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samineru comments on Ego syntonic thoughts and values - Less Wrong
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Tossing a coin would arguably be better. I expect I wouldn't have any kind of trouble with that arrangement if played against someone similar to myself in that respect. With someone sufficiently like myself, I can imagine even discussing the question of whose life should be retained and coming to an agreement (I'm not sure about the potential for a severe psychological backlash from this method of decision-making, but it's probably a less significant consideration than the difference between a better and worse choice for who to save).
(This is an isolated remark unrelated to the topic of the post.)
The problem with flipping the coin is that it already asks you to make an uncomfortable judgement about the values of John and Lisa's lives. You have already said "Our lives' values are comparable enough to merit coin-flipping". Acting out of the view of the partner gives the sort of mutual plausibly deniability that allows you to both percieve that the partner values your more than themselves.