Why would we not have the “Direction” component standardized to have unit norm?
I think what the OP is getting at is that the space of endeavors has a bunch of privileged directions of high impact, and your impact depends on (1) how good your aim is and (2) how hard you shoot. So it’d be something like magnitude times the sum of cosine similarities with each high-impact vector; or perhaps just the magnitude if we use the high-impact vectors as the basis.
Also, “Magnitude” is probably the wrong term for the component in question; it seems to mean “how much you achieve”, but that’s actually what “Impact” is measuring! And indeed, impact is very much a function of the direction in which you’re going. “Magnitude” should instead be “Effort” or “Short-Term Profit” or something.
(Yes, I truly believe that nitpicking this toy model is the best use of my time right now.)
I think the point wasn’t having a unit norm, it was that impact wasn’t defined as directional, so we’d need to remove the dimensionality from a multidimensionally defined direction.
So to continue the nitpicking, I’d argue impact = || Magnitude * Direction ||, or better, ||Impact|| = Magnitude * Direction, so that we can talk about size of impact. And that makes my point in a different comment even clearer—because almost by assumption, the vast majority of those with large impact are pointed in net-negative directions, unless you think either a significant proportion of directions are positive, or that people are selecting for it very strongly, which seems not to be the case.
I was thinking something potentially similar. This is super nitpicky, but the better equation would be impact = Magnitude * ||Direction||
Why would we not have the “Direction” component standardized to have unit norm?
I think what the OP is getting at is that the space of endeavors has a bunch of privileged directions of high impact, and your impact depends on (1) how good your aim is and (2) how hard you shoot. So it’d be something like magnitude times the sum of cosine similarities with each high-impact vector; or perhaps just the magnitude if we use the high-impact vectors as the basis.
Also, “Magnitude” is probably the wrong term for the component in question; it seems to mean “how much you achieve”, but that’s actually what “Impact” is measuring! And indeed, impact is very much a function of the direction in which you’re going. “Magnitude” should instead be “Effort” or “Short-Term Profit” or something.
(Yes, I truly believe that nitpicking this toy model is the best use of my time right now.)
I think the point wasn’t having a unit norm, it was that impact wasn’t defined as directional, so we’d need to remove the dimensionality from a multidimensionally defined direction.
So to continue the nitpicking, I’d argue impact = || Magnitude * Direction ||, or better, ||Impact|| = Magnitude * Direction, so that we can talk about size of impact. And that makes my point in a different comment even clearer—because almost by assumption, the vast majority of those with large impact are pointed in net-negative directions, unless you think either a significant proportion of directions are positive, or that people are selecting for it very strongly, which seems not to be the case.