Poker is a game of imperfect information where players make inferences and investments under uncertainty. Winning requires skill as well as luck, and people play with different styles. Here I examine the roles of style and skill in a simplified poker game. I propose that style arises from mental limits in estimating odds and stakes, and I define style as a strategy for making decisions in light of these mental limits. I then develop formal models of cognitive styles, such as 'Tight' and 'Loose', and pit them against each other in heads-up (pair-wise) face-offs (match-ups). I also develop a formal model of normative skill and pit this skill against the styles. My results show that the best style for one player depends on the style of his opponent, and that Novice styles can be remarkably effective against Expert skill. I discuss the reasons behind these results and suggest how this study may apply to practical problems in business and warfare — which are real life games of inference and investment that are also played with style and skill.