auction equilibrium characterization

This paper examines the impact of introducing a “Best and Final Offer” (BAFO) mechanism in sequential procurement auctions on game-theoretic equilibria and allocative efficiency. We consider an extensive-form game with multiple fully informed sellers and a single buyer possessing private information about its valuation—modeled as an arbitrary monotone function. Method: We formulate and solve a subgame-perfect equilibrium (SPE) model for this setting. Contribution/Results: We provide the first rigorous proof that BAFO guarantees full efficiency—i.e., socially optimal allocation—in all SPEs, without requiring strong assumptions such as gross substitutability. Furthermore, although allocations are identical across all SPEs, the buyer’s procurement cost varies significantly, revealing a novel trade-off between efficiency and cost. This result constitutes a theoretical breakthrough in procurement mechanism design, offering both foundational insights and practical implications for auction-based sourcing.

This study investigates how participants’ understanding of equilibrium strategies transfers across different mechanisms. To this end, it introduces the concept of “strategic analogy,” which extends traditional notions of strategic equivalence by simultaneously remapping both actions and types. The paper develops a knowledge representation framework grounded in payoff comparisons to formally characterize strategic understanding. Integrating tools from mechanism design, equilibrium analysis, and knowledge representation, the work demonstrates that, provided a clear correspondence between actions and types is established, equilibrium reasoning can be effectively transferred across strategically analogous mechanisms. The proposed framework applies broadly to settings such as single-item auctions, scoring auctions, and nonlinear pricing with capacity constraints, offering both a theoretical foundation and practical pathways for cross-mechanism strategic transfer.

Existing methods for computing equilibria in sequential auctions with continuous action and value spaces are limited to single-round settings and fail to address the challenges of infinite subgames, absence of optimal substructure, and dynamic coupling between beliefs and strategies. Method: We propose a customized game abstraction framework incorporating public belief states, integrated with dynamic programming and Bayesian updating. We further establish a utility-loss upper-bound decomposition theorem to enable rigorous verification. Contribution/Results: This work presents the first verifiable computation of pure-strategy ε-perfect Bayesian equilibria (ε-PBE) in sequential auctions with continuous spaces. Our approach provides theoretically guaranteed convergence and controllable error bounds. Experiments confirm correctness on established equilibrium benchmarks and successfully compute novel equilibria for multi-round combinatorial auctions—demonstrating both scalability and practical applicability.

This paper investigates cooperative pricing between a platform and independent sellers in a revenue-sharing Bertrand game. Addressing the limitation of conventional models that neglect distributional collaboration, we develop an extended model wherein the revenue-sharing ratio is endogenously determined and sellers possess outside options. Using game-theoretic analysis and Nash equilibrium characterization, we establish— for the first time—that under specific cost structures and revenue-sharing parameters, introducing independent sellers can simultaneously increase both the incumbent manufacturer’s profit and consumer surplus, thereby overturning the conventional efficiency–profit trade-off. We further derive necessary and sufficient conditions for the existence and uniqueness of multiple equilibrium types, and demonstrate that the revenue-sharing mechanism exerts a non-monotonic effect on market efficiency, firm profits, and social welfare. Our findings provide theoretical foundations and actionable insights for cooperative pricing and revenue-allocation mechanism design in platform economies.

Analyzing complex multi-agent games—such as auctions, cybersecurity scenarios, and competitive games—is hindered by the absence of tractable analytical models and intractable equilibrium computation. Method: This paper proposes a general empirical game-theoretic analysis (EGTA) framework tailored to black-box, ultra-large-scale environments. It systematically integrates interactive game sampling, empirical equilibrium computation, responsive strategy learning, Monte Carlo simulation, and machine learning–aided modeling—thereby overcoming limitations of declarative modeling. Crucially, it unifies sampling strategy design, equilibrium discovery, and model compression into a coherent subproblem structure, while leveraging machine learning to accelerate strategy-space compression and equilibrium approximation. Results: Experiments demonstrate that the framework significantly improves modeling fidelity and computational scalability of EGTA for non-differentiable, high-dimensional, and analytically inexpressible games. It establishes a reproducible, data-driven paradigm for strategic reasoning in complex, real-world settings.

This work addresses the long-standing challenge of Nash equilibrium existence in infinite games, which traditionally relies on strong assumptions and lacks a unified framework. By abandoning countable additivity and introducing finitely additive mixed strategies, the paper establishes—under the most general setting—that a Nash equilibrium exists for any nonempty set of players and any bounded utility functions. This result unifies existing equilibrium existence theorems and demonstrates that the equilibrium correspondence is nonempty, compact-valued, and upper hemicontinuous. The proof synthesizes tools from finitely additive measure theory, analysis of upper hemicontinuous correspondences, and finite approximation techniques, thereby enabling direct equilibrium analysis of infinite games previously considered intractable.

This study investigates multi-player discrete-bidding graph games, where token ownership is determined each turn via auction. It extends classical two-player bidding games to a multi-player coalition setting, integrating game-theoretic analysis, graph game models, and discrete budget mechanisms. The work establishes that such games are determined under mild tie-breaking rules, proves the universal existence of pure-strategy Nash equilibria for qualitative objectives, and demonstrates that the decision problem of determining winning strategies is PSPACE-hard—even when budgets are encoded in unary—marking a stark contrast to the NP ∩ coNP complexity known for the two-player case.

This study addresses the moral hazard problem in auctions wherein bidders lack ex ante incentives to improve quality due to sunk costs. To mitigate this issue, the authors propose embedding a post-performance quality-based reward mechanism into auction contracts. By developing a unified framework that integrates linear incentive contracts with various auction formats, they employ Bayesian Nash equilibrium analysis, asymptotic methods, and the revenue equivalence theorem to demonstrate that, as the number of bidders grows large, the optimal reward coefficient converges to the auctioneer’s marginal benefit from quality. Under symmetric equilibrium, this mechanism substantially enhances the auctioneer’s expected revenue and achieves full pass-through of quality value, exhibiting both theoretical robustness and practical efficacy in incentivizing quality improvement.

This study investigates a mean-payoff bidding game played by two agents on a graph, where the right to move a token is determined each round through an auction, generating an infinite path whose long-run average payoff defines the players’ utilities. Focusing on the actual trajectories induced when both players employ adversarial optimal strategies, the work provides the first formal analysis of such non-antagonistic dynamics and establishes that, under certain conditions, the resulting trajectories eventually become periodic. By integrating tools from game theory, automata theory, and explicit constructions of optimal strategies, the paper addresses the complex dynamics arising in infinite state spaces and presents an efficient algorithm to compute the mean-payoff utilities for each player along the eventual periodic trajectory.