On Mon, 2004-10-04 at 07:55, Eric Cavalcanti wrote: 
> On Sun, 2004-10-03 at 16:56, Stathis Papaioannou wrote:
> > Hal Finney writes:
> > 
> > > Stathis Papaioannou writes:
> > > >Here is another example which makes this point. You arrive before two
> > > >adjacent closed doors(...) However, this cannot be right,
> > > >because you tossed a coin, and you are thus
> > > >equally likely to find yourself in either room when the light goes on.

> > >Again the problem is that you are not a typical member of the room unless
> > >the mechanism you used to choose a room was the same as what everyone
> > >else did.  And your description is not consistent with that.
> > >This illustrates another problem with the lane-changing example, which
> > >is that the described mechanism for choosing lanes (choose at random)
> > >is not typical.  Most people don't flip a coin to choose the lane they
> > >will drive in.

I agree with both of these remarks. Of course that is not
the mechanism for choosing lanes, and that may be the
problem with the whole argument. As John M pointed out,
a highway is a very complex system, and we are treating
it in a model that might just not correspond to the real
thing.

On the other hand, trying to think of an idealized model
might not tell us anything about the specific problem we
are talking about, but can shed some light on a general
class of (at least gedanken) problems.

> > >Suppose we modify it so that you are handed a biased coin, a coin
> > >which 
> > >will come up heads or tails with 99% vs 1% probability.  You know about
> > >the bias but you don't know which way the bias is.  You flip the coin
> > >and walk into the room.  Now, I think you will agree that you have a
> > >good reason to expect that when you turn on the light, you will be in
> > >the more crowded room.  You are now a typical member of the room so the
> > >same considerations that make one room more crowded make it more likely
> > >that you are in that room.
In this case, I agree with the expectation you might have
in the end. 
> > Yes, this is correct. The "typical observer" must be typical in the way he 
> > makes the choice of room or lane. With the traffic example, given that there 
> > are slower and faster lanes on most roads, even in the absence of road works 
> > or accidents, this may mean that for whatever reason the typical driver on 
> > that day is more likely to choose the slower lane on entering the road. If 
> > this is so, then a winning strategy for getting to your destination faster 
> > could be to pick the lane with the most immediate appeal, then reflect on 
> > this (having participated in the present discussion) and choose a 
> > _different_ lane. This is analogous to counter-cyclical investing in the 
> > stock market, where you deliberately try to do the opposite of what the 
> > typical investor does.

I think I start to understand Bostrom's argument. In case we
don't know the mechanism that took us to the point we are;
and if it is reasonable to make the assumption that we just
followed the same mechanism as everyone else; and in case
this mechanism is biased towards one of the lanes;
THEN I can think of myself as a typical driver in the road,
such that it would be more likely for me to be at the slower
lane. 

The weak link of this argument is the third premise, though.
But provided that the mechanism is biased, and affects everyone
equally, there are grounds for Bostrom's reasoning.

> > But there may be a problem with the above argument. Suppose everyone really 
> > did flip a perfectly fair coin to decide which lane of traffic to enter. It 
> > is then still very most that one lane would be more crowded than the other 
> > at any given time, purely through chance. Now, every driver might reason, 
> > "everyone including me has flipped a coin to decide which lane to enter, so 
> > there is nothing to be gained by changing lanes". However, most of the 
> > drivers reasoning thus would, by chance, be in the more crowded lane, and 
> > therefore most would in fact be better off changing lanes.

In the road case, as Stathis points out above, it is possible
to make one of the lanes be more crowded than the other merely
by each driver randomly choosing a lane by chance. In fact,
it is very *unlikely* that both lanes would have the same speed
with such a pure random mechanism. 

But in this case, there is no basis for the third premise above.
Suppose, in the room problem, that instead of a biased coin,
everyone tossed a fair coin, as in Stathis original problem, and
enters a room by the decision of the coin. If the number of people
is large enough, it is highly likely that one of the rooms will
be more crowded. But as you enter one of the rooms, you have no
reason to believe that you are in the more crowded room, **even
though you followed the same mechanism as everyone else**, in
constrast with Stathis original problem (where there were already
1000/10 people in each room).

And this is the case where this problem is most paradoxical.
We are very likely to have one of the lanes more crowded than
the other; most of the drivers reasoning would thus, by chance,
be in the more crowded lane, such that they would benefit from
changing lanes; even though, NO PARTICULAR DRIVER would benefit
from changing lanes, on average. No particular driver has basis
for infering in which lane he is. In this case you cannot reason
as a random sample from the population.

Eric.

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