# Egalitarian rule

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{{Short description|Rawlsian decision rule for social choice}}
In [social choice](/source/social_choice_theory) and [operations research](/source/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](/source/Egalitarianism) philosophy. It also corresponds to [John Rawls](/source/John_Rawls)' principle of maximizing the welfare of the worst-off individual.<ref name=":0">{{Cite book|last=Sen|first=Amartya|url=https://www.degruyter.com/document/doi/10.4159/9780674974616/html|title=Collective Choice and Social Welfare|date=2017-02-20|publisher=Harvard University Press|isbn=978-0-674-97461-6|language=en|doi=10.4159/9780674974616}}</ref>

== Definition ==
Let <math>X</math> be a set of possible `states of the world' or `alternatives'. Society wishes to choose a single state from <math>X</math>.  For example, in a [single-winner election](/source/single-winner_election), <math>X</math> may represent the set of candidates; in a [resource allocation](/source/resource_allocation) setting, <math>X</math> may represent all possible allocations.

Let <math>I</math> be a finite set, representing a collection of individuals.  For each <math>i \in I</math>, let <math>u_i:X\longrightarrow\mathbb{R}</math> be a ''[utility function](/source/utility)'', describing the amount of happiness an individual ''i'' derives from each possible state.

A ''[social choice rule](/source/social_choice_theory)'' is a mechanism which uses the data <math>(u_i)_{i \in I}</math> to select some element(s) from <math>X</math> which are `best' for society. The question of what 'best' means is the basic question of [social choice theory](/source/social_choice_theory). The '''egalitarian rule''' selects an element <math>x \in X</math> which maximizes the ''minimum utility'', that is, it solves the following optimization problem: 

{{center|<math> \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](/source/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](/source/Amartya_Sen) in 1970,<ref name=":0" /> and discussed in depth in many later books.<ref>{{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|url-access=subscription}}</ref><ref>{{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}}</ref><ref>{{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}}</ref><ref name="moulin">{{Cite Moulin 2004}}</ref>{{rp|sub.2.5}} <ref>{{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}}</ref>

== Properties ==

=== Conditions for Pareto efficiency ===
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). 

=== Pigou-Dalton property ===
The leximin rule satisfies the [Pigou–Dalton principle](/source/Pigou%E2%80%93Dalton_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:<ref name="moulin" />{{Rp|266}}

# 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](/source/fair_division). In this setting, the set <math>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: 

* [Division of a single homogeneous resource](/source/Fair_division_of_a_single_homogeneous_resource); 
* [Fair subset sum problem](/source/Fair_subset_sum_problem);
* [Egalitarian cake-cutting](/source/Egalitarian_cake-cutting); 
* [Egalitarian item allocation](/source/Egalitarian_item_allocation); 
*Egalitarian (leximin) bargaining;<ref>{{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|url-access=subscription}}</ref> 
*Leximin rule for [cake sharing](/source/cake_sharing). 

== See also ==

* [Utilitarian rule](/source/Utilitarian_social_choice_rule) - a different rule that emphasizes the sum of utilities rather than the smallest utility.
*[Proportional-fair rule](/source/Proportional-fair_rule) - a different rule that emphasizes the product of utilities rather than the smallest utility.
*[Max-min fair scheduling](/source/Max-min_fair_scheduling) - max-min fairness in process scheduling.
*[Regret (decision theory)](/source/Regret_(decision_theory))
*[Wald's maximin model](/source/Wald's_maximin_model)

== References ==
{{Reflist}}
*
Category:Egalitarianism
Category:Fairness criteria

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Adapted from the Wikipedia article [Egalitarian rule](https://en.wikipedia.org/wiki/Egalitarian_rule) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Egalitarian_rule?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.