Fair division

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$\newcommand{\Fcal}{\mathcal{F}}$ $\newcommand{\Ical}{\mathcal{I}}$ In the fair division problem, there is a finite set $N$ of agents and a set $M$ of items. We need to distribute the items among the agents fairly.

Formally, we are given as input a fair division instance $\Ical = (N, M, V, w)$. Here $w = (w_i)_{i \in N}$ is a collection of positive numbers that sum to 1, and $V = (v_i)_{i \in N}$ is a collection of functions, where $v_i: 2^M → \mathbb{R}$ and $v_i(∅) = 0$ for each $i$. $v_i$ is called agent $i$'s valuation function, and $w_i$ is called agent $i$'s entitlement.

Our task is to find a fair allocation. An allocation $A = (A_i)_{i \in N}$ is a collection of pairwise-disjoint subsets of $M$ such that $\bigcup_{i=1}^n A_i = M$. The set $A_i$ is called agent $i$'s bundle in $A$. Multiple definitions (a.k.a. notions) of fairness have been proposed.

For any non-negative integer $t$, we denote $\{1, \ldots, t\}$ by $[t]$. Without loss of generality, we assume $N = [n]$.

Settings and special cases

Indivisible vs divisible vs cake-cutting:
- In the indivisible setting, also called discrete fair division, $M$ is a finite set.
- In the cake cutting setting, each valuation function $v_i$ is a function over a sigma-algebra $(M, \Fcal)$. Furthermore, for any set $S \in \Fcal$, both of the following hold:
  - for any $\alpha \in [0, 1]$, there exists a set $T$ such that $v_i(T) = \alpha v_i(S)$.
  - for any positive numbers $\beta_1, \beta_2, \ldots, \beta_k$, there exists a partition $(P_1, \ldots, P_k)$ of $M$ such that $v_i(P_j)/\beta_j$ has the same value for all $j \in [k]$.
- The divisible setting is a special case of cake cutting. Here we have $m$ homogenous goods that can be split into pieces. Here a bundle is represented as a vector $x \in [0,1]^m$, where $x_j$ is the fraction of good $j$ in the bundle. Each agent $i$'s valuation function $v_i: [0,1]^m \to \mathbb{R}$ tells us how much they value each bundle $x \in [0,1]^m$. An allocation is a sequence of $n$ bundles that sum to $\{1\}^m$.
- The contiguous cake-cutting setting is a special case of cake-cutting where $M = [0, 1]$ and each bundle $A_i$ must be an interval. Furthermore, for any interval $[a, b]$, both of the following hold:
  - for any $\alpha \in [0, 1]$, there exist $c, d \in [a, b]$ such that $v_i([a, c]) = v_i([d, b]) = \alpha v_i([a, b])$,
  - for any positive numbers $\beta_1, \beta_2, \ldots, \beta_k$, there exist numbers $a = c_0 < c_1 < c_2 < \ldots < c_{k-1} < c_k = b$, such that $v_i([c_{j-1}, c_j])/\beta_j$ has the same value for all $j \in [k]$.
Goods vs chores vs mixed manna:
- $G \subseteq M$ is called a set of goods for agent $i$ iff for all $S \subseteq G$ and $T \subseteq M$, we have $v_i(S \mid T) \ge 0$.
- $C \subseteq M$ is called a set of chores for agent $i$ iff for all $S \subseteq C$ and $T \subseteq M$, we have $v_i(S \mid T) \le 0$. For any set $X \subseteq M$, we define $d_i(X) := -v_i(X)$.
- When $M$ may not be a set of goods or a set of chores, we call $M$ a set of mixed manna.
- $M$ is said to be doubly monotone for agent $i$ if $M = G \cup C$, where $G$ is a set of goods for $i$ and $C$ is a set of chores for $i$.
Equal vs unequal entitlements: When $w_i = 1/n$ for all $i \in [n]$, we say that the agents have equal entitlements. This is such an important special case that we will assume equal entitlements unless specified otherwise. The case where entitlements are unequal is called weighted fair division.
Identical vs non-identical valuations: If $v_1 = v_2 = \ldots = v_n$, we say that the agents have identical valuations. In this case, we denote agent $i$'s valuation function by $v$ instead of $v_i$.
Complete vs partial allocations: A partial allocation is an $n$-tuple $A = (A_1, \ldots, A_n)$ of disjoint subsets of $M$ (i.e., their union may not be $M$). Sometimes, we are allowed to output partial allocations, but there may be other constraints to ensure that too many items don't remain unallocated. In the goods setting, the set of unallocated items is called the charity bundle.

Other special cases are obtained by constraining the set of valuation functions or the number of agents.

Fairness notions

A fairness notion $F$ is a function that takes as input a fair division instance $\Ical$, a (partial) allocation $A$, and an agent $i$, and outputs either true or false. When $F(\Ical, A, i)$ is true, we say that allocation $A$ is $F$-fair to agent $i$, or that agent $i$ is $F$-satisfied by allocation $A$. Allocation $A$ is $F$-fair if it is $F$-fair to all agents.

A notion $F$ of fairness is said to be feasible if for every fair division instance, there exists an $F$-fair allocation.

We say that a notion $F_1$ of fairness implies another notion $F_2$ of fairness if every $F_1$-fair allocation is also an $F_2$-fair allocation.

Non-real valuations

We usually let the co-domain of valuation functions be $\mathbb{R}$, but sometimes we may assume that all valuation functions have a common co-domain $U$, where $U$ is a totally-ordered vector space. A common choice is $U = \mathbb{R}^k$, for some $k \in \mathbb{N}$, where the total order is given by lexicographic ordering (this is also how tuples are compared in Python). Unless specified otherwise, assume $U = \mathbb{R}$.

Fair division

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Settings and special cases

Fairness notions

Non-real valuations

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