Agentic AI architectures augment LLMs with external tools, unlocking strong capabilities. However, tool use is not always beneficial; some calls may be redundant or even harmful. Effective tool use, therefore, hinges on a core LLM decision: whether to call or not call a tool when performing a task. This decision is particularly challenging for web search tools, where the benefits of external information depend on the model's internal knowledge and its ability to integrate potentially noisy tool responses. We introduce a principled framework inspired by decision-making theory to evaluate web search tool-use decisions along three key factors: necessity, utility, and affordability. Our analysis combines two complementary lenses: a normative perspective that infers true need and utility from an optimal allocation of tool calls, and a descriptive perspective that infers the model's self-perceived need and utility from their observed behaviors. We evaluate six open and one closed-source frontier models under two harnesses, one conditioning on only the current turn and its search results, the other on the full execution traces, across four web-search tools and three tasks. In every setting, we find that a model's perceived need and utility are frequently misaligned with the true need and utility. Building on this framework, we train lightweight estimators of need and utility from the models' hidden states. These estimators drive simple controllers that improve decision quality and yield stronger task performance than the self-perceived baseline for most of the open-source models.