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GARP and Afrait theorem assume that the alternative $x\in\mathbb R_+$ is always positive. In some economic contexts, such as financial choices, the attribute can be negative.

I wonder if we can naturally relax the $\mathbb R_+$ constraint. i.e. let $x\in\mathbb R$,

GARP: if $p_i⋅x_{i+1}≤p_i⋅x_i$ for $i∈\{1,...,N−1\}$ and $p_N⋅x_1≤p_N⋅x_N$, then those inequalities must be equalities.

At a first glance, the generalization to negative number seems plausible and not problematic.

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    $\begingroup$ I think if you allow prices to be negative in GARP, then requiring $x\in\mathbb R_+$ is without loss of generality. $\endgroup$ – Herr K. May 2 at 16:54
  • $\begingroup$ @HerrK. You mean allowing $x\in\mathbb R$ and $p\in\mathbb R$? $\endgroup$ – High GPA May 2 at 22:39
  • $\begingroup$ I mean $x\in\mathbb R_+$ and $p\in \mathbb R$. $\endgroup$ – Herr K. May 2 at 22:44

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