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Seeking Advantage

The sample space ( N, {Vi}, p, M, {Rj}, f ):

We want points in which Vf > Vp.

Where are they?

They aren't in the part of the sample space where f = p,

because there, Vf = Vp.

The points we want have to be in the part where f != p.

We can divide the f != p section into three parts:

Vf > Vp : the ones we want

Vf = Vp : ones we don't care about

Vf < Vp : the ones we don't want

Let's say we had two points in which N, {Vi}, p, M, and {Rj} were the same,

but in one, f = p, and in the other, f != p, like this;

Could there be a rule, based on N, M, and {Rj}, that would help us pick the one with the greater Vf?

No. There is no such rule that would help more often than hinder.


Because for each such pair of points where the rule would lead us to a point in Vf > Vp,

there is another pair of points in which N, {Vi}, M and {Rj} are the same,

but the values of p and f are reversed,

and for this pair, the rule would lead us to the point in Vf < Vp.

There's one point in Vf < Vp for every point in Vf > Vp,

so a rule based on N, M, and {Rj} would lead you astray as often as not.