TripleWin: Fixed-Point Equilibrium Pricing for Data-Model Coupled Markets

TL;DR

TripleWin introduces a data–model coupled market that prices dataset–model use and buyer-specific model access simultaneously via bidirectional quotations. By proving the joint operator is a standard interference function ($SIF$), the paper guarantees a unique fixed point and global convergence, enabling a simple, distributed implementation. The approach leverages Shapley-based data valuation and an effective revenue aggregation to ensure fair cost distribution and robust transaction success across multi-buyer and multi-seller settings. Empirically, TripleWin demonstrates efficient convergence, fairness alignment with data contributions, and resilience to demand and cost variability, offering a principled alternative to broker-centric pricing in data-model marketplaces.

Abstract

The rise of the machine learning (ML) model economy has intertwined markets for training datasets and pre-trained models. However, most pricing approaches still separate data and model transactions or rely on broker-centric pipelines that favor one side. Recent studies of data markets with externalities capture buyer interactions but do not yield a simultaneous and symmetric mechanism across data sellers, model producers, and model buyers. We propose a unified data-model coupled market that treats dataset and model trading as a single system. A supply-side mapping transforms dataset payments into buyer-visible model quotations, while a demand-side mapping propagates buyer prices back to datasets through Shapley-based allocation. Together, they form a closed loop that links four interactions: supply-demand propagation in both directions and mutual coupling among buyers and among sellers. We prove that the joint operator is a standard interference function (SIF), guaranteeing existence, uniqueness, and global convergence of equilibrium prices. Experiments demonstrate efficient convergence and improved fairness compared with broker-centric and one-sided baselines. The code is available on https://github.com/HongrunRen1109/Triple-Win-Pricing.

Figures (7)

• Figure 1: Workflow of the data–model coupled market cleared by TripleWin. (1) Buyers submit bids prices. (2) Producers select datasets and request licenses. (3) Data sellers post asks prices. (4) Models are trained on licensed data. (5) Trained models are delivered to buyers. TripleWin clears dataset licensing and model sales simultaneously by aligning buyer-side and data-side quotations.
• Figure 2: Bidirectional pricing in the coupled data--model market. Left: data sellers post $p_{D\to M}$ and buyers bid $p_{B\to M}$; the market maps these to a supply quote$\mathcal{Q}_{B\to M}$ and a demand quote$\mathcal{Q}_{D\to M}$, moderated by producer margins $\delta_M$ and overheads $(\kappa_D,\kappa_M)$ (dashed arrows indicate internal cost transmission). At equilibrium, posted prices match quotes ($p=v$). Right: price signals propagate downstream (data $\rightarrow$ model $\rightarrow$ buyer) and upstream (buyer $\rightarrow$ model $\rightarrow$ data).
• Figure 3: Cobweb diagram for the TripleWin fixed-point update in the $(x,y)$ plane with $x=p_{B\to M}$ (buyer price) and $y=p_{D\to M}$ (seller price). The solid curve is the data-side quotation $y=\mathcal{Q}_{D\to M}(x)$. The dashed curve is the buyer-side quotation drawn as a locus in the plane, $x=\mathcal{Q}_{B\to M}(y)$. Starting from an initial $x_0$ on the horizontal axis, the iteration alternates the vertical update $y_{t+1}=\mathcal{Q}_{D\to M}(x_t)$ and the horizontal update $x_{t+1}=\mathcal{Q}_{B\to M}(y_{t+1})$, producing the staircase arrows that converge to $(x^\star,y^\star)$, where $x^\star=\mathcal{Q}_{B\to M}(y^\star)$ and $y^\star=\mathcal{Q}_{D\to M}(x^\star)$.
• Figure 4: Fairness alignment between Shapley contributions and realized data–revenue shares under different total buyer weights $\rho\in\{0.4,0.6,0.8,0.99\}$. Each panel plots the realized revenue share $p_{D_i\to M_j}/\sum_i p_{D_i\to M_j}$ against the corresponding Shapley value $\mathrm{SV}_{i\mid j}$ for all datasets and models. Legends report mean per–model Spearman correlations. TripleWin (red triangles) aligns almost perfectly with Shapley contributions as $\rho$ increases, while one–sided baselines remain weakly correlated.
• Figure 5: Evolution of buyer surplus and seller profit across propagation stages. Each subpanel shows normalized distributions for the four methods at four stages: initial market, one market price propagation, five market price propagation, and convergence. Grey boxes correspond to buyer surplus and hatched boxes to seller profit. Only TripleWin achieves balanced and stable incidence for both sides at convergence, consistent with a unique tri–sided fixed point.

Theorems & Definitions (22)

• Definition 1: SIF
• Theorem 1: Main Result
• proof
• Theorem 2: Existence of Fixed Point Equilibrium Price
• proof
• Corollary 1: Existence of a fixed point
• Corollary 2: Uniqueness of the fixed point
• Corollary 3: Optimal pricing
• Proposition 1: Monotonicity in the offsets
• proof