The final form of "integrity," of course, is the one they're usually trying to talk about, which is game trust. Predictability. When you say you're going to do one thing, I can count on the fact that you're going to do that one thing. It's not just that you tell me the truth about your intentions at a particular point; it's that if you tell me you're going to vote one way, you will reliably vote that way. The opposite of this is the "flip" or "flop" or "backstab." This is so similar to a concept of actual trust that people rely on in their day-to-day lives -- if you agree to complete a project at a particular time, I can count on the fact that you're going to do it; if you agree not to sleep with someone else while we're dating, I can count on the fact that you won't -- that people confuse it with actual trust, which is a character issue, even though that's not what it is. The person whose game play is inferior when there is a collapse of game trust is the trusting person, not the trusted person. You're under no obligation to be predictable. That's not part of the rules, and everyone knows it isn't part of the rules. People have been saying one thing and doing something else since the beginning of the game, including saying "I won't say one thing and do something else," and later changing their minds. It's just that being good at identifying people whose behavior you can predict -- as Yul did with Jonathan -- is part of being a good player. When you know what people are going to do, you can affect it infinitely more easily.
I guess the real last definition of integrity, which perhaps is the one that's actually in the back of these people's heads, is one that amounts to "people with integrity don't do anything to keep me from winning." I mean, normally, you have a right to resent people who do something specifically designated to keep you from getting something you want. If someone sandbags you with regard to a promotion, or intentionally hits on your date, or tries to make it so you'll be late to an interview, you normally have a right to be pissed off. The problem is that Survivor is like poker or bumper cars, in that (1) it is zero-sum; and (2) everyone agrees at the beginning, at least tacitly, that they will all try to win. They aren't allowed to keep it from being zero-sum. They aren't allowed to do what an actual social group might actually do, which is split the money. They are forced to have no purpose in the game other than self-interest. It is impossible to play this game in a way that intentionally improves someone else's odds of winning without decreasing everyone else's odds of winning, including your own. This is why "integrity" stuff about alliances doesn't work. The idea is, "We formed an alliance, and you stabbed me in the back." The problem is that those decisions aren't symmetrical. If I make a sacrifice for you on the understanding that you also make sacrifices for me (I pick you up and take you to work, because I know you'll do the same for me, and maybe we even talk about that), then later, I have a chit to call in. But when you form an alliance, it's in everyone's interest. If it weren't, you wouldn't do it. Nobody gets into an alliance thinking that it damages their position and they're making a sacrifice for the other people in the alliance. But later, at the time when the alleged "backstabber" makes the flip, it's because now, his interests are better served by ditching the alliance, meaning that by staying with it, he damages himself. You can't ask people to do something that costs them something in return for something you did that cost you nothing. That's not even. That's why they were so pissed off to find out that Yul had the idol -- at some level, they realized that Jonathan really did have to flip. They were really in the situation they didn't want to be in, where Jonathan did the only right thing by flipping on them, and he even did something that probably helped Candice.