Platform Equilibrium: Analayzing Social Welfare in Online Market Places

TL;DR

This work develops a theoretical framework for Platform Equilibrium in markets with unit-demand buyers and unit-supply sellers, where sellers can join a platform to access more buyers at a transaction fee α. Prices clear through a Walrasian competitive equilibrium, and the platform’s revenue motive interacts with seller participation to shape welfare. In homogeneous-goods markets, a pure equilibrium exists and is computable in polynomial time, while unregulated platforms can cause welfare to degrade to Θ(log(min(n,m))) of the optimum; imposing a cap α yields a tight PoA of (2−α)/(1−α) and a nontrivial welfare guarantee for buyers and sellers, with extensions to multiple platforms, production costs, and additive valuations. The results rely on a lattice of competitive prices, opportunity-path insights, and a Bayesian reduction to handle mixed equilibria, yielding practical welfare implications for platform regulation in digital marketplaces.

Abstract

We introduce the theoretical study of a Platform Equilibrium in a market with unit-demand buyers and unit-supply sellers. Each seller can join a platform and transact with any buyer or remain off-platform and transact with a subset of buyers whom she knows. Given the constraints on trade, prices form a competitive equilibrium and clears the market. The platform charges a transaction fee to all on-platform sellers, in the form of a fraction of on-platform sellers' price. The platform chooses the fraction to maximize revenue. A Platform Equilibrium is a Nash equilibrium of the game where each seller decides whether or not to join the platform, balancing the effect of a larger pool of buyers to trade with, against the imposition of a transaction fee. Our main insights are: (i) In homogeneous-goods markets, pure equilibria always exist and can be found by a polynomial-time algorithm; (ii) When the platform is unregulated, the resulting Platform Equilibrium guarantees a tight $Θ(log(min(m, n)))$-approximation of the optimal welfare in homogeneous-goods markets, where $n$ and $m$ are the number of buyers and sellers respectively; (iii) Even light regulation helps: when the platform's fee is capped at $α\in[0,1)$, the price of anarchy is 2-$α$/1-$α$ for general markets. For example, if the platform takes 30 percent of the seller's revenue, a rather high fee, our analysis implies the welfare in a Platform Equilibrium is still a 0.412-fraction of the optimal welfare. Our main results extend to markets with multiple platforms, beyond unit-demand buyers, as well as to sellers with production costs.

Figures (6)

• Figure 1: An example with no Platform Equilibrium in pure strategies at $\alpha=\frac{1}{2}$. Black solid lines are direct links, as captured by $N(i)$ for buyer $i$, and blue dotted lines indicate missing links. Buyer values are annotated adjacent to each edge, and all any value that is omitted is zero.
• Figure 2: A homogeneous goods market with price of anarchy $H_n$. There are no off-platform edges. Buyer $b_1$ has value $n+\epsilon$ for any seller, and buyer $b_i$, $i=2,\ldots,n$, has value $n/i$ for any seller.
• Figure 3: An example with a tight Bound for PoA for an $\alpha$ transaction fee, for any $\alpha \in [0,1)$. Black solid lines are direct links, as captured by $N(i)$ for buyer $i$, and blue dotted lines indicate missing links. Buyer values are annotated adjacent to each edge, and all any value that is omitted is zero.
• Figure 4: A market with infinite price of anarchy for any $\alpha\in[0,1]$. Black solid lines are direct links, as captured by $N(i)$ for buyer $i$, and blue dotted lines indicate missing links. Buyer values are annotated adjacent to each edge, and all any value that is omitted is zero.
• Figure 5: An $n$-buyer-$n$-seller general valuation market where $PoA=n$. Black solid lines are direct links, as captured by $N(i)$ for buyer $i$, and blue dotted lines indicate missing links. Buyer values are annotated adjacent to each edge. $x$ is a large constant. For the first seller $v_{11}=\frac{nx}{(n-1)(x+n)}+\frac{n}{x+n}, v_{21}=\frac{nx}{n-1}$. For sellers $j=2,...,n-1$, $v_{j+1,j}=\frac{n}{n-1}x, v_{jj}=\frac{n^2x}{(n-1)(x+n)}+(i-1)\epsilon$. For the last seller $v_{1n}=1,v_{jn}=x$ for $j=2,3,...,n$ All other values are zero.

Theorems & Definitions (80)

• Theorem 1: Informal Version of Proposition \ref{['prop:no_pure']} and Theorems \ref{['thm:PE_pure']}
• Theorem 2: Informal Version of Theorem \ref{['thm:poa_upper_bound_homo']} and Theorems \ref{['thm:poa_lower_bound_homo']}
• Theorem 3: Informal Version of Theorem \ref{['thm:pure_poa']}, \ref{['thm:mixed_poa']} and \ref{['thm:poa_tight']}
• Definition 2.1: Competitive Equilibrium
• Theorem 2.2: First Welfare Theorem
• Theorem 2.3: Second Welfare Theorem gul1999walrasian
• Theorem 2.4: Lattice structure for competitive prices gul1999walrasian
• Theorem 2.5: Characterization of competitive prices gul1999walrasian
• Definition 2.6: Pure Platform Equilibrium
• Definition 2.7: Mixed Platform Equilibrium