| Περίληψη: |
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 pest control, radiation therapy, and economic sanctions, 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 exhibit 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 geometrically symmetric strategies where both players balance interval choices between the boundaries and the center. 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. • Alignment games: modeling an agent trying to align with a hidden state. • Payoffs from misalignment capture commission vs omission errors. • Motivated by problems in pest management, sanctions enforcement, radiation therapy. • Optimal strategies for several variants of the game can be derived succinctly. • Related to geometric games and Matching Pennies. [ABSTRACT FROM AUTHOR] |