Revealed Preference Relations ============================= .. _revealed: 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}.