Social:Egalitarian rule

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Short description: Rawlsian decision rule for social choice

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

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

[math]\displaystyle{ \max_{x\in X} \min_{i\in I} u_i(x). }[/math]

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 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:[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 [math]\displaystyle{ X }[/math] 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:

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

  1. ↑ 1.0 1.1 Sen, Amartya (2017-02-20) (in en). Collective Choice and Social Welfare. Harvard University Press. doi:10.4159/9780674974616. ISBN 978-0-674-97461-6. https://www.degruyter.com/document/doi/10.4159/9780674974616/html. 
  2. ↑ D'Aspremont, Claude; Gevers, Louis (1977). "Equity and the Informational Basis of Collective Choice". The Review of Economic Studies 44 (2): 199–209. doi:10.2307/2297061. ISSN 0034-6527. https://www.jstor.org/stable/2297061. 
  3. ↑ Kolm, Serge-Christophe (2002) (in en). Justice and Equity. MIT Press. ISBN 978-0-262-61179-4. https://books.google.com/books?id=HyctVz6tRbQC&dq=S.-C.+Kolm,+Justice+et+%C3%89quit%C3%A9,+Cepremap,+CNRS+Paris,+1972,+English+translation:+Justice+and+Equity,+MIT+Press,+1998.&pg=PA3. 
  4. ↑ Moulin, Herve (1991-07-26) (in en). Axioms of Cooperative Decision Making. Cambridge University Press. ISBN 978-0-521-42458-5. https://books.google.com/books?id=mK6nEvHnqQIC&dq=H.+Moulin%2C+Axioms+of+Cooperative+Decision+Making%2C+Cambridge+University+Press%2C+1988.&pg=PR11. 
  5. ↑ 5.0 5.1 Herve Moulin (2004). Fair Division and Collective Welfare. Cambridge, Massachusetts: MIT Press. ISBN 9780262134231. 
  6. ↑ Bouveret, Sylvain; Lemaître, Michel (2009-02-01). "Computing leximin-optimal solutions in constraint networks" (in en). Artificial Intelligence 173 (2): 343–364. doi:10.1016/j.artint.2008.10.010. ISSN 0004-3702. 
  7. ↑ Nicosia, Gaia; Pacifici, Andrea; Pferschy, Ulrich (2017-03-16). "Price of Fairness for allocating a bounded resource" (in en). European Journal of Operational Research 257 (3): 933–943. doi:10.1016/j.ejor.2016.08.013. ISSN 0377-2217. https://www.sciencedirect.com/science/article/abs/pii/S0377221716306282. 
  8. ↑ Imai, Haruo (1983). "Individual Monotonicity and Lexicographic Maxmin Solution". Econometrica 51 (2): 389–401. doi:10.2307/1911997. ISSN 0012-9682. https://www.jstor.org/stable/1911997. 



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