AnyPrice share

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$\newcommand{\defeq}{:=}$ $\newcommand{\APS}{\operatorname{APS}}$ $\newcommand{\dAPS}{\operatorname{dAPS}}$ $\newcommand{\pAPS}{\operatorname{pAPS}}$ $\newcommand{\argmin}{\mathop{\mathrm{argmin}}}$ $\newcommand{\argmax}{\mathop{\mathrm{argmax}}}$ $\newcommand{\Scal}{\mathcal{S}}$ $\newcommand{\phat}{\widehat{p}}$ $\newcommand{\Ghat}{\widehat{G}}$ $\newcommand{\Chat}{\widehat{C}}$ $\newcommand{\Shat}{\widehat{S}}$ Let $([n], [m], V, w)$ be a fair division instance with indivisible items. Let $\Delta_m \defeq \{x \in \mathbb{R}_{\ge 0}^m: \sum_{i=1}^m x_i = 1\}$. For any $x \in \mathbb{R}^m$ and $S \subseteq [m]$, let $x(S) \defeq \sum_{j \in S} x_j$.

The AnyPrice share has two definitions: a price-based definition (a.k.a. primal definition) and a dual definition. These definitions are equivalent.

Primal Definition

Agent $i$'s AnyPrice Share is defined as \[ \APS_i \defeq \min_{p \in \mathbb{R}^m}\;\max_{S \subseteq [m]: p(S) ≤ w_ip([m])} v_i(S). \] Here $p$ is called the price vector.

A vector $p^* \in \mathbb{R}^m$ is called an optimal price vector if \[ p^* \in \argmin_{p \in \mathbb{R}^m}\;\max_{S \subseteq [m]: p(S) ≤ w_ip([m])} v_i(S). \]

Lemma 1: For agent $i$, let $G$ be the set of goods and $C$ be the set of chores. Then for some optimal price vector $\phat$, we have $\phat_j ≥ 0$ for all $j \in G$ and $\phat_j ≤ 0$ for all $j \in C$.

When all items are goods, by Lemma 1, we can assume without loss of generality that $p$ is non-negative and $p([m]) = 1$. Hence, \[ \APS_i = \min_{p \in \Delta_m}\;\max_{S \subseteq [m]: p(S) ≤ w_i} v_i(S). \]

When all items are chores, by Lemma 1, we can assume without loss of generality that $p$ is non-positive and $p([m]) = -1$. Hence, \[ -\APS_i = \max_{q \in \Delta_m}\;\min_{S \subseteq [m]: q(S) ≥ w_i} d_i(S). \]

Dual Definition

For any $z \in \mathbb{R}$, let $\Scal_z \defeq \{S \subseteq [m]: v_i(S) ≥ z\}$. Then $\APS_i$ is the largest value $z$ such that \[ \exists x \in \mathbb{R}_{\ge 0}^{\Scal_z}, \sum_{S \in \Scal_z} x_S = 1 \textrm{ and } \left(\forall j \in [m], \sum_{S \in \Scal_z: j \in S} x_S = w_i\right). \]

Proof of Lemma 1

Let $p^*$ be an optimal price vector. Let $\Ghat \defeq \{g \in G: p^*_g < 0\}$ and $\Chat \defeq \{c \in C: p^*_c > 0\}$.

The high-level idea is that if we change the price of $\Ghat \cup \Chat$ to 0, we get potentially better prices.

Define $\phat \in \mathbb{R}^m$ as \[ \phat_j \defeq \begin{cases} 0 & \textrm{ if } j \in \Ghat \cup \Chat \\ p^*_j & \textrm{ otherwise} \end{cases}, \] and let \[ \Shat \in \argmax_{S \subseteq [m]: \phat(S) ≤ w_i\phat([m])} v_i(S) .\] Since $\phat(\Shat \cup \Ghat \setminus \Chat) = \phat(\Shat)$ and $v_i(\Shat \cup \Ghat \setminus \Chat) ≥ v_i(\Shat)$, we can assume without loss of generality that $\Ghat \subseteq \Shat$ and $\Chat \cap \Shat = \emptyset$. \[ p^*(\Shat) - w_ip^*([m]) = (\phat(\Shat) - w_i\phat([m])) - (1-w_i)(-p^*(\Ghat)) - w_ip^*(\Chat) \le 0. \] Hence, \[ \max_{S \subseteq [m]: p^*(S) ≤ w_ip^*([m])} v_i(S) \ge v_i(\Shat) = \max_{S \subseteq [m]: \phat(S) ≤ w_i\phat([m])} v_i(S), \] so $\phat$ is also an optimal price vector.

Proof of Equivalence of Primal and Dual Definitions

Let $\pAPS_i$ and $\dAPS_i$ be agent $i$'s AnyPrice shares given by the primal and dual definitions, respectively. We will show that for any $z \in \mathbb{R}$, $\pAPS_i ≥ z$ iff $\dAPS_i ≥ z$. This would prove that $\pAPS_i = \dAPS_i$.

$\dAPS_i ≥ z$ iff the following linear program has a feasible solution: \[ \min_{x \in \mathbb{R}_{\ge 0}^{\Scal_z}} 0 \textrm{ where } \sum_{S \in \Scal_z} x_S = 1 \textrm{ and } \left(\forall j \in [m], \sum_{S \in \Scal_z: j \in S} x_S = w_i\right). \] Its dual is \[ \max_{p \in \mathbb{R}^m, r \in \mathbb{R}} r - w_ip([m]) \textrm{ where } p(S) ≥ r \textrm{ for all } S \in \Scal_z. \] The dual LP is feasible since $(0, 0)$ is a solution. Furthermore, if $(p, r)$ is feasible for the dual LP, then $(\alpha p, \alpha r)$ is also feasible, for any $\alpha ≥ 0$. Hence, by strong duality of LPs, the primal LP is feasible iff all feasible solutions to the dual have objective value at most 0.

For a given $p$, the optimal $r$ is $\min_{S \in \Scal_z} p(S)$. Hence, the dual LP is bounded iff for all $p \in \mathbb{R}^m$, \[ \min_{S \in \Scal_z} p(S) ≤ w_ip([m]). \] Furthermore, \begin{align} & \forall p \in \mathbb{R}^m, \min_{S \in \Scal_z} p(S) ≤ w_ip([m]) \\ &\iff \forall p \in \mathbb{R}^m, \exists S \subseteq [m] \textrm{ such that } p(S) ≤ w_ip([m]) \textrm{ and } v_i(S) ≥ z \\ &\iff \left(\min_{p \in \mathbb{R}^m} \max_{S \subseteq [m]: p(S) ≤ w_ip([m])} v_i(S)\right) ≥ z \\ &\iff \pAPS_i ≥ z. \end{align} Hence, $\dAPS_i ≥ z \iff \pAPS_i ≥ z$.

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Depth: 4
Number of transitive dependencies: 7

Transitive dependencies:

/sets-and-relations/countable-set
σ-algebra
Set function
Fair division
Optimization: Dual and Lagrangian
Dual of a linear program
Linear programming: strong duality (incomplete)