Abstract
This survey considers approaches to fair division from diverse disciplines, including mathematics, operations research and economics, all of which place fair division within the same general structure. There is some divisible item, or some set of items (which may be divisible or indivisible) over which a set of agents has preferences in the form of value functions, preference rankings, etc. Emphasizing the algorithmic part of the literature, we will look at procedures for fair division, and from an axiomatic point of view will investigate important properties that might be satisfied or violated by such procedures. Links to the analysis of aggregating individual preferences (See the chapter by Nurmi, this Volume) and bargaining theory (see the chapter by kibris, this Kibris, Volume) will become apparent. Strategic and computational aspects are also considered briefly. In mathematics, the fair division literature focuses on cake-cutting algorithms. We will provide an overview of the general framework, discuss some specific problems that have been studied in depth, such as pie-cutting and cake-cutting, and present several procedures. Then, moving to an informationally scarcer framework, we will consider the problem of sharing costs or benefits. Based on an example from the Talmud, various procedures to divide a fixed resource among a set of agents with different claims will be discussed. Certain changes to the framework, e.g. the division of costs or variable resources, will then be investigated. The survey concludes with a short discussion of fair division issues from the viewpoint of economics.
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Notes
- 1.
I am very grateful to Steven Brams, Andreas Darmann, Daniel Eckert, Marc Kilgour and Michael Jones for providing helpful comments.
- 2.
See also Young (1994b) for an excellent book-length treatment of various fair division aspects.
- 3.
A brief survey over some parts ot the mathematics literature on fair division has recently been provided by Brams, (2006).
- 4.
See Weller (1985) for a general approach to fair division of measurable spaces.
- 5.
Those are the properties most often used in the literature. However, there do exist many other properties in this literature, e.g. strengthenings or weakenings of the above properties (see e.g. Barbanel, 2005).
- 6.
This is like following the maximin solution concept as used in non-cooperative game theory.
- 7.
Su (1999) uses Sperner’s lemma to show the existence of an envy-free cake division under the assumption that the players prefer a piece with mass to no piece (i.e. players being “hungry”) and preference sets being closed.
- 8.
- 9.
The amount of individual envy of a player is determined by the number of other players she envies.
- 10.
This could also be seen as the equitability property for unequal entitlements.
- 11.
Other thresholds besides \(\frac{1}{n}\) could be used, especially if we have a situation in which the players have unequal entitlements.
- 12.
A certain similarity to maximin behavior can be observed (see also Crawford (1980) for the use of maximin behavior in economic models).
- 13.
E.g. the simple divide and choose method leads to an allocation that is both envy-free and efficient (for n – 1 cut procedures). However, for non-identical preferences (actually, non-equivalent 50–50 points, i.e. the point which divides the cake into two pieces of exactly the same value for that player), whoever is the divider will envy the chooser for being in the position of receiving a value of more than 50% of the cake whereas the divider can only guarantee a value of 50% to herself. The fairness problem involved in that has been discussed e.g. by Crawford (1977).
- 14.
For a detailed discussion of ranking sets of items based on a ranking of the items see Barbera et al. (2004).
- 15.
The procedure has a certain similarity to the use of the greedy algorithm in knapsack problems. See Kellerer et al. (2004) for an extensive treatment of knapsack problems.
- 16.
- 17.
See also Taylor and Zwicker (1999).
- 18.
An envy-free split is trivial if each player values its subset at exactly 50%.
- 19.
A closely linked problem is the housemates problem, where there are n rooms rent by n housmates. Each housmate bids for every single room and finally pays a price for the room he or she gets (Su, 1999). Allocating the rooms according to standard principles such as proportionality might lead to unattractive rents, e.g. paying more than one’s bid, being paid to take a room, etc. Brams and Kilgour (2001) developed a procedure which somehow avoids many problems arising with other allocation procedures.
- 20.
- 21.
Other examples stem from medicine where a restricted amount of medicine needs to be divided among sick people with possibly different chances of survival. Also every tax system somehow has to solve the same problem, as the cost, i.e. the total tax necessary to run the state, needs to be raised from the taxpayers whose claims are their different income levels.
- 22.
The analog to this in cake-cutting would be that if \(\mu_i(\cdot)=\mu_j(\cdot)\) for some \(i,j\in N\), then the value of the pieces they receive should be identical.
- 23.
This is a sort of analog to house monotonicity in apportionment theory which is of interest w.r.t. the Alabama paradox. See Balinski and Young (2001).
- 24.
It also has received other names in the literature such as Maimonides’ rule (Young, 1994b).
- 25.
In the surplus case, i.e. the resource line being beyond the claims point, the proportional rule would become most beneficial to the agents with higher claims.
- 26.
The proportional, uniform losses and uniform gains methods are parametric methods. The first two of them belong to an important subclass of parametric methods, namely equal sacrifice methods. These are of relevance in taxation, where the x i would represent taxable income and r the total aftertax income, making the difference x N – r the total tax raised. Given that, one can see that the three rules are important candidates for tax functions, with the proportional rule being both, progressive (average taxes do not decrease with income) and regressive (average taxes do not increase with income). Actually, the uniform gains method is the most progressive and the uniform losses method the most regressive among those rules satisfying fair ranking (see Moulin, 2003).
- 27.
This has an obvious connection to the Shapley value.
- 28.
The Talmudic method and the Random Priority method are both self-dual, however the Talmudic method is the only consistent extension of the contested garment method. See Moulin (2002).
- 29.
An important aspect of those rules that relates this topic to cooperative game theory is the fact that both of them have well known couterparts in cooperative game theory. Aumann and Maschler (1985) proved that the Talmudic method allocates the resources according to the nucleolus (of the appropriate games) and the Random-Priority method allocates the resources according to the Shapley value (of the appropriate games). Actually, it was via those counterparts that the Talmudic method has eventually been found. Thomson (2008) evaluates certain of the above rules by looking at two families of rules. Among other things, he looks at duality aspects of the rules and offers characterisation results for consistent rules.
- 30.
This could be seen as the counterpart of the move from cake cutting to the division of indivisible items as in the previous section.
- 31.
Maniquet (2003) provides a characterization of the Shapley value in queuing problems, combining classical fair division properties such as equal treatment of equals with properties specific to the scheduling model. He shows that the Shapley value solution stands out as a very equitable one among queuing problems.
- 32.
This is somehow based on the assumption of single-peaked preferences. See Thomson (2007b) for a discussion.
- 33.
Depending on what the cost function looks like, this suggests upper and lower bounds on cost shares. For c being convex, the stand-alone lower bound \(y_i\geq c(x_i)\) says that no agent can benefit from the presence of other users of the technology. In this sense we could think of other agents creating a negative externality. The opposite argumentation works for c being concave, creating a positive externality. Other bounds properties are discussed in the literature and used for characterization results. See Moulin (2002).
- 34.
A further change in the framework would require the individual demands to be binary, i.e. \(x_i\in\{0,1\}\). This moves us towards the model of cooperative games with transferable utility. The most famous method within this framework is the Shapley value (see Shapley, 1953). See also Moulin (2002, 2003) for a discussion.
- 35.
If, instead of a cost structure, one uses preferences on the graph, a different framework arises in which the aggregate satisfaction of the agents determines the distribution network. Hence, this closely links this area with social choice theory. See e.g. Darmann et al. (2009).
- 36.
In what follows we will slightly abuse the notation and define the cost of a spanning tree T as \(c(T)\equiv\sum_{(ij)\in T}c(ij)\).
- 37.
The important thing is, that the structure of the problem implies that the domain of the allocation rule will be smaller than the domain in a more general cost sharing problem. This actually helps in creating allocation rules satisfying certain desirable properties, something that is impossible for larger domains (see e.g. Young, 1994a).
- 38.
- 39.
Different models occur depending on the set \(\mathcal{R}\), i.e. what the preferences look like (e.g. quasi-linear preferences) and the exact specification of ω.
- 40.
No-envy has the clear counterpart of envy-freeness used in cake-cutting. Other concepts related to no-envy – but not discussed here – do exist, such as average no envy, strict no-envy, balanced envy, etc. (see Thomson, 2007b).
- 41.
Observe that no preference information is used for this property.
- 42.
For any two vectors \(y,y'\in\Re^{ln}_+\), we use yRy ′ for saying \(y_iR_iy'_i\ \textrm{for\ all}\ i\in N \).
- 43.
We can also create equity criteria for groups. This somehow is in the spirit of core properties from other areas. Many of the above properties can be translated into this framework, e.g. equal-division core of e, group envy-freeness, etc.
- 44.
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Klamler, C. (2010). Fair Division. In: Kilgour, D., Eden, C. (eds) Handbook of Group Decision and Negotiation. Advances in Group Decision and Negotiation, vol 4. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9097-3_12
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