We explore the influence of cognitive ability and judgment on strategic behavior in the beauty contest game (where the Nash equilibrium action is zero). Using the level-k model of bounded rationality, cognitive ability and judgment both predict higher level strategic thinking. However, individuals with better judgment choose zero less frequently, and we uncover a novel dynamic mechanism that sheds light on this pattern. Taken together, our results indicate that fluid (i.e., analytical) intelligence is a primary driver of strategic level-k thinking, while facets of judgment that are distinct from fluid intelligence drive the lower inclination of high judgment individuals to choose zero.