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<title>Social welfare function</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Social welfare function</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">For the practical application of social choice to elections, see <a href="Electoral_system" title="Electoral system">Electoral system</a>.</div>

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</style><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><td class="sidebar-pretitle">A joint <a href="Portal%3APolitics" title="Portal:Politics">Politics</a> and <a href="Portal%3AEconomics" title="Portal:Economics">Economics</a> series</td></tr><tr><th class="sidebar-title-with-pretitle" style="border-top:1px #fafafa solid; border-bottom:1px #fafafa solid; background:#efefef; background: var(--background-color-interactive, #efefef); color: var(--color-base, #000); padding:0.2em;"><a href="Social_choice_theory" title="Social choice theory">Social choice</a> and <a href="Electoral_system" title="Electoral system">electoral systems</a></th></tr><tr><td class="sidebar-image"></td></tr><tr><td class="sidebar-above">

<div class="hlist"><ul><li><a href="Social_choice_theory" title="Social choice theory">Social choice</a></li><li><a href="Mechanism_design" title="Mechanism design">Mechanism design</a></li><li><a href="Comparative_politics" title="Comparative politics">Comparative politics</a></li><li><a href="Comparison_of_voting_rules" title="Comparison of voting rules">Comparison</a></li><li><a href="List_of_electoral_systems" title="List of electoral systems">List</a><span class="nowrap"> </span>(<a href="List_of_electoral_systems_by_country" title="List of electoral systems by country">By country</a>)</li></ul></div></td></tr><tr><td class="sidebar-content" style="text-align:left;">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#efefef; border-top:1px solid;background: var(--background-color-interactive, #efefef); color: var(--color-base, #000);;color: var(--color-base)"><a href="Single-member_district" title="Single-member district">Single-winner methods</a></div><div class="sidebar-list-content mw-collapsible-content"><b>Single vote – <a href="Plurality_voting" title="Plurality voting">plurality</a> methods</b>

<ul><li><a href="First-past-the-post_voting" title="First-past-the-post voting">First preference plurality (FPP)</a></li>

<li><a href="Two-round_system" title="Two-round system">Two-round</a> (<abbr style="font-size:85%" title=""><a href="American_English" title="American English">US</a>:</abbr> <a href="Nonpartisan_primary" title="Nonpartisan primary">Jungle primary</a>)

<ul><li><a href="Partisan_primary" class="mw-redirect" title="Partisan primary">Partisan primary</a></li></ul></li>

<li><a href="Instant-runoff_voting" title="Instant-runoff voting">Instant-runoff</a>

<ul><li><abbr style="font-size:85%" title=""><a href="British_English" title="British English">UK</a>:</abbr> Alternative vote (AV)</li>

<li><abbr style="font-size:85%" title=""><a href="American_English" title="American English">US</a>:</abbr> Ranked-choice (RCV)</li></ul></li>

<li><a href="Party_block_voting" class="mw-redirect" title="Party block voting">Party block voting</a></li>

<li><a href="Plurality_block_voting" title="Plurality block voting">Plurality block voting</a></li></ul>

<hr>

<p><b><a href="Condorcet_method" title="Condorcet method">Condorcet methods</a></b>

</p>

<ul><li><a href="Tideman_alternative_method" title="Tideman alternative method">Condorcet-IRV</a></li>

<li><a href="Round-robin_voting" title="Round-robin voting">Round-robin voting</a>

<ul><li><a href="Minimax_Condorcet_method" title="Minimax Condorcet method">Minimax</a></li>

<li><a href="Kemeny_method" title="Kemeny method">Kemeny</a></li>

<li><a href="Schulze_method" title="Schulze method">Schulze</a></li>

<li><a href="Ranked_pairs" title="Ranked pairs">Ranked pairs</a></li>

<li><a href="Maximal_lottery" class="mw-redirect" title="Maximal lottery">Maximal lottery</a></li></ul></li></ul>

<hr>

<p><b><a href="Positional_voting" title="Positional voting">Positional voting</a></b>

</p>

<ul><li><a href="First-preference_plurality" class="mw-redirect" title="First-preference plurality">Plurality</a> (<abbr style="font-size:85%" title=""><a href="Sequential_elimination_method" title="Sequential elimination method">el.</a></abbr> <a href="Instant-runoff_voting" title="Instant-runoff voting">IRV</a>)</li>

<li><a href="Borda_count" title="Borda count">Borda count</a> (<abbr style="font-size:85%" title=""><a href="Sequential_elimination_method" title="Sequential elimination method">el.</a></abbr> <a href="Baldwin's_method" class="mw-redirect" title="Baldwin's method">Baldwin</a>, <abbr style="font-size:85%" title=""><a href="Highest_median_voting_rules" title="Highest median voting rules">Mdn.</a></abbr> <a href="Bucklin_voting" title="Bucklin voting">Bucklin</a>)</li>

<li><a href="Anti-plurality_voting" title="Anti-plurality voting">Antiplurality</a> (<abbr style="font-size:85%" title=""><a href="Sequential_elimination_method" title="Sequential elimination method">el.</a></abbr> <a href="Coombs_method" class="mw-redirect" title="Coombs method">Coombs</a>)</li></ul>

<hr>

<p><b><a href="Rated_voting" title="Rated voting">Cardinal voting</a></b>

</p>

<ul><li><a href="Score_voting" title="Score voting">Score voting</a></li>

<li><a href="Approval_voting" title="Approval voting">Approval voting</a></li>

<li><a href="Highest_median_voting_rules" title="Highest median voting rules">Majority judgment</a></li>

<li><a href="STAR_voting" title="STAR voting">STAR voting</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content" style="text-align:left;">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#efefef; border-top:1px solid;background: var(--background-color-interactive, #efefef); color: var(--color-base, #000);;color: var(--color-base)"><a href="Proportional_representation" title="Proportional representation">Proportional representation</a></div><div class="sidebar-list-content mw-collapsible-content"><b><a href="Party-list_proportional_representation" title="Party-list proportional representation">Party-list</a></b>

<ul><li><a href="Apportionment_(politics)" title="Apportionment (politics)">Apportionment</a>

<ul><li><a href="Highest_averages_method" title="Highest averages method">Highest averages</a></li>

<li><a href="Largest_remainder_method" class="mw-redirect" title="Largest remainder method">Largest remainders</a></li>

<li><a href="National_remnant" title="National remnant">National remnant</a></li>

<li><a href="Biproportional_apportionment" title="Biproportional apportionment">Biproportional</a></li></ul></li>

<ul><li><a href="Closed_list" title="Closed list">Closed list</a></li>

<li><a href="Panachage" title="Panachage">Panachage</a></li>

<li><a href="Localized_list" title="Localized list">Localized list</a></li></ul></li></ul>

<hr>

<p><b><a href="Electoral_quota" title="Electoral quota">Quota-remainder methods</a></b>

</p>

<li><a href="Schulze_STV" title="Schulze STV">Schulze STV</a></li>

<li><a href="Quota_Borda_system" title="Quota Borda system">Quota Borda</a></li></ul>

<hr>

<p><b><a href="Approval-based_committee" class="mw-redirect" title="Approval-based committee">Approval-based committees</a></b>

</p>

<ul><li><a href="Proportional_approval_voting" title="Proportional approval voting">Thiele's method</a></li>

<li><a href="Phragmen's_voting_rules" title="Phragmen's voting rules">Phragmen's method</a></li>

<li><a href="Expanding_approvals_rule" title="Expanding approvals rule">Expanding approvals rule</a></li>

<li><a href="Method_of_equal_shares" title="Method of equal shares">Method of equal shares</a></li></ul>

<hr>

<p><b><a href="Fractional_social_choice" title="Fractional social choice">Fractional social choice</a></b>

</p>

<ul><li><a href="Direct_representation" title="Direct representation">Direct representation</a>

<ul><li><a href="Interactive_representation" title="Interactive representation">Interactive representation</a></li>

<li><a href="Liquid_democracy" title="Liquid democracy">Liquid democracy</a></li></ul></li>

<li><a href="Fractional_approval_voting" title="Fractional approval voting">Fractional approval voting</a></li>

<li><a href="Maximal_lottery" class="mw-redirect" title="Maximal lottery">Maximal lottery</a></li>

<li><a href="Random_ballot" title="Random ballot">Random ballot</a></li></ul>

<hr>

<p><b><a href="Semi-proportional_representation" title="Semi-proportional representation">Semi-proportional representation</a></b>

</p>

<ul><li><a href="Cumulative_voting" title="Cumulative voting">Cumulative</a>

<li><a href="Limited_voting" title="Limited voting">Limited voting</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content" style="text-align:left;">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#efefef; border-top:1px solid;background: var(--background-color-interactive, #efefef); color: var(--color-base, #000);;color: var(--color-base)"><a href="Mixed_electoral_system" title="Mixed electoral system">Mixed systems</a></div><div class="sidebar-list-content mw-collapsible-content"><b>By results of combination</b>

<ul><li><a href="Mixed-member_majoritarian_representation" title="Mixed-member majoritarian representation">Mixed-member majoritarian</a></li>

<li><a href="Mixed-member_proportional_representation" title="Mixed-member proportional representation">Mixed-member proportional</a></li></ul>

<hr><b>By mechanism of combination</b>

<ul><li><b>Non-<a href="Compensation_(electoral_systems)" title="Compensation (electoral systems)">compensatory</a></b>

<ul><li><a href="Parallel_voting" title="Parallel voting">Parallel (superposition)</a></li>

<li><a href="Coexistence_(electoral_systems)" title="Coexistence (electoral systems)">Coexistence</a></li>

<li>Conditional</li>

<li><a href="Majority_bonus_system" title="Majority bonus system">Fusion (majority bonus)</a></li></ul></li>

<li><b><a href="Compensation_(electoral_systems)" title="Compensation (electoral systems)">Compensatory</a></b>

<ul><li>Seat linkage system

<li><a href="Vote_linkage_mixed_system" class="mw-redirect" title="Vote linkage mixed system">Vote linkage system</a>

<ul><li><a href="Scorporo" title="Scorporo">Negative vote transfer</a></li>

<li><a href="Mixed_ballot_transferable_vote" title="Mixed ballot transferable vote">Mixed ballot</a></li></ul></li></ul></li>

<li><a href="Mixed_electoral_system" title="Mixed electoral system">Supermixed systems</a>

<ul><li><a href="Dual-member_proportional_representation" class="mw-redirect" title="Dual-member proportional representation">Dual-member proportional</a></li>

<li><a href="Rural%E2%80%93urban_proportional_representation" title="Rural–urban proportional representation">Rural–urban proportional</a></li>

<li><a href="Majority_jackpot_system" title="Majority jackpot system">Majority jackpot</a></li></ul></li></ul>

<hr>

<p><b>By ballot type</b>

</p>

<ul><li><a href="Mixed_single_vote" title="Mixed single vote">Single vote</a>

<ul><li><a href="Double_simultaneous_vote" title="Double simultaneous vote">Double simultaneous vote</a></li></ul></li>

</tr><tr><td class="sidebar-content" style="text-align:left;">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#efefef; border-top:1px solid;background: var(--background-color-interactive, #efefef); color: var(--color-base, #000);;color: var(--color-base)"><a href="Pathological_(mathematics)#Voting" title="Pathological (mathematics)">Paradoxes and pathologies</a></div><div class="sidebar-list-content mw-collapsible-content"><b>Spoiler effects</b>

<ul><li><a href="Spoiler_effect" title="Spoiler effect">Spoiler effect</a></li>

<li><a href="Independence_of_clones" class="mw-redirect" title="Independence of clones">Cloning paradox</a></li>

<li><a href="Condorcet_winner_criterion" title="Condorcet winner criterion">Frustrated majorities paradox</a></li>

<li><a href="Center_squeeze" title="Center squeeze">Center squeeze</a></li></ul>

<hr>

<p><b>Pathological response</b>

</p>

<ul><li><a href="Perverse_response" class="mw-redirect" title="Perverse response">Perverse response</a></li>

<li><a href="Best-is-worst_paradox" class="mw-redirect" title="Best-is-worst paradox">Best-is-worst paradox</a></li>

<li><a href="No-show_paradox" class="mw-redirect" title="No-show paradox">No-show paradox</a></li>

<li><a href="Multiple_districts_paradox" class="mw-redirect" title="Multiple districts paradox">Multiple districts paradox</a></li></ul>

<hr>

<p><b><a href="Strategic_voting" title="Strategic voting">Strategic voting</a></b>

</p>

<ul><li><a href="Sincere_favorite_criterion" title="Sincere favorite criterion">Lesser evil voting</a></li>

<li><a href="Strategic_voting#Exaggeration" title="Strategic voting">Exaggeration</a></li>

<li><a href="Truncation_(voting)" class="mw-redirect" title="Truncation (voting)">Truncation</a></li>

<li><a href="Turkey-raising" class="mw-redirect" title="Turkey-raising">Turkey-raising</a></li>

<li><a href="Wasted_vote" title="Wasted vote">Wasted vote</a></li></ul>

<hr>

<p><b>Paradoxes of <a href="Majority_rule" title="Majority rule">majority rule</a></b>

</p>

<ul><li><a href="Tyranny_of_the_majority" title="Tyranny of the majority">Tyranny of the majority</a></li>

<li><a href="Discursive_dilemma" class="mw-redirect" title="Discursive dilemma">Discursive dilemma</a></li>

<li><a href="Condorcet_paradox" title="Condorcet paradox">Conflicting majorities paradox</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content" style="text-align:left;">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#efefef; border-top:1px solid;background: var(--background-color-interactive, #efefef); color: var(--color-base, #000);;color: var(--color-base)"><a href="Social_choice_theory" title="Social choice theory">Social and collective choice</a></div><div class="sidebar-list-content mw-collapsible-content"><b><a href="Proof_of_impossibility" title="Proof of impossibility">Impossibility theorems</a></b>

<ul><li><a href="Arrow's_impossibility_theorem" title="Arrow's impossibility theorem">Arrow's theorem</a></li>

<li><a href="Condorcet_paradox" title="Condorcet paradox">Majority impossibility</a></li>

<li><a href="Moulin's_impossibility_theorem" class="mw-redirect" title="Moulin's impossibility theorem">Moulin's impossibility theorem</a></li>

<li><a href="McKelvey%E2%80%93Schofield_chaos_theorem" title="McKelvey–Schofield chaos theorem">McKelvey–Schofield chaos theorem</a></li>

<li><a href="Gibbard's_theorem" title="Gibbard's theorem">Gibbard's theorem</a></li></ul>

<hr>

<p><b>Positive results</b>

</p>

<ul><li><a href="Median_voter_theorem" title="Median voter theorem">Median voter theorem</a></li>

<li><a href="Condorcet's_jury_theorem" title="Condorcet's jury theorem">Condorcet's jury theorem</a></li>

<li><a href="May's_theorem" title="May's theorem">May's theorem</a></li>

<li><a href="Arrow's_theorem" class="mw-redirect" title="Arrow's theorem">Condorcet dominance theorems</a></li>

<li>Harsanyi's utilitarian theorem</li>

<li><a href="Vickrey-Clarke-Groves_mechanism" class="mw-redirect" title="Vickrey-Clarke-Groves mechanism">VCG mechanism</a></li>

<li><a href="Quadratic_voting" title="Quadratic voting">Quadratic voting</a></li></ul></div></div></td>

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<p>In <a href="Welfare_economics" title="Welfare economics">welfare economics</a> and <a href="Social_choice_theory" title="Social choice theory">social choice theory</a>, a <b>social welfare function</b>—also called a <b>social</b> <b>ordering</b>, <b>ranking</b>, <b>utility</b>, or <b>choice</b> <b>function</b>—is a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> that ranks a set of social states by their desirability. Each person's preferences are combined in some way to determine which outcome is considered better by society as a whole.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> It can be seen as mathematically formalizing <a href="Jean-Jacques_Rousseau" title="Jean-Jacques Rousseau">Rousseau</a>'s idea of a <a href="General_will" title="General will">general will</a>.

</p><p>Social choice functions are studied by <a href="Economist" title="Economist">economists</a> as a way to identify socially-optimal decisions, giving a procedure to rigorously define which of two outcomes should be considered better for society as a whole (e.g. to compare two different possible <a href="Income_distribution" title="Income distribution">income distributions</a>).<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> They are also used by <a href="Democracy" title="Democracy">democratic</a> governments to choose between several options in <a href="Election" title="Election">elections</a>, based on the preferences of voters; in this context, a social choice function is typically referred to as an <a href="Electoral_system" title="Electoral system">electoral system</a>.

</p><p>The notion of social utility is analogous to the notion of a utility function in <a href="Consumer_choice" title="Consumer choice">consumer choice</a>. However, a social welfare function is different in that it is a mapping of <i>individual</i> utility functions onto a single output, in a way that accounts for the judgments of everyone in a society.

</p><p>There are two different notions of social welfare used by economists:

</p>

<ul><li><a href="Ordinal_utility" title="Ordinal utility"><b>Ordinal</b></a> (or <a href="Ranked_voting" title="Ranked voting">ranked voting</a>) functions only use <a href="Ordinal_utility" title="Ordinal utility">ordinal</a> information, i.e. whether one choice is better than another.</li>

<li><a href="Cardinal_utility" title="Cardinal utility"><b>Cardinal</b></a> (or <a href="Rated_voting" title="Rated voting">rated voting</a>) functions also use <a href="Cardinal_utility" title="Cardinal utility">cardinal</a> information, i.e. how much better one choice is compared to another.</li></ul>

<p><a href="Arrow's_impossibility_theorem" title="Arrow's impossibility theorem">Arrow's impossibility theorem</a> is a key result on social welfare functions, showing an important difference between social and consumer choice: whereas it is possible to construct a <a href="Rational_choice_theory" class="mw-redirect" title="Rational choice theory">rational</a> (non-self-contradictory) decision procedure for consumers based only on ordinal preferences, it is impossible to do the same in the social choice setting, making any such ordinal decision procedure a <a href="Second_best" class="mw-redirect" title="Second best">second-best</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Terminology_and_equivalence">Terminology and equivalence</h2></div>

<p>Some authors maintain a distinction between three closely related concepts:

</p>

<ol><li>A social <i>choice</i> function selects a single best outcome (a single candidate who wins, or multiple if there happens to be a tie).</li>

<li>A social <i>ordering</i> function lists the candidates, from best to worst.</li>

<li>A social <i>scoring</i> function maps each candidate to a number representing their quality. For example, the standard social scoring function for <a href="First-preference_plurality" class="mw-redirect" title="First-preference plurality">first-preference plurality</a> is the total number of voters who rank a candidate first.</li></ol>

<p>Every social ordering can be made into a choice function by considering only the highest-ranked outcome. Less obviously, though, every social choice function is also an ordering function. Deleting the best outcome, then finding the new winner, results in a runner-up who is assigned second place. Repeating this process gives a full ranking of all candidates.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p><p>Because of this close relationship, the three kinds of functions are often conflated by <a href="Abuse_of_terminology" class="mw-redirect" title="Abuse of terminology">abuse of terminology</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Example">Example</h3></div>

<p>Consider an <a href="Instant-runoff_voting" title="Instant-runoff voting">instant-runoff</a> election between Top, Center, and Bottom. Top has the most first-preference votes; Bottom has the second-most; and Center (positioned <a href="Political_spectrum" title="Political spectrum">between the two</a>) has the fewest first preferences.

</p>

</caption>

<th>

</th>

<th>Round 1

</th>

<th>Round 2

</th></tr>

<tr>

<th>Top

</th>

<td>40

</td>

</td></tr>

<tr>

<th>Center

</th>

</td>

<td><i>Eliminated</i>

</td></tr>

<tr>

<th>Bottom

</th>

<td>34

</td>

<td>47

</td></tr></tbody></table>

<p>Under instant-runoff voting, Top is the winner. Center is eliminated in the first round, and their second-preferences are evenly split between Top and Bottom, allowing Top to win.

</p><p>To find the second-place finisher, we find the winner if Top had not run. In this case, the election is between Center and Bottom.

</p>

<th>Runner-up

</th>

<th>Round 1

</th></tr>

<tr>

<th>—

</th>

<td><i>Excluded</i>

</td></tr>

<tr>

<th>Center

</th>

</td></tr>

<tr>

<th>Bottom

</th>

<td>34

</td></tr></tbody></table>

<p>(Note that the finishing order is not the same as the elimination order for <a href="Sequential_elimination_method" title="Sequential elimination method">sequential elimination methods</a>: despite being eliminated first, Center is the runner-up in this election.)

</p>

<div class="mw-heading mw-heading2"><h2 id="Ordinal_welfare">Ordinal welfare</h2></div>

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<p>In a 1938 article, <a href="Abram_Bergson" title="Abram Bergson">Abram Bergson</a> introduced the term <i>social welfare function,</i> with the intention "to state in precise form the value judgments required for the derivation of the conditions of maximum economic welfare." The function was real-valued and <a href="Differentiable_function" title="Differentiable function">differentiable</a>. It was specified to describe the society as a whole. Arguments of the function included the quantities of different commodities produced and consumed and of <a href="Factors_of_production" title="Factors of production">resources</a> used in producing different commodities, including labor.

</p><p>Necessary general conditions are that at the maximum value of the function:

</p>

<ul><li>The marginal "dollar's worth" of welfare is equal for each individual and for each commodity</li>

<li>The marginal "dis-welfare" of each "dollar's worth" of labor is equal for each commodity produced of each labor supplier</li>

<li>The marginal "dollar" cost of each unit of resources is equal to the marginal value productivity for each commodity.</li></ul>

<p>Bergson argued that <a href="Welfare_economics" title="Welfare economics">welfare economics</a> had described a standard of economic efficiency despite dispensing with <i>interpersonally-comparable</i> <a href="Cardinal_utility" title="Cardinal utility">cardinal utility</a>, the hypothesization of which may merely conceal value judgments, and purely subjective ones at that.

</p><p>Earlier neoclassical welfare theory, heir to the classical <a href="Utilitarianism" title="Utilitarianism">utilitarianism</a> of <a href="Jeremy_Bentham" title="Jeremy Bentham">Bentham</a>, often treated the <a href="Law_of_diminishing_marginal_utility" class="mw-redirect" title="Law of diminishing marginal utility">law of diminishing marginal utility</a> as implying interpersonally comparable utility. Irrespective of such comparability, income or wealth <i>is</i> measurable, and it was commonly inferred that redistributing income from a rich person to a poor person tends to increase total utility (however measured) in the society. But Lionel Robbins (<a href="An_Essay_on_the_Nature_and_Significance_of_Economic_Science" title="An Essay on the Nature and Significance of Economic Science">1935</a>, ch. VI) argued that how or how much utilities, as mental events, change relative to each other is not measurable by any empirical test, making them <a href="Unfalsifiable" class="mw-redirect" title="Unfalsifiable">unfalsifiable</a>. Robbins therefore rejected such as incompatible with his own philosophical <a href="Behaviorism" title="Behaviorism">behaviorism</a>.

</p><p>Auxiliary specifications enable comparison of different social states by each member of society in preference satisfaction. These help define <i><a href="Pareto_efficiency" title="Pareto efficiency">Pareto efficiency</a></i>, which holds if all alternatives have been exhausted to put at least one person in a more preferred position with no one put in a less preferred position. Bergson described an "economic welfare increase" (later called a <i>Pareto improvement</i>) as at least one individual moving to a more preferred position with everyone else indifferent. The social welfare function could then be specified in a <i>substantively</i> individualistic sense to derive Pareto efficiency (optimality). <a href="Paul_Samuelson" title="Paul Samuelson">Paul Samuelson</a> (2004, p. 26) notes that Bergson's function "could derive Pareto optimality conditions as <i>necessary</i> but not sufficient for defining interpersonal normative equity." Still, Pareto efficiency could also characterize <i>one</i> dimension of a particular social welfare function with distribution of commodities among individuals characterizing <i>another</i> dimension. As Bergson noted, a welfare improvement from the social welfare function could come from the "position of some individuals" improving at the expense of others. That social welfare function could then be described as characterizing an equity dimension.

</p><p>Samuelson (<a href="Foundations_of_Economic_Analysis" title="Foundations of Economic Analysis">1947</a>, p. 221) himself stressed the flexibility of the social welfare function to characterize <i>any</i> one ethical belief, Pareto-bound or not, consistent with:

</p>

<ul><li>a complete and transitive ranking (an ethically "better", "worse", or "indifferent" ranking) of all social alternatives and</li>

<li>one set out of an infinity of welfare indices and cardinal indicators to characterize the belief.</li></ul>

<p>As Samuelson (1983, p. xxii) notes, Bergson clarified how production and consumption efficiency conditions are distinct from the interpersonal ethical values of the social welfare function.

</p><p>Samuelson further sharpened that distinction by specifying the <i>welfare function</i> and the <i>possibility function</i> (1947, pp. 243–49). Each has as <a href="Function_(mathematics)#The_vocabulary_of_functions" title="Function (mathematics)">arguments</a> the set of utility functions for everyone in the society. Each can (and commonly does) incorporate Pareto efficiency. The possibility function also depends on technology and resource restraints. It is written in implicit form, reflecting the <i>feasible</i> locus of utility combinations imposed by the restraints and allowed by Pareto efficiency. At a given point on the possibility function, if the utility of all but one person is determined, the remaining person's utility is determined. The welfare function ranks different hypothetical <i>sets</i> of utility for everyone in the society from ethically lowest on up (with ties permitted), that is, it makes interpersonal comparisons of utility. Welfare maximization then consists of maximizing the welfare function subject to the possibility function as a constraint. The same welfare maximization conditions emerge as in Bergson's analysis.

</p>

<td>For a two-person society, there is a graphical depiction of such welfare maximization at the first figure of <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060215000915/http://cepa.newschool.edu/het/essays/paretian/paretosocial.htm#swf">Bergson–Samuelson social welfare functions</a>. Relative to <a href="Consumer_theory" class="mw-redirect" title="Consumer theory">consumer theory</a> for an <i>individual</i> as to two commodities consumed, there are the following parallels:

<ul><li>The respective hypothetical utilities of the two persons in two-dimensional utility space is analogous to respective quantities of commodities for the two-dimensional commodity space of the indifference-curve <i>surface</i></li>

<li>The Welfare function is analogous to the indifference-curve <i>map</i></li>

<li>The Possibility function is analogous to the budget constraint</li>

<li>Two-person welfare maximization at the tangency of the highest Welfare function curve on the Possibility function is analogous to tangency of the highest indifference curve on the budget constraint.</li></ul>

</td></tr></tbody></table>

<p><a href="Kenneth_Arrow" title="Kenneth Arrow">Kenneth Arrow</a>'s <a href="Social_Choice_and_Individual_Values" title="Social Choice and Individual Values">1963 book</a> demonstrated the problems with such an approach, though he would not immediately realize this. Along earlier lines, Arrow's version of a social welfare function, also called a 'constitution', maps a set of individual orderings (<a href="Ordinal_utility_function" class="mw-redirect" title="Ordinal utility function">ordinal utility functions</a>) for everyone in society to a social ordering, which ranks alternative social states (such as which of several candidates should be elected).

</p><p>Arrow found that contrary to the assertions of <a href="Lionel_Robbins" title="Lionel Robbins">Lionel Robbins</a> and other <a href="Behaviorism" title="Behaviorism">behaviorists</a>, dropping the requirement of real-valued (and thus <a href="Cardinal_utility" title="Cardinal utility">cardinal</a>) social orderings makes <a href="Decision_theory" title="Decision theory">rational</a> or <a href="Coherence_(philosophical_gambling_strategy)" class="mw-redirect" title="Coherence (philosophical gambling strategy)">coherent</a> behavior at the social level impossible. This result is now known as <a href="Arrow's_impossibility_theorem" title="Arrow's impossibility theorem">Arrow's impossibility theorem</a><i>.</i> Arrow's theorem shows that it is impossible for an ordinal social welfare function to satisfy a standard axiom of <a href="Decision_theory" title="Decision theory">rational behavior</a>, called <a href="Independence_of_irrelevant_alternatives" title="Independence of irrelevant alternatives">independence of irrelevant alternatives</a>. This axiom says that changing the value of one outcome should not affect choices that do not involve this outcome. For example, if a customer buys apples because he prefers them to blueberries, telling them that cherries are on sale should not make them buy blueberries instead of apples.

</p><p><a href="John_Harsanyi" title="John Harsanyi">John Harsanyi</a> later strengthened this result by showing that if societies must make decisions under <a href="Uncertainty" title="Uncertainty">uncertainty</a>, the unique social welfare function satisfying <a href="Coherence_(philosophical_gambling_strategy)" class="mw-redirect" title="Coherence (philosophical gambling strategy)">coherence</a> and <a href="Pareto_efficiency" title="Pareto efficiency">Pareto efficiency</a> is the <a href="Utilitarian_rule" title="Utilitarian rule">utilitarian rule</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Cardinal_welfare">Cardinal welfare</h2></div>

<p>A <b>cardinal social welfare function</b> is a function that takes as input numeric representations of individual utilities (also known as <a href="Cardinal_utility" title="Cardinal utility">cardinal utility</a>), and returns as output a numeric representation of the collective welfare. The underlying assumption is that individuals utilities can be put on a common scale and compared. Examples of such measures include <a href="Life_expectancy" title="Life expectancy">life expectancy</a> or per capita income.

</p><p>For the purposes of this section, income is adopted as the measurement of utility.

</p><p>The form of the social welfare function is intended to express a statement of objectives of a society.

</p><p>The <a href="Utilitarian" class="mw-redirect" title="Utilitarian">utilitarian</a> or <a href="Benthamite" class="mw-redirect" title="Benthamite">Benthamite</a> social welfare function measures social welfare as the total or sum of individual utilities:

</p>

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<annotation encoding="application/x-tex">{\displaystyle W=\sum _{i=1}^{n}Y_{i}}</annotation>

</semantics>

</math></span><img src="./9cdd125db470bdc943e794f45de58816f2d6bc55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:11.426ex; height:6.843ex;" alt="{\displaystyle W=\sum _{i=1}^{n}Y_{i}}" loading="lazy"></span></dd></dl>

<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

</semantics>

</math></span><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"></span> is social welfare and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y_{i}}</annotation>

</semantics>

</math></span><img src="./d57be496fff95ee2a97ee43c7f7fe244b4dbf8ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.15ex; height:2.509ex;" alt="{\displaystyle Y_{i}}" loading="lazy"></span> is the income of individual <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span> among <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> individuals in society. In this case, maximizing the social welfare means maximizing the total income of the people in the society, without regard to how incomes are distributed in society. It does not distinguish between an income transfer from rich to poor and vice versa. If an income transfer from the poor to the rich results in a bigger increase in the utility of the rich than the decrease in the utility of the poor, the society is expected to accept such a transfer, because the total utility of the society has increased as a whole. Alternatively, society's welfare can also be measured under this function by taking the average of individual incomes:

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<annotation encoding="application/x-tex">{\displaystyle W={\frac {1}{n}}\sum _{i=1}^{n}Y_{i}={\overline {Y}}}</annotation>

</semantics>

</math></span><img src="./982da8fa218e47fa1a58d41c75d856da28270274.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.306ex; height:6.843ex;" alt="{\displaystyle W={\frac {1}{n}}\sum _{i=1}^{n}Y_{i}={\overline {Y}}}" loading="lazy"></span></dd></dl>

<p>In contrast, the max-min or Rawlsian social welfare function (based on the philosophical work of <a href="John_Rawls" title="John Rawls">John Rawls</a>) measures the social welfare of society on the basis of the welfare of the least well-off individual member of society:

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<annotation encoding="application/x-tex">{\displaystyle W=\min(Y_{1},Y_{2},\cdots ,Y_{n})}</annotation>

</semantics>

</math></span><img src="./972afe30e5f8b2e80d9a4954beb2e033c2a5be18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.809ex; height:2.843ex;" alt="{\displaystyle W=\min(Y_{1},Y_{2},\cdots ,Y_{n})}" loading="lazy"></span></dd></dl>

<p>Here maximizing societal welfare would mean maximizing the income of the poorest person in society without regard for the income of other individuals.

</p><p>These two social welfare functions express very different views about how a society would need to be organised in order to maximize welfare, with the first emphasizing total incomes and the second emphasizing the needs of the worst-off. The max-min welfare function can be seen as reflecting an extreme form of <a href="Uncertainty_aversion" class="mw-redirect" title="Uncertainty aversion">uncertainty aversion</a> on the part of society as a whole, since it is concerned only with the worst conditions that a member of society could face.

</p><p><a href="Amartya_Sen" title="Amartya Sen">Amartya Sen</a> proposed a welfare function in 1973:

</p>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle W_{\mathrm {Gini} }={\overline {Y}}(1-G)}</annotation>

</semantics>

</math></span><img src="./a9c19d2b2431fb1f81414149755f0db3366497af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.445ex; height:3.509ex;" alt="{\displaystyle W_{\mathrm {Gini} }={\overline {Y}}(1-G)}" loading="lazy"></span></dd></dl>

<p>The average per capita income of a measured group (e.g. nation) is multiplied with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1-G)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (1-G)}</annotation>

</semantics>

</math></span><img src="./cdba75972051b4052da5210c98cb40d3b6334acc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.639ex; height:2.843ex;" alt="{\displaystyle (1-G)}" loading="lazy"></span> where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math></span><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span> is the <a href="Gini_index" class="mw-redirect" title="Gini index">Gini index</a>, a relative inequality measure. James E. Foster (1996) proposed to use one of <a href="Anthony_Barnes_Atkinson" class="mw-redirect" title="Anthony Barnes Atkinson">Atkinson</a>'s Indexes, which is an entropy measure. Due to the relation between Atkinsons entropy measure and the <a href="Theil_index" title="Theil index">Theil index</a>, Foster's welfare function also can be computed directly using the Theil-L Index.

</p>

<msub>

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</msub>

<mover>

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<annotation encoding="application/x-tex">{\displaystyle W_{\mathrm {Theil-L} }={\overline {Y}}\mathrm {e} ^{-T_{L}}}</annotation>

</semantics>

</math></span><img src="./feb7b86da0dc7e02a082ddb9ad162c53cbda9637.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.315ex; height:3.343ex;" alt="{\displaystyle W_{\mathrm {Theil-L} }={\overline {Y}}\mathrm {e} ^{-T_{L}}}" loading="lazy"></span></dd></dl>

<p>The value yielded by this function has a concrete meaning. There are several possible incomes which could be earned by a <i>person</i>, who randomly is selected from a population with an unequal distribution of incomes. This welfare function marks the income, which a randomly selected person is most likely to have. Similar to the <a href="Median" title="Median">median</a>, this income will be smaller than the average per capita income.

</p>

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<annotation encoding="application/x-tex">{\displaystyle W_{\mathrm {Theil-T} }={\overline {Y}}\mathrm {e} ^{-T_{T}}}</annotation>

</semantics>

</math></span><img src="./cb51aa4f16fb63df59c2b355394e4770965b895e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.505ex; height:3.343ex;" alt="{\displaystyle W_{\mathrm {Theil-T} }={\overline {Y}}\mathrm {e} ^{-T_{T}}}" loading="lazy"></span></dd></dl>

<p>Here the Theil-T index is applied. The inverse value yielded by this function has a concrete meaning as well. There are several possible incomes to which a <i>Euro</i> may belong, which is randomly picked from the sum of all unequally distributed incomes. This welfare function marks the income, which a randomly selected Euro most likely belongs to. The inverse value of that function will be larger than the average per capita income.

</p>

<div class="mw-heading mw-heading3"><h3 id="Axioms_of_cardinal_welfarism">Axioms of cardinal welfarism</h3></div>

<p>Suppose we are given a <a href="Preference_relation" title="Preference relation">preference relation</a> <i>R</i> on utility profiles. <i>R</i> is a weak <a href="Total_order" title="Total order">total order</a> on utility profiles—it can tell us, given any two utility profiles, if they are indifferent or one of them is better than the other. A reasonable preference ordering should satisfy several axioms:<sup id="cite_ref-moulin2004_4-0" class="reference"><a href="#cite_note-moulin2004-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 66–69">: 66–69 </span></sup>

</p><p>1. <a href="Monotonicity_criterion" class="mw-redirect" title="Monotonicity criterion"><b>Monotonicity</b></a>: if the utility of one individual increases, while all other utilities remain equal, <i>R</i> should strictly prefer the second profile. For example, it should prefer the profile (1, 4, 4, 5) to (1, 2, 4, 5). Such a change is called a <a href="Pareto_optimality" class="mw-redirect" title="Pareto optimality">Pareto improvement</a>.

</p><p>2. <a href="Anonymity_(social_choice)" class="mw-redirect" title="Anonymity (social choice)"><b>Symmetry</b></a>: <a href="Permutation" title="Permutation">reordering or relabeling</a> the values in the utility profile should not change the output of <i>R</i>. This axiom formalizes the idea that every person should be treated equally in society. For example, <i>R</i> should be indifferent between (1, 4, 4, 5) and (5, 4, 4, 1), since these profiles are reorders of each other.

</p><p>3. <b>Continuity</b>: for every profile <i>v</i>, the set of profiles weakly better than <i>v</i> and the set of profiles weakly worse than <i>v</i> are <a href="Closed_set" title="Closed set">closed sets</a>.

</p><p>4. <b>Independence of unconcerned agents:</b> <i>R</i> should be independent of individuals whose utilities have not changed. For example, if <i>R</i> prefers (2, 2, 4) to (1, 3, 4), it also prefers (2, 2, 9) to (1, 3, 9); the utility of agent 3 should not affect the comparison between two utility profiles of agents 1 and 2. This property can also be called <b>locality</b> or <b>separability</b>. It allows us to treat allocation problems in a local way, and separate them from the allocation in the rest of society.

</p><p>Every preference relation with properties 1–4 can be represented as by a function <i>W</i> which is a sum of the form:

</p>

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<annotation encoding="application/x-tex">{\displaystyle W(u_{1},\dots ,u_{n})=\sum _{i=1}^{n}w(u_{i})}</annotation>

</semantics>

</math></span><img src="./7a02209d10fb8aef8b16b5e78988b0a249cde01e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.798ex; height:6.843ex;" alt="{\displaystyle W(u_{1},\dots ,u_{n})=\sum _{i=1}^{n}w(u_{i})}" loading="lazy"></span></dd></dl>

<p>where <i>w</i> is a continuous increasing function.

</p>

<div class="mw-heading mw-heading3"><h3 id="Harsanyi's_theorem">Harsanyi's theorem</h3></div>

<p>Introducing one additional axiom—the nonexistence of <a href="Coherence_(philosophical_gambling_strategy)" class="mw-redirect" title="Coherence (philosophical gambling strategy)">Dutch Books</a>, or equivalently that social choice behaves according to the <a href="Von_Neumann%E2%80%93Morgenstern_utility_theorem" title="Von Neumann–Morgenstern utility theorem">axioms of rational choice</a>—implies that the social choice function must be the <a href="Utilitarian_rule" title="Utilitarian rule">utilitarian rule</a>, i.e. the weighting function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w(u)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w(u)}</annotation>

</semantics>

</math></span><img src="./cf182c8908d77e22c7cc7ea47938fa82fc3a8458.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.803ex; height:2.843ex;" alt="{\displaystyle w(u)}" loading="lazy"></span> must be equal to the utility functions of each individual. This result is known as Harsanyi's utilitarian theorem.

</p>

<div class="mw-heading mw-heading3"><h3 id="Non-utilitarian">Non-utilitarian</h3></div>

<p>By Harsanyi's theorem, any non-utilitarian social choice function will be incoherent; in other words, it will agree to some bets that are unanimously opposed by every member of society. However, it is still possible to establish properties of such functions.

</p><p>Instead of imposing rational behavior on the social utility function, we can impose a weaker criterion called <b>independence of common scale</b>: the relation between two utility profiles does not change if both of them are multiplied by the same constant. For example, the utility function should not depend on whether we measure incomes in cents or dollars.

</p><p>If the preference relation has properties 1–5, then the function <i>w</i> must be the <a href="Isoelastic_utility" title="Isoelastic utility">isoelastic function</a>:

</p><p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {c^{1-\eta }-1}{1-\eta }}}">

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {c^{1-\eta }-1}{1-\eta }}}</annotation>

</semantics>

</math></span><img src="./91f6de5d318b1d0eae65e21766cd304f7f79fb7d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.006ex; height:6.176ex;" alt="{\displaystyle {\frac {c^{1-\eta }-1}{1-\eta }}}" loading="lazy"></span>

</p><p>This family has some familiar members:

</p>

<ul><li>The limit when <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta \to -\infty }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \eta \to -\infty }</annotation>

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</math></span><img src="./2e7605cb315526f4fa32becdab5a6a496dee7c38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.915ex; height:2.509ex;" alt="{\displaystyle \eta \to -\infty }" loading="lazy"></span> is the <i>leximin</i> ordering.</li>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \eta =0}</annotation>

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</math></span><img src="./2dc20bdff327bfeab94c37936d4c1a05dfbd9784.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.43ex; height:2.676ex;" alt="{\displaystyle \eta =0}" loading="lazy"></span> we get the <a href="Nash_bargaining_solution" class="mw-redirect" title="Nash bargaining solution">Nash bargaining solution</a>—maximizing the product of utilities.</li>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \eta =1}</annotation>

</semantics>

</math></span><img src="./9b65b387d2e1c91321cafa675da485eabb20ca82.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.43ex; height:2.676ex;" alt="{\displaystyle \eta =1}" loading="lazy"></span> we get the <a href="Utilitarian" class="mw-redirect" title="Utilitarian">utilitarian</a> welfare function—maximizing the sum of utilities.</li>

<li>The limit when <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta \to \infty }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \eta \to \infty }</annotation>

</semantics>

</math></span><img src="./4d776fb31f3b03db5f7752ca8127dd2db6560235.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.107ex; height:2.343ex;" alt="{\displaystyle \eta \to \infty }" loading="lazy"></span> is the <i>leximax</i> ordering.</li></ul>

<p>If we require the <b><a href="Pigou%E2%80%93Dalton_principle" title="Pigou–Dalton principle">Pigou–Dalton principle</a></b> (that inequality is not a positive good) then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \eta }</annotation>

</semantics>

</math></span><img src="./e4d701857cf5fbec133eebaf94deadf722537f64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }" loading="lazy"></span> in the above family must be at most 1.

</p>

<ul><li><a href="Aggregation_problem" title="Aggregation problem">Aggregation problem</a></li>

<li><a href="Arrow's_impossibility_theorem" title="Arrow's impossibility theorem">Arrow's impossibility theorem</a></li>

<li><a href="Community_indifference_curve" title="Community indifference curve">Community indifference curve</a></li>

<li><a href="Distribution_(economics)" title="Distribution (economics)">Distribution (economics)</a></li>

<li><a href="Economic_welfare" class="mw-redirect" title="Economic welfare">Economic welfare</a></li>

<li><a href="Extended_sympathy" title="Extended sympathy">Extended sympathy</a></li>

<li><a href="Gorman_polar_form" title="Gorman polar form">Gorman polar form</a></li>

<li><a href="Justice_(economics)" class="mw-redirect" title="Justice (economics)">Justice (economics)</a></li>

<li><a href="Liberal_paradox" title="Liberal paradox">Liberal paradox</a></li>

<li><a href="Production-possibility_frontier" class="mw-redirect" title="Production-possibility frontier">Production-possibility frontier</a></li>

<li><a href="Social_choice_theory" title="Social choice theory">Social choice theory</a></li>

<li><a href="Welfare_economics" title="Welfare economics">Welfare economics</a></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</style><div class="reflist reflist-columns references-column-width">

<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="Amartya_K._Sen" class="mw-redirect" title="Amartya K. Sen">Amartya K. Sen</a>, 1970 [1984], <i>Collective Choice and Social Welfare</i>, ch. 3, "Collective Rationality." p. 33, and ch. 3*, "Social Welfare Functions." <a rel="nofollow" class="external text" href="https://www.citeulike.org/user/rlai/article/681900">Description.</a></span>

</li>

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</style><cite id="CITEREFTresch2008" class="citation book cs1">Tresch, Richard W. (2008). <i>Public Sector Economics</i>. New York: PALGRAVE MACMILLAN. p. 67. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-230-52223-7</bdi>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFQuesada2002" class="citation journal cs1">Quesada, Antonio (2002). <a rel="nofollow" class="external text" href="https://ideas.repec.org//a/ebl/ecbull/eb-02d70006.html">"From social choice functions to dictatorial social welfare functions"</a>. <i>Economics Bulletin</i>. <b>4</b> (16): <span class="nowrap">1–</span>7.</cite></span>

</li>

<li id="cite_note-moulin2004-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-moulin2004_4-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFHerve_Moulin2004" class="citation book cs1">Herve Moulin (2004). <i>Fair Division and Collective Welfare</i>. Cambridge, Massachusetts: MIT Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780262134231</bdi>.</cite></span>

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li><a href="Kenneth_Arrow" title="Kenneth Arrow">Kenneth J. Arrow</a>, 1951, 2nd ed., 1963, <i><a href="Social_Choice_and_Individual_Values" title="Social Choice and Individual Values">Social Choice and Individual Values</a></i> <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-300-01364-7</bdi></li>

<li><a href="Abram_Bergson" title="Abram Bergson">Abram Bergson</a> (Burk),"A Reformulation of Certain Aspects of Welfare Economics," <i>Quarterly Journal of Economics</i>, 52(2), February 1938, 310–34</li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20060215000915/http://cepa.newschool.edu/het/essays/paretian/paretosocial.htm#swf">Bergson–Samuelson social welfare functions</a> in Paretian welfare economics from the New School.</li>

<li>James E. Foster and <a href="Amartya_Sen" title="Amartya Sen">Amartya Sen</a>, 1996, <i>On Economic Inequality</i>, expanded edition with annexe, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-19-828193-5</bdi>.</li>

<li><a href="John_C._Harsanyi" class="mw-redirect" title="John C. Harsanyi">John C. Harsanyi</a>, 1987, “interpersonal utility comparisons," <i><a href="The_New_Palgrave%3A_A_Dictionary_of_Economics" class="mw-redirect" title="The New Palgrave: A Dictionary of Economics">The New Palgrave: A Dictionary of Economics</a></i>, v. 2, 955–58</li>

<li><cite id="CITEREFPattanaik2008" class="citation cs2"><a href="Prasanta_Pattanaik" title="Prasanta Pattanaik">Pattanaik, Prasanta K.</a> (2008), "social welfare function <i>(definition)</i>", in <a href="Steven_N._Durlauf" class="mw-redirect" title="Steven N. Durlauf">Durlauf, Steven N.</a>; <a href="Lawrence_E._Blume" title="Lawrence E. Blume">Blume, Lawrence E.</a> (eds.), <i>The new Palgrave dictionary of economics</i> (2nd ed.), Basingstoke, Hampshire New York: Palgrave Macmillan, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780333786765</bdi>.</cite></li></ul>

<dl><dd><dl><dd>Also available as: <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1057/9780230226203.1567">a journal article.</a></dd></dl></dd></dl>

<ul><li><a href="Jan_de_Van_Graaff" class="mw-redirect" title="Jan de Van Graaff">Jan de Van Graaff</a>, 1957, "Theoretical Welfare Economics", 1957, Cambridge, UK: Cambridge University Press.</li>

<li><a href="Lionel_Robbins" title="Lionel Robbins">Lionel Robbins</a>, 1935, 2nd ed.. <i><a href="An_Essay_on_the_Nature_and_Significance_of_Economic_Science" title="An Essay on the Nature and Significance of Economic Science">An Essay on the Nature and Significance of Economic Science</a></i>, ch. VI</li>

<li>____, 1938, "Interpersonal Comparisons of Utility: A Comment," <i>Economic Journal</i>, 43(4), 635–41</li>

<li><a href="Paul_A._Samuelson" class="mw-redirect" title="Paul A. Samuelson">Paul A. Samuelson</a>, 1947, Enlarged ed. 1983, <i><a href="Foundations_of_Economic_Analysis" title="Foundations of Economic Analysis">Foundations of Economic Analysis</a></i>, pp. xxi–xxiv & ch. VIII, "Welfare Economics," <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-674-31301-1</bdi></li>

<li>_____, 1977. "Reaffirming the Existence of 'Reasonable' Bergson–Samuelson Social Welfare Functions," <i>Economica</i>, N.S., 44(173), p <a rel="nofollow" class="external text" href="https://www.jstor.org/pss/2553553">pp. 81</a>–88. Reprinted in (1986) <i>The Collected Scientific Papers of Paul A. Samuelson</i>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=UKeJEc46R9AC&pg=PA47=gbs_atb">47–54.</a></li>

<li>_____, 1981. "Bergsonian Welfare Economics", in S. Rosefielde (ed.), <i>Economic Welfare and the Economics of Soviet Socialism: Essays in Honor of Abram Bergson</i>, <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, Cambridge, pp. 223–66. Reprinted in (1986) <i>The Collected Scientific Papers of Paul A. Samuelson</i>, pp. 3 <a rel="nofollow" class="external text" href="https://books.google.com/books?id=UKeJEc46R9AC&pg=PA225=gbs_atb">–46.</a></li>

<li><a href="Amartya_Sen" title="Amartya Sen">Sen, Amartya K.</a> (1963). "Distribution, Transitivity and Little's Welfare Criteria," <i>Economic Journal</i>, 73(292), <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2228209">pp. 771</a>–78.</li>

<li>_____, 1970 [1984], <i>Collective Choice and Social Welfare</i> <a rel="nofollow" class="external text" href="https://www.citeulike.org/user/rlai/article/681900">(description)</a>, ch. 3, "Collective Rationality." <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-444-85127-5</bdi></li>

<li>_____ (1982). <i>Choice, Welfare and Measurement</i>, MIT Press. <a rel="nofollow" class="external text" href="https://scholar.harvard.edu/sen/publications/choice-welfare-and-measurement">Description</a> and scroll to chapter-preview <a rel="nofollow" class="external text" href="https://books.google.com/books?id=u8GPYeT1qAUC&q=false">links.</a></li>

<li><a href="Kotaro_Suzumura" title="Kotaro Suzumura">Kotaro Suzumura</a> (1980). "On Distributional Value Judgments and Piecemeal Welfare Criteria," <i>Economica</i>, 47(186), p <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2553231">pp. 125</a>–39.</li>

<li>_____, 1987, “social welfare function," <i>The New Palgrave: A Dictionary of Economics</i>, v. 4, 418–20</li></ul>

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</style></div><div role="navigation" class="navbox authority-control" aria-label="Navbox391" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Authority control databases: National </th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4190150-2">Germany</a></span></li></ul></div></td></tr></tbody></table></div><div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Electoral_systems390" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Electoral_systems390" style="font-size:114%;margin:0 4em"><a href="Electoral_system" title="Electoral system">Electoral systems</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div><i>Part of the <a href="Portal%3APolitics" title="Portal:Politics">politics</a> and <a href="Portal%3AEconomics" title="Portal:Economics">Economics</a> series</i></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Single-winner_voting_system" class="mw-redirect" title="Single-winner voting system">Single-winner</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Approval_voting" title="Approval voting">Approval voting</a>

<ul><li><a href="Combined_approval_voting" title="Combined approval voting">Combined approval voting</a></li>

<li><a href="Unified_primary" title="Unified primary">Unified primary</a></li></ul></li>

<li><a href="Borda_count" title="Borda count">Borda count</a></li>

<li><a href="Bucklin_voting" title="Bucklin voting">Bucklin voting</a></li>

<li><a href="Condorcet_methods" class="mw-redirect" title="Condorcet methods">Condorcet methods</a>

<ul><li><a href="Copeland's_method" title="Copeland's method">Copeland's method</a></li>

<li><a href="Dodgson's_method" title="Dodgson's method">Dodgson's method</a></li>

<li><a href="Kemeny_method" title="Kemeny method">Kemeny method</a></li>

<li><a href="Minimax_Condorcet_method" title="Minimax Condorcet method">Minimax Condorcet method</a></li>

<li><a href="Nanson's_method" title="Nanson's method">Nanson's method</a></li>

<li><a href="Ranked_pairs" title="Ranked pairs">Ranked pairs</a></li>

<li><a href="Schulze_method" title="Schulze method">Schulze method</a></li></ul></li>

<li><a href="Exhaustive_ballot" title="Exhaustive ballot">Exhaustive ballot</a></li>

<li><a href="First-past-the-post_voting" title="First-past-the-post voting">First-past-the-post voting</a></li>

<li><a href="Instant-runoff_voting" title="Instant-runoff voting">Instant-runoff voting</a>

<ul><li><a href="Coombs'_method" title="Coombs' method">Coombs' method</a></li>

<li><a href="Contingent_vote" title="Contingent vote">Contingent vote</a></li>

<li><a href="Supplementary_vote" class="mw-redirect" title="Supplementary vote">Supplementary vote</a></li></ul></li>

<li><a href="Majority_rule" title="Majority rule">Simple majoritarianism</a></li>

<li><a href="Plurality_voting_system" class="mw-redirect" title="Plurality voting system">Plurality</a></li>

<li><a href="Positional_voting_system" class="mw-redirect" title="Positional voting system">Positional voting system</a></li>

<li><a href="Score_voting" title="Score voting">Score voting</a></li>

<li><a href="STAR_voting" title="STAR voting">STAR voting</a></li>

<li><a href="Two-round_system" title="Two-round system">Two-round system</a></li>

<li><a href="Graduated_majority_judgment" title="Graduated majority judgment">Graduated majority judgment</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Proportional_representation" title="Proportional representation">Proportional</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Systems</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Mixed-member_proportional_representation" title="Mixed-member proportional representation">Mixed-member</a></li>

<li><a href="Mixed_single_vote#Proportional_systems" title="Mixed single vote">Mixed single vote</a></li>

<li><a href="Party-list_proportional_representation" title="Party-list proportional representation">Party-list</a></li>

<li><a href="Proportional_approval_voting" title="Proportional approval voting">Proportional approval voting</a></li>

<li><a href="Rural%E2%80%93urban_proportional_representation" title="Rural–urban proportional representation">Rural–urban</a></li>

<li><a href="Sequential_proportional_approval_voting" title="Sequential proportional approval voting">Sequential proportional approval voting</a></li>

<li><a href="Single_transferable_vote" title="Single transferable vote">Single transferable vote</a>

<li><a href="Hare%E2%80%93Clark_electoral_system" title="Hare–Clark electoral system">Hare–Clark</a></li>

<li><a href="Schulze_STV" title="Schulze STV">Schulze STV</a></li></ul></li>

<li><a href="Spare_vote" title="Spare vote">Spare vote</a></li>

<li><a href="Indirect_single_transferable_voting" title="Indirect single transferable voting">Indirect single transferable voting</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Allocation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Highest_averages_method" title="Highest averages method">Highest averages method</a>

<ul><li><a href="Sainte-Lagu%C3%AB_method" title="Sainte-Laguë method">Webster/Sainte-Laguë</a></li>

<li><a href="D'Hondt_method" title="D'Hondt method">D'Hondt</a></li></ul></li>

<li><a href="Largest_remainders_method" class="mw-redirect" title="Largest remainders method">Largest remainders method</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Quotas</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Droop_quota" title="Droop quota">Droop quota</a></li>

<li><a href="Hagenbach-Bischoff_quota" class="mw-redirect" title="Hagenbach-Bischoff quota">Hagenbach-Bischoff quota</a></li>

<li><a href="Hare_quota" title="Hare quota">Hare quota</a></li>

<li><a href="Imperiali_quota" title="Imperiali quota">Imperiali quota</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Mixed_electoral_system" title="Mixed electoral system">Mixed</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Parallel_voting" title="Parallel voting">Parallel voting</a></li>

<li><a href="Additional_member_system" class="mw-redirect" title="Additional member system">Additional member system</a></li>

<li><a href="Alternative_vote_plus" title="Alternative vote plus">Alternative vote plus</a></li>

<li><a href="Mixed_single_vote" title="Mixed single vote">Mixed single vote</a></li>

<li><a href="Mixed_ballot_transferable_vote" title="Mixed ballot transferable vote">Mixed ballot transferable vote</a></li>

<li><a href="Scorporo" title="Scorporo">Scorporo</a></li>

<li><a href="Vote_linkage_mixed_system" class="mw-redirect" title="Vote linkage mixed system">Vote linkage mixed system</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Semi-proportional_representation" title="Semi-proportional representation">Semi-proportional</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Single_non-transferable_vote" title="Single non-transferable vote">Single non-transferable vote</a></li>

<li><a href="Limited_voting" title="Limited voting">Limited voting</a></li>

<li><a href="Cumulative_voting" title="Cumulative voting">Cumulative voting</a></li>

<li><a href="Satisfaction_approval_voting" title="Satisfaction approval voting">Satisfaction approval voting</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Criteria</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Condorcet_winner_criterion" title="Condorcet winner criterion">Condorcet winner criterion</a></li>

<li><a href="Condorcet_loser_criterion" title="Condorcet loser criterion">Condorcet loser criterion</a></li>

<li><a href="Consistency_criterion" title="Consistency criterion">Consistency criterion</a></li>

<li><a href="Independence_of_clones_criterion" title="Independence of clones criterion">Independence of clones</a></li>

<li><a href="Independence_of_irrelevant_alternatives" title="Independence of irrelevant alternatives">Independence of irrelevant alternatives</a></li>

<li><a href="Independence_of_Smith-dominated_alternatives" title="Independence of Smith-dominated alternatives">Independence of Smith-dominated alternatives</a></li>

<li><a href="Later-no-harm_criterion" title="Later-no-harm criterion">Later-no-harm criterion</a></li>

<li><a href="Majority_favorite_criterion" class="mw-redirect" title="Majority favorite criterion">Majority criterion</a></li>

<li><a href="Majority_loser_criterion" title="Majority loser criterion">Majority loser criterion</a></li>

<li><a href="Monotonicity_criterion" class="mw-redirect" title="Monotonicity criterion">Monotonicity criterion</a></li>

<li><a href="Mutual_majority_criterion" title="Mutual majority criterion">Mutual majority criterion</a></li>

<li><a href="Participation_criterion" title="Participation criterion">Participation criterion</a></li>

<li><a href="Plurality_criterion" class="mw-redirect" title="Plurality criterion">Plurality criterion</a></li>

<li><a href="Resolvability_criterion" class="mw-redirect" title="Resolvability criterion">Resolvability criterion</a></li>

<li><a href="Reversal_symmetry" title="Reversal symmetry">Reversal symmetry</a></li>

<li><a href="Smith_criterion" class="mw-redirect" title="Smith criterion">Smith criterion</a></li>

<li><a href="Seats-to-votes_ratio" title="Seats-to-votes ratio">Seats-to-votes ratio</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Ballot" title="Ballot">Ballot</a></li>

<li><a href="Election_threshold" class="mw-redirect" title="Election threshold">Election threshold</a></li>

<li><a href="First-preference_votes" class="mw-redirect" title="First-preference votes">First-preference votes</a></li>

<li><a href="Liquid_democracy" title="Liquid democracy">Liquid democracy</a></li>

<li><a href="Spoilt_vote" title="Spoilt vote">Spoilt vote</a></li>

<li><a href="Sortition" title="Sortition">Sortition</a></li>

<li><a href="Unseating" title="Unseating">Unseating</a></li>

<li><a href="Wasted_vote" title="Wasted vote">Wasted vote</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Comparison</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Comparison_of_voting_rules" title="Comparison of voting rules">Comparison of voting systems</a></li>

<li><a href="List_of_electoral_systems_by_country" title="List of electoral systems by country">Voting systems by country</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><b><a href="Portal%3APolitics" title="Portal:Politics">Portal</a></b> — <b>Project</b></div></td></tr></tbody></table></div><div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1066933788">

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</style></div><div role="navigation" class="navbox" aria-labelledby="Economics681" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Economics681" style="font-size:114%;margin:0 4em"><a href="Economics" title="Economics">Economics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Economics#Theoretical_research" title="Economics">Theoretical</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Microeconomics" title="Microeconomics">Microeconomics</a>

<ul><li><a href="Decision_theory" title="Decision theory">Decision theory</a></li>

<li><a href="Price_theory" class="mw-redirect" title="Price theory">Price theory</a></li>

<li><a href="Game_theory" title="Game theory">Game theory</a></li>

<li><a href="Contract_theory" title="Contract theory">Contract theory</a></li>

<li><a href="Mechanism_design" title="Mechanism design">Mechanism design</a></li></ul></li>

<li><a href="Macroeconomics" title="Macroeconomics">Macroeconomics</a></li>

<li><a href="Mathematical_economics" title="Mathematical economics">Mathematical economics</a></li>

<li><a href="Complexity_economics" title="Complexity economics">Complexity economics</a></li>

<li><a href="Computational_economics" title="Computational economics">Computational economics</a>

<ul><li><a href="Agent-based_computational_economics" title="Agent-based computational economics">Agent-based computational economics</a></li></ul></li>

<li><a href="Behavioral_economics" title="Behavioral economics">Behavioral economics</a></li>

<li><a href="Pluralism_in_economics" title="Pluralism in economics">Pluralism in economics</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Economics#Empirical_research" title="Economics">Empirical</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Econometrics" title="Econometrics">Econometrics</a>

<ul><li><a href="Economic_statistics" title="Economic statistics">Economic statistics</a></li></ul></li>

<li><a href="Experimental_economics" title="Experimental economics">Experimental economics</a></li>

<li><a href="Economic_history" title="Economic history">Economic history</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Applied_economics" title="Applied economics">Applied</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">