# Fairness

**DOI:**https://doi.org/10.1007/978-3-319-32903-1_31-1

Given a vector \({{\mathbf x}}\in {\mathbb {R}}^{n}_{+}\), where *x*_{i} is the resource allocated to user *i*, *how fair* is it? Consider two allocations among three users: **x** = [1, 2, 3] and **y** = [1, 10, 100]. Among the large variety of choices for quantifying fairness, it is possible to have fairness values such as 0.33 or 0.86 for **x** and 0.01 or 0.41 for **y**: **x** is viewed as 33 times more fair than **y**, or just twice as fair as *y*. How many such “viewpoints” are there? What would disqualify a quantitative metric of fairness? Can they all be constructed from a set of simple statements taken as true for the sake of subsequent inference?

Fairness of **x** can be quantified through a fairness measure, which is a function *f* that maps **x** into a real number. These measures are sometimes referred to as diversity indices in statistics. Various fairness measures have been proposed throughout the years. These range from simple ones, e.g., the ratio between the smallest and the largest entries of **x**, to more sophisticated functions, e.g., Jain’s index and the entropy function. Some of these fairness measures map **x** to a normalized range between 0 and 1, where 0 denotes the minimum fairness, 1 denotes the maximum fairness (often corresponding to an **x** where all *x*_{i} are the same), and a larger value indicates more fairness. For example, min-max ratio (Marson and Gerla, 1982) is given by the maximum ratio of any two user’s resource allocation, while Jain’s index (Jain et al., 1984) computes a normalized square mean. How are these fairness measures are related? Is one measure “better” than any other? What other measures of fairness may be useful?

An alternative method that has gained attention in the networking research community since (Kelly et al., 1998; Mo and Walrand, 2000) is the optimization-theoretic approach of *α*-fairness and the associated utility maximization problem. Given a set of feasible allocations, a maximizer of the *α*-fair utility function satisfies the definition of *α*-fairness. Two well-known examples are as follows: a maximizer of the log utility function (*α* = 1) is proportionally fair, and a maximizer of the *α*-fair utility function as *α* →*∞* is max-min fair. More recently, *α*-fair utility functions have been connected to divergence measures (Uchida and Kurose, 2009). In Bonald and Massoulie (2001); Massoulie and Roberts (2002), the parameter *α* was viewed as a fairness measure in the sense that a fairer allocation is one that is the maximizer of an *α*-fair utility function with larger *α* – although the exact role of *α* in trading-off fairness and throughput can sometimes be surprising (Tang et al., 2006). While it is often believed that *α* →*∞* is more fair than *α* = 1, which is in turn more fair than *α* = 0, it remains unclear what it means to say, for example, that *α* = 3 is more fair than *α* = 2.

- (A)
A system-wide, global measure: (A1)

*f*(**x**) where*f*is our fairness function, or (A2)*f*(*U*_{1},*U*_{2}, …,*U*_{n}) where*U*_{i}is a utility function for each user that may depend on the entire**x**. - (B)
Individual, global measures: the set of {

*f*_{i}(**x**)} (each user*i*cares about the entire allocation**x**). - (C)
Individual, local measures: the set of \(\{ \tilde {f}_i(x_i)\}\) (user

*i*only cares about her resource via some function \(\tilde {f}_i\)).

### Renyi entropy.

- 1.
Symmetry.

- 2.
Continuity.

- 3.
Normalization.

- 4.
Additivity.

- 5.
Mean-value property.

### Lorenz Curve.

*P*

_{x}(

*y*) be the cumulative distribution of a resource allocation

**x**. Its Lorenz curve

*L*

_{x}, defined by

**x**and has used to characterize the social welfare distributions and relative income differences in economics. In 2001, an axiomatic characterization of Lorenz curve orderings is proposed based on a set of four axioms (Aaberge, 2001):

- 1.
Order. (The ordering is transitive and complete.)

- 2.
Dominance. (The ordering is Shur-concavity.)

- 3.
Continuity.

- 4.
Independence.

### Cooperative Economic Theories.

In economics, a number of theories have been developed to study the collective decisions of groups. Many of these theories have also been uniquely associated with sets of axioms Moulin (1991), including two well-known axiomatic constructions: the Nash bargaining solution in 1950 (Nash, 1950) and the Shapley value in 1953 (Shapley, 1953).

- 1.
Invariance to affine transformation.

- 2.
Pareto optimality.

- 3.
Independence of irrelevant alternatives (IIA).

- 4.
Symmetry.

- 1.
Pareto Optimality.

- 2.
Symmetry.

- 3.
Dummy.

- 4.
Additivity.

### Ultimatum Game.

Cake-Cutting and Fair Division: In ultimatum game (Nowak et al., 2000), player A divides a resource into two parts: one part for herself and the other for player B. Player B can then choose to accept the division or reject it, in which case neither player receives any resource. Without a prior knowledge about player B’s reaction, player A may divide anywhere between [0:5; 0:5] and [1; 0]. Running this game as a social experiment in different cultures have led to debates about the exact implications of the results on people’s perception on fairness: how fair does it take for player B’s perception, and player A’s guess of that, to accept player A’s division? This has also been contrasted with the perception of fairness in the related dictator game (Ben-Ner et al., 2004; Bolton et al., 1998), where player B has no option but to accept the division by player A.

A classic generalization of ultimatum game that has received increasing attention in the past decade is the cake-cutting problem. As reviewed in books (Barbanel, 2005; Brams and Taylor, 1996), the cake is a measure space, and each player uses a countably additive, non-atomic measure to evaluate different parts of the cake. Among the work studying the cake-cutting problem, e.g., (Brams and Taylor, 1995), the primary focus has been on two criteria: efficiency (Pareto optimality) and fairness (envy-freeness). Achievability results for four users or more are still challenging.

Fairness in cake-cutting and fair division is traditionally defined as envy-freeness. It is a binary summary based on each individual’s local evaluation: the allocation of cake is fair if no user wants to trade her piece with another piece. In (Brams et al., 2006), this restrictive viewpoint on fairness is expanded to include proportional allocation of the left-over piece after each user gets 1=nth of the cake (in her own evaluation). It is shown that Pareto optimality and proportional sense of fairness may not be compatible for three players or more.

### **Rawls’ Theory of Justice and Distributive Fairness.**

*principles*(axioms stated in English),

- 1.
“Each person is to have an equal right to the most extensive scheme of equal basic liberties compatible with a similar scheme of liberties for others.”

- 2.
“Social and economic inequalities should be arranged so that they are both (a) to the greatest benefit of the least advantaged persons, and (b) attached to offices and positions open to all under conditions of equality of opportunity.”

The first principle governs the distribution of liberties and has priority over the second principle. One may interpret it as a principle of distributive fairness in allocating limited resources among users. Next, the first part of Rawls second principle concerns with the distribution of opportunity, while the second part is the celebrated “difference principle”: an approach different from strict egalitarianism (since it is on the absolute value of the least advantaged user rather than the relative value) and utilitarianism (when narrowly interpreted where the utility function does not capture fairness).

### Normative Economics and Welfare Theory.

Rawls theory also has intricate interactions with normative economics, where many results are analytic in nature [68]. In addition to stochastic dominance, Arrow’s impossibility theorem, and the cooperation game theories of Nash and of Shapley, there are several major branches (Atkinson, 1970; Kolm, 1969; Sen, 1973). Another set is the ethical axioms of transfers. For example, the Pigou-Dalton principle states that inequality decreases via Robin Hood operation that does not reverse relative ranking. The principle of proportional transfers states that what the donor gives and the beneficiary receives should be proportional to their initial positions.

Bergson-Samuelson social welfare function (Dworkin, 1981; Bergson, 1938) *W*(*U*_{1}(**x**), …, *U*_{n}(**x**)) aims at enabling complete and consistent social welfare judgment on top of individual preference-based utility functions *U*_{i}. Kolm’s theory of fair allocation (Kolm, 1972) uses the criterion of equity as no-envy, and it is well- known that competitive equilibrium with equal budget is the only Pareto-efficient and envy-free allocation if preferences are sufficiently diverse and form a continuum (Varian, 1976).

### Sociology and Psychology: Inequality Indices.

Quantifying inequality/injustice/unfairness using individual, local measures has been pursued in sociology. For example, Jasso in 1980 (Jasso, 1980) advocated justice evaluation index as log of the ratio between actual allocation and just allocation. Allocation can be done either in quantity or in quality (in which case ranking quantifies the quality allocation). Many properties were derived in theory and experimented with in data about income distribution in different countries (Jasso, 2007). In particular, probability distribution of the index is induced by the probability distribution of the allocation. This index is derived based on two principles and three laws in the paper, including equal allocation maximizes justice, and aggregate justice is arithmetic mean of individual ones.

In Jasso (1999), two other injustice indexes, JI1 and JI2, were developed based on the above. One interesting feature is that JI1 differentiates between under-reward and over-reward as two types of injustice. Another useful feature is the decomposition of the total amount of perceived injustice into injustice due to scarcity and injustice due to inequality. They are further unified with Atkinson’s measure of inequality (Atkinson, 1970): 1 minus the ratio of geometric mean and arithmetic mean. At the heart of these indices is the approach of taking combinations of arithmetic and geometric means of an allocation to quantify the spread.

### Lan-Chiang fairness theory.

*α*-fair utility functions are continuous and strictly increasing in each entry of

**x**; thus, its maximization results in Pareto optimal resource allocations. On the other hand, scale-invariant fairness measures (ones that map

**x**to the same value as a normalized

**x**) are unaffected by the magnitude of

**x**, and an allocation that does not use all the resources can be as fair as one that does. Can the two approaches be unified? To address the above questions, Lan and Chiang develop an axiomatic approach to fairness measures (Lan et al., 2010). It is shown that a set of five axioms, each of which simple and intuitive, thus accepted as true for the sake of subsequent inference, can lead to a useful family of fairness measures, i.e.:

- 1.
Continuity.

- 2.
Homogeneity.

- 3.
Saturation.

- 4.
Partition.

- 5.
Starvation.

*g*: any increasing and continuous functions that lead to a well-defined “mean” function (i.e., from any Kolmogorov-Nagumo function Kolmogoroff 1930; Aczél 1948). Using power functions with exponent

*β*as the generator function, a unique family of fairness measures are derived, i.e.:

*F*

_{β,λ}(

**x**) for 1∕

*λ*= 0 and

*β*< 1; inverse of

*p*-norm is another subclass of

*F*

_{β,λ}(

**x**) for

*β*≤−1; and

*α*-utility is obtained for 1∕

*λ*=

*β*∕(1 −

*β*) and

*β*> 0. New fairness measures are revealed corresponding to other ranges of

*β*and

*λ*. The degree of homogeneity 1∕

*λ*determines how

*F*

_{β,λ}(

**x**) scales as throughput increases, while

*β*provides trade-off between “resolution” and “strictness” of the fairness measure. The unification is illustrated in Fig. 1, including all known fairness measures that are global (i.e., mapping a given allocation vector to a single scalar) and decomposable (i.e., subsystems fairness values can be somehow collectively mapped into the overall systems fairness value).

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