No, regardless of UI, I think the dilemma of having to buy yourself the least popular one to get an extra t-shirt is important. And what would happen to your buisness model if everybody liked (and ordered) t-shirt A but nominated t-shirt B? You would have to ship two t-shirts to everybobdy, whereas your current plan guarantees that no more than 20% of customers get a free t-shirt --- which also makes it more challenging to play the game.
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No, regardless of UI, I think the dilemma of having to buy yourself the least popular one to get an extra t-shirt is important. And what would happen to your buisness model if everybody liked (and ordered) t-shirt A but nominated t-shirt B? You would have to ship two t-shirts to everybobdy, whereas your current plan guarantees that no more than 20% of customers get a free t-shirt --- which also makes it more challenging to play the game.