# Forced-Choice Models (no outside option)¶

## Utility Maximization / Rational Choice¶

[Samuelson, 1938; Houthakker, 1950; Uzawa, 1956; Arrow, 1959; Richter, 1966; Chambers and Echenique, 2016]

### Strict¶

A general choice dataset \(\mathcal{D}\) on a set of alternatives \(X\) is explained by
**(strict) utility maximization** if there is a strict linear
order \(\succ\) on \(X\) such that for every menu \(A\) in \(\mathcal{D}\)

### Non-strict¶

A general choice dataset \(\mathcal{D}\) on a set of alternatives \(X\) is explained by
**(non-strict) utility maximization** if there is a weak order
\(\succsim\) on \(X\) such that for every menu \(A\) in \(\mathcal{D}\)

**and**

Tip

When analysing other models that generalize utility maximization/rational choice, 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/rational choice always be included in all model-estimation tasks.

Tip

When “Utility Maximization - Swaps” is selected, Prest computes the “Swaps” index that is analyzed in Apesteguia and Ballester [2015].

*Note:* this is only possible for forced- and single-valued choice datasets.

## Incomplete-Preference Maximization: Undominated Choice¶

[Schwartz, 1976; Bossert, Sprumont, and Suzumura, 2005; Eliaz and Ok, 2006]

### Strict¶

A general choice dataset on a set of alternatives \(X\) is explained by
**(strict) undominated choice** if there is a strict
partial order \(\succ\) on \(X\) such that for every menu \(A\) in \(\mathcal{D}\)

### Non-strict¶

A general choice dataset on a set of alternatives \(X\) is explained by
**(non-strict) undominated choice** if there is an incomplete preorder \(\succsim\) on \(X\) such
that for every menu \(A\) in \(\mathcal{D}\)

**and**

## Incomplete-Preference Maximization: Partially Dominant Choice (forced)¶

A general choice dataset \(\mathcal{D}\) on a set of alternatives \(X\) is explained by
**partially dominant choice (forced)** if there exists a strict partial order \(\succ\) on \(X\)
such that for every menu \(A\) in \(\mathcal{D}\)

## Top-Two Choice¶

[Eliaz, Richter, and Rubinstein, 2011]

A general choice dataset \(\mathcal{D}\) on a set of alternatives \(X\) is explained by
**top-two choice** if there exists a strict linear order \(\succ\) on \(X\)
such that for every menu \(A\) in \(\mathcal{D}\)

## Sequentially Rationalizable Choice¶

[Manzini and Mariotti, 2007; Dutta and Horan, 2015; de Clippel and Rozen, 2021]

A general choice dataset \(\mathcal{D}\) on a set of alternatives \(X\) is explained by
**sequentially rationalizable choice** if there exist
two strict partial orders \(\succ_1\), \(\succ_2\) on \(X\) such that for every menu
\(A\) in \(\mathcal{D}\)

where, for any \(A\subseteq X\),

Tip

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