You are seven people hanging out in a student dormitory. You have very little money but you all manage to pitch-in (different amounts of money) to order a pizza from one of the better pizza places in town. A splurge. The smallest person weighs 50kg (115 pounds) and the biggest is 100kg (230 pounds). Some of you are fat, some of you are almost anorexic. Some of you haven't eaten all day...others of you have had some instant noodles in your own rooms. The anorexic has a 50/50 chance of purging her stomach later that night. One person is on the rugby team and has been ordered by his coach to fill up on carbs. One of the students is the leader of the dormitory (responsible for discipline and rule setting) and because he has a little salary for it he paid for half the pizza. One of the students just moved in a couple weeks ago and is just getting to know everyone. For one of the students it was his birthday two days ago and a few of the people there forgot to give him a present (but did buy him a can of beer for today). 

So...

All seven people have a slice of which most are more or less the same size. There is one slice left over. The host asks if anyone wants the last slice. Everyone murmers "meh" or "I don't have to have it". But several people desperately want it. No one is going to answer any questions that are clearly asked to help decide who should get the last pizza slice.

Keep in mind...some people will resist or laugh at the idea of cutting it up into seven small pieces (though that doesn't mean it is not a solution).

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Instead of intuitively answering the question (what first comes to mind or what seems obvious) try to answer this question through the following paradigm (keep in mind that the paradigms aren't the best ways to summarize each moral system but will suffice here).

An answer that avoids the most suffering and brings the greatest pleasure.

An answer that addresses the consequences of the decision rather than what is fair.

The selfish approach (the tactics that should be used by each agent if they only really care most about their own self interests and nothing else).

A radical altruistic approach (the most selfless decision that can be made).

An answer that is based on the unwritten rules (or tacit contracts) that each of these seven people likely have between one another.

A categorical rule based approach (an answer that comes from the most broadest moral principles in which exceptions should be avoided).

The anonymous dictatorial approach (what you would do if the seven agreed to do whatever decision you made...an anonymous decision).

Extreme etiquette (not what will necessarily satisfy anyone but will avoid the most social tension or social difficulty).

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and finally

If you had to chose a political approach...which would make most sense? 

The host choses (dictator)

A vote (democracy)

A discussion (consensus)

The loudest most obnoxious annoying idiot debater or argument wins (Fox News politics)

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It's interesting that you say people will resist the idea of cutting it into 7 small pieces? Why do you think that? Is it because it seems farcical because the pieces will be too small? 

For a political approach I would go for a (limited time) discussion followed by a vote. Without the discussion I would feel that people's vote would be knee-jerk and uninformed. Perhaps each person would get carte blanche during the discussion to say A) why they should have it and B) who they think should have it and why.

Oh some people would definitely find that rediclous (cutting a small slice into 7 smaller pieces) or would rather just have someone eat it than make things complicated or uncomfortable.

What do you think would be the best approach under "extreme etiquette"?

I was actually going to say that, silly though it may be, the 7 ways approach would be best for etiquette. This is because it pre-emptively shuts down any opportunity to argue over who is more deserving because that criteria has no effect on the outcome. There may be hidden resentments but socially it should be smooth.

So I guess waiting till someone goes to the bathroom and then the other 6 spitting on the slice and offering it to the victim when they come back would not be the best "etiquette" like solution?

I hadn't even thought of that. That must be a testament to your twisted mind. ;-).

While I'm no big fan of Ayn Rand, her view on compromise makes a lot of sense: a compromise is a situation in which there are no winners because no one comes out of it fully satisfied.

I'm yet to know a content or happy person who doesn't know how to compromise nor a well content or happy person who compromises all the time. 

Good for you There are several billion other people who also have opinions and beliefs.

What does that have to do with what I said? What's your point?

The ;truth isn't settled by opinion polls or one person's reflections on his observations.

What makes you think it would be 7 pieces?

Everyone murmers "meh" or "I don't have to have it"

That's gonna cut it down to more like 4 pieces... totally reasonable imo.

First ask if the guy who paid half wants it. Then (unfortunately) my default personality kicks in, which is to defer decisions to others; watch and learn who does what. Punt. See how long the last piece just sits there.

Oh, but don't watch too closely... see if someone tries to sneak a grab at it.

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