# Consistency Criteria for General Datasets¶

For every subject whose choices are in the dataset Prest can compute, view and export the total number of violations for each of the axioms/criteria of choice consistency that are listed below.

Note

Much of the terminology and notation that follows is introduced and explained in the Datasets and Revealed Preference Relations sections.

## Weak Axiom of Revealed Preference - WARP¶

For any two distinct alternatives \(x,y\) in \(X\)

Note

Prest reports two WARP counts for general datasets: **WARP (pairs)** and **WARP (all)**.

**WARP (pairs)** is the number of *pairs of menus* that are implicated in a WARP violation.

**WARP (all)** is the total number of WARP violations.

For example, the data \(C(\{x,y,z\})=\{x,y\}\) and \(C(\{x,z\})=\{z\}\) is associated with a WARP (pairs) count of 1 and a WARP (all) count of 2, the latter involving alternatives \(x,z\) and \(y,z\), respectively.

## Congruence¶

For any two distinct alternatives \(x,y\) in \(X\)

Note

In Prest, Congruence violations of length 2 coincide with the **WARP (all)** count.

## Strict Choice Consistency¶

For any two distinct alternatives \(x,y\) in \(X\)

## Strict Binary Choice Consistency¶

For any two distinct alternatives \(x,y\) in \(X\)

## Binary Choice Consistency¶

For any two distinct alternatives \(x,y\) in \(X\)

Tip

**To use the consistency-analysis feature:** right-click on the dataset of interest [e.g. “DatasetX.csv”] in the workspace and select *“Analysis -> Consistency analysis”*.

**To view the consistency-analysis output:** right-click on the Prest-generated dataset [“DatasetX.csv (consistency)”] in the workspace and then click on “View”.

**To export the consistency-analysis output (in .xslx or .csv format):** right-click on the Prest-generated dataset [“DatasetX.csv (consistency)”]
in the workspace, click on “Export”, and then select one of the following options:

**Summary**: lists the total number of violations of each axiom (per subject).**Congruence violations (wide)**: lists the number of Congruence violations, decomposed by cycle length.**Strict general cycles (wide)**: lists the number of Strict Choice Consistency violations, decomposed by cycle length.**Strict binary cycles (wide)**: lists the number of Strict Binary Choice Consistency violations, decomposed by cycle length.**Binary cycles (wide)**: lists the number of Binary Choice Consistency violations, decomposed by cycle length.

## Additional Features: Inconsistent Tuples¶

### Inconsistent tuples of alternatives¶

By right-clicking on the dataset and then selecting *“Analysis -> Inconsistent tuples of alternatives”*, Prest computes and enumerates
all distinct pairs, triples, quadruples, …, \(n\)-tuples of alternatives that have led to a Congruence violation, and groups them according to the size of \(n\).

Following the same steps as above, this output can be viewed within Prest or exported to a .csv or .xslx file.

Tip

If the same menu \(A\) appears more than once for the same subject in \(\mathcal{D}\),
Prest allows for **merging the choices** made at this menu in the different observations.

For example, if the dataset \(\mathcal{D}\) is such that \(A_1=A_5=\{w,x,y\}\) and \(C(A_1)=\{x\}\), \(C(A_5)=\{y\}\) for the same subject, then \(\mathcal{D}\) would be altered after the merging operation so that the menu \(A_1=A_5:=A\) appears only once, and with \(C(A)=\{x,y\}\) being the subject’s new choice at this menu.

**To use this feature:** right-click on the dataset of interest [e.g. “DatasetX.csv”]
in the workspace and select *“Analysis -> Merge options at the same menu”*. The resulting merged dataset appears in the workspace [“DatasetX.csv (merged)”] and can then be analysed separately
for consistency analysis or model estimation after the potential “noisiness” of choice data has been accounted for in this way through multi-valued choice.

**Remark:** *If the merging operation is applied on a non-forced-choice dataset where a subject has chosen an alternative from menu* \(A\) *in one or more instances and has deferred choice/opted for the outside option
in at least another, then the merged dataset will feature menu* \(A\) *appearing twice: one where* \(C(A)\) *comprises all alternatives in* \(A\) *that were chosen at least once; and one where* \(C(A)=\emptyset\).

**An example of a dataset that may help as an illustration for these merging features is available** here.

Note

**We provide an** example general dataset with default alternatives **and** an example general dataset without default alternatives, **that can be analysed for consistency as described above**.