It is a formal mathematical representation of the egalitarian philosophy.
It also corresponds to John Rawls' principle of maximizing the welfare of the worst-off individual.
Society wishes to choose a single state from
be a finite set, representing a collection of individuals.
be a utility function, describing the amount of happiness an individual i derives from each possible state.
A social choice rule is a mechanism which uses the data
In this case, the egalitarian rule often uses the leximin order, that is: subject to maximizing the smallest utility, it aims to maximize the next-smallest utility; subject to that, maximize the next-smallest utility, and so on.
Then state x is leximin-optimal, since its utility profile is (2,4) which is leximin-larger than that of y (9,1) and z (1,8).
The leximin rule for social choice was introduced by Amartya Sen in 1970,[1] and discussed in depth in many later books.
[2][3][4][5]: sub.2.5 [6] The leximin rule is Pareto-efficient if the outcomes of every decision are known with certainty.
However, by Harsanyi's utilitarian theorem, any leximin function is Pareto-inefficient for a society that must make tradeoffs under uncertainty: There exist situations in which every person in a society would be better-off (ex ante) if they were to take a particular bet, but the leximin rule will reject it (because some person might be made worse off ex post).