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I am reading a JPE paper "Let's agree all dictatorship are equally bad" from Uzi Segal (2000).

I have a small problem with this paper. We have the following Utility Opportunity Set (UOS):

$$ S(w)=\{(u_{1}(L_{1}),\dots,u_{n}(L_{n}))\} $$

where $L_{1},\dots,L_{n}$ are the individual lotteries induced from $L \in £(w)$, and $£(w)$ is the set of lotteries over possible resources allocations $w$.

If $g={w^*}$ is a singleton, the assumption "Indifference by pure and mixed lotteries" in the paper implies linear parallel indifference curves in the UOS by standard techniques. However, I do not understand why when $w^*$ has a range (different possible values), the induced order on each $S(w^*)$ may be flat (if the policy everywhere else is fixed).

Another one have read the paper before? Any help would be appreciated!

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