Barter Exchange with Shared Item Valuations

TL;DR

A randomized algorithm is developed that achieves optimal utility in expectation and where, i) for any agent, with probability 1 their received value is at least their given value minus v^* where v^* is said agent's most valuable owned and wished-for item, and ii) each agent's given and received values are equal in expectation.

Abstract

In barter exchanges agents enter seeking to swap their items for other items on their wishlist. We consider a centralized barter exchange with a set of agents and items where each item has a positive value. The goal is to compute a (re)allocation of items maximizing the agents' collective utility subject to each agent's total received value being comparable to their total given value. Many such centralized barter exchanges exist and serve crucial roles; e.g., kidney exchange programs, which are often formulated as variants of directed cycle packing. We show finding a reallocation where each agent's total given and total received values are equal is NP-hard. On the other hand, we develop a randomized algorithm that achieves optimal utility in expectation and where, i) for any agent, with probability 1 their received value is at least their given value minus $v^*$ where $v^*$ is said agent's most valuable owned and wished-for item, and ii) each agent's given and received values are equal in expectation.

Figures (3)

• Figure 1: A VBM instance for a $\mathsf{BarterSV}$ instance with $\mathcal{N} = \{1, 2\}$, $\mathcal{I} = \{ a, b, c, d \}$, $H_1 = \{ a, b\}$, $W_1 = \{ c, d\}$, $H_2 = \{ c\}$, $W_2 = \{ a, d \}$, $H_3 = \{ d\}$, $W_3 = \{b, c\}$, $v_a = 100$, $v_b = v_c = v_d = 1$, and $w_e=1$ for all $e \in E$
• Figure 2: An example CCC $\mathcal{P} = \left< \ell_{1,a} \leadsto r_{2,a}, r_{2,d} \leadsto r_{1,d}\right>$ based on the VBM from \ref{['fig:vbm']}. All edges and path endpoint vertices, i.e., $\ell_{1,a}$, $r_{2,a}$, $r_{1, d}$, and $r_{2,d}$, must be floating. The vertex $\ell_{3,d}$. The rounding step proceeds in one of two ways denoted by orange and green text, chosen at random. With the probability $\alpha / (\alpha + \beta)$, displayed in orange, the orange modifications to $x_e$'s take place. Otherwise, with probability $\beta / (\alpha + \beta)$, the green modifications take place. In the orange rounding event agent 1 gives away an additional $v_a \alpha / v_a = \alpha$ value and gains an additional $v_d \alpha / v_d = \alpha$ value therefore $D_i$ for agent 1 does not change after this rounding step. It is easy to see the same occurs for both agents, in either -- orange or green -- rounding event.
• Figure 3: The bipartite graph corresponding to the $\mathsf{BarterSV}$ instance in \ref{['ex:worstcase']}is pictured. Blue and orange vertices correspond to agents $1$ and $2$, respectively. The optimal LP solution is $x = [ 0.5,0.5,1,1]$.

Theorems & Definitions (28)

• Definition 1
• Lemma 1.1
• Corollary 1
• Lemma 1.2
• Remark 1
• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 5.1
• Lemma 5.2