Non-Forced-Choice Models (feasible outside option)

Utility Maximization with an Outside Option

Strict

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

\[\begin{split}C(A) = \left\{ \begin{array}{ll} \mathcal{B}_{\succ}(A), & \text{if $x\succ x^*$ for $\{x\}= \mathcal{B}_\succ(A)$}\\ &\\ \emptyset, & \text{otherwise}\\ \end{array} \right.\end{split}\]

where

\[\mathcal{B}_{\succ}(A):=\Big\{x\in A: x\succ y\; \text{for all $y\in A\setminus\{x\}$}\Bigr\}\]

is the strictly most preferred alternative in \(A\) according to \(\succ\).

Non-strict

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

\[\begin{split}C(A) = \left\{ \begin{array}{ll} \mathcal{B}_{\succsim}(A), & \text{if $x\succ x^*$ for all $x\in \mathcal{B}_\succsim(A)$}\\ &\\ \emptyset, & \text{otherwise}\\ \end{array} \right.\end{split}\]

and

\[x\sim y\;\; \text{for distinct}\; x,y\; \text{in}\; X\]

where

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

is the set of weakly most preferred alternatives in \(A\) according to \(\succsim\).


Overload-Constrained Utility Maximization

Strict

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

\[\begin{split}C(A) = & \left\{ \begin{array}{ll} \mathcal{B}_{\succ}(A), & \text{if $|A|\leq n$}\\ &\\ \emptyset, & \text{otherwise} \end{array} \right.\end{split}\]

where

\[\mathcal{B}_{\succ}(A):=\Big\{x\in A: x\succ y\; \text{for all $y\in A\setminus\{x\}$}\Bigr\}\]

is the strictly most preferred alternative in \(A\) according to \(\succ\).

Non-strict

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

\[\begin{split}C(A) = & \left\{ \begin{array}{ll} \mathcal{B}_{\succsim}(A), & \text{if $|A|\leq n$}\\ &\\ \emptyset, & \text{otherwise} \end{array} \right.\end{split}\]

and

\[x\sim y\;\; \text{for distinct}\; x,y\; \text{in}\; X\]

where

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

is the set of weakly most preferred alternatives in \(A\) according to \(\succsim\).


Incomplete-Preference Maximization: Maximally Dominant Choice

Strict

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

\[\begin{split}C(A) = \left\{ \begin{array}{ll} \mathcal{B}_{\succ}(A), & \text{if $\mathcal{B}_\succ(A)\neq\emptyset$}\\ &\\ \emptyset, & \text{otherwise}\\ \end{array} \right.\end{split}\]

where

\[\mathcal{B}_{\succ}(A):=\Big\{x\in A: x\succ y\; \text{for all $y\in A\setminus\{x\}$}\Bigr\}\]

is the (possibly non-existing) strictly most preferred alternative in \(A\) according to \(\succ\).

Non-strict

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

\[\begin{split}C(A) = \left\{ \begin{array}{ll} \mathcal{B}_{\succsim}(A), & \text{if $\mathcal{B}_{\succsim}(A)\neq\emptyset$}\\ &\\ \emptyset, & \text{otherwise}\\ \end{array} \right.\end{split}\]

and

\[x\sim y\;\; \text{for distinct}\; x,y\; \text{in}\; X\]

where

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

is the (possibly empty) set of the weakly most preferred alternatives in \(A\) according to \(\succsim\).


Incomplete-Preference Maximization: Partially Dominant Choice (non-forced)

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

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