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Formal Verification of Diffusion Auctions

Rustam Galimullin, Munyque Mittelmann, Laurent Perrussel

TL;DR

This paper tackles the formal verification of diffusion auctions where sellers leverage social networks to recruit participants. It develops two logics, $L^n$ and its strategic extension $SL^n$, to capture diffusion dynamics and strategic incentives within diffusion-augmented mechanism models, and provides model-checking procedures for these logics. The authors establish that $ ext{MC}(L^n)$ lies in $P$, while $ ext{MC}(SL^n)$ is PSPACE-complete, and that the strategy-existence problem is NP-complete. They also introduce a multi-seller diffusion framework with a SMF style placement and diffusion updates, and study the expressive power of the coalitional modality, illustrating both the potential and limitations of formal verification in diffusion auctions. The work opens avenues for extending the framework to probabilistic settings, buyer strategies, and multi-item diffusion, aiming to bridge mechanism design with formal verification on networked markets.

Abstract

In diffusion auctions, sellers can leverage an underlying social network to broaden participation, thereby increasing their potential revenue. Specifically, sellers can incentivise participants in their auction to diffuse information about the auction through the network. While numerous variants of such auctions have been recently studied in the literature, the formal verification and strategic reasoning perspectives have not been investigated yet. Our contribution is threefold. First, we introduce a logical formalism that captures the dynamics of diffusion and its strategic dimension. Second, for such a logic, we provide model-checking procedures that allow one to verify properties as the Nash equilibrium, and that pave the way towards checking the existence of sellers' strategies. Third, we establish computational complexity results for the presented algorithms.

Formal Verification of Diffusion Auctions

TL;DR

This paper tackles the formal verification of diffusion auctions where sellers leverage social networks to recruit participants. It develops two logics, and its strategic extension , to capture diffusion dynamics and strategic incentives within diffusion-augmented mechanism models, and provides model-checking procedures for these logics. The authors establish that lies in , while is PSPACE-complete, and that the strategy-existence problem is NP-complete. They also introduce a multi-seller diffusion framework with a SMF style placement and diffusion updates, and study the expressive power of the coalitional modality, illustrating both the potential and limitations of formal verification in diffusion auctions. The work opens avenues for extending the framework to probabilistic settings, buyer strategies, and multi-item diffusion, aiming to bridge mechanism design with formal verification on networked markets.

Abstract

In diffusion auctions, sellers can leverage an underlying social network to broaden participation, thereby increasing their potential revenue. Specifically, sellers can incentivise participants in their auction to diffuse information about the auction through the network. While numerous variants of such auctions have been recently studied in the literature, the formal verification and strategic reasoning perspectives have not been investigated yet. Our contribution is threefold. First, we introduce a logical formalism that captures the dynamics of diffusion and its strategic dimension. Second, for such a logic, we provide model-checking procedures that allow one to verify properties as the Nash equilibrium, and that pave the way towards checking the existence of sellers' strategies. Third, we establish computational complexity results for the presented algorithms.

Paper Structure

This paper contains 12 sections, 9 theorems, 9 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Contribution
  3. Diffusion Auctions With Multiple Sellers
  4. Strategic Properties of Diffusion Auctions
  5. Nash Equilibrium
  6. Model Checking and Strategy Existence
  7. Reasoning About Sellers' Strategies
  8. Expressivity and Model Checking
  9. Logics for auctions
  10. DEL and social network logics
  11. Conclusion
  12. Technical Appendix

Key Result

Proposition 1

Let $\varphi \in \mathcal{L}^n$ and a mechanism $M$ be given. Then we have that $M,a \models \varphi$ iff $M_{\mathit{fin}},a \models \varphi$.

Figures (5)

  • Figure 1: Mechanisms $M_1$ (left) and $M_2$ (right).
  • Figure 2: Mechanism $M$ (left) and updated mechanism $M^{\sigma:\alpha}$ (right). For the seller $s$, her name is $\sigma$ and her budget in $M$ is 5. For buyer $a$, her name is $\alpha$, and $(3, 1,5)$ denotes the fact that $Bdg(a) = 3$, $V(a) = 1$, and $I(a, s) = 5$. Similarly, for other agents. The new link in $M^\alpha$ is dashed.
  • Figure 3: Mechanism $M$ and its updates. In the mechanisms, there are two sellers, $s_1$ and $s_2$, and six buyers, $a,b,...,f$. Agents' names are shown near the state label. New links are dashed. The allocation of the sold items is depicted by $\heartsuit$.
  • Figure 4: Mechanism $M$ for the 3-SAT instance $\Psi$ with one new link being dashed, and other new links being omitted for readability.
  • Figure 5: Mechanisms $M_1$ (left) and $M_2$ (right).

Theorems & Definitions (27)

  • Definition 1
  • Example 1
  • Definition 2
  • Definition 3: SMF Auction
  • Example 2
  • Definition 4
  • Definition 5
  • Example 3
  • Proposition 1
  • Theorem 1
  • ...and 17 more