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| {{More footnotes|date=August 2011}}
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| '''Fair division''' is the problem of dividing a set of goods between several people, such that each person receives his/her due share. This problem arises in various real-world settings:
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| auctions, divorce settlements, electronic spectrum and [[frequency allocation]], airport traffic management, or exploitation of [[Earth Observation Satellite]]s. This is an active research area in [[Mathematics]], [[Economics]] (especially [[Social choice]] theory), [[Game theory]], [[Dispute resolution]], and more.
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| There are many different kinds of fair division problems, depending on the nature of goods to divide, the criteria for fairness, the nature of the players and their preferences, and other criteria for evaluating the quality of the division.
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| == Definitions ==
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| There is a set <math>X</math>, and a group of <math>n</math> players, <math>P_1, P_2, ... P_n</math>. A '''division''' is a partition of <math>X</math> to <math>n</math> disjoint subsets: <math>X = X_1 \amalg X_2 \amalg \cdots \amalg X_n</math>, one subset per player.
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| === What is divided? ===
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| The set <math>X</math> can be of several types: | |
| * X may be a finite set of '''indivisible''' items, for example: <math>X = \{piano, car, apartment\}</math>, such that each item should be given entirely to a single person.
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| * X may be an infinite set representing a '''divisible''' resource, for example: money, or a cake. Mathematically, a divisible resource is often modeled as a subset of a real space, for example, the section [0,1] may represent a long narrow cake, that has to be cut into parallel pieces. The unit circle may represent an apple pie.
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| Additionally, the set to be divided may be:
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| * '''homogeneous''' - such as money, or -
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| * '''heterogeneous''' - such as a cake, that may have different ingredients, different icings, etc.
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| Finally, it is common to make some assumptions about whether the items to be divided are:
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| * '''desirable''' - such as a car or a cake, or -
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| * '''undesirable''' - such as house chores.
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| The problem of dividing a set of ''indivisible and heterogeneous'' items is also called '''fair allocation''' or '''fair assignment'''.
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| The problem of dividing a ''divisible, heterogeneous and desirable'' resource is also called '''fair [[cake-cutting]]'''.
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| The problem of dividing a set of ''heterogeneous and undesirable'' items is also called '''fair [[Chore division]]'''.
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| Combinations are also possible, for example:
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| * When dividing [[inheritance]], or dividing household property during [[divorce]], it is common to have both ''desirable indivisible heterogeneous'' items, ''desirable divisible heterogeneous'' property such as land, and ''desirable divisible homogeneous'' property such as money.
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| * In the [[housemates problem]], several friends rent a house together, and they have to both allocate the rooms in the apartment (a set of ''indivisible, heterogeneous, desirable'' goods), and divide the rent to pay (''divisible, homogeneous, undesirable'' good). This problem is also called '''room assignment-rent division'''.
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| === What is fair? ===
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| The question of fair division becomes interesting when the goods to divide are heterogeneous. According to the [[Subjective theory of value]], there cannot be an objective measure of the value of each item. Therefore, ''objective fairness'' is not possible, as different people may assign different values to each item. Empirical experiments on how people define the concept of fairness <ref>{{cite doi|10.1007/BF00297056}}</ref> lead to inconclusive results.
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| Therefore, most current research on fairness focuses on concepts of ''subjective fairness''. Each of the <math>n</math> people, <math>P_i (i = 1..n)</math> is assumed to have a personal, subjective ''utility function'' or ''value function'', <math>v_i</math>, which assigns a numerical value to each subset of <math>X</math>. Usually the functions are assumed to be normalized, so that every person values the empty set as 0 (<math>v_i (\empty) = 0</math> for all i), and the entire set of items as 1 (<math>v_i (X) = 1</math> for all i) if the items are desirable, and -1 if the items are undesirable. Examples are:
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| * If <math>X</math> is the set of indivisible items {piano, car, apartment}, then Alice may assign a value of 1/3 to each item, which means that each item is important to her just the same as any other item. Bob may assign the value of 1 to the set {car, apartment}, and the value 0 to all other sets except X; this means that he wants to get only the car and the apartment together; the car alone or the apartment alone, or each of them together with the piano, is worthless to him.
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| * If <math>X</math> is a long narrow cake (modeled as the section [0,1]), then, Alice may assign each subset a value proportional to its length, which means that she wants as much cake as possible, regardless of the icings. Bob may assign value only to subsets of [0.4, 0.6], for example, because this part of the cake contains cherries, and Bob only cares about cherries.
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| Now we can define several concepts of ''subjective fairness'':
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| * '''[[proportional (fair division)|proportional]]''' division, also called '''simple fair division''' means, that every person gets at least his due share, ''according to his own value function'', i.e., each of the <math>n</math> people gets a subset of X which he values as at least <math>1/n</math>:
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| ** <math>v_i(X_i) \ge 1/n</math> for all i.
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| * '''[[envy-free]]''' division means, that every person gets a share that he values at least as much as all other shares, i.e.:
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| ** <math>v_i(X_i) \ge v_i(X_j)</math> for all i and j.
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| * '''equitable''' or '''exact''' division means, that every person feels exactly the same happiness, i.e., got exactly the same value:
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| ** <math>v_i(X_i) = v_j(X_j)</math> for all i and j.
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| The problem is easier when recipients have different measures of value of the parts of the resource: in the "cake cutting" version, one recipient may like marzipan, another prefers cherries, and so on—then, and only then, the ''n'' recipients may get even more than what would be one ''n''-th of the value of the "cake" for each of them. On the other hand, the presence of different measures opens a vast potential for many challenging questions and directions of further research.
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| Most of what is normally called a fair division is not considered so by the theory because of the use of [[arbitration]]. This kind of situation happens quite often with mathematical theories named after real life problems. The decisions in the [[Talmud]] on [[entitlement (fair division)|entitlement]] when an estate is [[bankrupt]] reflect some quite complex ideas about fairness,<ref>[http://www.elsevier.com/framework_aboutus/Nobel/Nobel2005/nobel2005pdfs/aum16.pdf Game Theoretic Analysis of a bankruptcy Problem from the Talmud]
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| Robert J. Aumann and Michael Maschler. Journal of Economic Theory 36, 195-213 (1985)</ref> and most people would consider them fair. However they are the result of legal debates by rabbis rather than divisions according to the valuations of the claimants.
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| [[Image:Berlin Blockade-map.svg|thumb|200px|right|Berlin divided by the [[Potsdam Conference]]]]
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| ==Assumptions==
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| Fair division is a [[mathematical theory]] based on an idealization of a real life problem. The real life problem is the one of dividing [[Good (economics)|goods]] or [[resources]] fairly between people, the 'players', who have an [[entitlement]] to them. The central tenet of fair division is that such a division should be performed by the players themselves, maybe using a [[mediation|mediator]] but certainly not an [[arbitration|arbiter]] as only the players really know how they value the goods.
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| The theory of fair division provides explicit criteria for various different types of fairness. Its aim is to provide procedures ([[algorithm]]s) to achieve a fair division, or prove their impossibility, and study the properties of such divisions both in theory and in real life.
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| The assumptions about the valuation of the goods or resources are:
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| * Each player has their own opinion of the value of each part of the goods or resources
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| * The value to a player of any allocation is the sum of his valuations of each part. Often just requiring the valuations be [[weakly additive]] is enough.
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| * In the basic theory the goods can be divided into parts with arbitrarily small value.
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| Indivisible parts make the theory much more complex. An example of this would be where a car and a motorcycle have to be shared. This is also an example of where the values may not add up nicely, as either can be used as transport. The use of money can make such problems much easier.
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| The criteria of a fair division are stated in terms of a players valuations, their level of [[Entitlement (fair division)|entitlement]], and the results of a fair division procedure. The valuations of the other players are not involved in the criteria. Differing entitlements can normally be represented by having a different number of proxy players for each player but sometimes the criteria specify something different.
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| In the real world of course people sometimes have a very accurate idea of how the other players value the goods and they may care very much about it. The case where they have complete knowledge of each other's valuations can be modeled by [[game theory]]. Partial knowledge is very hard to model. A major part of the practical side of fair division is the devising and study of procedures that work well despite such partial knowledge or small mistakes.
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| A fair division [[Algorithm|procedure]] lists actions to be performed by the players in terms of the visible data and their valuations. A valid procedure is one that guarantees a fair division for every player who acts rationally according to their valuation. Where an action depends on a player's valuation the procedure is describing the [[strategy]] a rational player will follow. A player may act as if a piece had a different value but must be consistent. For instance if a procedure says the first player cuts the cake in two equal parts then the second player chooses a piece, then the first player cannot claim that the second player got more.
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| What the players do is:
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| * Agree on their criteria for a fair division
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| * Select a valid procedure and follow its rules
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| It is assumed the aim of each player is to maximize the minimum amount they might get, or in other words, to achieve the [[minimax|maximin]].
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| Procedures can be divided into finite and continuous procedures. A finite procedure would for instance only involve one person at a time cutting or marking a cake. Continuous procedures involve things like one player [[moving-knife procedure|moving a knife]] and the other saying stop. Another type of continuous procedure involves a person assigning a value to every part of the cake.
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| == Criteria for a fair division ==
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| There are a number of widely used criteria for a fair division. Some of these conflict with each other but often they can be combined. The criteria described here are only for when each player is entitled to the same amount.
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| * A [[proportional (fair division)|proportional]] or simple fair division guarantees each player gets his fair share. For instance if three people divide up a cake each gets at least a third by their own valuation.
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| * An [[envy-free]] division guarantees no-one will want somebody else's share more than their own.
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| * An [[exact division]] is one where every player thinks everyone received exactly their fair share, no more and no less.
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| * An efficient or [[Pareto optimal]] division ensures no other allocation would make someone better off without making someone else worse off. The term efficiency comes from the [[economics]] idea of the [[efficient market]]. A division where one player gets everything is optimal by this definition so on its own this does not guarantee even a fair share.
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| * An [[Equity (economics)|equitable]] division is one where the proportion of the cake a player receives by their own valuation is the same for every player. This is a difficult aim as players need not be truthful if asked their valuation.
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| ==Two players==
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| For two people there is a simple solution which is commonly employed. This is the so-called [[divide and choose]] method. One person divides the resource into what they believe are equal halves, and the other person chooses the "half" they prefer. Thus, the person making the division has an incentive to divide as fairly as possible: for if they do not, they will likely receive an undesirable portion. This solution gives a [[proportional (fair division)|proportional]] and [[envy-free]] division.
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| The article on [[divide and choose]] describes why the procedure is not equitable. More complex procedures like the [[adjusted winner procedure]] are designed to cope with indivisible goods and to be more equitable in a practical context.
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| Austin's [[moving-knife procedure]]<ref>A.K. Austin. ''Sharing a Cake''. Mathematical Gazette 66 1982</ref> gives an exact division for two players. The first player positions a knife over the left side of the cake. He moves the knife to the right and when either player says to stop, they receive the left piece of cake. This produces an envy-free decision.
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| The [[surplus procedure]] (SP) achieves a form of equitability called proportional equitability. This procedure is strategy proof and can be generalized to more than two people.<ref name="Brams2006">{{cite journal
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| | last = Brams
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| | first = Steven J.
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| | coauthors = Michael A. Jones and Christian Klamler
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| | title = Better Ways to Cut a Cake
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| | journal = [[Notices of the American Mathematical Society]]
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| |date=December 2006
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| | volume = 53
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| | issue = 11
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| | pages = pp.1314–1321
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| | url = http://www.ams.org/notices/200611/fea-brams.pdf
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| | format = [[PDF]]
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| | accessdate = 2008-01-16
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| | authorlink = Steven Brams }}</ref>
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| ==Many players== | |
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| Fair division with three or more players is considerably more complex than the two player case.
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| [[Proportional (fair division)|Proportional]] division is the easiest and the article describes some procedures which can be applied with any number of players. Finding the minimum number of cuts needed is an interesting mathematical problem.
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| [[Envy-free]] division was first solved for the 3 player case in 1960 independently by [[John Selfridge]] of [[Northern Illinois University]] and [[John Horton Conway]] at [[Cambridge University]]. The best algorithm([[Selfridge–Conway discrete procedure]]) uses at most 5 cuts.
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| The [[Brams-Taylor procedure]] was the first cake-cutting procedure for four or more players that produced an [[envy-free]] division of cake for any number of persons and was published by [[Steven Brams]] and [[Alan D. Taylor|Alan Taylor]] in 1995.<ref>{{cite journal| title=An Envy-Free Cake Division Protocol | author1=Steven J. Brams | author2=Alan D. Taylor | journal=[[The American Mathematical Monthly]] | volume=102 | issue=1 | pages=9–18 |date=January 1995 | jstor=2974850 | publisher=[[Mathematical Association of America]] | doi=10.2307/2974850 | authorlink1=Steven Brams | authorlink2=Alan D. Taylor }}</ref> This number of cuts that might be required by this procedure is unbounded. A bounded moving knife procedure for 4 players was found in 1997.
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| There are no discrete algorithms for an exact division even for two players, a moving knife procedure is the best that can be done. There are no exact division algorithms for 3 or more players but there are 'near exact' algorithms which are also envy-free and can achieve any desired degree of accuracy.
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| A generalization of the [[surplus procedure]] called the equitable procedure (EP) achieves a form of equitability. Equitability and envy-freeness can be incompatible for 3 or more players.<ref name="Brams2006" />
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| ==Variants== | |
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| Some cake-cutting procedures are ''discrete'', whereby players make cuts with a [[knife]] (usually in a sequence of steps). [[Moving-knife procedure]]s, on the other hand, allow continuous movement and can let players call "stop" at any point.
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| A variant of the fair division problem is [[chore division]]: this is the "dual" to the cake-cutting problem in which an undesirable object is to be distributed amongst the players. The canonical example is a set of chores that the players between them must do. Note that "I cut, you choose" works for chore division. A basic theorem for many person problems is the Rental Harmony Theorem by Francis Su.<ref>{{cite journal |author=Francis Edward Su |title=Rental Harmony: Sperner's Lemma in Fair Division |journal=Amer. Math. Monthly |volume=106 |year=1999 |pages=930–942 |url=http://www.math.hmc.edu/~su/papers.dir/rent.pdf |doi=10.2307/2589747 |issue=10}}</ref> An interesting application of the Rental Harmony Theorem can be found in the international trade theory.<ref>{{cite journal |author=Shiozawa, Y. A |title=New Construction ofa Ricardian Trade Theory |journal=Evolutionary and Institutional Economics Review | volume=3 |issue=2 |year=2007 |pages=141–187}}</ref>
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| [[Sperner's Lemma]] can be used to get as close an approximation as desired to an envy-free solutions for many players. The algorithm gives a fast and practical way of solving some fair division problems.<ref>Francis Edward Su. Cited above. (based on work by Forest Simmons 1980)</ref><ref>{{cite web|url=http://www.math.hmc.edu/~su/fairdivision/calc/ |title=The Fair Division Calculator}}</ref><ref>{{cite web |url=http://www.maa.org/mathland/mathtrek_3_13_00.html |title=A Fair Deal for Housemates |author=Ivars Peterson |work=MathTrek |date=March 13, 2000}}</ref>
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| The division of property, as happens for example in divorce or inheritance, normally contains indivisible items which must be fairly distributed between players, possibly with cash adjustments (such pieces are referred to as ''atoms'').
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| A common requirement for the division of land is that the pieces be [[connectedness|connected]], i.e. only whole pieces and not fragments are allowed. For example the division of Berlin after World War 2 resulted in four connected parts.<ref>{{cite book| title=Fair division: from cake-cutting to dispute resolution |author1=Steven J. Brams |author2=Alan D. Taylor |page=38 |isbn=978-0-521-55644-6 |publisher=Cambridge University Press |year=1996}}</ref>
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| A consensus halving is where a number of people agree that a resource has been evenly split in two, this is described in [[exact division]].
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| == History ==
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| According to [[Sol Garfunkel]], the cake-cutting problem had been one of the most important open problems in 20th century mathematics,<ref>Sol Garfunkel. More Equal than Others: Weighted Voting. For All Practical Purposes. COMAP. 1988</ref> when the most important variant of the problem was finally solved with the [[Brams-Taylor procedure]] by [[Steven Brams]] and [[Alan D. Taylor|Alan Taylor]] in 1995.
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| [[Divide and choose]]'s origins are undocumented. The related activities of [[bargaining]] and [[barter]] are also ancient. [[Negotiation]]s involving more than two people are also quite common, the [[Potsdam Conference]] is a notable recent example.
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| The theory of fair division dates back only to the end of the second world war. It was devised by a group of [[Poland|Polish]] mathematicians, [[Hugo Steinhaus]], [[Bronisław Knaster]] and [[Stefan Banach]], who used to meet in the [[Scottish Café]] in Lvov (then in Poland). A [[proportional (fair division)]] division for any number of players called 'last-diminisher' was devised in 1944. This was attributed to Banach and Knaster by Steinhaus when he made the problem public for the first time at a meeting of the [[Econometric Society]] in Washington D.C. on 17 September 1947. At that meeting he also proposed the problem of finding the smallest number of cuts necessary for such divisions.
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| [[Envy-free]] division was first solved for the 3 player case in 1960 independently by [[John Selfridge]] of [[Northern Illinois University]] and [[John Horton Conway]] at [[Cambridge University]], the algorithm was first published in the 'Mathematical Games' column by [[Martin Gardner]] in [[Scientific American]].
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| Envy-free division for 4 or more players was a difficult open problem of the twentieth century. The first cake-cutting procedure that produced an [[envy-free]] division of cake for any number of persons was first published by [[Steven Brams]] and [[Alan D. Taylor|Alan Taylor]] in 1995.
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| A major advance on equitable division was made in 2006 by [[Steven Brams|Steven J. Brams]], Michael A. Jones, and Christian Klamler.<ref name="Brams2006" />
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| == In popular culture ==
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| * In ''[[Numb3rs]]'' season 3 episode "One Hour", Charlie talks about the cake-cutting problem as applied to the amount of money a kidnapper was demanding.
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| * [[Hugo Steinhaus]] wrote about a number of variants of fair division in his book ''Mathematical Snapshots''. In his book he says a special three-person version of fair division was devised by G. Krochmainy in Berdechów in 1944 and another by Mrs L Kott.<ref>Mathematical Snapshots. H.Steinhaus. 1950, 1969 ISBN 0-19-503267-5</ref>
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| * [[Martin Gardner]] and [[Ian Stewart (mathematician)|Ian Stewart]] have both published books with sections about the problem.<ref>aha! Insight. Martin. Gardner, 1978. ISBN ISBN 978-0-7167-1017-2</ref><ref>How to cut a cake and other mathematical conundrums. Ian Stewart. 2006. ISBN 978-0-19-920590-5</ref> Martin Gardner introduced the chore division form of the problem. Ian Stewart has popularized the fair division problem with his articles in ''[[Scientific American]]'' and ''[[New Scientist]]''.
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| * A ''[[Dinosaur Comics]]'' strip is based on the cake-cutting problem.<ref>http://www.qwantz.com/archive/001345.html</ref>
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| == See also ==
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| {{multicol}}
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| * [[Alan D. Taylor]]
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| * [[Brams–Taylor procedure]]
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| * [[Equity (economics)]]
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| * [[Game theory]]
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| * [[Justice (economics)]]
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| * [[International trade]]
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| * [[Knapsack problem]]
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| * [[Nash bargaining game]]
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| * [[Pizza theorem]]
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| {{multicol-break}}
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| * [[Sperner's lemma]]
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| * [[Spite]]
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| * [[Steven Brams]]
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| * [[Topological combinatorics]]
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| * [[Tragedy of the anticommons]]
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| * [[Tragedy of the commons]]
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| * [[Weakly additive]]
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| {{multicol-end}}
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| == References ==
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| {{reflist|2}}
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| == Further reading ==
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| * Steven J. Brams and Alan D. Taylor (1996). ''Fair Division - From cake-cutting to dispute resolution'' Cambridge University Press. ISBN 0-521-55390-3
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| * T.P. Hill (2000). "Mathematical devices for getting a fair share", ''American Scientist'', Vol. 88, 325-331.
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| * Jack Robertson and William Webb (1998). ''Cake-Cutting Algorithms: Be Fair If You Can'', AK Peters Ltd, . ISBN 1-56881-076-8.
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| ==External links==
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| * [http://3quarksdaily.blogs.com/3quarksdaily/2005/04/3qd_monday_musi.html Short essay about the cake-cutting problem] by S. Abbas Raza of ''3 Quarks Daily''.
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| * [http://www.colorado.edu/education/DMP/fair_division.html Fair Division] from the Discrete Mathematics Project at the University of Colorado at Boulder.
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| * [http://www.math.hmc.edu/~su/fairdivision/calc/ The Fair Division Calculator] (Java applet) at Harvey Mudd College
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| * [http://www.cut-the-knot.org/Curriculum/SocialScience/LoneDivider.shtml Fair Division: Method of Lone Divider]
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| * [http://www.cut-the-knot.org/Curriculum/SocialScience/Markers.shtml Fair Division: Method of Markers]
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| * [http://www.cut-the-knot.org/Curriculum/SocialScience/SealedBids.shtml Fair Division: Method of Sealed Bids]
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| * Vincent P. Crawford (1987). "fair division," ''The [[New Palgrave: A Dictionary of Economics]]'', v. 2, pp. 274–75.
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| * [[Hal R. Varian|Hal Varian]] (1987). "fairness," ''The New Palgrave: A Dictionary of Economics'', v. 2, pp. 275–76.
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| * Bryan Skyrms (1996). ''The Evolution of the Social Contract'' Cambridge University Press. ISBN 978-0-521-55583-8
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| {{game theory}}
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| {{DEFAULTSORT:Fair Division}}
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| [[Category:Fair division| ]]
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| [[Category:Game theory]]
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| [[Category:Welfare economics]]
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