Progressive Second-Price Auction

• Progressive Second-Price Auction is a decentralized auction mechanism that iteratively updates local bids to allocate divisible resources in real time.
• It uses a formal methodology with clearing prices, pro rata allocation, and adaptive reserve pricing to maintain incentive compatibility and market stability.
• The mechanism supports asynchronous updates and local equilibrium convergence, demonstrating robustness against network latency and strategic bidding.

A Progressive Second-Price Auction (PSPA) is a decentralized market mechanism for resource allocation and real-time pricing of divisible goods, governed by local interactions and iterative agent strategies. Unlike classic Vickrey or uniform-price auctions, PSPA is characterized by dynamic, bid-driven updates within a network, yielding incentive-compatible and robust equilibrium properties. The methodology generalizes to bipartite multi-auction settings and supports asynchronous updates, strategic bidder responses, reserve price adaptations, and explicit modeling of latent network conditions.

1. Mechanism Design and Mathematical Formalization

PSPA operates over a network comprising buyers and sellers, each seller possessing a divisible resource of total quantity $Q^j$. Buyers may simultaneously participate in multiple local auctions. Bids are submitted as pairs $s_i^j(t) = (q_i^j(t), p_i^j(t))$, quantifying the desired quantity and the unit price per auction round $t$ (Blazek et al., 24 Nov 2025). Sellers aggregate bids, determine allocations, and clear prices via:

• Clearing Price: The minimum price $\chi^j(t)$ at which the cumulative awarded quantity meets supply:

$$\chi^j(t) = \min\left\{y: \sum_{i:\,p_i^j(t)>y} a_i^j(t) \ge Q^j\right\}$$

• Allocation Rule: Under price-level ties, residual quantity $R^j(y, t)$ is distributed pro rata:

$$a_i^j(t) = \min\left\{q_i^j(t),\, \frac{q_i^j(t)}{\sum_{\ell:p_\ell^j=y}q_\ell^j(t)} R^j(y, t)\right\}$$

• Reserve Updating: Sellers monotonically raise the reserve price to ensure stability, e.g.,

$$p_*^j(t+1) = \max\{p_*^j(t), \overline{p}^j(t)+\epsilon\}$$

• Payment and Utility: Buyer’s externality-driven payment and quasi-linear utility:

$$c_i(s) = \int_{0}^{a_i(s)} P^j(z,\,s^j_{-i})\,dz, \quad u_i(s) = \theta_i\left( \sum_j a_i^j \right) - c_i(s)$$

The PSPA framework inherits truthfulness from Vickrey–Clarke–Groves mechanisms: buyers cannot profit by misreporting bids (Blazek et al., 24 Nov 2025, Blazek et al., 21 Nov 2025).

2. Bid Graphs, Influence Sets, and Projection Operators

The multi-auction structure is encoded as a bipartite graph, where edges represent active bids. The set $$I_\mathrm{active}(t) = \{(i,j) \in B \times L: q_i^j(t) > 0\}$$ indexes nonzero bids and auction connectivity (Blazek et al., 24 Nov 2025).

Projections define local neighborhoods and influence propagation:

• $\pi: I_\mathrm{active}(t) \to B$, $\varpi: I_\mathrm{active}(t) \to L$
• One-hop Influence Sets:

$$\Lambda_L^{(1)}(j, t) = (P \circ Q)(j), \qquad \Lambda_B^{(1)}(i, t) = (Q \circ P)(i)$$

• Expanded Influence Sets: Recursively define $n$-hop neighborhoods ($\Lambda^{(n)}$) governing phase transitions in allocation.

This projection-based structure supports decentralized PSP auctions, enabling the system to propagate allocation and price changes across the network without requiring global knowledge (Blazek et al., 24 Nov 2025).

3. Partial Orderings, Market Shifts, and Phase Dynamics

On each seller, active bids $B^j(t)$ admit a partial order by price. Critical intervals $\overline{p}^j(t)<p_*^j(t)<\underline{p}^j(t)$ bracket the clearing price and delimit margin-driven recomputation events:

• Demand Shortfall: $\sum_{i \in B^j(t)} a_i^j(t) < Q^j$
• Bid Overtake: A new bid $p_{i^*}^j(t) > \overline{p}^j(t)$ displaces the marginal winner

Market shifts are thus encoded as local changes in partial orders that trigger phase transitions and saturation phenomena within one-hop shells (Blazek et al., 24 Nov 2025).

4. Saturated Shells and Equilibrium Structure

A seller shell $\Sigma = \Lambda_L^{(1)}(j, t)$ is saturated if no participant can improve utility via unilateral deviation:

$$u_i(t) \ge u_i'(t) \quad \forall\, i \in B^j(t),\ \forall\ \text{feasible}\;s_i'(t)$$

• Local Price Ladder: Nested reserve and bid prices preserve monotonicity across shells:

$p_*^k \le p_\ell^* < p_*^j \le p_i^*$

• Shell-By-Shell Saturation: Inductive propagation through asynchronously added vertices deterministically covers the connected component, enforcing equilibrium (Blazek et al., 24 Nov 2025).

A plausible implication is that this layered saturation tightly constrains the strategy space and prevents instability, oscillation, or cycles in decentralized multi-auction networks.

5. Incentive Compatibility, Nash Convergence, and Truthfulness

PSPA enforces incentive compatibility via the second-price externality rule, ensuring that allocations and payments are locally VCG-type. The structured partial order and saturated shell framework guarantee that no local deviation is profitable—oscillations are eliminated.

Global $\varepsilon$-Nash Theorem: Under bounded-delay, asynchronous updates, and small $\varepsilon$, decentralized PSPA dynamics converge deterministically to an $\varepsilon$-Nash equilibrium. This holds absent global auction graph information and is robust to asynchronous local steps (Blazek et al., 24 Nov 2025, Blazek et al., 21 Nov 2025).

The scheme thus achieves stability, deterministic coverage, and convergence in complex multi-auction scenarios.

6. Intra-Round Dynamics and Asynchronous Update Protocols

Within each round, sellers independently execute allocation steps indexed by micro-time $\tau_k$:

• Seller Step: Select the highest outstanding bidder, allocate $a_i^j(\tau_k)$, update reserve $p_*^j(\tau_{k+1})$, recompute influence shell.
• Joint Constraints: Buyers adjust feasible quantities across auctions:

$$q_i^j(\tau_{k+1}) = Q_i(t) - \sum_{j' \in L_i(t)} a_i^{j'}(\tau_k),\quad \forall j \in L_i(t)$$

• Influence Propagation: Asynchronous seller actions, coupled by buyer constraints, manifest as local re-orderings that maintain price ladder and preserve shell saturation.

Consequently, after finitely many steps, the network reaches a local equilibrium with no further profitable updates—a decentralized, robust outcome in the presence of asynchronous actors (Blazek et al., 24 Nov 2025).

7. Robustness: Latency, Strategic Behavior, and Dynamic Reserve Optimization

Empirical and theoretical analysis demonstrates high resilience of PSPA to network latency, asynchronous bid evaluation, and strategic bidder manipulation. Reserve prices just below the clearing price reliably stabilize seller revenue while maintaining allocative efficiency; truthful $\varepsilon$-best replies ensure almost welfare-maximizing equilibria (Blazek et al., 21 Nov 2025).

Moreover, reinforcement learning approaches in PSPA settings (e.g., the CLUB algorithm) address reserve price design under MDP dynamics, strategic misreporting, and noise estimation. These mechanisms achieve sublinear revenue regret and efficiently balance exploration and exploitation (Ai et al., 2022). Strategic buffer periods and simulation-based estimation curb manipulation and improve reserve adaptation in multi-phase environments.

This suggests that the progressive second-price paradigm is broadly applicable to diverse decentralized allocation tasks, supporting both static and adaptive resource environments while preserving incentive compatibility and equilibrium stability.