Give away your chief asset before the contest even begins. Second, there is of course no such thing as choosing to be systematically non-rational. That would be a condition beyond insanity, a kind of permanent primal scream. We can, however, certainly be more or less rational in the sense of trying to limit or enhance the extent that other factors override our rationality. But all anti-rationalists use rationality as a matter of course.
Rationality is, after all, just another name for how people think.
We therfore express our allegiances and determine our actions partly based on our rationality, but also partly based on additional things like desire, fear, and habit. Indeed, neither rationality as a whole nor just logic can be dispensed for long without disastrous results. Thus, if dispensing with rationality was the anti-rationalist agenda, their cause would not only be unwise but also foredoomed. Indeed, while it lacks philosophical profundity, this is a very effective plea for an anti-rationalist to make to enforce that his or her critics never utter an effective contrary sound. The claim of science is that we should use our experience and the experiences of others, our intuitions and the intuitions of others, and even our fears and guesses and the fears and guesses of others, all mediated, however, by logic and the rules of evidence.