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.^[1]

Definition

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

Let ${\displaystyle I}$ be a finite set, representing a collection of individuals. For each ${\displaystyle i\in I}$, let ${\displaystyle u_{i}:X\longrightarrow \mathbb {R} }$ 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 ${\displaystyle (u_{i})_{i\in I}}$ to select some element(s) from ${\displaystyle X}$ 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 ${\displaystyle x\in X}$ which maximizes the minimum utility, that is, it solves the following optimization problem:

${\displaystyle \max _{x\in X}\min _{i\in I}u_{i}(x).}$

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,^[1] and discussed in depth in many later books.^{[2][3][4][5]: sub.2.5  [6]}

Properties

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:^[5]: 266

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 ${\displaystyle X}$ 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;^[7]
• Egalitarian cake-cutting;
• Egalitarian item allocation.
• Egalitarian (leximin) bargaining.^[8]

See also

• Utilitarian rule - a different rule that emphasizes the sum of utilities rather than the smallest utility.
• Proportional-fair rule
• Max-min fair scheduling - max-min fairness in process scheduling.
• Regret (decision theory)
• Wald's maximin model

References