What You Earn from What You Learn: Characterization and Complexity of the Value of Information

Abstract

We examine the value of information in extensive-form decision problems: the largest multiplicative gain that a decision maker can obtain from an information refinement, taken over all payoff environments. This measures the structural importance of information disentangled from any particular payoff specification, and generalizes the notion of value of recall recently introduced by Berker et al. (AAAI'25). We establish a unifying characterization of the value of information under any information refinement, uncovering an intriguing connection to the maximum fractional chromatic number over a certain family of conflict graphs. We then employ our characterization to elucidate the complexity of two central information refinements. First, for the value of perfect information in perfect-recall decision problems, we find that the relevant conflict graphs are cographs, so the fractional chromatic number collapses to the clique number. Even though the clique number is tractable for a given cograph, we show that the additional layer of optimization (over the family of graphs) makes computing the value of perfect information NP-complete. On the positive side, we prove fixed-parameter tractability in terms of the degree of imperfect information. Second, for the value of recall in imperfect-recall decision problems (with no absentmindedness), we first show that the value of recall can realize the fractional chromatic number of any graph. Interestingly, this implies that, unlike the value of perfect information, the value of recall can take fractional values. Building on this reduction, we then establish S2P-completeness. This addresses questions left open by Berker et al. (AAAI'25)---both on the characterization and the complexity of the value of recall---and shows that computing the value of recall is strictly harder than the value of perfect information.