# Predicting Online Item-Choice Behavior: A Shape-Restricted Regression Approach
## Appendix A Proofs
### Proof of Theorem 4  
**Appendix A.2.1**. : The \"Only if\" Part  
Suppose that \((\mathbf{u},\mathbf{v})\in E^{*}_{\textsc{US}}\). We then have \(\mathbf{v}\in\textsc{US}(\{\mathbf{u}\})\) from Definition 5 and Lemma 2. Thus, we consider the following two cases:  
Case 1: \(\mathbf{v}=\textsc{Up}(\mathbf{u},s)\) for Some \(s\in[1,n]\)  
For the sake of contradiction, assume \(u_{j}\in\{u_{s},u_{s}+1\}\) for some \(j\in[s+1,n]\). If \(u_{j}=u_{s}\), we set \(\mathbf{w}=\textsc{Up}(\mathbf{u},j)\) and then have \(\mathbf{v}=\textsc{Swap}(\mathbf{w},s,j)\). If \(u_{j}=u_{s}+1\), we set \(\mathbf{w}=\textsc{Swap}(\mathbf{u},s,j)\) and then have \(\mathbf{v}=\textsc{Up}(\mathbf{w},j)\). This implies that \(\mathbf{u}\prec_{\textsc{US}}\mathbf{w}\prec_{\textsc{US}}\mathbf{v}\), which contradicts \((\mathbf{u},\mathbf{v})\in E^{*}_{\textsc{US}}\) due to condition (C2) of Lemma 2. Case 2: \(\mathbf{v}=\textsc{Swap}(\mathbf{u},s,t)\) for Some \((s,t)\in[1,n]\times[1,n]\)  
For the sake of contradiction, assume \(u_{j}\in[u_{s},u_{t}]\) for some \(j\in[s+1,t-1]\). If \(u_{s}<u_{j}<u_{t}\), we set \(\mathbf{w}_{1}=\textsc{Swap}(\mathbf{u},j,t)\) and \(\mathbf{w}_{2}=\textsc{Swap}(\mathbf{w}_{1},s,j)\) and then have \(\mathbf{v}=\textsc{Swap}(\mathbf{w}_{2},j,t)\). If \(u_{j}=u_{s}\), we set \(\mathbf{w}=\textsc{Swap}(\mathbf{u},j,t)\) and then have \(\mathbf{v}=\textsc{Swap}(\mathbf{w},s,j)\). If \(u_{j}=u_{t}\), we set \(\mathbf{w}=\textsc{Swap}(\mathbf{u},s,j)\) and then have \(\mathbf{v}=\textsc{Swap}(\mathbf{w},j,t)\). Each of these results contradicts \((\mathbf{u},\mathbf{v})\in E^{*}_{\textsc{US}}\) due to condition (C2) of Lemma 2. **Appendix A.2.2**. : The \"if\" Part  
We show that \((\mathbf{u},\mathbf{v})\in E^{*}_{\textsc{US}}\) in the following two cases:  
Case 1: Condition (US1) Is Fulfilled  
Condition (C1) of Lemma 2 is clearly satisfied. To satisfy condition (C2), we consider \(\mathbf{w}\in\Gamma\) such that \(\mathbf{u}\preceq_{\textsc{US}}\mathbf{w}\preceq_{\textsc{US}}\mathbf{v}\). From Lemma 1, we have \(\mathbf{u}\preceq_{\textsc{lex}}\mathbf{w}\preceq_{\textsc{lex}}\mathbf{v}\), implying that \(w_{j}=u_{j}\) for all \(j\in[1,s-1]\). Therefore, we cannot apply any operations to \(w_{j}\) for \(j\in[1,s-1]\) in the process of transforming \(\mathbf{w}\) from \(\mathbf{u}\) into \(\mathbf{v}\). We must apply the Up operation only once, because the value of \(\sum_{j=1}^{n}w_{j}\) remains the same after the Swap operation. Condition (US1) guarantees that for all \(j\in[s+1,n]\), \(w_{j}\) does not coincide with \(u_{s}+1\) even if Up\((\,\cdot\,,j)\) is applied. Therefore, \(\textsc{Swap}(\,\cdot\,,s,j)\) for \(j\in[s+1,n]\) never leads to \(w_{s}=u_{s}+1\). As a result, Up\((\,\cdot\,,s)\) must be performed. Other applicable Swap operations produce a sequence that cannot be converted into \(\mathbf{v}\). This means that \(\mathbf{w}=\mathbf{u}\) or \(\mathbf{w}=\textsc{Up}(\mathbf{u},s)=\mathbf{v}\). Case 2: Condition (US2) Is Fulfilled  
Condition (C1) of Lemma 2 is clearly satisfied. To satisfy condition (C2), we consider \(\mathbf{w}\in\Gamma\) such that \(\mathbf{u}\preceq_{\textsc{US}}\mathbf{w}\preceq_{\textsc{US}}\mathbf{v}\). From Lemma 1, we have \(\mathbf{u}\preceq_{\textsc{lex}}\mathbf{w}\preceq_{\textsc{lex}}\mathbf{v}\). This implies that \(w_{j}=u_{j}\) for all \(j\in[1,s-1]\), and that \(w_{s}\in[u_{s},u_{t}]\). Therefore, we cannot apply any operations to \(w_{j}\) for \(j\in[1,s-1]\) in the process of transforming \(\mathbf{w}\) from \(\mathbf{u}\) into \(\mathbf{v}\). To keep the value of \(\sum_{j=1}^{n}w_{j}\) constant, we can apply only the Swap operation. However, once the Swap operation is applied to \(w_{j}\) for \(j\in[t+1,n]\), the resultant sequence cannot be converted into \(\mathbf{v}\). We cannot adopt \(\mathbf{w}=\textsc{Swap}(\mathbf{u},s,j)\) for \(j\in[s+1,t-1]\) due to condition (US2).