Friday, 21 November 2014

D.G Saari dethroning Dictators & diminishing Dark Matter

 This is a Video where D.G Saari shows that assertions about 'missing' dark matter arise from bad maths. He tells us that his paper, though accepted as mathematically correct, was nevertheless rejected by a peer reviewed journal because the referee was professionally invested in MACHOs! However, one implication of his work is that the Universe might be infinite.

Saari ought to be a thorn in the side of lazy Social Choice type theorists who rely on bad maths to throw away information and conjure meaningless impossibility results out of thin air. In particular, Saari showed that Arrow's result arises by forbidding using information arising from the assumption of transitivity of agent preferences as a global constraint.

This paper of his shows the following-
Economic models as well as aggregation and decision problems with “holes” in the domain can be difficult to analyze because, unexpectedly, they are related to Arrow’s Impossibility Theorem: embedded within the model may be “topological dictators.” But, just as it is possible to remove the negative impact of Arrow’s dictator by recognizing that the problem is caused by not using crucial, available information (about voter preferences), the obstacles confronting these economic decision problems can be removed by identifying what kind of available information is not being used.

The problem here is that using information available from the assumption of transitivity of preferences means that the set of Choices can become infinite.

This is Arrow on the genesis of his General Possibility theorem-

Suppose only three equally qualified and capable people exist. A teaches Econ at a State School and earns 20,000. B, who has identical preferences to A, teaches Econ at a Private School and earns 18,000. A and B agree that B is worse off. C teaches at a worse Private School than B and earns 10,000. But, he is a snob. He thinks he is better off than A because he doesn't have to deal with proletarian scum. Arrow would say this puts paid to Hick's desire for 'interpersonal comparisons' based on purely ordinal considerations.
However, it seems there is some information available, from the assumption that agents are rational and have transitive preferences, which appears to be unused.
Consider the following. We decide to try to capture C's 'psychic benefit' by speaking of it as his Preference to have his own Preferences rather than someone else's. In this case, we know it is at least 10,000. Similarly A benefits by at least 10,000 by having his own preferences rather than being cursed with C's snobbish attitude.
Suppose B decides to have a meta-preference to be a snob like C. Then, suddenly, he is better off than A by at least 10,000.
A could turn the tables on B by explicitly embracing egalitarianism. He could say, 'the fact that I don't mind working in a State School is so valuable to me that it is worth twice whatever psychic gain B gets by his new meta-preference.'
Agent rationality, or preference transitivity, if considered as a global constraint, plausibly involves impredicativity of this type. But this means Saari's 'Intensity of Binary Independence' Decision rule can't gain purchase because the number of alternatives has become infinite.
But, maybe, that was always in the Math.

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