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Manfred comments on Open thread, August 5-11, 2013 - Less Wrong Discussion
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The dot product is just yer' regular old integral over the domain, weighted in some (unspecified) way.
The thing is though, the average product over the whole infinite space of possibilities isn't much use when it comes to intelligent agents. This is because only one outcome really happens, and intelligent agents will try to choose a good one, not one that's representative of the average. If two wedding planners have opposite opinions about every type of cake except they both adore white cake with raspberry buttercream, then they'll just have white cake with raspberry buttercream - the fact that the inner product of their cake functions is negative a bajillion doesn't matter, they'll both enjoy the cake.
Yeah, but Wedding Planner 1's deep vitriolic moral hatred of the lemon chiffon cake that delights Wedding Planner 2 that abused her as a young girl or Wedding Planner 2's thunderous personal objection to the enslavement of his family that went into making the cocoa for the devil's food cake that Wedding Planner 1 adores could easily make them refuse to share said delicious white cake with raspberry buttercream to the point where either would very happily destroy it to prevent the other from getting any. This seems suboptimal, though.