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Bipartiteness in Progressive Second-Price Multi-Auction Networks with Perfect Substitute

Jordana Blazek, Frederick C. Harris

TL;DR

This work develops a graph-theoretic, projection-based framework to analyze Progressive Second Price (PSP) auctions in decentralized, bipartite buyer–seller networks. By formalizing primary and expanded influence sets and introducing saturation shells, the authors derive conditions under which local auctions propagate stable, monotone bid updates and converge to coherent equilibria without centralized information. The methodology combines projection operators, partial ordering of bids, and intra-round resolution to capture asynchronous dynamics and phase transitions across multi-auction markets. Simulations demonstrate price synchronization and robust equilibrium formation as buyer overlap across auctions increases, highlighting practical potential for decentralized resource allocation in networks with limited information sharing.

Abstract

We consider a bipartite network of buyers and sellers, where the sellers run locally independent Progressive Second-Price (PSP) auctions, and buyers may participate in multiple auctions, forming a multi-auction market with perfect substitute. The paper develops a projection-based influence framework for decentralized PSP auctions. We formalize primary and expanded influence sets using projections on the active bid index set and show how partial orders on bid prices govern allocation, market shifts, and the emergence of saturated one-hop shells. Our results highlight the robustness of PSP auctions in decentralized environments by introducing saturated components and a structured framework for phase transitions in multi-auction dynamics. This structure ensures deterministic coverage of the strategy space, enabling stable and truthful embedding in the larger game. We further model intra-round dynamics using an index to capture coordinated asynchronous seller updates coupled through buyers' joint constraints. Together, these constructions explain how local interactions propagate across auctions and gives premise for coherent equilibria--without requiring global information or centralized control.

Bipartiteness in Progressive Second-Price Multi-Auction Networks with Perfect Substitute

TL;DR

This work develops a graph-theoretic, projection-based framework to analyze Progressive Second Price (PSP) auctions in decentralized, bipartite buyer–seller networks. By formalizing primary and expanded influence sets and introducing saturation shells, the authors derive conditions under which local auctions propagate stable, monotone bid updates and converge to coherent equilibria without centralized information. The methodology combines projection operators, partial ordering of bids, and intra-round resolution to capture asynchronous dynamics and phase transitions across multi-auction markets. Simulations demonstrate price synchronization and robust equilibrium formation as buyer overlap across auctions increases, highlighting practical potential for decentralized resource allocation in networks with limited information sharing.

Abstract

We consider a bipartite network of buyers and sellers, where the sellers run locally independent Progressive Second-Price (PSP) auctions, and buyers may participate in multiple auctions, forming a multi-auction market with perfect substitute. The paper develops a projection-based influence framework for decentralized PSP auctions. We formalize primary and expanded influence sets using projections on the active bid index set and show how partial orders on bid prices govern allocation, market shifts, and the emergence of saturated one-hop shells. Our results highlight the robustness of PSP auctions in decentralized environments by introducing saturated components and a structured framework for phase transitions in multi-auction dynamics. This structure ensures deterministic coverage of the strategy space, enabling stable and truthful embedding in the larger game. We further model intra-round dynamics using an index to capture coordinated asynchronous seller updates coupled through buyers' joint constraints. Together, these constructions explain how local interactions propagate across auctions and gives premise for coherent equilibria--without requiring global information or centralized control.

Paper Structure

This paper contains 22 sections, 4 theorems, 58 equations, 8 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background and Related Work
  3. The PSP Auction Mechanism
  4. Bounded Participation
  5. Residual Quantity and Allocation
  6. Exclusion--Compensation
  7. Valuation and Utility
  8. Influence Sets
  9. Primary (Direct) Influence Sets
  10. Expanded (Indirect) Influence Sets
  11. Projection Domains and Influence Operations
  12. Projection-Based Influence Propagation
  13. Partial Ordering and Market Shifts
  14. Influence Shells and Local Saturation
  15. Ordering and Influence Propagation
  16. ...and 7 more sections

Key Result

Lemma 3

Let the market be at time $t$ with seller $j$ and its saturated primary influence set $\Sigma := \Lambda^{(1)}_{\mathcal{L}}(j,t)$. Pick any neighbor $k\in\Sigma$ and two buyers i.e., $i$ bids on $j$, $k$ is another seller reached from $i$, and $\ell$ is a buyer that bridges further to $k$ but not to $j$. If the shell $\Lambda^{(1)}_{\mathcal{L}}(j,t)$ is saturated--no profitable deviation exists

Figures (8)

  • Figure 1: 3D matrix view: rows (buyers), columns (sellers), and $z$ encodes price tiers. The colored surface shows the buyer price; filled markers are active bids; open circles show marginal winners. The right panel shows a transition where a new high-tier participant appears at $j{=}1$, a demand shortfall removes a low cell, and a reconfiguration shifts activity at $j{=}2$.
  • Figure 2: Adjacency structure showing market connectivity between buyers and sellers.
  • Figure 3: Buyer 0 valuation curve and marginal diagnostics.
  • Figure 4: Adjacency and market connectivity for the 8 $\times$ 2 experiment. Connectivity is set at 50%.
  • Figure 5: Single buyer--seller utility surface for buyer 6 at seller 0. The surface plots $u_i(z_i^j,w_i^j)=\theta_i(z_i^j)-z_i^j w_i^j$ over quantity $z_i^j$ and unit price $w_i^j$, holding the opposing bids fixed at the snapshot.
  • ...and 3 more figures

Theorems & Definitions (9)

  • Definition 1: Bounded participation rule
  • Definition 2: Saturated Influence Shell
  • Lemma 3: Local Price Ladder Blocher2021
  • Proposition 4: Local Monotonicity
  • Remark 5: On the ordering of the PSP price map
  • Definition 6: Allocation Step $\tau_k$
  • Proposition 7: Saturated Shell
  • Corollary 8
  • Example 9: Market Shift Revealed by Partial Ordering