Information Acquisition with $\alpha$-Divergence Costs

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 (Mat\v{e}jka and McKay, 2015) while preserving its analytical tractability. The information cost is derived from the $\alpha$-divergence, which nests the KL-divergence ($\alpha=-1$), the reverse KL-divergence ($\alpha=1$), and the squared Hellinger distance ($\alpha=0$), and is represented in closed form via the $\alpha$-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 (Mat\v{e}jka 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 $\alpha$.