Social welfare in terms of preferences How does one define a social welfare in terms of individuals’ preferences $\succeq_i$?
If we have utility functions $u_i$ then a social welfare maximizing outcome $x$ is simply one that maximizes $\sum_i u_i(x)$. How does one specify a social welfare maximizing outcome in terms of preferences?
Usually one tries to construct a Social Welfare Function (SWF), i.e. a general rule how to aggregate individual preferences to a social preference relation. Then various axioms are formulated that a useful SWF should obey, and some such SWF is selected. A social welfare maximizing alternative would then be one which is weakly preferred to all other alternatives by society.
However, if there are at least three alternatives and the axioms include Universal Domain, the Pareto Criterion, and Independence of Irrelevant Alternatives, then Arrow's Impossibility Theorem shows that the only SWF obeying all three of these axioms is a dictatorship.