Connected Trading Cycles

TL;DR

The paper investigates incentive-compatible diffusion in housing markets on social networks, where invitation dynamics complicate stability and efficiency. It proves that no mechanism can be Pareto optimal, IC, and IR simultaneously, and introduces stable-cc and optimal-cc as tight IC-compatible benchmarks defined via complete components. Three mechanisms—SWN, LS, and CTC—are proposed, with SWN and LS achieving IC, IR, and stable-cc, while CTC attains the strongest guarantees by also achieving optimal-cc. The results delineate clear theoretical boundaries for networked house allocations and offer mechanisms capable of leveraging invitations without sacrificing core economic properties. Potential extensions include handling practical network constraints and developing distributed implementations for these mechanisms.

Abstract

Incentivizing the existing participants to invite new participants to join an auction, matching or cooperative game have been extensively studied recently. One common challenge to design such incentive in these games is that the invitees and inviters are competitors. To have such an incentive, we normally have to sacrifice some of the traditional properties. Especially, in a housing market (one kind of one-sided matching), we cannot maintain the traditional stability and optimality. The previous studies proposed some new matching mechanisms to have the invitation incentive (part of the incentive compatibility), but did not have any guarantee on stability and optimality. In this paper, we propose new notions of stability and optimality which are achievable with incentive compatibility. We weaken stability and optimality on a special structure (complete components) on networks. We first prove that the weakened notions are the best we can achieve with incentive compatibility. Then, we propose three mechanisms (Swap With Neighbors, Leave and Share, and Connected Trading Cycles) to satisfy the desirable properties. Connected Trading Cycles is the first mechanism to satisfy the best stability and optimality compatible with incentive compatibility.

Figures (10)

• Figure 1: A social network example. Preferences are $h_3 \succ_1 h_2 \succ_1 h_1, \ \ \ h_1 \succ_2 h_2 \succ_2 h_3, \ \ \ h_1 \succ_3 h_3 \succ_3 h_2$.
• Figure 2: A counterexample for the coexistence of optimal-wcc, IR, and IC. Preferences are $h_4 \succ_1 h_2 \succ_1 h_1, \ \ \ h_3 \succ_2 h_2, \ \ \ h_1 \succ_3 h_3, \ \ \ h_1 \succ_4 h_4$. Agents 1,2 are initial players in the matching. The red solid arrows represent favorite pointing. The red dashed arrows represent the second favorite pointing. The dashed agents are unqualified, so they are allocated their endowments.
• Figure 3: Preferences are $h_3 \succ_1 h_2 \succ_1 h_1, \ \ \ h_4 \succ_2 h_1 \succ_2 h_2, \ \ \ h_1 \succ_3 h_4 \succ_3 h_3, \ \ \ h_2 \succ_4 h_3 \succ_4 h_4$.
• Figure 4: The properties of our three mechanisms. LS and SWN are stable-cc, IC and IR. CTC reaches the boundaries.
• Figure 5: Preferences are $h_4 \succ_1 h_1 \succ_1 \cdots, \ \ \ h_3 \succ_2 h_2 \succ_2 \cdots, \ \ \ h_2 \succ_3 h_3 \succ_3 \cdots, \ \ \ h_1 \succ_4 h_4 \succ_4 \cdots$.

Theorems & Definitions (37)

• definition 1
• definition 2
• definition 3: Individual Rationality (IR)
• definition 4: Incentive Compatibility (IC)
• definition 5: Pareto Optimality (PO)
• definition 6: Stability
• theorem 1: Impossibility for PO, IC and IR
• theorem 2: Impossibility for stability and IC
• definition 7: Complete Component
• definition 8: Optimality under Complete Components (Optimal-cc)