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A couple of friends ##n = 1 \ldots N## of mine bet on the result of the next general election in Germany. We select the six most important political parties ##p = 1 \ldots 6##. For each party ##p## and each friend ##n## we have the forecast ##x_{pn}## and the official election result ##x_p##.
Now we calculate
$$D_n = \sum_p w(x_p) \, d(x_{pn} - x_p)$$
with a weight-function ##w## and a deviation-function ##d##. The winner is the guy with smallest ##D_p##.
My question is, what are the most reasonable functions?
It seems natural to set
$$d(x) = x^2$$
$$w(x) = 1$$
which corresponds to the Euclidean distance with equal weight for each party.
But of course other choices are conceivable, e.g.
$$w(x) = x^c$$
weighting bigger parties more than smaller parties.
Are there any reasonable arguments and choices for the weight-function ##w## and the deviation-function ##d##?
Now we calculate
$$D_n = \sum_p w(x_p) \, d(x_{pn} - x_p)$$
with a weight-function ##w## and a deviation-function ##d##. The winner is the guy with smallest ##D_p##.
My question is, what are the most reasonable functions?
It seems natural to set
$$d(x) = x^2$$
$$w(x) = 1$$
which corresponds to the Euclidean distance with equal weight for each party.
But of course other choices are conceivable, e.g.
$$w(x) = x^c$$
weighting bigger parties more than smaller parties.
Are there any reasonable arguments and choices for the weight-function ##w## and the deviation-function ##d##?
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