[pzhon]

To summarize, running it twice doesn't change EV. However, this explanation was terrible, incorrect, and not convincing to anyone who was not ready to be convinced.

Unfortunately, I don't think there is any really simple, convincing explanation (which is not a given, despite the controversy). One attempt is to say that burn cards don't affect equity, so the equity of the second time is the same as the equity of the first time. Running it twice takes the average of using burn cards or not. This may not be convincing for people who have no intuition about equity, or who think burn cards may hurt one player, but I think it is more commonly accepted that burn cards are harmless than that running it twice does not favor either player.

By the way, there are more than psychological effects. Running it twice may decrease the amount of money on the table, on average, since players will not rebuy after a split pot. Running it twice may keep the game going when a player is not willing to rebuy.