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Arrow impossibility theorem presupposes: that individual preferences can be expressed ordinally, but not cardinally, and that "Pareto", "relevance" (i.e. “ Pareto Efficiency” and “Independence of Irrelevant Alternatives”) and "no dictatorship" are necessary conditions for democratic elections. The first point stems from the ...

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Let $X$ be the nonempty set of alternatives, $\mathcal{P}_X$ the set of preference relations on $X$ and $N=\{1,\ldots,n\}$ a finite set of agents. Under the universal domain condition, Arrow's theorem concerns functions from the set $\mathcal{P}_X^N=\underbrace{\mathcal{P}_X\times \mathcal{P}_X\cdots\times\mathcal{P}_X}_{n\text{ times}}$ to $\mathcal{P}_X$. ...

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The utilitarian SWF assumes that individuals' utility functions are quasilinear in money. Thus, they violate Arrow's Unrestricted Domain axiom, but satisfy the other axioms. If you accept quasilinearity of utility as a (at least approximately) valid assumption, then you can work with this SWF.

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