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In this clip, Sammy Farha and Patrik Antonius haggle over running it several times, and are reluctant to reveal their cards before having found an agreement.
What is the reasoning behind this?
If I want to reduce my variance, why do my odds of winning the pot matter?
It is true that running it multiple times will not influence the expected value and therefore the hand of the other player does not matter in the decision to run it multiple times. As an example:
Player A has a probability of 0.9 to win the pot of €2000,- and Player B a probability of 0.1.
The expected value for a single run:
When we run it twice we get the following (exactly the same) expected values:
So in conclusion, you are right that the amount of times the game is run does not influence the expected value, but can reduce variance. The odds of who will be winning do not matter in this case. I suppose that in the clip they tried to make the situation more 'tense' for entertaining purposes.
What is the reasoning behind this?
If your goal is to reduce your variance, then a good number of times to run the cards is the LCD of your winning chances. For example, if you are 66% and he is 33%, you want to run it three times. If you are 75% and he is 25%, four times should be enough, etc. (it is "expected" that you will win the correct number of times). So seeing your cards might change the number of runs he agrees to (i.e. whether to give you less variance or more),
Furthermore, while revealing your cards shouldn't make a difference to your opponent, people aren't always logical (even players who use math heavily can have moments of weakness and irrationality).
Lastly, and this is just a general rule in poker, if you "know" your opponent wants you do X, it's usually the best idea to do the exact opposite (i.e. if it looks like they want you to call, fold... if they want you to fold, raise). It almost becomes instinct.
It's an old superstition, that whoever shows his hand first in a showdown, will lose.
It has nothing to do with running it multiple times, because as shown by the rest of the comments, this technique is only employed to decrease variance. Thus most poker pros, (including Antonius and Farha, as presented in televised games) usually run it more than once or even "chop the pot", especially when dealing with very large pot sizes.
Antonius seems to be a bit embarrassed with how the hand was developed (when Farha calls his all-in, it is obvious he is not happy) , so that might be an extra motive for not wanting to show his hand.
