Revealed Preference Relations

Revealed Preference in General Datasets

Consider a general dataset \(\mathcal{D}=\left\{\big(A_i,C(A_i)\bigr)\right\}_{i=1}^k\) and distinct choice alternatives \(x,y\) in \(X\).

The following definitions and notation will be used for the different ways in which \(x\) may be revealed preferred to \(y\) in \(\mathcal{D}\):

\(x\succsim^R y\) if there exists a menu \(A_i\) in \(\mathcal{D}\) such that \(x\in C(A_i)\) and \(y\in A_i\).

\(x\succ^R y\), if there exists a menu \(A_i\) in \(\mathcal{D}\) such that \(x\in C(A_i)\) and \(y\in A_i\setminus C(A_i)\) [i.e. \(y\in A_i\) and \(y\not\in C(A_i)\)].

\(x\succsim^{\widehat{R}} y\), if there exists a sequence of menus \(A_{(1)},\ldots, A_{(n)}\) in \(\mathcal{D}\) and a sequence of alternatives \(x_{(1)},\ldots,x_{(n)}\) in \(X\) such that \(x=x_{(1)}\), \(y=x_{(n)}\) and \(x_{(i)}\succsim^R x_{(i+1)}\) for all \(i=1,\ldots,n-1\).

\(x\succ^{\widehat{R}} y\), if there exists a sequence of menus \(A_{(1)},\ldots, A_{(n)}\) in \(\mathcal{D}\) and a sequence of alternatives \(x_{(1)},\ldots,x_{(n)}\) in \(X\) such that \(x=x_{(1)}\), \(y=x_{(n)}\) and \(x_{(i)}\succ^R x_{(i+1)}\) for all \(i=1,\ldots,n-1\).

\(x\succsim^B y\) if \(x\in C(\{x,y\})\).

\(x\succ^B y\) if \(\{x\}=C(\{x,y\})\).

\(x\succsim^{\widehat{B}}y\) if there is a sequence of alternatives \(x_{(1)},\ldots,x_{(n)}\in X\) such that \(x=x_{(1)}\), \(y=x_{(n)}\) and \(x_{(i)}\succsim^B x_{(i+1)}\) for all \(i=1,\ldots,n-1\).

\(x\succ^{\widehat{B}}y\) if there is a sequence of alternatives \(x_{(1)},\ldots,x_{(n)}\in X\) such that \(x=x_{(1)}\), \(y=x_{(n)}\) and \(x_{(i)}\succ^B x_{(i+1)}\) for all \(i=1,\ldots,n-1\).

Revealed Preference in Budgetary Datasets

Consider a budgetary dataset \(\mathcal{D}=\left\{(p^i,x^i)\right\}_{i=1}^k\) and consumption bundles \(x^i,x^j\) in \(\mathbb{R}^n_+\) such that \(i,j\leq k\).

The following definitions and notation will be used for the different ways in which \(x^i\) may be revealed preferred to \(x^j\) in \(\mathcal{D}\):

\(x^i\succsim^R x^j\) if \(p^ix^i\geq p^ix^j\).

\(x^i\succ^R x^j\) if \(p^ix^i>p^ix^j\).

\(x^i\succsim^{\widehat{R}} x^j\) if there exist observations \((p^l,x^l),\ldots,(p^{l+n},x^{l+n})\) in \(\mathcal{D}\) such that \(x^i=x^l\), \(x^j=x^{l+n}\) and \(p^lx^l\geq p^lx^{l+1}\), \(\ldots\), \(p^{l+n-1}x^{l+n-1}\geq p^{l+n-1}x^{l+n}\).