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I want to say that the preferences of agent 1 on a consumption set $X$ are the same as agent 2's and the inverse of agent 3's (hence 2's is also the inverse of 3's). How would I do this using the ≽ notation, or alternatively any other notation?

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  • $\begingroup$ What do you mean by "inverse"? That 3's ranking of elements in $X$ is exactly opposite of 2's? Or simply that 3's preference is different from 2's? $\endgroup$
    – Herr K.
    May 14 at 4:01

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For all pairs $x,x' \in X$ we have $$ x \preceq_1 x' \Longleftrightarrow x \preceq_2 x' $$ and $$ x \preceq_1 x' \Longleftrightarrow x \succeq_3 x'. $$

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