Good deck construction minimizes luck. I am about to enter my first tournament (Pauper). In addition to testing my deck against other decks in actual games, I simulate initial hands and first few cards using deckbox.org to help to fine tune it.
It is clear from simulation that I would have to mulligan at least a quarter of my hands. I would assume that beyond a certain mulligan percentage, a deck cannot possibly be competitive in a Pauper tournament.
While there may be special cases where a mulligan does not confer much of a disadvantage, let's assume for the purpose of this question that this deck is average in the sense that each time the initial hand is decreased by a card, it hurts chances of winning quite a bit.
Beyond what percentage of hands requiring mulligan would a deck be considered non-competitive? If someone has actually studied this question statistically, please provide a link to the study.