In the $500 NL game I play in, players are allowed to make a specific type of deal if they go all-in before the river: they can run the rest of the cards either two or three times and then split the pot in halves or thirds, respectively. The pieces of the pot are distributed according to who won each of the two or three iterations. (I gave an example in this post.)
It is strange and fascinating to listen to other players' philosophies on whether to make a deal and whether to run the board twice or three times. As far as I can see, there are only a few things that deserve any consideration:
1. What is my EV for each option? This can be answered mathematically: my EV is the same for all three options. This follows from the fact that EV(A+B) = EV(A)+EV(B), regardless of whether A and B are correlated. If we take A to be the first running of the cards, and B to be subsequent runnings of the cards, we can see that the EV does not depend on what type of deal we make.
2. What happens to my volatility for each option? This can also be answered mathematically: the volatility is lower if you run the board more times. This is the reason why running it more times might be worthwhile.
3. How much time is this going to take? If you are a winning player, you can lose money if too much time is wasted. Also, I would consider it bad table etiquette to needlessly waste the other players' time. Splitting the pot three ways can take a while. In fact, last week I won five-sixths of a pot because we ran the board three times and we chopped the last one, and this took a minute or two to sort out. Another thing that can waste time is the possibility that your opponent might take a while to decide whether to make a deal. I've seen such a decision take over five minutes.
4. How will this affect my table image? Being willing to run the board twice can change the way people play against me. The most significant difference is probably that some players may try to get all-in earlier in the hand in order to give themselves a chance to make a deal and chop the pot. They also may be more willing to call my all-in bets before the river. This is probably a good thing, since players are deviating away from trying to maximize their EV, but for certain players, it likely makes them play better (if they will now call with +EV hands that they would have otherwise folded).
Points 1 and 2 seem to me to be by far the most significant, but I also often consider point 3 and choose not to make a deal if the pot is small. I do this out of respect for the other players' time and because I figure it won't add too much volatility to my winnings. Point 4 is one I don't worry too much about, but it seems like it could be a reasonable consideration.
Most players know points 1 and 2 (minus the math), yet they often take longer than a minute to decide whether to make a deal. What could they be thinking about? Clearly it's not point 3 (time considerations), or they wouldn't waste everyone's time with their slow decisions. For most players, it's not point 4, either. The answer? Whim. This wouldn't be so bad if it weren't for the fact that they clearly consider these decisions to be excruciatingly important, and once they form their opinions on which option is best, they smugly ridicule players who make a different decision. Listening to them argue about it is like trying to watch one of those shows where pundits discuss politics. Nobody has any evidence to support their claims, but they are all quite satisfied in their certitude. For example:
"If I call a big bet with a draw, I'm trying to win the whole pot. Why run it twice? To get back half the pot? It's better to just fold and save your money than call and make a deal."
"If you initiate the thing, it's three times," meaning that if you put in the bet and are called, you should always run it three times, rather than twice.
Maybe I'll try to document more of these, but the point is that the decisions are made on a whim, and then supported with whatever "evidence" seems to fit.