[EM] Puzzle example X 1000

[Richard Moore]

MIKE OSSIPOFF wrote:

> >Then can you explain the following? I posted this at the end of my
> >previous message:
> >
> >"If we replace voters 1, 2, and 3 with three groups of 1000 voters
> >each, with the preferences and voting probabilities indicated in the
> >example, then I think the mean utility strategy would work (because
> >of the equal probabilities) even though it is not a true ZI case."
>
> What's to explain? For one thing, it's a completely different example.
> It isn't 0-info, but it isn't the initial example either. What the
> strategy is for the new example has no bearing on 0-info strategy.

Well, you seem to be missing the point. I'm arguing that your own
first choice influences any calculation about your second choice for
small numbers of voters. I'm giving a non-ZI case with perfect symmetry
in the probabilities, so all Pij (i != j) are identical, and I'm saying
that your first choice has little effect on those Pij, so you can ignore
that effect in deciding your second vote. (You said the same in your
last message: "But if there are lots of voters, then those 2 Pij can be
assumed to be the same, because a difference of 2 votes isn't
enough to change the probability density function by any appreciable
percentage.") Since the Pij are identical, the strategy in this case
is identical to ZI.

I'm then extending the example down to small populations. When the
population reaches four voters, you have my original example. But
we've already seen that ZI strategy fails for that example. So we
have to conclude that the Pij calculated from the external data only
will diverge from the Pij calculated from external data plus what you
know about your first choice.

Perhaps you think it is too much of a leap to say that this sort of
divergence would take place between ZI calculations of Pij and
calculations of Pij based on your knowledge of your own first
choice and no external data. But all I'm saying is that if you're
deciding whether to vote for B, then any information about any
other votes cast (whether by you or someone else) is relevant to
the statistics of the situation; only in a large population the
information about a single vote is so diluted as to be insignificant.

> This doesn't relate to the 0-info issue, but are you saying that each
> group of 1000 as a whole, flips a coin to decide which way to vote and they
> all vote the same way? Or are you saying that each of them flips
> a coin separately and decides individually based on that coin flip?

The latter was intended. The voters have similar tendencies but act
independently of each other.

> If it's the former, then your best strategy is to vote only for A.
> Solve it in the same way as in the initial example. Same method,
> different numbers, but same conclusion: Vote only for A.

> If it's the latter, then it would probably be quicker & easier for
> Joe than for you or me.

If your B vote has a probability k of helping B, it has a probability of
k/2 of hurting A and a probability k/2 of hurting C (B is just as likely
to be tied with A as with C). So your B vote's strategic value
(assuming B's utility is 70 as in the original example) would be:

-100k/2 + 70k = -50k + 70k = 20k

which is positive since k is positive. So vote AB.

> In any case, it's a completely different example, and so it has no
> bearing on 0-info strategy.

I wasn't making an assertion about ZI strategy per se, but rather
about the effects of one's own first choice on his or her later
choices. I believe this assertion is valid in both ZI and non-ZI
cases.

 -- Richard