[EM] Is this cyclic preference example not rational? (Was: SFC definition supports partial preference relations)

[Jobst Heitzig]

Dear James,

First of all, I do *not* think that we should make the same mistake as
early economists did in assuming or even requiring too much rationality!
Is it not a bit too arrogant to say to someone, "You are not allowed to
have these preferences since they don't seem rational to me"?

However, you asked me for an example of cyclic but still rational
individual preferences, so let us discuss this one:


EXAMPLE:
	Let's assume there are two issues X and Y I consider important. Issue X
is the more important one, and there are two candidates A and B who are
equally strong at this issue. Issue Y is somewhat less important for me,
and the two candidates B and C are strong at it, but C being the better
one for issue Y. We could summarize this in a table like this:

		candidate
	issue	A	B	C
	X	100%	100%	0%
	Y	0%	50%	100%
	other	...	...	...

Let us further assume that although issue X is the more important for
me, the winner can only be successful at it when there's not too much
public opposition to that issue. For example, this issue could be
increasing the immigration. Being successful at issue Y, on the other
hand, would not require much public support. This could be legislature
to increase national security, for instance.
	When I'm asked for my pairwise preferences concerning A, B, and C, I
will have to imagine situations in which the respective preference is
decisive. For example, my choice between A and B will be decisive when
both A and B have a majority against C but neither having a majority
against the other. In that situation, I would choose B since she stands
for both of my important issues. In case that, on the other hand, both A
and C have a majority against B, I would choose A since he stands for
the more important issue X.
	But now assume that both B and C have a majority against A. This would
mean considerably opposition to issue X! Under that assumption it could
easily be rational to drop issue X as impossible to accomplish.
Consequently, I would then choose C instead of B since she is better at
the remaining issue Y.


I must admit that is seems perfectly rational to me to have the cyclic
preferences B>A>C>B here.
	The essential point seems to be that, in a *group* choice situation, my
preferences between A and B, when defined operationally by the question
"which would I choose if my choice was decisive", could depend on the
preferences of the other group members!
	The intuitive reasoning that when A is "better" than B and B is
"better" than C, also A must be "better" than C is misleading here
because the operational definitions of "preference between A and B" and
"preference between B and C" refer to different situations! The question
about my "preference between A and B" refers to the situation where my
choice between these two would be decisive, whereas the question about
my "preference between B and C" refers to the situation where my choice
between these other two would be decisive -- and these are two different
(and probably even mutually exclusive) situations!
	One could try to reformulate the question about the pairwise
preferences so that this becomes more clear. The real questions are:

	1. Would I prefer the situation
		(i)   "A and B tied on top, A elected" or
		(ii)  "A and B tied on top, B elected".
	2. Would I prefer the situation
		(iii) "B and C tied on top, B elected" or
		(iv)  "B and C tied on top, C elected".
	3. Would I prefer the situation
		(v)   "C and A tied on top, C elected" or
		(vi)  "C and A tied on top, A elected".

For friends of runoff elections, we could replace "tied on top" by "in
the last runoff round". When discussing immunity against majority
complaints, we could instead ask, Would I prefer the situation "A in
office, B contesting A, A remaining in office" or "A in office, B
contesting A, A being replaced by B", and so on.
	However we put it, (ii) and (iii) are obviously different situations,
so I do not actually violate transitivity when voting (ii)>(i),
(iv)>(iii), and (vi)>(v), that is, B>A>C>B !


When we proceed further along this line of thought, it seems that it
would be even more adequate to ask not only for pairwise preferences but
even for an individual *choice function*. That would be a function  f
assigning to each nonempty subset  M  of the set of all options some
nonempty subset  f(M)  of  M.  The meaning of  f(M)  would be that if
the voter could choose from M, she would draw an option from  f(M)
uniformly at random. In other words,  f(M)  are those options the voter
would consider optimal but mutually incomparable if her choice between
the options of  M  was decisive. With too much candidates, this is
certainly too much information to ask for on a ballot. But with only
three candidates, one could easily ask the following four questions:
	
	- Assuming A and B get the most support
	  and your choice decides the election,
	  whom would you choose?
		O  A	O  B	O  toss a coin

	- Assuming A and C get the most support
	  and your choice decides the election,
	  whom would you choose?
		O  A	O  C	O  toss a coin

	- Assuming B and C get the most support
	  and your choice decides the election,
	  whom would you choose?
		O  B	O  C	O  toss a coin

	- Assuming all three candidates get equal support
	  and your choice decides the election,
	  whom would you choose?
		O  A	O  B	O  C
		O  toss a coin for A or B
		O  toss a coin for A or C
		O  toss a coin for B or C
		O  draw any of them at random

The longer I think about it, the more would I want to be asked in
exactly this way -- but don't take this too serious :-)

Awaiting your opposition,
Jobst