Interestingly, you can either check or bet a King or a Jack—this is poker, after all and in this case bluffing/slow-playing is as good as playing your cards straight. But holding a Queen is tricky: If you bet, your opponent folds with a Jack or raises with a King. Half the time, you win your opponent's one-chip ante, and half the time you lose your ante plus your bet.
This is not good. In fact, it's bad. You're losing twice as many chips as you're winning.
So you check.
Now your opponent only checks if holding the Jack and you win the one-chip ante. If your opponent bets he/she either has the King or is bluffing with the Jack. So calling this bet wins half the time (assuming your opponent is an ice-cold bluffer). If you call, you've got two chips versus two chips in a 50/50 pot; if you fold, you lose your ante every time.
So your best strategy when holding the Queen and playing first is to check and then call if necessary. Unfortunately, even this optimal strategy loses 1/18th each hand.
So you'd rather play first (see above).
For serious Game Theory geeks, here's the decision tree, with dominated strategies (the bad ones) already removed:
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