Alignment Games

2025-09-02

AI Summary

This paper addresses strategic decision-making under partial observability—arising in medical diagnosis, economic sanctions, and resource allocation—where intervention efficacy depends on alignment with latent states. We propose a novel zero-sum game model, termed *alignment games*, whose payoff is defined via the symmetric difference between the agent’s action set and the hidden state. The model explicitly captures the false-positive–false-negative trade-off through a cost function (penalizing misclassification) and a penalty function (penalizing missed detection). Theoretically, we show that the cost-to-penalty ratio governs the structure of optimal strategies. Closed-form Nash equilibria are derived for both continuous domains (e.g., the unit circle and unit interval) and discrete spaces, revealing threshold behavior and a phase transition wherein optimal strategies concentrate mass on high-cost locations. Our framework unifies and generalizes classical search games, providing an analytically tractable and interpretable foundation for strategic intervention under uncertainty.

Technology Category

• Game Theory and Economic Paradigms: Adversarial Learning
• Search and Optimization: Adversarial Search
• Reasoning under Uncertainty: Sequential Decision Making

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

This paper introduces alignment games, a new class of zero-sum games modeling strategic interventions where effectiveness depends on alignment with an underlying hidden state. Motivated by operational problems in medical diagnostics, economic sanctions, and resource allocation, this framework features two players, a Hider and a Searcher, who choose subsets of a given space. Payoffs are determined by their misalignment (symmetric difference), explicitly modeling the trade-off between commission errors (unnecessary action) and omission errors (missed targets), given by a cost function and a penalty function, respectively. We provide a comprehensive theoretical analysis, deriving closed-form equilibrium solutions that contain interesting mathematical properties based on the game's payoff structure. When cost and penalty functions are unequal, optimal strategies are consistently governed by cost-penalty ratios. On the unit circle, optimal arc lengths are direct functions of this ratio, and in discrete games, optimal choice probabilities are proportional to element-specific ratios. When costs are equal, the solutions exhibit rich structural properties and sharp threshold behaviors. On the unit interval, this manifests as a geometric pattern of minimal covering versus maximal non-overlapping strategies. In discrete games with cardinality constraints, play concentrates on the highest-cost locations, with solutions changing discontinuously as parameters cross critical values. Our framework extends the theory of geometric and search games and is general enough that classical models, such as Matching Pennies, emerge as special cases. These results provide a new theoretical foundation for analyzing the strategic tension between comprehensive coverage and precise targeting under uncertainty.

Problem

• Modeling strategic interventions with hidden state alignment
• Analyzing trade-offs between commission and omission errors
• Deriving equilibrium solutions for cost-penalty ratio optimization

Innovation

• Zero-sum games model strategic interventions with hidden states
• Closed-form equilibrium solutions based on cost-penalty ratios
• Geometric patterns and threshold behaviors in optimal strategies