Consistency Criteria for General Datasets

For every subject whose choices are in the dataset, Prest v0.9.11 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 \(X\)

\[x\succ^R y\;\; \Longrightarrow\;\; y\not\succsim^R x\]

Note

Prest v0.9.11 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 \(X\)

\[x\succ^R y\;\; \Longrightarrow\;\; y\not\succsim^{\widehat{R}} x\]

Note

In Prest v0.9.11, Congruence violations of length 2 coincide with the WARP (all) count.

Strict Choice Consistency

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

\[x \succ^{\widehat{R}} y\;\; \Longrightarrow\;\; y\not\succsim^R x\]

Strict Binary Choice Consistency

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

\[x\succ^{\widehat{B}} y\;\; \Longrightarrow\;\; y\not\succsim^B x\]

Binary Choice Consistency

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

\[x\succsim^{\widehat{B}} y\;\; \Longrightarrow\;\; y\not\succ^B x\]

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 menus

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

Note

The number of inconsistent pairs of menus coincides with the WARP (pairs) count.

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

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.