[Election-Methods] utility theory lesson

[Juho Laatu]

On Jan 3, 2008, at 10:44 , Jobst Heitzig wrote:

> http://lists.electorama.com/htdig.cgi/election-methods- 
> electorama.com/2007-February/019584.html

I reread (or actually scanned) Jobst Heitzig's excellent old essay  
through again. I sent some comments already back then but now I  
identified another style of describing the intuitively problematic  
examples.

This explanation is based on observing the events of the actual  
world. I use quotation marks ("x") to indicate that x is an event of  
the real world.

The basic idea is that expression p?a:b can mean p?"a":"b" or "p? 
a:b". The first variant compares two different alternative events in  
the target world from a viewpoint of an external observer. In the  
second version we have actually three possible events of the world:  
the lottery, "a" and "b". If "p?a:b" will happen, the lottery will  
happen with certainty and "a" and "b" conditionally, depending on the  
outcome of the lottery.

In the second variant the preferences and utilities depend not only  
on how one sees "a" and "b" but also on what one thinks about the  
lottery. The second variant works better in many real life examples.

The Archimedean property (Archi) is not always true. We can assume <<  
For all a,b,c: if "b"R"c", there is some p>0 such that (p?"a":"b") 
R"c" >> to be true but << ... ("p?a:b")R"c" >> need not be. Also e.g.  
validity of (Exp) may have similar restrictions. With this approach  
the basic rules of the lotteries in the essay hold for lotteries that  
are 'external' but for real life lotteries the rules are more complex.

If the lottery is a real world event then the person that  
intentionally takes part in this lottery will know this afterwards.  
The state of the world will thus not be changed by "a" or "b" alone  
but also by the fact that a lottery was arranged.

Example 1:
We have one piece of chocolate left.
a = I'll eat it
b = you'll eat it
For me u("a") = 10 and u("b") = 9 but u("p?a:b") (where p=0.5) = 20  
(neither of us feels guilty/greedy then).

If "a" or "b" will happen just naturally, e.g. so that the last piece  
will be taken quite automatically by a different person than the one  
that just took the one but last piece, then the u("p?a:b") would be  
closer to u(p?"a":"b").

Example 2:
a = next week I'll travel to Hawaii
b = next week I'll travel to North Pole
The lottery ("p?a:b") may happen right now or next week. I prefer  
lottery right now since then I know better what kind of clothes to  
buy for the trip.

Example 3:
I go to Las Vegas.
a = I win $100
b = I lose $1
p = 0,001
My money related utilities correspond to the number of dollars I  
have. But I play this game ("p?a:b") anyway repeatedly since I enjoy  
lotteries. (This game offers me nice opportunities for daydreaming.)

What do you think? Is this a valid or useful interpretation, or did I  
miss something essential here?

Juho