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Stuart_Armstrong comments on In the Pareto world, liars prosper - Less Wrong
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He can only lie about how much he values the point - not about how much the other player values it.
I may be missing something: for Figure 5, what motivation does Y have to go along with perceived choice (0.95, 0.4), given that in this situation Y does not possess the information possessed (and true) in the previous situation that '(0.95, 0.4)' is actually (0.95, 0.95)?
In Figure 2, (0.6, 0.6) appears symmetrical and Pareto optimal to X. In Figure 5, (0.6, 0.6) appears symmetrical and Pareto optimal to Y. In Figure 2, X has something to gain by choosing/{allowing the choice of} (0.95, 0.4) over (0.6, 0.6) and Y has something to gain by choosing/{allowing the choice of} (0.95, 0.95) over (0.6, 0.6), but in Figure 5, while X has something to gain by choosing/{allowing the choice of} (0.6, 0.4) over (0.5, 0.5), Y has nothing to gain by choosing/{allowing the choice of} (0.95, 0.4) over (0.6, 0.6).
Is there a rule(/process) that I have overlooked?
Going through the setup again, it seems as though in the first situation (0.95, 0.95) would be chosen while looking to X as though Y was charitably going with (0.95, 0.4) instead of insisting on the symmetrical (0.6, 0.6), and that in the second situation Y would insist on the seemingly-symmetrical-and-(0.6, 0.6) (0.4, 0.6) instead of going along with X's desired (0.6, 0.4) or even the actually-symmetrical (0.5, 0.5) (since that would appear {non-Pareto optimal}/{Pareto suboptimal} to Y).
As Stuart_Armstrong explains to me on a different thread, the decision process isn't necessarily picking one of the discrete outcomes, but can pick a probabilistic mixture of outcomes. (.6,.6) doesn't appear Pareto-optimal because it's dominated by, e.g., selecting (.95, .4) with probability p=.6/.95 and (0,1) with probability 1-p.
The point of the proof is that if there is an established procedure that takes as input people's stated utilities about certain choices, and outputs a Pareto outcome, then it must be possible to game it by lying. The motivations of the players aren't taken into account once their preferences are stated.
Rather than X or Y succeeding at gaming it by lying, however, it seems that a disinterested objective procedure that selects by Pareto optimalness and symmetry would then output a (0.6, 0.6) outcome in both cases, causing a -0.35 utility loss for the liar in the first case and a -0.1 utility loss for the liar in the second.
Is there a direct reason that such an established procedure would be influenced by a perceived (0.95, 0.4) option to not choose an X=Y Pareto outcome? (If this is confirmed, then indeed my current position is mistaken. <nods>)
(0.6, 0.6) is not Pareto. The "equal Pareto outcome" is the point (19/31,19/31) which is about (0.62,0.62). This is a mixed outcome, the weighted sum of (0,1) and (0.95,0.4) with weights 11/31 and 20/31. In reality, for y's genuine utility, this would be 11/31(0,1) + 20/31(0.95,0.95)=(19/31,30/31), giving y a utility of about 0.97, greater than the 0.95 he would have got otherwise.
<checks the 19/31 through y = (-0.6*20/19)x + 1; nods>
<nods with realisation at the selected x=y=19/31 point corresponding to a different location in the true depiction>
(Assuming that it stays on the line of 'what is possible', in any case a higher Y than otherwise, but finding it then according to the constant X--1 - ((19/31) * (1/19)), 30/31, yes...)
I confess I do not understand the significance of the terms mixed outcome and weighted sum in this context, I do not see how the numbers 11/31 and 20/31 have been obtained, and I do not presently see how the same effect can apply in the second situation in which the relative positions of the symmetric point and its (Pareto?) lines have not been shifted, but I now see how in the first situation the point selected can be favourable for Y! (This representing convincing of the underlying concept that I was doubtrful of.) Thank you very much for the time taken to explain this to me!