Information Acquisition with $\hat{I}\pm$-Divergence Costs

Abstract

Building on the $f$-information model of Bloedel et al. (2025), this paper introduces a one-parameter family of information acquisition models and characterizes optimal information acquisition. This family extends the mutual information model (Matějka and McKay, 2015) while preserving its analytical tractability. The information cost is derived from the $\hat{I}\pm$-divergence, which nests the KL-divergence ($\hat{I}\pm =-1$), the reverse KL-divergence ($\hat{I}\pm =1$), and the squared Hellinger distance ($\hat{I}\pm =0$), and is represented in closed form via the $\hat{I}\pm$-integration of Amari (2007). The optimal choice probabilities belong to the $q$-exponential family, which appears in nonextensive statistical mechanics (Tsallis, 1988) and in the $q$-logit model of traffic route choice (Nakayama, 2013). This family reduces to the modified logit in the mutual information case (Matějka and McKay, 2015). We further show that the relationship between payoffs and the set of actions chosen with positive probability in each state changes qualitatively across ranges of $\hat{I}\pm$.

Preliminary version. Comments are welcome