[sweetjazz]

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The more interesting question is this: what if position is important, i.e., the first player to the left of the button acts first, etc. It's clear that if everyone is playing optimally, because of the "gap principle", the first player doesn't quite need as strong a hand since the other players "know" that it is very unlikely that they can win if they hold a hand slightly better than the utg player's minimal opening hand (as long as x,y and N are reasonable).

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I haven't had a chance to check your math for the original problem, but I am pretty sure this intuition is wrong. While the fact that you cannot be reraised after coming in in early position allows you to play more hands, I still think you would want to be tighter than average because of information leakage (we're assuming everyone is playing optimally, so everyone knows everyone else's strategy).

If y is much larger than Nx, it is true UTG can often "steal the antes", but he will pay a significant price those times someone else has a stronger hand.

If y is small compared to Nx, then others players will simply be able to call with most numbers that are above UTGs minimum because of the good pot odds they will get.