization - Swaps” is selected in the Model estimation window, Prest computes the “Swaps” index of Apesteguia and Ballester [2015]. Note: this is only possible for datasets with nonempty and single choices at every menu.

Utility Maximization with an Outside Option [2]

There is a weak order \(\succsim\) on \(X\) and some “acceptability-threshold” alternative \(x^*\in X\) such that for every menu \(A\) in \(\mathcal{D}\)

\[\begin{split}\begin{array}{lllll} C(A) & = & \{x\in A: x\succsim y\;\; \text{for all $y\in A$}\} & \Longleftrightarrow & z\succ x^* \text{ for some } z\in A \\ C(A) & = & \emptyset & \Longleftrightarrow & x^*\succsim z \text{ for all } z\in A \end{array}\end{split}\]

Prest allows for testing either/both special cases of this model:

• Utility Maximization with an Outside Option (Strict): there are no distinct alternatives \(x\) and \(y\) such that \(x\sim y\).
• Utility Maximization with an Outside Option (Non-Strict): there are some distinct alternatives \(x\) and \(y\) such that \(x\sim y\).

Undominated Choice with Incomplete Preferences [3]

There is a strict partial order \(\succ\) on \(X\) such that for every menu \(A\) in \(\mathcal{D}\)

\[C(A) = \{x\in A: y\not\succ x\;\; \text{for all $y\in A$}\}\]

Prest allows for testing the following general and special versions of this model:

• Undominated Choice with Incomplete Preferences (Strict): the relation \(\succ\) is not the asymmetric part of a preorder \(\succsim\) on \(X\).
• Undominated Choice with Incomplete Preferences (Non-Strict): the relation \(\succ\) is the asymmetric part of a preorder \(\succsim\) on \(X\) and there are some distinct alternatives \(x\) and \(y\) such that \(x\sim y\).

Note

If a dataset is explained by non-strict undominated choice under some preorder \(\succsim\) with asymmetric and symmetric parts \(\succ\) and \(\sim\) where \(x\sim y\) is true for distinct alternatives \(x\) and \(y\), then it is also explained by strict undominated choice under strict partial order \(\succ\) where \(x\nsucc y\nsucc x\) for all such \(x\) and \(y\). The converse is not true in general.

Status-Quo-Biased Undominated Choice with Incomplete Preferences (Bewley) [4]

A general dataset with default/status quo alternatives \(\mathcal{D}\) is explained by this model if there exists a strict partial order \(\succ\) on \(X\) such that for every decision problem \((A,s)\) in \(\mathcal{D}\)

\[\begin{split}\begin{array}{llc} C(A,s)=\{s\} & \Longleftrightarrow & \text{$x\nsucc s$ for all $x\in A$}\\ & &\\ C(A,s)\neq \{s\} &\Longleftrightarrow & C(A,s)= \left\{ \begin{array}{lc} & z\nsucc x\; \text{for all $z\in A$}\\ x\in A: &\text{and}\\ & x\succ s \end{array} \right\} \end{array}\end{split}\]

Rational Shortlisting [5]

[experimental implementation]

There are two strict partial orders \(\succ_1\), \(\succ_2\) on \(X\) such that for every menu \(A\) in \(\mathcal{D}\)

\[\begin{split}\begin{array}{llll} |C(A)| & = & 1 \\ C(A) & = & M_{\succ_1}\Big(M_{\succ_2}(A)\Bigr) & \end{array}\end{split}\]

where

\[M_{\succ_i}(A) := \{x\in A: y\not\succ_i x\;\; \text{for all}\;\; y\in A\}\]

is the set of undominated alternatives in \(A\) according to \(\succ_i\) and \(|C(A)|=1\) means that \(C(A)\) contains exactly one alternative.

Tip

Prest currently supports only a Pass/Fail test for this model, with distance score output “0” and “>0”, respectively.

Dominant Choice with Incomplete Preferences [6]

There is an incomplete preorder \(\succsim\) on \(X\) such that for every menu \(A\) in \(\mathcal{D}\)

\[C(A) = \{x\in A: x\succsim y\;\; \text{for all $y\in A$}\}\]

In particular, \(C(A) = \emptyset\) \(\Longleftrightarrow\) for all \(x\in A\) there is \(y_x\in A\) such that \(x\not\succsim y_x\).

Prest allows for testing either/both special cases of this model:

• Dominant Choice with Incomplete Preferences (Strict): there are no distinct alternatives \(x\) and \(y\) such that \(x\sim y\).
• Dominant Choice with Incomplete Preferences (Non-Strict): there are some distinct alternatives \(x\) and \(y\) such that \(x\sim y\).

Partially Dominant Choice with Incomplete Preferences

Forced-Choice version [7]

There is a strict partial order \(\succ\) on \(X\) such that for every menu \(A\) in \(\mathcal{D}\)

\[\begin{split}\begin{array}{llc} C(A)=A & \Longleftrightarrow & x\nsucc y\;\; \text{and}\;\; y\nsucc x\;\; \text{for all}\;\; x,y\in A\\ & &\\ C(A)\subset A & \Longleftrightarrow & C(A)= \left\{ \begin{array}{lll} & & \hspace{-12pt} z\nsucc x\qquad \text{for all}\;\; z\in A\\ x\in A: & & \;\;\;\;\;\;\text{and}\\ & & \hspace{-12pt} x\succ y\qquad \text{for some}\;\; y\in A \end{array} \right\} \end{array}\end{split}\]

Free-Choice version [8]

There is a strict partial order \(\succ\) on \(X\) such that for every menu \(A\) in \(\mathcal{D}\) with at least two alternatives

\[\begin{split}\begin{array}{llc} C(A)=\emptyset & \Longleftrightarrow & x\nsucc y\;\; \text{and}\;\; y\nsucc x\;\; \text{for all}\;\; x,y\in A\\ & &\\ C(A)\neq\emptyset & \Longleftrightarrow & C(A)= \left\{ \begin{array}{lll} & & \hspace{-12pt} z\nsucc x\qquad \text{for all}\;\; z\in A\\ x\in A: & & \;\;\;\;\;\;\text{and}\\ & & \hspace{-12pt} x\succ y\qquad \text{for some}\;\; y\in A \end{array} \right\} \end{array}\end{split}\]

Note

In its distance-score computation of the free-choice version of this model, Prest penalizes deferral/choice of the outside option at singleton menus. Although this is not a formal requirement of the model, its predictions at non-singleton menus are compatible with the assumption that all alternatives are desirable, and hence that active choices be made at all singletons.

Overload-Constrained Utility Maximization [9]

There is a weak order \(\succsim\) on \(X\) and an integer \(n\) such that for every menu \(A\) in \(\mathcal{D}\)

\[\begin{split}\begin{array}{lllll} C(A) & = & \{x\in A: x\succsim y\;\; \text{for all $y\in A$}\} & \Longleftrightarrow & |A|\leq n \\ C(A) & = & \emptyset & \Longleftrightarrow & |A|>n \end{array}\end{split}\]

Prest allows for testing either/both special cases of this model:

• Overload-Constrained Utility Maximization (Strict): there are no distinct alternatives \(x\) and \(y\) such that \(x\sim y\).
• Overload-Constrained Utility Maximization (Non-Strict): there are some distinct alternatives \(x\) and \(y\) such that \(x\sim y\).

Footnotes

[1]

This is the textbook economic model of rational choice. Its revealed-preference implications have been studied extensively. Some important references include Samuelson (1938), Houthakker (1950), Uzawa (1956), Arrow (1959), Richter (1966) and the monograph by Chambers and Echenique (2016).

[2]

The model was formulated and analysed in this way in Gerasimou (2018, Section 3). If choice of the deferral/outside option at menu \(A\) is not captured as \(C(A)=\emptyset\) in the dataset but, instead, as \(C(A)=o\) for some alternative \(o\) that is feasible in every menu, then this model can be tested in Prest via the Utility Maximization model presented above. Encoding choice of the deferral/outside option as \(C(A)=\emptyset\) gives the user more flexibility because they can also test the dataset against some of the other models below where the \(C(A)=o\) way of encoding that choice is no longer equivalent to \(C(A)=\emptyset\).

[3]

See, among others, Schwartz (1976), Bossert, Sprumont, and Suzumura (2005), Eliaz and Ok (2006) and Ribeiro and Riella (2017).

[4]

See Bewley (2002), Mandler (2004), Masatlioglu and Ok (2005) and, for this specific formulation, Gerasimou (2016).

[5]

See Manzini and Mariotti (2007), Dutta and Horan (2015), de Clippel and Rozen (2021) and de Clippel and Rozen (2024).

[6]

See Gerasimou (2018, Section 2).

[7]

See Gerasimou (2016) and Qin (2017).

[8]

See Gerasimou (2016).

[9]

See Gerasimou (2018, Section 4).