[EM] Condorcet Jury Theorem

[Greg Nisbet]

http://en.wikipedia.org/wiki/Condorcet's_jury_theorem

Let's pretend for the moment that we are attempting to determine the
truth of propositions rather than deciding on policy (this matters,
since policy decisions can't be objectively right or wrong and alters
what the "credibility" function would be, as I will describe later)

now the condorcet jury theorem has a bunch of assumptions, but two of
them are relevant for the question I wish to pose to the community
today

1) objective truth exists. A jury's decision is either correct or
incorrect and by the condorcet jury theorem this probability
approaches one as teh jury size approaches infinity.

2) the condorcet jury theorem assumes that all the jury members vote
completely independently of each other.

now for the purposes of democracy (1) doesn't hold true as stated.
there's no such thing as a "correct" policy decision. I suppose we
could modify our notion of correct to mean "correct according to the
correct utility function" but that ultimately doesn't get us anywhere
... so I'll just pretend that we're voting on propositions rather than
policy decisions.

now (2) obviously does not hold in real life. voter's guesses are not
independent of each other. That's why we don't expect to be able to
guess difficult math problems like "P = NP" or the like by proposing
them to the general population and seeing what most people vote on.
Ignorance has patterns to it... people are wrong in non-random ways.

so then let's say that this jury is voting on many propositions. Let's
also assume that they all vote honestly so that you game theorists
don't yell at me. Now that we have that covered, the independence
assumption becomes easier to fulfill.

we can identify non-independence experimentally, more or less, by
identifying correlation between individual jury members and adjusting
their weight according to how "independent" or "not clone-y" their
opinions are.

I posit that they weight of an individual jury member should be
f(c(m))/c(m) with m being the member in question

with f being the "credibility function" as I shall define below and c
being the "expected number of clones" as I shall define below.

the definition of the credibility function is f is as follows. f tells
you how the "effective credibility" of the the opinions of a group of
clones depend on the group size. In the case of democracy f(n) = n. If
two million people believe p, that is considered "twice as credible"
as 1 million people believing p. However, intuitively, this feels
wrong. Most of the earth's population believes in the existence of a
deity, yet that does not make the proposition more credible (the
proposition being that at least one powerful interventionist deity
exists)... Each marginal person believing the truth of the proposition
does not contribute as much to the probability that it is correct, I
argue, as the last person did. the choice of credibility function is
exogenous to the problem.

we also need to define the "expected number of clones". the expected
number of clones is at least one, since each person is a clone of
themselves... and this helps us firmly establish a maximum weight of 1
for each individual jury member. yay. Now the choice of definition of
c(m) is also exogenous. it depends on what you consider to be likely
indicators that two people are in fact "clones" ... or more
accurately, the likely *extent* to which they are clones of each
other.

for a democracy, I would argue the credibility function is f(n) = n.
this has some nice consequences, each person has an equal weight (1)
... and whether or not a particular voter votes identically to another
voter has no impact at all on how much either of their opinions
influence the outcome...

however for this question, we aren't dealing with a pure democracy...
we're attempting to determine the truth of propositions given
individual predictors that are fallible in non-independent ways. in my
view, this justifies a credibility function that isn't f(n) = n

a jury is usually small enough that a credibility function of f(n) =
(n>0) is good... i.e. if two voters have the same exact opinions on
all propositions considered so far, they will each have a weight of
1/2. so, in effect, it does not matter how many individuals represent
a given belief set, the effective credibility will not be altered.

now we need to define the "expected number of clones" or c(m)... we
need a model for how clones work... i.e. how they agree or disagree
with each other and how likely different forms of error are given that
they really are clones. there is some variation regarding what c(m)
could be, but I don't suspect that it makes a huge amount of
difference provided that the expected number of clones function is
reasonable to begin with.

so yeah, that's what I am wondering about at the moment. Just as a
disclaimer, I'm very drunk right now, so that might be to blame if I
explain something badly or fail to articulate the idea I thought of in
a reasonable manner. Thank you all.