{"id":2543,"date":"2025-11-01T00:01:18","date_gmt":"2025-11-01T04:01:18","guid":{"rendered":"https:\/\/mathvoices.ams.org\/featurecolumn\/?p=2543"},"modified":"2025-11-03T09:04:32","modified_gmt":"2025-11-03T14:04:32","slug":"achieving-fairness","status":"publish","type":"post","link":"https:\/\/mathvoices.ams.org\/featurecolumn\/2025\/11\/01\/achieving-fairness\/","title":{"rendered":"Achieving Fairness"},"content":{"rendered":"

What may surprise you is the extent to which mathematicians (and computer scientists) have systematically studied issues involving fairness…<\/em><\/span><\/p>\n

Achieving Fairness<\/h1>\n

Joe Malkevitch
\nYork College (CUNY)<\/b><\/p>\n

Introduction<\/h2>\n<\/p>\n

The experience of being alive begins with the fact
\nthat no matter how hard one may try to put oneself in the position of
\nanother person, for each of us, we are the center of our
\nexistence. As a person ages from childhood we become aware there are
\npeople close to us like our parents or guardians, possible siblings, and other
\nfamily members. We quickly learn that there other people and that in
\nsome cases these people seem to be "better off" than we are.
\nDad seems to treat me better than sister Alice while Mom seems to treat
\nbrother Bob better than me or Alice, but Grandpa Charles seems to treat
\nus all equally well. We quickly develop a sense of whether we are being
\ntreated fairly and this, as we age extends to the markers that we label
\nourselves with—woman, living in the US but born in Somalia, etc.
\nHopefully, whatever groups we identify with we come to have
\nempathy for other human beings, in particular those that are further
\nremoved from those people we interact with regularly.<\/p>\n

\n\"A
\nThis child is looking for a fair way to share ice cream. Photo by Aaron Concannon<\/a>, CC BY-NC-ND 2.0<\/a>.<\/em>\n<\/div>\n

If you
\nhave a question about an illness, about what caused the recent flooding
\nin your neighborhood, why the price of eggs has suddenly changed and
\nthat your favorite cereal seems no longer easily available for purchase,
\nyou learn who are the "experts" on questions of this
\nparticular type. Similarly, if your toilet overflows or if as a home
\nowner your roof develops a leak, you learn in both of these cases it is
\nnot a carpenter or mathematician whose services are needed but in the
\nfirst case you should call a plumber and the second a roofer. But who
\nare the "experts" on fairness? Perhaps when you were young, if
\nyou felt that you were not being treated fairly you might
\nturn to a parent, grandparent, close friend, or a member of the clergy
\nfor advice about what to do about your concern. More abstractly, you
\nmight think that philosophers, lawyers, politicians, or judges might be
\npeople who had wisdom about fairness. Are there areas of academia where
\nthe issues of fairness are addressed? What may surprise you is the
\nextent to which mathematicians (and computer scientists) have
\nsystematically studied issues involving fairness. While the mathematical tools used to get
\ninsights into these fairness questions covers a large range, a
\nsurprisingly large amount of the scholarly fairness literature is
\nreadable by individuals with a relatively small amount of mathematical
\nbackground. Many of these fairness questions emerge from issues related
\nto trying to run countries committed to democracy in a way that respects
\nthe rights of its citizens and seeks input from society members with
\ndifferent views and values.<\/p>\n

Sample Fairness Questions\/Situations<\/h2>\n<\/p>\n

What follows are a small sample of
\nfairness questions\/situations that have been studied to varying degrees
\nby the mathematics community. Many of these settings overlap in the way
\nthat mathematics can be used in getting insights into the problem.
\nHistorically, progress on one kind of fairness problem has often brought
\nabout progress in how to be fair in other situations. New fairness
\nproblems lead to new methods useful in older problems and sometimes new
\nproblems lead to new methods and insights useful to problems studied in
\nthe past. Rarely does mathematical progress close off a field of
\nresearch. Each new result typically spawns new questions that foster
\nnew progress and in turn new problems to address.<\/p>\n

    \n
  1. Apportionment<\/li>\n

    The US Constitution requires that a census be
    \ntaken. The number of Senators that each state has is two regardless
    \nof the population of the state, as a way to recognize that geographic
    \nregions to some extent affect the interests of the people in rural and
    \nlow density parts of the US and those living in high population cities
    \nsuch as Los Angeles or Chicago. However, one might argue that it is not
    \nfair to the people of California that Hawaii and Idaho have two Senators
    \njust the way that California does.<\/p>\n

    In the other part of the US
    \nlegislative system, the House of Representatives, the     Fourteenth Amendment<\/a> directs, “Representatives shall be apportioned among the several states according to their respective numbers, counting the whole number of persons in each state.” However, there is the mathematical issue that if a
    \nstate has 10 percent of the US population it is currently entitled to 43.5 seats
    \nin the House of Representatives. Should it get 43, 44 or some other
    \ninteger number of seats? A fraction of a person can’t be sent to the
    \nHouse of Representatives. Mathematicians have developed a rich
    \ncollection of insights into different ways that seats might
    \nbe distributed.<\/p>\n

  2. Disaster relief <\/li>\n

    After a natural
    \ndisaster caused by a hurricane, flooding, earthquake or wildfire,
    \nsometimes funds are set aside to help those harmed to recover and
    \nrebuild their lives. If the claims made against the disaster relief fund
    \nexceed the funds available, what might be a fair way to distribute the
    \nfunds?<\/p>\n

  3. Food Banks<\/li>\n

    In many parts of the United States, there are organizations that
    \nhelp provide food to the hungry. These organizations sometimes are known as food banks. Food banks
    \nhave to decide how to allocate the food that they get to the
    \noutlets that actually distribute the food. The issue arises
    \nhow such allocations can be done fairly. (The     February 2021 Feature Column<\/a> discusses another mathematical question involving food banks: what is the best way to allocate volunteers?)<\/p>\n

  4. Estate division<\/li>\n

    In some situations, when a person dies the person’s
    \nheirs are to share the proceeds of the dead person’s estate. This estate
    \nmay be only money or it may be a mixture of money and other possessions—a house, art collection, rugs, etc. How can the estate be fairly
    \ndivided where there are several heirs? Similar issues arise in settling
    \nhow to distribute jointly owned items in a divorce settlement.<\/p>\n

  5. Fair matchings<\/li>\n

    Given two equal-sized groups of claimants,
    \nwhere each group has provided strict rankings of the members of the
    \nother group, when can the two groups be paired off so that the result is
    \nstable? Here stable means that no pair of claimants matched can do
    \nbetter by agreeing to swap the partners that they were
    \nassigned. One of the best-known examples of such a system is the system
    \nthat is used by hospitals who need to have residencies filled and many
    \nmedical school graduates who need a residency assignment to complete
    \ntheir medical school training. Other applications include assigning
    \nworkers to jobs. Mathematicians and other scholars have developed many
    \nvariants of the original work due to Dave Gale (1921-2008) and Lloyd
    \nShapely (1923-2016). Thus, there might be a difference in size between
    \nthe two groups to be matched or the rankings might not be strict and
    \nallow ties.<\/p>\n

  6. Fair games<\/li>\n

    While the history of using
    \nmathematics to get insight into games has many roots in many cultures,
    \nin relatively recent times the major catalyst in setting of interest in
    \ngames was the Hungarian-born mathematician John von Neumann (1903-1957).\n<\/p>\n

    <\/P><\/p>\n

    \"Black-and-white<\/p>\n

    Photo of John von
    \nNeumann,     by permission of Los Angeles National Security<\/a>.<\/DIV><\/p>\n

    John von Neumann collaborated with the economist Oskar Morgenstern (1902-1977)
    \nin writing a book entitled Games and Economic Behavior,<\/em> which set off a
    \ntorrent of interest in studying games and fairness questions.
    \nAs children and adults we often get to play various games. Many of these games involve a mixture of luck and
    \nskill. Mathematicians have looked at the issue of
    \ndetermining if various classes of games are fair. To make it
    \neasier to answer fairness questions about games, mathematicians work with
    \nstylized settings where competition occurs
    \nbetween two or more players. These games are often shown (modeled) as a
    \ntable with choices for one player (claimant) set out in rows, and the
    \nchoices for the other player tied to the choice of a column. An entry in
    \nthe table (matrix) such as $(-5, 6)$ means the choices of actions by the
    \nplayers result in the outcome of -5 to the row player, and an outcome of
    \n6 to the column player. What units are these payoffs in? Here we
    \ninterpret -5 as Row losing 5 dollars and 6 as winning 6 dollars. The
    \nstakes that change hands in the game might just as easily be chocolate bars or jelly
    \nbeans.<\/p>\n

    In the diagram below, the players Row and Column simultaneously
    \nchoose Row 1 or Row 2 and Column I or Column II, respectively—thus, if Row chooses Row 2 and Column chooses Column I, then Row
    \nloses 15 and Column wins 15. <\/p>\n

    <\/p>\n

    <\/p>\n

    <\/p>\n

      <\/TD>
    \nColumn I <\/TD>     		Column II <\/TD> <\/TR>
    Row 1 <\/TD> $(3, -3)$ <\/TD> $(-2,
    \n2)$<\/TD> <\/TR>
    Row 2 <\/TD> $(-15, 15)$ <\/TD> $(10, -10)$<\/FONT> <\/TD> <\/TABLE> <\/P><\/p>\n


    <\/DIV><\/p>\n


    This game is special because it is zero-sum, i.e.,
    \ngains by Row correspond on each play (the game may be played several
    \ntimes, one round after another) and the lose to Column add to zero. The
    \ngame might be played once or over and over again. When Row wins, Row
    \ngets either 3 or 10 while when Column wins Column gets 2 or 15. It might
    \nappear because 13 is less than 17 that Column has an advantage in this
    \ngame. But this game, if played many times, on average the amount of
    \nmoney that changes changes hands is zero, and most people would thus
    \nregard it as a fair game.<\/p>\n

  7. Fair elections
    \n(voting)<\/li>\n

    In many situations there are actions (or candidates to
    \nbe chosen) which must be selected by a group of people who have
    \ndifferent views. In such situations, one
    \ncan conduct a vote or an election. Each eligible voter is
    \nasked to produce a ballot which captures their views about the
    \nalternative actions (candidates or choices) and based on these ballots
    \nan election decision method or algorithm is decided on in
    \nadvance of the voting. Mathematicians (and political scientists,
    \nphilosophers and economists) have developed many different types of
    \nballots to evaluate or rank the candidates or actions and many different
    \ndecision methods, tied to the kind of ballot used to find the winner
    \n(winners) or declare a tie. Usually, the decision method is
    \ndesigned to be decisive, meaning that there is a single
    \noutcome selected by the voters as the choice for society.\n<\/p>\n

    Different election decision methods will select a single
    \nwinner based on this set of ballots:<\/p>\n

    \"The<\/DIV><\/p>\n

    Which candidate in your view most deserves to win this election?
    \nWhat method of choosing a winner will result in this candidate winning?
    \nWhat fairness properties does your algorithm (method) for picking a
    \nwinner obey?<\/p>\n

    Fair Allocation<\/h2>\n<\/p>\n

    What is an
    \nallocation situation?<\/p>\n

    A limited supply of
    \nitems or prizes (e.g. money, food, legislative seats, etc.) is
    \nto be distributed fairly to $m$ claimants, based on some
    \ncollection of claims each made by a single claimant. Different claimants
    \nmay have different views about what it means to be treated fairly, and
    \nthe different individuals or institutions that might be trying to be
    \nfair might not agree what constitutes being fair.<\/p>\n

    \n\"Photo
    \n
    \nHow should we allocate slices from a pizza with different toppings? (Photo by Hajime Nakano<\/a>, CC-BY 2.0<\/a>.)<\/em>\n<\/div>\n

    One of the
    \nfirst useful things that a mathematical analysis brings about is making
    \nsure that the terminology used in the problem setting is clear and
    \nprecise. How many claimants are there? What one might do could vary
    \nsignificantly depending on whether there were 2 claimants, 10 claimants,
    \nor 50,000 claimants. What is the nature of the prizes or items that are
    \nto be distributed? Though money is easily divisible, one can’t divide art works (paintings,
    \nrugs, books, houses) into fractional parts. It would not make much sense
    \nfor a painting by Van Gogh to be cut up into three parts of the same
    \narea. Sometimes there may be several identical non-subdividable items
    \nthat are to be distributed, but what is much more common is that there
    \nare indivisible items which can be sensibly assigned a value, and that
    \nwhile only one person gets each indivisible item, the claimants can get
    \na "fair share" of the items in terms of the total
    \nvalue of what was distributed. Sometimes all of the claimants and
    \nthe size of their claims is known in advance but in some problems, known
    \nas online situations, the claims arrive one at a time and one has to
    \nhave a scheme in which one tries to be fair in this environment. Another
    \nissue to be considered when evaluating fairness methods is how much
    \ncomputational difficulty there is in finding or computing the
    \nanswer, even when one can prove that there is a fair
    \nsolution. Issues of computational complexity come into play in addition
    \nto the question of whether one can prove that there is a fair solution.
    \nSometimes one feasible fair solution can be found quickly while a
    \ndifferent fair solution might not be practical to find. Sometimes
    \napproximate fair solutions can be proved to exist—no claimant is
    \ntreated significantly worse than another, and one can find such an
    \napproximate solution rapidly even when the number of claimants or the
    \nnumber of prizes (chores) is very large.<\/p>\n

    What characteristics
    \ndo the claimants have? Rarely is it the case that claimants
    \nare completely identical. Sometimes claimants are people, sometimes
    \nstates, sometimes corporations, etc. If the claimants are
    \npeople, some claimants may be old and some may be young; in
    \nfact, some claimants may be so young that they are being
    \nrepresented by someone (or a legal entity) other than
    \nthemselves. <\/p>\n

    Typically some kind of procedure or algorithm is
    \nused to assign the prizes to the claimants. There are many issues
    \nrelated to the nature of these procedures.<\/p>\n

    Does the procedure
    \nhave some kind of built-in bias? Some procedures may not
    \ntreat people of different genders equally. Some procedures may treat poor people
    \ndifferently from rich people, and there may be different approaches to
    \nbeing fair for people under the age of 18 and over the age of 75. Thus,
    \na commuter railroad may have one price for children under 5 and special
    \ndiscounts for people who are over 70.<\/p>\n

    For now, consider
    \nallocation problems where the items to be allocated can not be
    \nsubdivided. Each item must be assigned to one claimant.
    \nImagine we have a collection of claimants and that the items to be
    \nallocated have been assigned. The items being allocated could have
    \npositive value to the claimant or negative value
    \nto the claimant. Sometimes items to be allocated that have negative
    \nvalue are referred to as being "chores," while those with
    \npositive value are called "prizes." In some settings a mixture
    \nof prizes and chores are called manna<\/em>. These environments turn out to
    \nhave rather different challenges for obtaining fairness
    \nresults.<\/p>\n

    Theoretical mathematical analysis typically proceeds
    \nby having a collection of undefined terms, a collection of defined words
    \nbuilding on these undefined terms, and a collection of rules or axioms
    \ninvolve these undefined terms. Now one takes this framework
    \nand using the laws of logic makes deductions of theorems,
    \nbased on the axioms. In this spirit, here are some of the terms that have
    \nbeen defined to investigate fairness issues for allocation
    \nsituations.<\/p>\n

    a. Proportionality<\/p>\n

    In a fairness situation
    \nwhere one can assign to the prizes or chores a value, a distribution of
    \nthe items is said to be proportional if, when there are $n$
    \nclaimants, each claimant gets at least $1\/n$ of the total value of the
    \nmanna—chores or prizes.<\/p>\n

    b. Envy free<\/p>\n

    An
    \nenvy free assignment of items in a fairness situation occurs when each
    \nclaimant $i$ accepts the items assigned, rather than claimant $i$ saying they
    \nwould rather have the assignment that was given to claimant $j$. In some
    \ncases, claimant $i$ will not envy claimant $j$ when the amount $i$ gets is
    \nequal to the amount claimant $j$ got. If there are two claimants and only one
    \npositively valued indivisible prize to be distributed, then there is no
    \nsystem of allocating the prize which will be envy free—the claimant
    \nwho does not get the prize will always envy the person who does. For
    \nsome situations, however, an envy free solution will be possible, and
    \nthis is a desirable feature of fair allocations.<\/p>\n

    c. EF1<\/p>\n

    Since there are situations where envy free allocations to the
    \nclaimants are not possible, scholars have to consider the notation of
    \nhow the envy free requirement, as desirable as this unobtainable goal
    \nmight be, be relaxed so that a "fair" allocation can be
    \ncarried out.<\/p>\n

    The EF1 rule is obeyed when if one specific fixed
    \nitem is disregarded when analyzing whether or not a particular
    \nallocation is viewed as envy free. Thus, while claimant $i$ will envy
    \nclaimant $j$ when item $I$ is part of what is assigned to claimant $j$, if $i$
    \ndisregards that $I$ was given to $j$, $i$ is no longer envious of what
    \nclaimant $j$ got. <\/p>\n

    One can modify the EF1 "rule" or
    \naxiom in various ways. One way is known as EFX or EFx, where instead of
    \nconsidering envy with regard to a fixed particular item, one ranges over
    \nbeing envy free with regard to any single item rather than one
    \nparticular item. One can consider algorithms that produce envy free
    \nallocations when several items rather than one specific item $i$ or any
    \nspecific item is taken into consideration. Another approach is that of
    \ntaking a pair of items in an allocation and swapping them. Thus, if item
    \n$U$ was assigned to claimant $i$ and $V$ was assigned to claimant $j$ and the
    \ntotal allocation (the way all items were allocated) was envied by one of
    \nthe claimants, would that individual cease to envy the other person when
    \nthe item $U$ went to $j$ and item $V$ went to $i$? <\/p>\n

    Conclusion<\/h2>\n<\/p>\n

    The range of examples
    \nwhere fairness situations come into play is vast. Mathematicians and computer scientists have played a growing
    \nrole in trying to understand and solve fairness problems. Perhaps you
    \ncan provide some examples of situations where communities could get new mathematical insight into questions of fairness. <\/P><\/p>\n


    <\/DIV><\/p>\n

    References<\/B><\/p>\n

    Aziz, H., Rauchecker, G., Schryen, G., and
    \nWalsh, T. (2017). Approximation
    algorithms for max-min share
    \nallocations of indivisible chores and goods. In Proceedings of the
    \nThirty-First AAAI Conference on Artificial Intelligence, pages
    \n335–341.<\/p>\n

    M. Balinski and H. Young, Fair Representation,
    \n2nd edition, Brookings Institution, Washington, 2001.<\/p>\n

    Brams,
    \nS.J. and A.D. Taylor (1999) : “ The Win-Win Solution: Guaranteeing
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    \nBudish, E. (2011). The combinatorial assignment problem: Approximate
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    P. di Cortona, C. Manzi, A.
    \nPennisi, F. Ricca, and B. Simeone, Evaluation and Optimization of
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    Edelman, P. and
    \nP.C. Fishburn (2001): “ Fair Division of Indivisible Items
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    \nSciences, Vol.
    41, 327-347.<\/p>\n

    Gale, D., and L. Shapley
    \n(1962): "College Admissions and the Stability of
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    \nLuce, R. Duncan and H. Raiffa, Games and Decisions, Revised edition,
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    Malkevitch, J., Rationality and Game Theory<\/a>.<\/p>\n

    Evangelos Markakis. 2017. Approximation Algorithms and Hardness
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    Ariel D. Procaccia. 2016. Cake Cutting
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    Rapoport, A. and M. Guyer, D. Gordon, The 2×2
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    Jack M. Robertson and William A.
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    Roth, Alvin E., and Marilda Sotomayor, Two-sided
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    A. Sen, Inequality Reexamined, Harvard U. Press,
    \nCambridge, 1992.<\/p>\n

    Hugo Steinhaus. 1948. The Problem of Fair
    \nDivision. Econometrica 16 (1948), 101–104.<\/p>\n

    Straffin,
    \nP., Game Theory and Strategy, Mathematical Association of America,
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    Warut Suksompong. 2017. Fairly Allocating Contiguous
    \nBlocks of Indivisible Items. In Proceedings of the 10th
    \nInternational Symposium on Algorithmic Game Theory (SAGT’17).
    \n333–344.<\/p>\n","protected":false},"excerpt":{"rendered":"

    What may surprise you is the extent to which mathematicians (and computer scientists) have systematically studied issues involving fairness… Achieving Fairness Joe Malkevitch York College (CUNY) Introduction The experience of being alive begins with the fact that no matter how hard one may try to put oneself in the positionRead More →<\/a><\/span><\/p>\n","protected":false},"author":2,"featured_media":2037,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"advanced_seo_description":"","jetpack_seo_html_title":"","jetpack_seo_noindex":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_post_was_ever_published":false},"categories":[178,17,137],"tags":[212,141],"class_list":["entry","author-uwhitcher","post-2543","post","type-post","status-publish","format-standard","has-post-thumbnail","category-178","category-joseph-malkevitch","category-math-and-social-sciences","tag-fair-division","tag-voting"],"jetpack_featured_media_url":"https:\/\/i0.wp.com\/mathvoices.ams.org\/featurecolumn\/wp-content\/uploads\/sites\/2\/2024\/07\/cropped-FC1380x500x2.png?fit=1380%2C288&ssl=1","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/mathvoices.ams.org\/featurecolumn\/wp-json\/wp\/v2\/posts\/2543","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathvoices.ams.org\/featurecolumn\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathvoices.ams.org\/featurecolumn\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathvoices.ams.org\/featurecolumn\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/mathvoices.ams.org\/featurecolumn\/wp-json\/wp\/v2\/comments?post=2543"}],"version-history":[{"count":13,"href":"https:\/\/mathvoices.ams.org\/featurecolumn\/wp-json\/wp\/v2\/posts\/2543\/revisions"}],"predecessor-version":[{"id":2564,"href":"https:\/\/mathvoices.ams.org\/featurecolumn\/wp-json\/wp\/v2\/posts\/2543\/revisions\/2564"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/mathvoices.ams.org\/featurecolumn\/wp-json\/wp\/v2\/media\/2037"}],"wp:attachment":[{"href":"https:\/\/mathvoices.ams.org\/featurecolumn\/wp-json\/wp\/v2\/media?parent=2543"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathvoices.ams.org\/featurecolumn\/wp-json\/wp\/v2\/categories?post=2543"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathvoices.ams.org\/featurecolumn\/wp-json\/wp\/v2\/tags?post=2543"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}