# 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 $$p^lx^l\geq p^lx^{l+1}$$, $$\ldots$$, $$p^{l+n-1}x^{l+n-1}\geq p^{l+n-1}x^{l+n}$$.