Egalitarian rule

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 $i\; \backslash in\; I$, let $u\_i:X\backslash longrightarrow\backslash mathbb\{R\}$ 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 $(u\_i)\_\{i\; \backslash in\; I\}$ 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 $x\; \backslash in\; X$ which maximizes the minimum utility, that is, it solves the following optimization problem:

{{center|$\backslash max\_\{x\backslash in\; X\}\; \backslash min\_\{i\backslash in\; I\}\; u\_i(x).$}}

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.{{Cite journal|last1=D'Aspremont|first1=Claude|last2=Gevers|first2=Louis|date=1977|title=Equity and the Informational Basis of Collective Choice|url=https://www.jstor.org/stable/2297061|journal=The Review of Economic Studies|volume=44|issue=2|pages=199–209|doi=10.2307/2297061|jstor=2297061|issn=0034-6527}}{{Cite book|last=Kolm|first=Serge-Christophe|url=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|title=Justice and Equity|date=2002|publisher=MIT Press|isbn=978-0-262-61179-4|language=en}}{{Cite book|last=Moulin|first=Herve|url=https://books.google.com/books?id=mK6nEvHnqQIC&dq=H.+Moulin%2C+Axioms+of+Cooperative+Decision+Making%2C+Cambridge+University+Press%2C+1988.&pg=PR11|title=Axioms of Cooperative Decision Making|date=1991-07-26|publisher=Cambridge University Press|isbn=978-0-521-42458-5|language=en}}{{Cite Moulin 2004}}{{rp|sub.2.5}} {{Cite journal|last1=Bouveret|first1=Sylvain|last2=Lemaître|first2=Michel|date=2009-02-01|title=Computing leximin-optimal solutions in constraint networks|journal=Artificial Intelligence|language=en|volume=173|issue=2|pages=343–364|doi=10.1016/j.artint.2008.10.010|issn=0004-3702|doi-access=free}}

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).

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:{{Rp|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.

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;{{Cite journal|date=2017-03-16|title=Price of Fairness for allocating a bounded resource|url=https://www.sciencedirect.com/science/article/abs/pii/S0377221716306282|journal=European Journal of Operational Research|language=en|volume=257|issue=3|pages=933–943|arxiv=1508.05253|doi=10.1016/j.ejor.2016.08.013|issn=0377-2217|last1=Nicosia|first1=Gaia|last2=Pacifici|first2=Andrea|last3=Pferschy|first3=Ulrich|s2cid=14229329}}
• Egalitarian cake-cutting;
• Egalitarian item allocation.
• Egalitarian (leximin) bargaining.{{Cite journal|last=Imai|first=Haruo|date=1983|title=Individual Monotonicity and Lexicographic Maxmin Solution|url=https://www.jstor.org/stable/1911997|journal=Econometrica|volume=51|issue=2|pages=389–401|doi=10.2307/1911997|jstor=1911997|issn=0012-9682}}

• 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

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Category:Egalitarianism

Category:Fairness criteria