The most commonly employed paradigms for decision making under risk are expected utility, prospect theory, and regret theory. We examine the simple heuristic of maximizing the probability of being ahead, which in some natural economic situations may be in contradiction to all three of the above fundamental paradigms. We test whether this heuristic, which we call probability dominance (PD), affects decisions under risk. We set up head-to-head situations where all preferences of a given class (expected utility, original or cumulative prospect theory, or regret theory) favor one alternative yet PD favors the other. Our experiments reveal that 49% of subjects' choices are aligned with PD in contradiction to any form of expected utility or prospect theory maximization; 73% are aligned with PD as opposed to preferences under risk aversion and under original and cumulative prospect theory preferences; and 68% to 76% are aligned with PD contradicting preferences under regret theory. We conclude that probability dominance substantially affects choices and should therefore be incorporated into decision-making models. We show that PD has significant economic consequences. The PD heuristic may have evolved through situations of winner-take-all competition.