[EM] Venzke's election simulations

[Juho]

I agree that fro simple simulations simple models and functions may be  
well be enough. I thought about different possibilities a bit an  
drafted one function that captures quite well what I had in mind. One  
function hopefully says more than thousand words.

f(x,y)  =  c*max(x,y)  +  (1-c)*min(x,y)  +  min(c,1-c)*((2a-1)*x 
+(1-2a)*y)

This function takes two arguments, x and y, that are opinions in two  
dimensions.

I assumed that human opinions / preferences can be expressed as values  
on a fixed range. In this case x and y values are in range 0..1. This  
means that if one wants to describe values with infinite range one  
must first use some function to map them to the used fixed range. That  
question is thus separate from the question of how to combine opinions  
from two different opinion dimensions, that I intend to discuss here.  
Also the results of the function are in range 0..1.

There are two adjustable parameters, c and a. Parameter c describes if  
the voter/person wants puts more emphasis on getting good results in  
both dimension or if getting good results in one dimension is enough.  
If c=0 then the function becomes simply min(x,y). This means that the  
voter is maximally happy only if both targets (dimensions) are met  
fully. And she is unhappy if either of the dimensions fail. If c=1  
then the function becomes max(x,y), and the voter is happy if she gets  
good results in either of the dimensions, and is unhappy only if both  
targets fail. If c=0.5 and a=0.5 then the function is neutral with  
respect to min/max behaviour. The function becomes (x+y)/2 and both  
dimensions have the same impact on the output. (In this case these two  
dimensions could as well be merged into one dimension only.)

Parameter a indicates which one of the dimensions is more important.  
If a=1 then x has more weight than y, and vice versa for a=0. If c=0.5  
(neutral with respect to min/max behaviour) then the function becomes  
a*x + (1-a)*y, and parameter a simply tilts the plane in either  
direction (x has more impact and y has less impact or the other way  
around). I made the function such that parameter c is stronger than a  
in the sense that when parameter c approaches 0 or 1 then the  
influence of parameter a goes down and has finally no effect. Also  
other approaches are possible but this one seemed intuitive enough.

These two parameters seem to cover a somewhat sensible variation of  
possible human behaviour/utility patterns. What do you think? Any use  
in simulation of human behaviour or elsewhere?

One simple approach to handling more than two dimensions would be to  
build the multi-dimensional space using multiple this kind of two- 
parameter functions. For example if I want world peace and clean  
environment, and I think that neither one of them is very enjoyable  
without the other, or alternatively I might be also happy if I got  
enough money, then my utility function in some election could be as  
follows. I'll use notation f(c,a,x,y).

utility(i) = f( 0.9, 0.4, f( 0.1, 0.6, world_peace(i),  
clean_environment(i) ), money(i) )

where "world_peace(i)" = "utility of anticipated level of world peace  
if candidate i was elected" etc.

In short this means "(world peace and clean environment) ... or  
money". "And" corresponds to "min", and "or" to "max".

Juho


P.S. Use Mac and Grapher and set parameters to:
View / Frame Limits...    =>    x: 0...1    y: 0...1    z: 0...1
c = 0.2
a = 0.9
z = c*max(x,y)  +  (1-c)*min(x,y)  +  min(c,1-c)*((2a-1)*x+(1-2a)*y)

P.P.S. There's some resemblance to Fuzzy Logic



On Jun 9, 2010, at 5:21 PM, Kevin Venzke wrote:

> Hi Warren,
>
> --- En date de : Mar 8.6.10, Warren Smith <warren.wds at gmail.com> a  
> écrit :
>>>> 1. I think using
>> utility=-distance
>>> is not as realistic as something like
>>> utility=1/sqrt(1+distance^2)
>>>
>>> I claim the latter is more realistic both near 0
>> distance
>>> and near
>>> infinite distance.
>>
>>> Why would that be? Do you mean it's more intuitive?
>>
>> --because utility is not unboundedly large.  If a
>> candidate gets
>> further from you, utility does not get worse and worse
>> dropping to
>> -infinity.
>> No.   Eventually the candidate as he moves
>> away approaches the worst
>> he can be for you, which is, say, advocating your death,
>> and then
>> moving the candidate twice as far away doesn't make him
>> twice as bad
>> from your perspective, and 10X as far doesn't make him 10X
>> worse.  It
>> only makes him a little worse.
>>
>> So my formula behaves better near infinity.
>
> A difficulty with this is that you have to know where this reduction  
> in
> effect (of distance) occurs in comparison to where the voters are. In
> other words are there really voters who advocate policies so bad for  
> me
> that I can't feel any difference among them, while they can?
>
>> Also, near 0 distance, it seems plausible there is a smooth
>> generic
>> peak, like the valley in U, not in V which has a
>> corner.    Hence
>> again my formula more realistic near 0.
>> Why should there be a singularity at 0?  Shouldn't
>> utility depend
>> smoothly on location?
>> If it should, then you must refuse to permit corners.
>
> This seems to have the same difficulty, of where is the curve? Suppose
> the issue is how close the bus will drop me off to my stop. Maybe  
> there
> is a curve... Maybe 1 meter isn't twice as good as 2 meters. But maybe
> 1 mile is twice as good as 2 miles. Within a simulation it's not clear
> what we're talking about.
>
>> Incidentally the formula could be
>>   A/sqrt(B+distance^2)
>> where A and B are positive constants chosen to yield
>> reasonable results.
>>
>>
>>>> 2. It has been argued that L2 distance may not be
>> as
>>> realistic as L1 distance.
>>> L2=euclidean
>>> L1=taxicab
>>
>>> That's interesting. I wonder what arguments were
>> used.
>>
>> --well, it was claimed.   It's
>> debatable.   If I differ from you on 3
>> issues, that ought
>> to be 3X as bad as 1 issue, not sqrt(3) times as bad.
>> It seems to make some sense.
>
> Yes, I'm thinking it makes sense at the moment.
>
>>> Well, it would be better to cycle over some of the
>> locations, but taking
>> the average over all possible locations would not be very
>> good evidence
>> either, since not all locations are equally likely.
>>
>> --average over the correct nonuniform distribution of
>> location-tuples.
>> I admit, what that is, is not obvious :)
>> But eventually you'll have to summarize in one number, which means
>> you have to
>> do this.  With some luck it may turn out not to matter
>> too much which
>> distribution is chosen from among a few reasonable ones.
>
> I think it's more likely that rather than guess at the correct
> distribution, I would try to categorize all possible scenarios  
> according
> to noteworthy effects.
>
> It's pretty clear to me that if you just toss out candidates randomly,
> RangeNS will usually win. It just happens that in the scenarios I pick
> out as being of interest to me, RangeNS isn't usually winning. So I  
> would
> like to investigate this to find exactly what are the circumstances  
> that
> cause methods like Bucklin or DAC to prevail.
>
> Kevin Venzke
>
>
>
>
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