Disrupting Bipartite Trading Networks: Matching for Revenue Maximization

TL;DR

This work analyzes how a platform disrupting a buyer-seller network can maximize revenue when trades are driven by competitive equilibrium prices. It shows the general revenue-maximization problem is computationally hard, but identifies tractable special cases and develops a principled link between welfare gains and revenue, including a logarithmic revenue guarantee in markets with large welfare improvements. In homogeneous-goods markets, revenue maximization aligns with welfare, yielding exact or near-exact optimization and strong guarantees such as PRM=1. The results illuminate when platforms can extract substantial revenue while preserving social welfare and provide efficient algorithms for key market structures, with implications for platform design and welfare analysis in digital marketplaces.

Abstract

We model the role of an online platform disrupting a market with unit-demand buyers and unit-supply sellers. Each seller can transact with a subset of the buyers whom she already knows, as well as with any additional buyers to whom she is introduced by the platform. Given these constraints on trade, prices and transactions are induced by a competitive equilibrium. The platform's revenue is proportional to the total price of all trades between platform-introduced buyers and sellers. In general, we show that the platform's revenue-maximization problem is computationally intractable. We provide structural results for revenue-optimal matchings and isolate special cases in which the platform can efficiently compute them. Furthermore, in a market where the maximum increase in social welfare that the platform can create is $ΔW$, we prove that the platform can attain revenue $Ω(ΔW/\log(\min\{n,m\}))$, where $n$ and $m$ are the numbers of buyers and sellers, respectively. When $ΔW$ is large compared to welfare without the platform, this gives a polynomial-time algorithm that guarantees a logarithmic approximation of the optimal welfare as revenue. We also show that even when the platform optimizes for revenue, the social welfare is at least an $O(\log(\min\{n,m\}))$-approximation to the optimal welfare. Finally, we prove significantly stronger bounds for revenue and social welfare in homogeneous-goods markets.

Figures (5)

• Figure 1: A market where adding all platform edges is arbitrarily bad for revenue. There are no world edges. All nonzero valuations are indicated by dashed blue lines. Buyer $b_1$ has value $1$ for seller $s_1$, Buyer $b_i$ has value $i$ for sellers $s_{i-1}$ and $s_{i}$ for $i=2,\ldots,n$. All other valuations are zero.
• Figure 2: An example of a SWSH market. World edges are depicted in black, and the optimal set of platform edges is in blue. In revenue optimal matching, $s_1$ sells to $b_2$, $s_2$ sells to $b_1$, $s_3$ sells to $b_3$ and $s_4$ sells to $b_4$.
• Figure 3: The market used in the proof of Proposition \ref{['prop:tight_welfare_conversion']}. Buyers $B_k=\{b_1,b_2,...,b_k\}$ and dummy buyers $B^d_k=\{b^d_1,b^d_2,...,b^d_k\}$ are fully connected to sellers $S_k=\{s_1, s_2,...,s_k\}$ through world edges, denoted by solid black edges. Dummy sellers $S^d_k=\{s^d_1, s^d_2,...,s^d_k\}$ are not connected to any buyers through world edges. Buyer $b_i\in B_k$ values all sellers in $S_k$ at $1/i$. Buyer $b^d_i\in B^d_k$ values all sellers at $1$. All other valuations are zero. The maximum welfare the platform can add through platform edges is $\Delta W = H_k$, indicated in dashed blue edges. The optimal revenue is $1$, obtained by adding any non-empty subset of the blue edges.
• Figure 4: A two-buyer, two-seller market in which the $\mathrm{PRM}$ approaches $2$. The world edge is marked as a black solid edge. Buyer values are annotated next to edges. The welfare-optimal platform edge is in purple while the revenue-optimal set of platform edges is shown in blue.
• Figure :

Theorems & Definitions (117)

• Theorem 1: Informal Version of Theorems \ref{['thm:nphard']} and \ref{['thm:vc_apx_proof']}
• Theorem 2: Informal Version of Theorems \ref{['thm: poly_AMOS']} and \ref{['thm:poly_identity']}
• Theorem 3: Informal Version of Theorem \ref{['thm:gen_welfare_conversion']}
• Theorem 4: Informal Version of Theorem \ref{['thm:poa_upper_bound']}
• Theorem 5: Informal Version of Theorems \ref{['thm:hom_conversion']} and \ref{['thm:hom_poa']}
• Definition 2.1: Competitive Equilibrium on a Buyer-Seller Network
• Theorem 2.2: First Welfare Theorem
• Theorem 2.3: gul1999walrasian
• Definition 2.4: The Platform's Problem
• Example 2.1