Non-Forced-Choice Models (feasible outside option) ================================================== Utility Maximization with an Outside Option ------------------------------------------- [:cite:authors:gerasimou18, :cite:year:gerasimou18] 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} .. math:: 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. where .. math:: \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} .. math:: 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. .. centered:: and .. math:: x\sim y\;\; \text{for distinct}\; x,y\; \text{in}\; X where .. math:: \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 ----------------------------------------- [:cite:authors:gerasimou18, :cite:year:gerasimou18] 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} .. math:: C(A) = & \left\{ \begin{array}{ll} \mathcal{B}_{\succ}(A), & \text{if $|A|\leq n$}\\ &\\ \emptyset, & \text{otherwise} \end{array} \right. where .. math:: \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} .. math:: C(A) = & \left\{ \begin{array}{ll} \mathcal{B}_{\succsim}(A), & \text{if $|A|\leq n$}\\ &\\ \emptyset, & \text{otherwise} \end{array} \right. .. centered:: and .. math:: x\sim y\;\; \text{for distinct}\; x,y\; \text{in}\; X where .. math:: \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 ------------------------------------------------------------- [:cite:authors:gerasimou18, :cite:year:gerasimou18] 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} .. math:: C(A) = \left\{ \begin{array}{ll} \mathcal{B}_{\succ}(A), & \text{if $\mathcal{B}_\succ(A)\neq\emptyset$}\\ &\\ \emptyset, & \text{otherwise}\\ \end{array} \right. where .. math:: \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} .. math:: C(A) = \left\{ \begin{array}{ll} \mathcal{B}_{\succsim}(A), & \text{if $\mathcal{B}_{\succsim}(A)\neq\emptyset$}\\ &\\ \emptyset, & \text{otherwise}\\ \end{array} \right. .. centered:: and .. math:: x\sim y\;\; \text{for distinct}\; x,y\; \text{in}\; X where .. math:: \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) -------------------------------------------------------------------------- [:cite:authors:gerasimou18, :cite:year:gerasimou16a] 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} with at least two alternatives .. math:: \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} .. note:: In its distance-score computation of this model, Prest penalizes deferral/choice of the outside option at singleton menus. Although this is not a formal requirement of the model, its predictions at non-singleton menus are compatible with the assumption that all alternatives are desirable, and hence that active choices be made at all singletons.