Egalitarian rule

In social choice and operations research, the egalitarian rule (also called the max-min rule or the Rawlsian rule) is a rule saying that, among all possible alternatives, society should pick the alternative which maximizes the minimum utility of all individuals in society. 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.

Definition
Let $$X$$ be a set of possible `states of the world' or `alternatives'. Society wishes to choose a single state from $$X$$. For example, in a single-winner election, $$X$$ may represent the set of candidates; in a resource allocation setting, $$X$$ may represent all possible allocations.

Let $$I$$ be a finite set, representing a collection of individuals. For each $$i \in I$$, let $$u_i:X\longrightarrow\mathbb{R}$$ 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 $$(u_i)_{i \in I}$$ to select some element(s) from $$X$$ which are `best' for society. The question of what 'best' means is the basic question of social choice theory. The egalitarian rule selects an element $$x \in X$$ which maximizes the minimum utility, that is, it solves the following optimization problem:

$ \max_{x\in X} \min_{i\in I} u_i(x).$

Leximin rule
Often, there are many different states with the same minimum utility. For example, a state with utility profile (0,100,100) has the same minimum value as a state with utility profile (0,0,0). 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.

For example, suppose there are two individuals - Alice and George, and three possible states: state x gives a utility of 2 to Alice and 4 to George; state y gives a utility of 9 to Alice and 1 to George; and state z gives a utility of 1 to Alice and 8 to George. 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 egalitarian rule strengthened with the leximin order is often called the leximin rule, to distinguish it from the simpler max-min rule.

The leximin rule for social choice was introduced by Amartya Sen in 1970, and discussed in depth in many later books.

Pareto inefficiency
The leximin rule is Pareto-efficient if the outcomes of every decision are known with perfect 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).

Pigou-Dalton property
The leximin rule satisfies the Pigou–Dalton principle, that is: if utility is "moved" from an agent with more utility to an agent with less utility, and as a result, the utility-difference between them becomes smaller, then resulting alternative is preferred.

Moreover, the leximin rule is the only social-welfare ordering rule which simultaneously satisfies the following three properties:


 * 1) Pareto efficiency;
 * 2) Pigou-Dalton principle;
 * 3) Independence of common utility pace - if all utilities are transformed by a common monotonically-increasing function, then the ordering of the alternatives remains the same.

Egalitarian resource allocation
The egalitarian rule is particularly useful as a rule for fair division. In this setting, the set $$X$$ represents all possible allocations, and the goal is to find an allocation which maximizes the minimum utility, or the leximin vector. This rule has been studied in several contexts:


 * Division of a single homogeneous resource;
 * Fair subset sum problem;
 * Egalitarian cake-cutting;
 * Egalitarian item allocation.
 * Egalitarian (leximin) bargaining.