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Spoiler for
So the question is "in what circumstance would C give up, and in what circumstances would he not?"
First, let's bring back the first answer, which we both agree on. In the case where giving up is not allowed, then A wins and the game goes as follows.
Turn 1- A: I don't know
Turn 2- B: I don't know
Turn 3- C: I don't know
Turn 4- A: I don't know
Turn 5- B: I don't know
Turn 6- C: I don't know
Turn 7- A: My number is 3.
Obviously, C only gets to answer twice. Once at turn 3 and once at turn 6. The rest of this post will show C has no strategy to win, therefore he has an incentive to give up.
Note that since 'Give up' means 'I don't know' + 'I forfeit my right to play further', an 'I don't know' is the same as 'Give up' on turn 5 or 6, since B and C don't get to play further in any instance.
I think we both agree so far that turn 1 and 2 should both be 'I don't know'. So, let's look at turn 3. C has no idea what his hat number is, so he can't win. If C chooses to give up, the the game goes as follows
A: I don't know
B: I don't know
C: I give up
A: I don't know
B: My number is 3.
If C chooses not to give up but to pass on turn 3, then the game goes as follows
Turn 1- A: I don't know
Turn 2- B: I don't know
Turn 3- C: I don't know
Turn 4- A: I don't know
Turn 5- B: I don't know/Give up (identical choices at this turn)
Turn 6- C: I don't know/Give up (identical choices at this turn)
Turn 7- A: My number is 3.
So essentially, C got no winning options if all hats are 3, so technically he would be required by the rules to give up on turn 3. Meaning that the game would proceed as follows
A: I don't know
B: I don't know
C: I give up
A: I don't know
B: My number is 3.
Almost there...
Spoiler for
Save for two things. First, a clarification of the problem statement: the game continues until each logician has either named his number or given up. So more than one of them can potentially "win". Secondly: would C give up if A had a 2 instead of a 3? 
