Note

Links to two example general datasets that can be analysed for choice consistency as described later on this page:

1. Without default/status quo alternatives

Deterministic Consistency Criteria for General Datasets¶

For datasets where each menu appears once, Prest can compute, view and export the total number of violations of each of the axioms/criteria of deterministic choice consistency listed below, for every subject in the dataset.

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

Contraction Consistency¶

For any alternative \(x\) in \(X\) and menu \(A\) in \(\mathcal{D}\)

\[x \in A \subset B \text{ and } x\in C(B)\;\; \Longrightarrow\;\; x\in C(A)\]

Note

Prest reports two Contraction-Consistency counts for general datasets: Contraction Consistency (pairs) and Contraction Consistency (all).

Contraction Consistency (pairs) is the number of pairs of menus that are implicated in a Contraction Consistency violation.

Contraction Consistency (all) is the total number of Contraction Consistency violations.

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

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 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 / Strong Axiom of Revealed Preference¶

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

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

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 \(X\)

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

Binary Choice Transitivity¶

For any sequence of distinct alternatives \(x_1,\ldots,x_k\) in \(X\)

\[x_1\succsim^B x_2,\; \ldots, x_{k-1}\succsim^B x_k\;\;\;\; \text{and}\;\;\; (x_1,x_k)\in\mathcal{D}\;\; \Longrightarrow\;\; x_1\succsim^B x_k\]

Binary Choice Consistency¶

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

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

Strict Binary Choice Consistency¶

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

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

Note

The difference between (Strict) Binary Choice Consistency and Binary Choice Transitivity is that, under the same antecedent, the latter will also count \(C(\{x_1,x_k\})=\emptyset\) as a violation whereas the former will not.

Tip

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

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

To export the deterministic-consistency output (in .xslx or .csv format): right-click on the Prest-generated dataset [“DatasetX.csv (deterministic 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).
Contraction Consistency violations: lists the number of Contraction Consistency (pairs) (all) violations.
WARP violations: lists the number of WARP (pairs) (all) violations.
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.
Binary intransitivities (wide): lists the number of Binary Choice Transitivity violations, decomposed by length of the violating sequence.
Binary cycles (wide): lists the number of Binary Choice Consistency violations, decomposed by cycle length.
Strict binary cycles (wide): lists the number of Strict Binary Choice Consistency violations, decomposed by cycle length.

Additional Features: Cyclic Tuples¶

Cyclic tuples of menus¶

By right-clicking on the dataset and then selecting “Analysis -> Cyclic 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 cyclic 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.

Cyclic tuples of alternatives¶

By right-clicking on the dataset and then selecting “Analysis -> Cyclic 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.

An example dataset that illustrates these merging features is available here.