Barter Exchange with Bounded Trading Cycles

Abstract

Consider a barter exchange problem over a finite set of agents, where each agent owns an item and is also associated with a (privately known) wish list of items belonging to the other agents. An outcome of the problem is a (re)allocation of the items to the agents such that each agent either keeps her own item or receives an item from her (reported) wish list, subject to the constraint that the length of the trading cycles induced by the allocation is up-bounded by a prespecified length bound k. The utility of an agent from an allocation is 1 if she receives an item from her (true) wish list and 0 if she keeps her own item (the agent incurs a large dis-utility if she receives an item that is neither hers nor belongs to her wish list). In this paper, we investigate the aforementioned barter exchange problem from the perspective of mechanism design without money, aiming for truthful (and individually rational) mechanisms whose objective is to maximize the social welfare. As the construction of a social welfare maximizing allocation is computationally intractable for length bounds k \geq 3, this paper focuses on (computationally efficient) truthful mechanisms that approximate the (combinatorially) optimal social welfare.We also study a more general version of the barter exchange problem, where the utility of an agent from participating in a trading cycle of length 2 \leq \ell \leq k is lambda(\ell), where λis a general (monotonically non-increasing) length function. Our results include upper and lower bounds on the guaranteed approximation ratio, expressed in terms of the length bound k and the length function λ. On the technical side, our main contribution is an algorithmic tool that can be viewed as a truthful version of the local search paradigm. As it turns out, this tool can be applied to more general (bounded size) coalition formation problems.

Figures (8)

• Figure 1: The $k$-cycle graph depicted in Figure \ref{['figure:example:cycle-graph-marked']} (where the agents in each node are listed according to their cyclic order) corresponds to the wish list vector represented by the digraph depicted in Figure \ref{['figure:example:cycle-digraph-marked']} under the length bound $k = 4$. The nodes marked in Figure \ref{['figure:example:cycle-graph-marked']} form an IS that corresponds to the exchange (i.e., set of disjoint trading cycles) marked in Figure \ref{['figure:example:cycle-digraph-marked']}. This IS turns out to be an optimal solution for the $k$-maxCGIS problem, hence the corresponding exchange is an optimal solution for the $k$-BE problem.
• Figure 2: The expansion improvement rule $r_{E}$ (\ref{['figure:expansion1']}--\ref{['figure:expansion3']}) and the all-for-$q$ improvement rule $r_{q}$ (\ref{['figure:all-for-q1']}--\ref{['figure:all-for-q3']}).
• Figure 3: The $k$-cycle graph $G_{\text{bad}} = (V, E)$ whose nodes are depicted by the $q + 1$ blue (solid) rectangles and the $2 q + 3$ red (dashed) rectangles, defined over a set of $n = 6 q + 9$ agents depicted by the black diamond shapes, so that the diamond shape corresponding to agent $i \in [n]$ is contained in the rectangle corresponding to node $v \in V$ if (and only if) $i \in \alpha(v)$.
• Figure 4: The agent digraph of an $(h, v)$-comb instance, where $2 \leq h < v \leq k$ and $\lambda(v) < \lambda(h)$, that consists of $h$ 'black agents' and $h (v - 1)$ 'white agents'. The (directed) cycles in the agent digraph are depicted by the stretched gray ellipses. Any truthful mechanism must output the exchange $\pi_{H}$ defined so that $\mathcal{C}_{\pi_{H}} = \{ c_{H} \}$, where $c_{H}$ is the 'horizontal cycle' that consists of the black agents; the social welfare of this exchange is $\mathrm{SW}(\pi_{H}) = h \cdot \lambda(h)$. In contrast, the social welfare of the exchange $\pi$, defined so that $\mathcal{C}_{\pi}$ includes all 'vertical cycles', is $\mathrm{SW}(\pi) = h \cdot v \cdot \lambda(v)$. Maximizing over $h$ and $v$, we obtain a lower bound of $\max_{2 \leq h < v \leq k \, : \, \lambda(h) > \lambda(v)} \left\{ \tfrac{v \cdot \lambda(v)}{\lambda(h)} \right\}$ on the approximation ratio.
• Figure 5: The agent digraph of a double $(h, v)$-comb instance that consists of $2 h - 1$ 'black agents' and $(2 h - 1) (v - 1)$ 'white agents'. The (directed) cycles in the agent digraph are depicted by the stretched gray ellipses.

Theorems & Definitions (61)

• Example
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• proof
• Definition : improvement rule, loyal, INPA, efficient
• Definition : local search algorithm