Egalitarian social choice 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 $u_i:X\longrightarrow\mathbb$ 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 $(u_i)_$ 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:

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 utilty, 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.

Properties

Pareto efficiency

The ''max-min'' rule may not necessarily lead to a Pareto efficient outcome. For example, it may choose a state which leades to a utility profile (3,3,3), while there is another state leading to a utility profile (3,4,5), which is a Pareto-improvement.

In contrast, the ''leximin'' rule always selects a Pareto-efficient outcome. This is because any Pareto-improvement leads to a leximin-better utility vector: if a state y Pareto-dominates a state x, then y is also leximin-better than x.

Pigou-Dalton property

The leximin rule satisfies the Pigou–Dalton principle, that is: if utility is "moved" from an agent with less utility to an agent with more 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:

# Pareto efficiency;
# Pigou-Dalton principle;
# 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.

See also

* Utilitarian rule - a different rule, that emphasizes the sum of utilities rather than the smallest utility.
* Proportional-fair rule - a rule that tries to balance the efficiency of the utilitarian rule and the fairness of the egalitarian rule.
* Max-min fair scheduling - max-min fairness in process scheduling.

References

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Egalitarianism
Fairness criteria