Open Community
Post to this Blog
« April 2008 »
S M T W T F S
1 2 3 4 5
6 7 8 9 10 11 12
13 14 15 16 17 18 19
20 21 22 23 24 25 26
27 28 29 30
You are not logged in. Log in
Group One
Group Two
Lycos Search
Social Choice and Beyond
Thursday, April 3, 2008
Range Voting and Arrow's General Impossibiliity Theorem
Mood:  incredulous
Now Playing: Wagner: Der Ring des Nibelungen
Topic: Range Voting

The position taken by the Center for Range Voting regarding Arrow's General Impossibility Theorem is, I believe, that it doesn't apply to utility based systems and, therefore, range voting "escapes" Arrow's conclusion. I believe a much stronger position should be taken: that Arrow intends that his General Impossibilty Theorem DOES apply to range voting and that range voting refutes it. Certainly, Arrow's Theorem is valid given certain assumptions, but I don't believe that it's "general." Instead it should be called Arrow's Special Impossibility Theorem since it only applies to ranking procedures. Arrow certainly considers utility based systems and claims that they are invalid given his acceptance criteria. Consider the following quote from pp. 32-33 of "Social Choice and Individual Values":

"It may be of interest, however, to consider a particular rule for assigning utility indicators to individual orderings. Assume that the individual orderings for probability distributions over alternatives obey the axioms of von Neumann and Morgenstern; then there is a method of assigning utilities to the alternatives, unique up to a linear transformation, which has the property that the probability distributions over alternatives are ordered by the expected value of utility. Assume that for each individual there is always one alternative which is preferred or indifferent to all other conceivable alternatives and one to which all other alternatives are preferred or indifferent. Then, for each individual, the utility indicator can be defined uniquely among the previously defined class, which is unique up to a linear transformation, by assigning the utility 1 to the BEST CONCEIVABLE alternative and 0 to the WORST CONCEIVABLE alternative. This assignment of values is designed to make individual alternatives interpersonally comparable.

"It is not hard to see that the suggested assignment of utilities is extremely unsatisfactory. Suppose there are altogether three alternatives and three individuals. Let two of the individuals have the utility 1 for alternative x, .9 for y, and 0 for z; and let the third individual have the utility 1 for y, .5 for x and 0 for z. According to the above criterion, y is preferred to x. Clearly, z is a very undesirable alternative since each individual regards it as worst. If z were blotted out of existence, it should not make any difference to the final outcome; yet, under the proposed rule for assigning utilities to alternatives, doing so would cause the first two individuals to have utility 1 for x and 0 for y, while the third individual has utility 0 for x and 1 for y, so that the ordering by sum of utilities would cause x to be preferred to y.

"A simple modification of the above argument shows that the proposed rule does not lead to a sum-of-utilities social welfare function consistent with Condition 3. Instead of blotting z out of existence, let the individual orderings change in such a way that the first two individuals find z indifferent to x and the third now finds z indifferent to y, while the relative positions of x and y are unchanged in all individual orderings. Then the assignment of utilities to x and y becomes the same as it became in the case of blotting out z entirely, so that again the choice between x and y is altered, contrary to Condition 3.

"The above result appears to depend on the particular method of choosing the units of utility. But this is not true, although the paradox is not so obvious in other cases. The point is, in general, that the choice of two particular alternatives to produce given utilities (say 0 and 1) is an arbitrary act, and this arbitrariness is ultimately reflected in the failure of the implied social welfare function to satisfy one of the conditions laid down."

Clearly, Arrow is setting up the rules so as to produce NORMALIZED range votes which violates his own assumption that 1 should correspond to the BEST POSSIBLE alternative and 0 to the WORST POSSIBLE alternative. Had he actually used UNNORMALIZED range voting, the ratings for x and y would not have changed and this would have lead to a contradiction of his theory.

However, his theory would still apply to ranking procedures. My point is that Arrow's GENERAL Impossibility Theorem is not general at all but in actuality is a SPECIAL Impossibility Theorem. This deflates Arrow's balloon and should encourage those seeking new forms in political science and economics. I think the Center  for Range Voting should maintain that range voting invalidates or refutes Arrow's GENERAL Impossibility Theorem.


Posted by jclawrence at 2:35 PM PDT
Updated: Thursday, April 3, 2008 2:41 PM PDT
Friday, November 30, 2007
A Districtless Congress
Mood:  a-ok
Topic: Social Choice

The US is divided up into political districts. Each voter gets to vote for one member of the House of Representatives and two members of the Senate in this bicameral national assembly. There are 435 voting members of the House so there are 435 congressional districts. Cap1 The member from your district supposedly represents your interests, but none of the others do. In the Senate the political districts are the states. There are 100 senators, two from each state. The two senators from your state supposedly represent your interests but none of the other 98 do. So 1/435 or 0.22% of the members of the house represent your interests, and 2/100 or 2% of the members of the senate represent your interests.

This is a pathetic situation, and it's even worse if you did not vote for any of the congressman or senators who supposedly represent you. Say you're a Democrat and the congressman elected from your district (whom you didn't vote for) is a Republican. Then arguably you have no representation at all in the House. The same could be said of the Senate if you didn't vote for either of the senators who actually got elected. In other countries where they use other methods for making up the national assembly or congress like, for example, proportional representation, the percentage of the members representing each voter's interests is much higher. For example, if 28% of the electorate (including you) voted for the Green Party, then 28% of the seats in the national assembly would be Green Party members.

In a districtless congress each voter would vote for each representative, and each representaive would represent the interests of all voters. For example, if there are 300 seats in the congress and 500 candidates running for those seats there would be 500!/(300!)(200!) possible congresses or ways that this congress could be made up. In theory each voter could list each possible congress in order of his/her preferences, and then all the voters' specifications could be amalgamated to get the one congress that best represented the electorate as a whole. The problem is that it would be impractical for each voter to study the qualifications of each candidate and then come up with a list of all possible congresses. It would be too much work. However, there are ways to expedite this process as explained in more detail here. If each voter just listed the candidates (instead of the congresses) in order of preference, this list could be translated by software into an ordered list of congresses. Senate Furthermore, the list of candidates could be simplified by using the recommend- ations of the voter's political party or other trusted experts in part or in whole. A customized list could be generated by taking eclectic recommend- ations cafeteria style. Or there could be different lists available to the voter depending on the voter's profile related to his/her political objectives. A simple questionnaire given to the voter could generate a list according to the voter's predilections. There are a lot of different ways any particular voter's list could be generated with the voter having complete control and the final say.

One way of amalgamating the list information is by range voting. Using this method each possible congress would be given a numerical rating, and the ratings for each congress would be added up over all the individual voters to determine the winner - the one with the highest overall social rating. There is no need to rank the possible congresses in order over the entire electorate since only the top rated one would be chosen. Therefore, Kenneth Arrow's model for social choice and his impossibility results as presented in Social Choice and Individual Values are invalid.  In fact Arrow's model which calls for a full social ranking doesn't apply to most political as well as most economic situations. The only thing it seems to apply to is combining judges' rankings in Olympic figure skating where it is important to know not only first place but also second and third. In political and economic situations it's necessary only to know the top rated or first place result.


Posted by jclawrence at 2:48 PM PST
Sunday, September 23, 2007
Fractal Voting
Mood:  happy
Now Playing: John Fedchock New York Big Band
Topic: Fractal Voting

Fractal Voting (FV) is a voting method I developed. It is one of a class of utilitarian methods similar to range voting (RV) in some respects. Other utilitarian methods are approval voting (AV) and Evaluative Voting (EV). RV, AV and EV are all special cases of FV. Utilitarian voting methods involve a ranking of candidates or alternatives from most preferred to least preferred. In general there are two types of rankings: ordinal and cardinal. Ordinal ranking involves a list such as ABCD where the position of the letter, for example, indicates its preference ranking in this case A is preferred to B is preferred to C etc. Cardinal ranking also indicates how much A is preferred to B and how much B is preferred to C etc. This "how much" is also referred to as preference intensity. With utilitarian methods  numbers are usually asssigned to the candidates and the differences between the numerical rankings of two candidates indicate the preference intensity. One of the differences between FV and RV is that voters don't assign numbers to the candidates but indicate  preference intensities graphically.

Fractal Voting consists of a graphical user interface (GUI) or we could call it a graphical voter (GVI) interface (we will use the terms voter and user interchangably) and underlying software which translates the voter's preferences into numerical values which can then be used to sum up the votes for  each candidate and  determine the winner. Range Voting involves a system defined scale such as 0-99, 0-9 etc. The voter assigns a number contained in this scale to each candidate, and then the numbers are summed for each candidate to determine the winner. With Fractal the voter determines the scale and the scale can be more finely determined or less finely determined in different segments as the voter wishes. For example, if the voter wants to differentiate among candidates near the top of the scale more finely, he or she can subdivide that portion of the scale more finely in order to make these distinctions. The fineness of the scale is called the sensitivity level.  The sensitivity level, in general, will differ for each voter depending on how finely a voter can distinguish between two candidates or alternatives. This concept can be generalized to include fine distinctions between tastes or smells for example. A person might be asked to distinguish and rank several wines. Some people would be able to distinguish them very finely and others would only be able to make rough distinctions, say between good and bad.

With RV, the system defines the sensitivity level which is the same for all voters. RV with a scale 0-99 has a higher sensitivity level than RV  with a scale 0-9 and allows finer distinctions among candidates to be made. For example, candidates A and B might each be given ratings of 5 with RV (0-9) but given ratings of 51 and 57 with range (0-99). So finer distinctions can be made the more levels are available. With Fractal, the user or voter has complete control over the sensitivity which is variable over the whole scale. For instance, at the beginning of the voting process, the first thing a voter would do is to choose the number of levels he or she would like to start out with. This might be just two - good and bad. So there would be two "buckets" if you will, the good bucket and the bad bucket. All candidates in the good bucket would be indistinguishable from or indifferent to each other.

However, before choosing the number of levels or buckets, the voter would first choose their most preferred or favorite candidate or candidates and least preferred candidate or candidates. Then all others would be relative to those. So initially a screen would be presented to the voters with two buckets - one for most preferred and one for least preferred. The voter would drag appropriate candidates onto these buckets from a list arbitrarily located on the right side of the screen. Then the voter would choose how many buckets or levels to start with. Please note that, if the voter chose 100 buckets and didn't go any deeper or finer than that in terms of sensitivity level, the voting method would be the same as RV. After the initial choice of number of buckets, that number of buckets appears on the screen as well as the buckets at either end denoting most preferred and least preferred. Now the voter drags other candidates from the list onto the buckets. Then the voter has the option of clicking on any one of the buckets and further subdividing this segment of the  scale. Let's say that the voter started with 10 buckets and in all but one bucket there is only one candidate. In one bucket there are 4 candidates. Thhe voter may choose to click on that bucket and then choose to subdivide that bucket alone into, for instance, 4 finer levels. These buckets then appear on the screen along with the list of the candidates who were in the original bucket. The voter then drags candidates from this list onto one of the buckets that represent subdividions of the original bucket. This process can be repeated indefinitely leading to finer and finer distinctions. When the voter is satisfied he or she can terminate the process and submit his or her vote.

The final vote can be printed out as a paper ballot showing an overall scale subdivided as the voter has indicated and all candidates listed in order of preference, preference intensity and fineness of distinction or sensitivity. The underlying software can be implemented in terms of a push down stack where the first word in the stack contains the number of words in the stack. Initially, this would be 2 for most preferred and least preferred. These words might contain the numbers 1 and 0, respectively. As the voter adds levels or buckets, words are added to the stack.  The stack would be popped up to the level where the voter indicates that he or she wishes to add levels, and then the number of levels added that the voter has indicated. For instance, if the voter initially wants to order candidates just in terms of good and bad (a binary decision), two words would be added to the stack between the words corresponding to least preferred and most preferred. One might contain the number 1/4 (corresponding to the mid-point of the "bad" bucket) and one might contain 3/4 (corresponding to the mid-point of the "good" bucket). Continuing on in this way, the buckets are each defined in terms of a numerical value in a process that is totally transparent to the voter who just has to deal with a simple GUI and repeat the same process over and over to as many levels as he or she wishes. Then each candidate is associated with a pointer that points to the appropriate numerical value in the stack.

Since the process is the same for the voter no matter how deeply he wishes to proceed in terms of sensitivity level, we call this method Fractal Voting. Think of it as branches on a tree some of which are subdivided into smaller branches which are further subdivided and so on. At each stage the voter performs the same steps so the process is simple and intuitive for the voter.  This is the essence of the fractal process: no matter what the depth, the procedure is the same. At completion each candidate will be asssociated with a pointer which represents his numerical rank. The pointer will point to a word in the stack which will contain a value between 0 and 1 which represents the intensity  of that rank. Note that, unlike RV, the voter never has to assign numerical values to candidates making the provess simpler and more intuitive akin to punching a hole on a ballot or putting a check mark next to a candidate. When all voters have submitted their ballots, the numerical values associated with each candidate are summed and the one(s) with the highest value win(s).

The advantages of Fractal over Range are the following:

1) There are no "partial strength" votes. A partial strength vote is submitted in Range when a voter does not pin his most (least) preferred candidates to the limits of the range.

2) The voter has a simpler and more intuitive while at the same time more sophisticated interface which allows him or her more options in the voting procedure.

3) The voter can choose his or her own sensitivity level and can continue to refine this as the voting process continues.

4) The voter can go into detail selectively in those parts of the overall ranking that concern him or her while doing a rough ranking in other parts of the overall scale.

5) The voter need not be concerned with numbers at all, but only with a visual on-screen representation of the preference rankings and intensities.

Fractal Voting lends itself to delegable proxies since various parts of the tree could be designated and filled in by trusted parties who have pre-voted and whose results are only a mouse click away. For instance, the voter could select certain candidates, indicate he wished to make a proxy vote and then select Ted Kennedy from a list of proxies. Then these candidates would be added to the screen in exactly the way that Ted Kennedy had previously indicated he would vote. This method would lend itself either to touch screen or computer screen voting. Security of the vote could be guaranteed by different methods, but this is really a separate issue. Issuance of a paper ballot and receipt would be a start.

In summary Fractal Voting is a generalization of AV, EV and RV and a voter could choose to vote in any of these styles if so desired. It is a utilitarian voting method since the placing of each candidate on a line in order of preference ranking and intensity reveals the voter's utility for each candidate in some sense. Social utility could be measured for each candidate by simply adding up the numerical values asssociated with that candidate in the stack over all voters. This would not represent a social utility in an absolute sense but in a relative sense. The voter is allowed to make either fine or rough distinctions among the candidates according to his or her sensitivity levels and/or knowledge of the candidates, and also to rely on the advice of trusted experts who have studied the issues and/or candidates more closely. Both the GUI and the underlying software are easily implemented.


Posted by jclawrence at 11:08 AM PDT
Saturday, July 14, 2007
Discussion on Range Voting Blog between Mr. Lomax and Myself
Topic: Range Voting

This discussion took place on the Range Voting blog:

 --- In http://groups.yahoo.com/group/RangeVoting/, Abd ul-Rahman Lomax <abd@...> wrote:
>
> At 01:15 PM 7/4/2007, jclawrence2 wrote:
> >--- In http://groups.yahoo.com/group/RangeVoting/, Abd ul-Rahman Lomax <abd@>
> >
> > > If we image that everyone has this rubber band,
> > > then stretching it until all are the same length
> > > is equivalent to normalizing the extremes of
> > > attraction and aversion. This is what I referred
> > > to as the first normalization. A particular
> > > election does not utilize the full length of the
> > > band, but only a portion of it; the utilized length varies from
> >voter to voter.
> >
> >This is where we differ.
>
> We don't differ, at least not in this. Rather, Mr. Lawrence's concept
> is incomplete. If reality is A + B, and one person states that, and
> another says, "reality is B," do they differ? Perhaps. But only if
> the second says, "There is no A."
>
> Mr. Lawrence has well explained it here:
>
> >  I would make the max and min points of each
> >voter's rubber bands correspond to each voter's most favored and
> >least favored candidates, respectively.
>
> This is the second normalization that I referred to. If I apply that
> transformation, I end up with what Mr. Lawrence uses.

There are two realities here, and we need to be clear which one we are talking about. One is

the actual process that an actual voter goes through when he is composing a range ballot for

an actual election. The second is the computer simulation. Model A (Abd's model) assumes that

each voter will place each candidate on a cardinal preference scale from his particular

absolute worst to his particular absolute best candidate. Then the voter will normalize this

scale to the max and min of the range voting scale, i.e., 99 and 0 if that is the chosen

range with the absolute best at 99 and the absolute worst at 0. This results, however, in a

partial strength vote. If the voter wishes to have a full strength vote he will normalize

again so that his most favored actual candidate is at 99 and his least favored is at 0. Note,

however, that if a voter wants a full strength vote, he need only derive his cardinal voting

preferences without the absolute scale and normalize once to the max and min of the voting

range.


Notice that  Model B (my model) assumes that each actual voter will place each candidate on a

cardinal preference scale where the only meaning is the preference orderings and preference

intensities or the length of the gaps between candidates. There is no meaning attached to the

most favored or least favored candidate's position on this scale relative to an absolute

scale. There is no meaning attached to the strength or weakness of his preferences or

utilities. Then the range votes are derived by normalizing each voter's preference or utility

scale to the max and min range votes, i.e., 99 to the max candidate and 0 to the min

candidate (if that is the scale used).

The second reality is the computer simulation. Here we randomly generate a utility for each

voter-candidate pair. In Model A we first have to randomly generate the end points

corresponding to the actual (as opposed to the absolute) best and worst candidates. If the

minimum utility is 0% for the absolute worst candidate (and 100% for the absolute best) we

need to come up with some percentage for the actual worst and some percentage for the actual

best. These are the end points. Since we have no information to go on, we need to generate

these numbers randomly. So we have to assume some probability distribution over end points.

This introduces an arbitrary assumption. Then once these end points are determined, we have

to randomly generate the utilities which attach to the candidate-voter pairs. These utilities

are equally likely or uniformly distributed over the range from actual worst to actual best.

For instance, if the actual worst utility for voter 1 is a% and the actual best utility for

voter 1 is b%, then voter 1 would have a randomly generated equally likely utility for each

actual candidate somewhere between a and b. These utilities would then be used to compute

Bayesian regret. The actual numerical range votes would be determined by assigning b% 99

votes and a% 0 votes (if that is the vote range) and interpolating in between for the other

candidates.

For model B I propose a different computer simulation. A uniform distribution of utilities is

assumed over all the candidates because each candidate is equally likely to have a utility at

any point on the scale between 0% and 100%. For each voter one candidate (randomly chosen)

will be placed at 0% and one at 100%. For the rest of the candidates a utility is generated

for the voter-candidate pairs which is assumed to have a uniform distribution between 0% and

100%. No end points need be computed. The maximum possible social utility would be achieved

if every voter got his first place choice as winner. This is in general not achievable except

in a special case. The actual social utility would be computed for the computer generated

voter-candidate-utilities. It will depend on the statistical distribution of utilities over

the voter-candidate pairs. The regret would then be the difference between the social utility

if everyone got their first place choice as winner minus the actual social utility. This

would then be averaged over many simulated elections. This figure could then be used as a

figure of merit and compared to other voting methods.

>
> >  You would make the end points
> >correspond to each voter's ideal best and ideal worst candidates,
> >respectively, who are not even on the ballot. This gives rise to two
> >models. Let's call them model A (your model) and model B (my model.
>
> Model A precedes model B.

No, Model A exists in its own right and Model B exists in its own right.

> Let's consider the rubber band as something
> that exists internally for the voter, entirely aside from the
> particular election. The voter then comes to consider a set of
> candidates, and places the candidates on the band. Does the voter
> utilize the entire band?
>
> Normally, not. The "entire band" i[n] model A.
>
> Now, the actual internal reality is something like a combination of
> these two. People don't have fixed ratings for candidates, rather, in
> fact, they do, quite likely, end up with model B, and fairly
> directly. But these bands, fixed by ratings of the extreme candidates
> in the actual candidate set, are not related to each other.
>
> Warren, in his utility simulations, if I'm correct, generates random
> utilities for each candidate, placing them on a *complete* band.

If these utilities are generated in such a way that they tend to cluster around the midpoint

of the band and fall off towards the extremes of absolute best and absolute worst, then it 

is assumed that the utility for the average candidate for each voter is at the midpoint of

the band. In general this is not true.
 
> If the voters vote those untransformed utilities, we get minimized
> Bayesian Regret. But, in fact, we expect that many or most voters
> will normalize the votes, by tying the extreme candidate positions to
> 0% and 100% Range votes. This is a normalization, what I called the
> second normalization, and it loses information, particularly by,
> potentially, distorting weak preferences, making them seem strong.
> This is how utility maximization leaks from standard Range. It's
> still quite good, but it can be better.

I would argue that in reality the only data the system has regarding the utility of the

voters in an actual election are the range votes themselves. You don't have any utility

information on any absolute scale. You can only measure social utility in any meaningful way

relative to the actual votes cast, and the figure of merit would be the summation over all

voters of  the difference between maximum and actual individual voter utilities. For example,

if max utility is 100% and the candidate actually elected represented a utility of c% (c

greater than or equal to 0) for voter i, then summation over i would represent social utility

and 100% minus this sum would equal Bayesian regret.

The simulation for Model B then would be entirely analagous to what could be measured in an

actual election. Assuming only full strength votes, each voter's max and min candidates would

be chosen randomly and then the other utilities would be randomly generated assuming a normal

probability distribution from 0% to 100%.

>
> > > If the voters take the utilities from this
> > > normalized band and express them linearly, we end
> > > up with something like maximized social utility.
> > > But this is not how voters will actually vote.
> > > This is "honest true utility" voting, and it
> > > does, in fact, happen informally in deliberative
> > > process in functional societies. People *do*
> > > express mild preferences as mild preferences,
> > > when they are being honest and open with each
> > > other. And, yes, doing so means that those
> > > preferences will not be considered with the same
> > > weight as strongly expressed preferences. Which is exactly as it
> >should be.
> >
> >I don't agree. Some people have strong preferfences about everything.
>
> So? Some people only have strong preferences when they have clear
> knowledge. In a democracy, we allow people to determine for
> themselves what their preferences are, and how strong they are.

Even assuming honesty, there is no good reason for anyone to submit anything less than a full

strength vote. If a voter has limited knowledge, he can defer to a recommendation from

someone he trusts. Isn't that what your work on delegable proxy is about, Abd? He can

delegate portions of his vote to someone he chooses. Not assuming honesty, only the savvy,

sophisticated and "in the know" voters would vote a full strength vote, and they would tend

to get their way at the expense of the weaker voters. Is this what you really want: a system

that lets the more sophisticated exploit the less sophisticated? I would think that you would

want to eliminate such tendencies rather than enshrine them. That's what we have today!

>
> >Does that mean they should always get their way?
>
> If everyone else has weak preferences, yes.

I think this point is debatable. In a democratic voting system one of the hallmarks is

equality of inputs.Letting people with stronger preferences have more sway in the voting

system is tantamount to giving them more votes and people with weaker preferences less votes.

>
> >  For instance, my
> >girlfriend and I go to the movies. She has her preference for which
> >movie we should see and I have mine. But she always claims to really,
> >really, REALLY want (and need!) to see her movie.
>
> You reveal a great deal about yourself.

Not really. See below.

>
> >  The world is going
> >to come to an end if she doesn't get to see her preferred movie while
> >I would like to see my preferred movie but I can't claim the world is
> >going to end if I don't get to see it. Does this mean that we should
> >always see the movie of her choice? I don't think so. I think
> >everyone's preferences should count as much as everyone else's.
> >That's the principal of fairness.
>
> Then what in the world are you doing supporting an SU method, which
> makes strong preferences count for more than weak?

Actually, I support Model B which makes all voter's inputs have equal strength.

>
> And, by the way, I'd suggest resolving some issues with your
> girlfriend. You could spend the rest of your life in misery if you don't.

This was an entirely fictitious example, but I appreciate your concern, Abd, for my mental

health.

> >
> > > >  After all utility is
> > > >simply a way of attaching numbers to preference info. Then if you
> > > >wanted to set one end point equal to 100 range votes and the other
> > > >end equal to 1 range vote, you could figure out the numerical
> > > >relationships for all alternatives in between as far as voting is
> > > >concerned.
> > >
> > > Sure. However, unless by "end point" you mean the
> > > maximum and minimum utilities for the set of
> > > candidates in the election,
> >
> >I do.
> >
> >this becomes the
> > > second normalization.
> >
> >In model A this is the second normalization. In model B this is the
> >one and only normalization.
>
> Sure. And thus it floats with the candidate set, and the full range
> for each voter differs in preference strength from the full range for
> every other voter. This results in loss of SU accuracy.

It depends on your interpretation of SU. I submit that SU as measured based on the actual

votes in an actual election is the only meaningful SU. And the simulation for Model A does

not correspond to reality. It assumes knowledge of what goes on in a voter-bot's mind. And

the figure of merit is higher the more votes you assume are less than full strength. I think

it's a positive thing when all voters vote full strength for reasons mentioned above.

Therefore, the figure of merit should be higher not lower! So I have problems with the

validity of the simulation.
 
> >  > >  Actually, you could do this without ever stretching the
> > > >rubber bands to the same length. Just make the end points (1,100)
> >for
> > > >each individual voter. The range votes will always come out the
> >same
> > > >whether or not a linear transformation is applied to any or all
> > > >individual preference info.
> > >
> > > That is "stretching the rubber bands to the same
> > > length." What we are doing is expressing position
> > > relative to the extent of the rubber band.
> > >
> >
> > > > > Absolute Vote utilities (which are Absolute utilities
> >normalized to
> > > >a
> > > > > scale from absolute best to absolute worst of the whole
> >universe of
> > > > > choices, not just those appearing in an election)? Then these
> >are
> > > > > "converted" to Range votes. By whom? How?
> >
> >Model A would have the voters specify cardinal preferences on an
> >absolute scale where the max and min end points are the absolute best
> >and absolute worst candidates (who are not in general on the ballot)
> >FOR EACH VOTER. Thanks for clarifying that.
>
> You are welcome. If you read the material on the simulations, you
> would realize that the way the preferences and ratings are generated does this.

For reasons stated above, I don't agree that the simulations are valid.

>
> >  Model B would have the
> >voters specify cardinal preferences on a scale where the max and min
> >end points are each voter's most preferred and least preferred
> >candidates of the candidates actually in the race. These could be
> >converted to range votes by the voter or by the system. Perhaps if
> >the voter's cardinal preferences were expressed by placing dots on a
> >line or by moving a slider on a computer display, this in itself
> >would constitute the "vote." From there the system would process the
> >information.
>
> Model B, as stated, does not allow voters to weaken their vote. Why?
> What is the social benefit? Or, for that matter, the individual benefit?

Equality of vote strength is a hallmark of democracy. And it doesn't allow savvy voters to

get their way at the expense of "suckers" who vote a less than full strength vote.

> > > >I don't think there is a need for an absolute scale, just a
> > > >specification by each voter on some scale.
> > >
> > > "Need" according to what purpose? If we want to
> > > maximize social utility, or the overall
> > > satisfaction of society, we are attempting to
> > > maximize something that we assume is comparable.

If each vote has equal strength, then the satisfactions of each voter are comparable and

social utility is computed in a different way. Model B has a different definition of social

utility than does Model A. Model B still is utility maximizing - just a different definition

of utility.

> >
> >Model A attempts to maximize social utility. I have doubts as to the
> >validity of that since it favors those with intense preferences as in
> >the movie example I cited above.
>
> This is a common argument against Range Voting. Now, please explain
> why people with strong preferences should *not* be favored.

Every person deserves to have his vote have equal strength in a democracy. You think the

smarter people should be favored more because they know more and, therefore, have stronger

preferences. This is an elitist position. Even an imbecile deserves to have his vote count as

much as an Einstein in a democracy. He can use your methods and delegate his vote to a proxy,

for example. Only ignorant people would vote less than a full strength vote. Smarter people

would tend to get their way more often, and this does not mean that society would have more

utility only that smart people would have more utility. This is paternalistic.

>
> Strong preferences can come from various sources. On the negative
> side, they can come from bigotry, prejudice, ignorance, combined with
> arrogance. But on the positive side, they can come from knowledge.
> Imagine an omniscient voter. The voter *knows* what the various
> outcomes will do. And, if we assume that this voter cares about what
> happens, the voter will have very clear and very strong preferences.

Elitist. Paternalistic.

>
> There is a common moralistic judgement about strong preferences, and
> almost automatic assumption that they represent fanaticism and are
> thus to be ignored. I find it fascinating.
>
> Lawrence shows that he wants to get his way sometimes. Why? If he
> really doesn't care, and his girlfriend does, why should he insist?
> But, of course, he *does* care. Sometimes. And sometimes, then, he
> would have to express his real preference strength. He really does
> want to see *this* movie.
>
> Of course, he could go without her.

And sometimes I do. For instance, I saw "Sicko" the other night. She stayed home because she

thought it would be depressing.

>
> The situation he described is a setup. He is, he believes,
> easy-going, accommodating. She is hysterical and demanding.
>
> Yet, in fact, he is *not* easy-going. He resents her demands.

We usually work it out. We're going tonight to see "La Vie En Rose" together, and she will go

to see most chick flicks by herself. But, as I said, the example was entirely fictitious. A

less ad hominem response would have been more appreciated.

>
> What I ultimately came to in considering Range Voting is that we
> should assume that preference strengths expressed are real, that is,
> we should respect them, but we should also understand that the
> limitations of the system and the context mean that the expressed
> preferences can be distorted for various reasons. The enemies of
> Range use the possibility of distortion as a reason to not use the
> system, and they write of[f] strategic voting (voting approval style)
> with contempt, as if it is morally wrong. It's odd. Because it is
> morally wrong, the argument must go if expressed completely -- they
> never do it -- for a person to exaggerate preferences, we should use
> a system that equates all preferences, thus exaggerating some while
> devaluing others.

I'm not arguing against range voting. After all the only real difference between Model A and

Model B in an actual election is that in the former "weak" votes would be allowed and in the

latter all votes would have the same strength. Does that really invalidate Model B from being

considered range voting? And then I would argue for a different method for measuring voter

satisfaction or social utility or Bayesian regret, a different way of simulating elections

that is more or less equivalent to the measurements that could be taken for computing social

utility in an actual election.

>
> Does Lawrence know about Borda Count? It seems that he wants a ranked
> method....

Actually, I favor Range over Borda, but Range where all votes have the same strength.

>
> >Model A also favors "idealists"
> >whose preferences tend to span the full range from absolute best to
> >absolute worse as opposed to "realists" whose preferences tend not to
> >approach the extremes of absolute best and absolute worst. I'm
> >refering here to "honest" voters and not to strategic voters who
> >would tend to represent themselves as idealists.
>
> Lawrence has totally misunderstood what is going on. "Model A" is a
> simulation of what actually goes on in the mind of the voter.

I don't think what goes on in the mind of a voter is to place all actual candidates on some

absolute scale. Rather there are methods and algorithms using actual candidates for deriving

preferences and preference intensities. I think an actual voter would use these or just

interpolate between actual best and actual worst seat of the pants style.

>It's not a voting method, it's a simulation method. It doesn't favor anyone.
>
> *Voting* the "Model A" preferences (full range, including all
> possible virtual candidates), would be idealistic. But I certainly
> was not proposing that anyone vote that way. If we want to study
> utility maximization, however, we need to know some kind of
> commensurable ratings of the candidates, otherwise we are measuring a
> sometimes drastically distorted utility, where the mildest of
> preferences can loom as large as a live-or-die preference.

I'm assuming that in an actual election anything illegal could not be an alternative or

candidate. For instance, it could not be a legal election if one candidate ran on a platform

of genocide for some minority if he won. These shouldn't be life or death alternatives. And

the way the alternatives are structured has a lot to do with whether you might get freaky

results. So, yes, I would favor equal vote strengths for all voters regardless of whether

some voters thought they should have their way because their preferences were more intense

than others, and I would compute social utility from the baseline of equal vote strengths and

not based on an absolute scale.

>
> >What I am attempting to do in Model B is to take the arbitrariness
> >out of the process. The first step of that would be to eliminate the
> >first normalization in Model A. An appropriate metric for Model B can
> >be chosen for the purposes of comparing voting systems, but I would
> >probably shy away from claiming that it maximized social utility.

Actually, I would say it maximized social utility as defined.

> >I might say it maximized voter preference satisfaction but I'm
> >speculating a little. Don't nail me on this!
>
> What Lawrence is trying to do is to eliminate the information that
> shows us that pure Range is not absolutely ideal.

It depends on what you consider to be ideal, and you have to consider what can be actually

measured as opposed to what is a figment of your imagination.

>Indeed, he,
> apparently, would force voters to vote in such a way as to guarantee
> that utility is not maximized.

It depends on how you measure utility.

>It's actually quite arbitrary.

Well, I think Model A is arbitrary because it depends on voters using less than a full

strength vote in order to reveal what their utilities truly are. In Model B utilities are

revealed by the actual votes themselves and social utility is a function of the distribution

of preferences and preference intensities over the voters.

>
> What we prefer is to allow voters to decide how to vote. We [have] given
> them a tool, a means of expression, and we tell them how the
> information they express will be used. The *meaning* of it? We do not
> tell them that. They may tell each other that, various people will
> tell them that a 100% vote means this or that, and a 50% vote means
> this or that, but the system doesn't care what the votes "mean."
>
> It just adds up the numbers, or engages in other analyses.
>
> For example, if it is what I'm calling Range PW, the Range winner is
> determined by adding up the votes for each candidate. The candidate
> with the highest vote total is the Range winner, we might call this
> candidate the "nominee." Then the ballots are recounted, if
> necessary, to determine if any candidate is preferred to the nominee,
> i.e., beats the nominee in a pairwise ranked election using the Range
> ballots as data. It's fairly simple counting; because every pair
> involves the nominee, unlike Condorcet analysis, where every pair
> must be totalled. If no candidate beats the nominee, the nominee is
> elected. Otherwise there is a runoff election between the nominee and
> the candidate beating the nominee; since the Range winner is
> *usually* not beaten pairwise, this should be rare. (The question of
> what to do if two or more candidates beat the nominee, which should
> be extraordinarily rare, I'm not addressing at this point, though a
> complete method would have to consider that.)
>
> >I assume that the candidate that maximizes the amount of money in the
> >voters' (collective) hands should win the election. I would think
> >this would hold for Model B but not for Model A which would tend to
> >maximize the money in the hands of the idealists at the expense of
> >the realists, i.e., those who enjoyed the money more at the expense
> >of those who enjoyed the money less.
>
> Nonsense. Lawrence simply makes up arguments as he goes. We were
> discussing an economic Range election, and assuming honest voters.
> How much they enjoy their gains is irrelevant;

In social utility how much someone enjoys something certainly is relevant. Personal utility

is a function of enjoyment, satisfaction and a few other things.

> they simply express
> the value of each outcome. Why would one assume that this would
> concentrate wealth? What does "idealism" have to do with it?
>
> The election was not asking the voters how much they would enjoy
> their gains, but what those gains would be. It's an economic model,
> and it's useful because it shows how Range *does*, with honest
> voters, actually maximize overall social benefit. There is still a
> potential distribution problem, but that's a separate complication.
>
> (In theory, some outcome could supposedly make a single individual
> fabulously wealthy, but for this outcome to prevail, it must not
> steal this wealth from the many. Range is not zero-sum. In the
> imagination of Lawrence, the "idealists", presumably many of them or
> else it would not work, lose money and this is transferred to the
> necessarily few "realists," though it was completely unstated how
> personal philosophy was related at all.)
>
> Model A involves maximizing the *actual cash* in the voters hands,
> thus there is more to spread around. Model B maximizes differential
> cash. Model B might indeed take from the rich and give to the poor,
> because it may well overvalue cash in the hands of the poor (since
> the difference looms larger). The problem is that the overall wealth
> of society can thus decline. You can only soak the rich so far! And
> the benefit to the poor is necessarily small, since there are so many
> poor. It might be educational to consider what would happen if *all*
> the assets of the very rich were distributed. How much per person?
>
> And that's a one-time distribution. It wouldn't come again.

Could we stick to the basics? Bringing economic models into this is getting pretty far afield

especially when it isn't clear to me what your economic model is all about.

>
> >Well, it is another possible model. Call it Model C. In this model
> >everyone agrees on an "absolute best" and "absolute worst" instead of
> >their being different for each voter as in Model A.
>
> I don't think that Model A has been understood. Absolute best and
> absolute worst are *equated* for all voters in Model A.

This is not what you said above when you said that absolute best and absolute worst were

peculiar to each voter in Model A, i.e., each voter had his own absolute best and absolute

worst candidates not necessarily the same for each voter:

<quote:> >Model A would have the voters specify cardinal preferences on an
> >absolute scale where the max and min end points are the absolute best
> >and absolute worst candidates (who are not in general on the ballot)
> >FOR EACH VOTER. Thanks for clarifying that.
>
> You are welcome.
<end of quote>

> Every voter
> is presumed to have a measuring stick; the stick is of finite length.
> While we may imagine that it is a different length for each voter, we
> express preferences on that stick in terms of percentage of the
> length. So the extreme end of the stick in one direction is the
> absolutely most pleased that the voter could possibly be. There is no
> more satisfaction possible. And the other extreme is the worst
> possible outcome.
>
> The assumption is that human emotion is bounded. You can only get so
> happy and so unhappy. We rarely experience the extremes on the stick,
> so we might say that "I was so happy I could have burst!" But we
> don't burst. And we can be so unhappy that we decide to end it all.
> It can feel incredibly bad! But if something stops us from ending it,
> we were not crushed by the feelings, they were finite, it only seemed
> that they were endless and that the situation was hopeless.
>
> Anyway, the simulations assign utilities on this scale, for the
> candidates in the election. There is no assumption made that one
> candidate must occupy each end of the scale, for there is no basis
> for assuming that; in real life, such extreme preferences would be
> rare (I was about as opposed to Bush as possible by 2004, but I did
> not feel like committing suicide when he was officially elected, if
> we can call it that.)
>
> Thus the range of utilities for the candidate set vary from voter to
> voter. If voters were to vote the actual utilities, social utility is
> maximized.

I would argue that voters cannot even measure or compute their actual utilities only that

they can derive preferense and preference intensities according to some algorithm or seat of

the pants style. Social utility cannot be maximized based on something that's a figment of

the imagination.

> But we do not expect that they will. Rather, they will
> normalize to expand the range of ratings to fill the available range
> of votes. Some may not do this completely; these are "weak votes."
> They are allowed, and they are socially useful. But they are not
> recommended unless the voters preferences are truly weak. If there
> are two candidates, and you *really* like both of them, there is no
> reason why you should not vote 100 for both. It's effectively an
> abstention, but it expresses support and confidence in them, and
> allows others who *might* have preferences (for better or for worse,
> i.e., based on ignorance or knowledge) to prevail.

If a voter likes all the candidates extremely well, he can give them all 100s. If he dislikes

them all extremely and equally he can give them all 0s. Or give them all 50s. This tells

something about the voter but it doesn't affect the election.

> > > >  The problem with
> > > >that is not everytbody could agree on what the absolute scale
> >should
> > > >be. One person's pleasure may be another person's pain. Therefore,
> >it
> > > >seems preferable to me to take each person's specification of
> > > >preferences at face value and, for voting purposes, assign the same
> > > >value to every person's max candidate and vice versa for min.
> > >
> > > It may seem preferable to Lawrence because, for
> > > reasons which escape me, though I can speculate,
> > > he doesn't understand what is going on.
> >
> >The reason is *it takes the arbitrariness out of it!*
>
> It *adds* arbitrariness to it! But it appears that Lawrence isn't
> going to see this.

One man's arbitrariness is another's underlying principle and vice versa.

> > > What has been done, I believe, is to establish an
> > > internal set of utilities that take up positions,
> > > to use his analogy, on the rubber band. These
> > > utilities are distributed in a way that
> > > simulates, much better than mere random positions
> > > equally spread along the band, how people are
> > > really going to feel about candidates. By the
> > > nature of the simulation -- and in correspondence
> > > with reality -- candidates would only rarely
> > > approach the ends of the band.
> >
> >In Model A. In Model B they are the ends.
>
> And then comes a new candidate, declared at the last minute, better
> or worse than all the others. Suddenly the "end" of the band isn't the end!

This is a red herring. It's only the final vote that counts, and voters can reevaluate their

positions and change their votes if a candidate enters or leaves before the actual vote. If a

voter's vote changes before he actually votes, so be it.

>
> That doesn't happen with Model A. The new candidate simply takes up a
> new position on the band. All the other relationships remain the same.
>
> Model A is independent of irrelevant alternatives. Model B is
> sensitive to them. If we have a Range election, and A is going to
> beat B, introducing C can cause B to beat A, if the ratings are normalized.
>
> >[...] But if we peg
> > > the scales to the min and max candidate, the
> > > utilities for each candidate *will* vary with the
> > > candidate set (that is, a new max or a new min will shift the
> >utilities).
> >
> >True, but utilities will not vary if candidates which aren't max or
> >min enter or leave.
>
> Nor does it make any difference if Mars is retrograde. The point was
> that introducing a new candidate *can* shift utilities in such a way
> as to change the outcome *without* the new winner being the
> introduced candidate.
>
> If utilities are normalized.
>
> And we do expect them to be normalized, usually.
>
> I start few threads. One that I did start, fairly recently, was about
> "the obvious defect in Range Voting," and why it was also the best method.
>
>
> >Thank you, Abd, for clarifying this. I was assuming Model C which,
> >however, is a viable model and the one I think Warren uses in the
> >simulator.
>
> Warren uses Model A.

Maybe we should ask Warren.

> How the voters vote varies with the simulation.
> Using the utilities directly from Model A is what generates minimum
> Bayesian Regret, if I'm correct.
>
> Lawrence has confused voting strategy with the measurement of how the
> method works. What he calls "Method B" is a voting strategy, one of
> many possible.

No, Model B is a slightly different model than Model A. It has nothing to do with strategy.

Model A and the simulation thereof and the measurement of merit thereof is totally dependent

on strategy in the sense that it assumes a higher figure of merit if more voters vote

honestly, i.e., weakly, thereby ceding the strength of their vote to the more opportunistic,

dishonest and savvy voters.

>
> It's not entirely clear to me what Lawrence means by Model C. There
> is no external reference point, and, indeed, it is simply an
> assumption that max and min possible preference for everyone is to be
> equated. This is Model A, no more, no less.
>
> > > The "honest true utilities" are positions on the
> > > band, which includes the possibly unused extremes.
> >
> >That's Model A. Model B assumes that the end points are the max and
> >min candidates for each voter.
>
> Yes. However, Model B is a voting strategy, not a method for
> determining utility.

The sole difference between Models A and B is that Model A would allow "weak" votes and Model

B would make all votes full strength thereby guaranteeing equality of voter input for all

voters.
 
> > > In other words, there is something better than
> > > pure Range Voting. And Lawrence seems to be
> > > consistently avoiding acknowledging this, even
> > > though it is shown in the simulations.
> >
> >No problem with acknowledging there might be something better than
> >pure range voting. I'm just trying to pare down the process to the
> >bare essentials and not try to deal with all the sophisticated
> >variations right now.
>
> I think it would profit Lawrence to understand how the simulations
> are done. What is "the process?" There is a process used in the
> simulations for assigning utilities. There is a process for
> converting these to Range votes. The former can be varied, though I'm
> not sure how much it has been done, I think Warren uses an issue
> distance model for assigning utilities.

For now, can't we keep this simple? A randomly computed utility for each voter-candidate

pair? This is unnecessarily complicating the discussion.

The latter varies with the
> voting strategy. The only "strategy-free" conversion is the direct
> expression of the originally assigned utilities. Model A, to use the
> name we have been giving it here. Other voting strategies are used,
> including what Lawrence calls Model B. It's one of a number of them.

See above "The sole difference..." It has nothing to do with strategy.
 
> >If the voters just submitted their cardinal preference specifications
> >without numbers attached, i.e., dots on a line or sliders on a
> >computer display, then it would be moot whether they were submitting
> >a full or partial strength ballot. Again I'm trying to pare down to
> >essentials. I think the full or partial strength ballot issue is a
> >side issue I'm not really concerned with at this point.
>
> If there are sliders on a computer display, like slide
> potentiometers, then it is *not* irrelevant whether they submit a
> "full or partial strength ballot." It affects the outcome, and
> specifically it affects the power of the voter's vote. To avoid this
> -- some consider it a problem, I don't -- some would normalize all
> the votes. That is what Lawrence had suggested, and he seems to be
> continuing it here. The sliders would presumably correspond to a
> rating, so numbers would be involved, even if the voter didn't see
> the actual numbers. Normalizing all the votes is complex, far more
> complex than simple Range addition of votes. It would be difficult to
> do manually.

I don't know. It seeems fairly simple to construct a ballot on which a voter would place his

max and min candidates and then interpolate in between for the others. Both manual and

electronic versions would be straightforward.

>


Posted by jclawrence at 10:37 AM PDT
Updated: Saturday, July 14, 2007 10:42 AM PDT
Thursday, June 7, 2007
A Voting Method Combining Range Voting with Maximizing Social Utility
Mood:  happy
Topic: Social Choice

With various voting methods, there is a method for individuals to vote, and then there is a method for combining the votes to determine the outcome of the election. For instance, with the Borda Count, all the candidates are ranked by the voters with the top ranked getting a number of points equal to the number of candidates and the bottom ranked getting 1 point. Then the votes are counted by counting the total points for each candidate, and the candidate with the most points wins. Range Voting is similar except the top ranked candidate can be assigned an arbitrary number of points usually determined by the ground rules of the election. Also 2 or more candidates can be assigned the same number of points. Then the points are counted for each candidate with the one getting the most points declared the winner. 

Computer simulations by the Center for Range Voting have shown that Range Voting is superior to other methods in that it minimizes Bayesian Regret. Bayesian Regret measures the difference between the social utility produced by the candidate whom, if he had won, would have maximized social utility, and the social utility produced by the winner as computed by the voting system used, in this case, Range Voting. The social utility is the sum over the individual utilities of all the voters. This presupposes that there is a meaningful measure of individual utility which is a foregone conclusion as far as the Center for Range Voting is concerned.

I would argue that, although there are many definitions of utility and the definition of utility used by the Range Voters is basically preference utility, it is, nevertheless, a meaningful form of utility. Each voter's utility is essentially revealed by his vote. In Range Voting with a range from 1 to 100, for example, if a voter rated some candidate a 100 and then that candidate won the election, that voter's individual utility would have been maximized. The number 100 may not have any meaning in itself, but just because it is the maximum point value that can be assigned in this example, the voter would be considered to have achieved maximum individual utility. On the other hand, if a voter rated the winner of an election as a 1, minimum utility would have been achieved by that individual.

My suggestion is this. Instead of summing point values over all individuals for all the candidates and then declaring the winner as the one with the highest point total, compute the social utility which would be the sum of the individual utilities for each candidate and declare as the winner the candidate who maximized social utility. Obviously, this system would minimize Bayesian Regret over all other systems! An individual's utility for any candidate would correspond to the point value assigned to that candidate. This system could be used for Borda Voting, Approval Voting, Plurality Voting or Range Voting. In fact, Range Voting is the generalization of Borda, Approval and Plurality. Any voter could submit his vote as a Borda, Approval, Plurality or Range Vote within the confines of Range Voting. For instance, with a range from 1 to 100, if a voter wished to be a plurality voter all he would have to do is vote 100 for some candidate and 1 for all others. For Borda, he would equally space his point values from 1 to 100 and then assign them in order of his preferences. For Approval Voting he would assign 100 points to all those candidates he approved of and 1 point to all others. Finally, for Range Voting, he would distribute point values among the candidates corresponding to his preference intensities.

Another consideration is strategic voting. Some feel that Approval or Plurality Voting within Range Voting is strategic, that really the voter has an "honest" distribution of point values over the candidates but then maximizes some and minimizes others. But how do you know that, or, more to the point, how can you assume that for the purposes of computer simulations? Maybe the maximin voter truly feels that this vote represents his true utility distribution. All the voting "system" knows is what the voter reveals by his submitted vote. You really can't tell if the vote is a strategic vote or not, so why worry about it, and why berate some method because the social utility is assumed to be lower than it would have been if all voters had voted "honestly."

The maximum social utility that can be achieved is a function of the distribution of utilities among the individual voters, the domain, if you will. Some distributions (or elements of the domain) will produce a greater social utility than others. How much social utility that can possibly be achieved depends on the distribution of utilities among the voters.

Another objection is that the computation of the maximum social utility for any election is much more complex than simply counting up the points. This is true, but it can be done and it was done in the computer simulations done by the Center for Range Voting. Otherwise, it wouldn't have been possible to calculate Bayesian Regret. In fact, these calculations can be precomputed and stored much in the way Google precomputes search results in order to speed up the search process. In addition shortcuts in the computation process may be discovered.


Posted by jclawrence at 6:32 PM PDT
Updated: Thursday, June 7, 2007 7:24 PM PDT
Monday, June 4, 2007
Arrow's Borda Count Example
Mood:  caffeinated
Now Playing: CNN
Topic: Social Choice

 

The following quotation is from "Social Choice and Individual Values," by Kenneth Arrow. We want to examine the "reasonableness" of Arrow's example of the Borda count which he maintains violates his Condition3: Independence of Irrelevant Alternatives.                                                                                                                                          

 

THE INDEPENDENCE OF IRRELEVANT ALTERNATIVES             27

CONDITION 3: Let R1', ... , Rn' and R1', ... , Rn'  be two sets of individual orderings and let G(S) and G'(S) be the corresponding social choice func­tions. If, for all individuals i and all x and y in a given environment S,

x Ri Y if and only if x Ri'y, then G(S) and G'(S) are the same (independ­ence of irrelevant alternatives).

The reasonableness of this condition can be seen by consideration of the possible results in a method of choice which does not satisfy Condi­tion 3, the rank-order method of voting frequently used in clubs.2 With a finite number of candidates, let each individual rank all the candidates, i.e., designate his first-choice candidate, second-choice candidate, etc. Let preassigned weights be given to the first, second, etc., choices, the higher weight to the higher choice, and then let the candidate with the highest weighted sum of votes be elected. In particular, suppose that there are three voters and four candidates, x, y, z, and w. Let the weights for the first, second, third, and fourth choices be 4, 3, 2, and 1, respectively. Suppose that individuals 1 and 2 rank the candidates in the order x, y, z, and w, while individual 3 ranks them in the order z, w, x, and y. Under the given electoral system, x is chosen. Then, certainly, if y is deleted from the ranks of the candidates, the system applied to the remaining candidates should yield the same result, espe­cially since, in this case, y is inferior to x according to the tastes of every individual; but, if y is in fact deleted, the indicated electoral system would yield a tie between x and z.

A similar problem arises in ranking teams in a contest which is essen­tially individual, e.g., a foot race in which there are several runners from each college, and where it is desired to rank the institutions on the basis of the rankings of the individual runners. This problem has been studied by Professor E. V. Huntington,3 who showed by means of an example that the usual method of team scoring in those circum­stances, a method analogous to the rank-order method of voting, was inconsistent with a condition analogous to Condition 3, which Hunting­ton termed the postulate of relevancy.

The condition of the independence of irrelevant alternatives implies that in a generalized sense all methods of social choice are of the type of

2 This example was suggested by a discussion with G. E. Forsythe, National Bureau of Standards.

'E. V. Huntington, "A Paradox in the Scoring of Competing Teams," Science, Vol. 88, September 23, 1938, pp. 287-288. I am indebted for this reference to J. Marschak.

 

Now we examine this example in some detail.

Let's define social utility as the sum of the individual utilities where the individual utility for an alternative is equal to the point value of that alternative according to the individual's rating. For example, in the first case individuals 1 and 2 assign a point value of 4 to alternative x, 3 to alternative y, 2 to alternative z and 1 to alternative w. Individual 3 assigns a point value of 4 to alternative z, 3 to alternative w, 3 to alternative x and 1 to alternative y. The social utility then is 10 (4+4+2) for alternative x, 7 (3+3+1) for y, 8 (2+2+4) for z and 5 (1+1+3) for w. So the winner is x.

Now we consider the case in which y is removed from the election. With the Borda count, the social utility for x is 7 (3+3+1); for z: 7 (2+2+3); for w: 4 (1+1+2). As Arrow observes, there is a tie between alternatives x and z. Although Arrow thinks that the result should still be x, if you interpret the situation that z is now considered first by one individual and second by 2 while x is considered first by 2 and third by 1, a tie between x and z is not unreasonable!

However, let us consider the above example using Range Voting instead of the Borda count. With Range Voting and point assignments between 1 and 4, the point assignments for each individual remain the same. Then the maximum social utility is 12 in both cases (including y and excluding y). The result in the first case is the same: x wins with a social utility of 10. With y removed, the social utility remains the same since the individual point values remain the same. So x still wins with a social utility of 10! With Range Voting the election is truly independent of irrelevant alternatives at least in this example. This is explored more fully in my paper, "Social Choice, Information Theory and the Borda Count."


Posted by jclawrence at 1:19 PM PDT
Updated: Monday, June 4, 2007 1:39 PM PDT
Tuesday, April 18, 2006
Arrow's R Notation
Mood:  energetic
Now Playing: Cross Blogged with Will Blog for Food
Topic: Social Choice
In the arcane world of social choice, a man by the name of Kenneth Arrow looms large. In 1951 he published a book, "Social Choice and Individual Values," in which he supposedly proved that social choice is impossible. But what is social choice? Let us say we have a society composed of N individuals numbered 1,2,3, ... . Those individuals have to order a set of M alternatives with their most preferred alternative being their first choice etc. Let's indicate the alternatives as a, b, c, ... . Then a social welfare function accepts the individual orderings as inputs and produces as output the social ordering which is an ordering of the alternatives that applies to the whole society.

If individual 1 prefers a to b, we write aP1b. If society prefers a to b, we write aPb. So far so good. But we also want to provide for the case in which an individual is indifferent between a and b. We write this aI1b and aIb, respectively. Arrow's analysis then combines these two relationships into a relationship he denotes as R which means "prefers or is indifferent to" so aR1b means individual 1 prefers a to b or is indifferent between a and b. Arrow's rationale for this is the following: "Instead of working with two relations, it will be slightly more convenient to use a single relation, 'preferred or indifferent.'" (p. 12) (emphasis added)

Arrow then goes on to postulate two axioms. Axiom 1 states that either xRy or yRx and he notes that this does not exclude the possibility that both xRy AND yRx. Axiom 2 has to do with transitivity which will not concern us here. Again Arrow states (p. 13): "Axioms 1 and 2 do not exclude the possibility that for some distinct x and y, both xRy and yRx. A strong ordering on the other hand, [one with only preferences and without indifferences] is a ranking in which no ties are possible." This is blatant nonsense. One could have half the population with xPy and half with yPx [strong orderings] and that certainly would represent a tie so a tie is possible. What Arrow is implying without coming out and saying it directly is that in his world a tie between two alternatives is to be represented as a social indifference. This is completely arbitrary and limits his entire analysis.

One must assume that in Arrow's world each individual will submit his input in terms of R. That is individual 1 would submit aR1b, aR1c etc. until all pairwise comparisons have been made. For now we will go along with Arrow's demand that only pairwise comparisons need to be submitted. It can be assumed that individuals are not permitted to submit a comparison using the indifference relation since then what would be the purpose of introducing R to make the analysis "slightly more convenient." The whole idea of "slightly more convenient" is to reduce the number of relations from 2 (P and I) to 1 (R). However, Arrow proposes (without saying so) to use the I relation in the social choice to cover the case of a tie. Therefore, the social choice could be aRb, bRa or aIb.

Now the idea of the social welfare function (or of any function for that matter) is to connect each element of the domain (consisting of all possible combinations of individual choices) to an element of the range (consisting of all possible social choices). There are a great number of possible functions. Each function will hook up elements of the domain with elements of the range differently. The important thing is that each possible element of the domain is hooked up to one and only one element of the range. Arrow implies that any element of the domain that represents a tie (such as half the population having aRb and half having bRa) should be hooked up with the range element aIb. Respectfully, I disagree with this approach for the following reason: the half of the population that has aRb could actually prefer a to b (no one is indifferent), and the half of the population that has bRa could actually prefer b to a. That represents a tie to be sure, but society is hardly indifferent between the two alternatives. Arrow has confused a tie with an indifference! By so doing he has guaranteed that his analysis will yield the result that no social choice is possible.

Secondly, I would like to point out that individual information is lost when an individual submits his input as aR1b or "I prefer a to b or I'm indifferent between a and b." The system does not know which, and this introduces ambiguity at the outset. Not only that, but say an individual is indifferent between a and b. He has two ways to express it! He can submit either aR1b or bR1a. The resulting analysis becomes meaningless as the system knows not how many of the individual aRb's represent indifferences and how many of them represent preferences. Ditto for the individual bRa's! There can be no meaningful social welfare function given these kinds of inputs.

Therefore, I suggest that Arrow's approach is not acceptable and that his conclusion that social choice is impossible is invalid. A more rigorous approach is necessary involving the possibility of ties between orderings as elements of the range. One possibility of dealing with these ties is to randomly choose among them which I think my friend, Ben, at Oxford is considering as a doctoral these.

For more on this subject, please see my blog Will Blog for Food.

Posted by jclawrence at 1:44 PM PDT
Updated: Tuesday, April 18, 2006 1:50 PM PDT
Tuesday, February 28, 2006
Issues for a Social Choice Based Political-Economic System (3): Advertising
Mood:  a-ok
Topic: Social Choice
Would there be advertising in a social choice based economy? There wouldn't be any need for advertising since advertising is done only to entice you to buy a particular product. In a capitalist society, each company is privately owned and it's in its self-interest to get you to buy their products. This incentive would not hold in a social choice based economy or Preferensism. Instead of being enticed to buy, consumers would initiate the buying process by checking out the products available, the ratings of these products and the pricing. This can be conveniently done online where even today the entire range of products ratings, suppliers and pricing is available.

In Preferensism it is of no concern to society as a whole which products are chosen since whether or not certain business enterprises prosper or flounder is of no concern except for the fact that businesses that don't produce well or don't produce products that people want will be dismantled while other businesses will be expanded or started. Society is neutral and disinterested in which particular products are bought and sold. The workers, however, cannot lose their jobs since everyone has equal access to the job market. They will be reabsorbed in other businesses.

The motive for advertising today is to get you to buy a product or service so that the provider of that product or service can prosper - not because it is the best product or service for you. In Preferensism, the goal would be to assist the consumers in making the decisions that are best for them. Preferensist society is completely neutral as to which decisions are actually made by consumers because there are no vested interests. People don't have to fear losing their jobs. Enterprises don't need to fear going bankrupt. There would be no such thing as bankruptcy only disassembling and assembling of enterprises that would be done strictly in terms of the relevance of those particular businesses to society.

How to be an intelligent consumer should be taught in school since it is something people do almost every day of their lives. Resisting advertising and doing your own research into what is actually best for you is something that is possible today. The internet has websites that do ratings of products, price comparisons, ratings of suppliers etc. Advertising is no longer necessary in order to inform consumers as to what's available. A Google query on a particular generic term will bring up all the products that are available in that area.

Posted by jclawrence at 5:08 PM PST
Monday, February 20, 2006
Social Choice and Capitalism
Mood:  incredulous
Topic: Social Choice
In another post we compared a social choice based society or preferensism to socialism. The basic dissimilarity is in the ownership of private property. In socialism most of the property is socially owned. In preferensism there are less restrictions on property ownership. It can be both individually and socially owned. The basic similarity is that individuals are compensated for their work without the need for profit. So the work involved to produce goods and services consumed both individually and collectively redounds to the credit of both workers and property owners with each citizen having an equal say in how the work and compensation for it are distributed. In this way it is similar to socialism in the sense that socialism stands for "to each according to his work." Socialism does not imply equality of compensation since some people are capable of more and better work than others. The same can be said of preferensism. However, in preferensism property owners get a say in the outcome equal to every other citizen's say whereas in socialism there are no property owners, and everyone is a worker. In preferensism some might be strictly workers, some may be worker-property owners and some may be strictly property owners.

The similarity of preferensism with capitalism is the emphasis on innovation, invention and individual initiative. Innovators and inventors should definitely be rewarded. Intellectual property such as patents, music, artwork and literary works should be rewarded. The dissimilarity with capitalism is that owners of property would not have the sole determination of how that property would be priced in the marketplace. Prices are set according to an amalgamation of all citizens' preference lists.

In preferensism corporations would not be privately owned. They would be set up and dismantled according to the demand for their produced items. The realm of private property would include real estate, natural resources, intellectual property and anything else that can be accumulated. However corporations or businesses as entities in themselves would not be considered private property since only individual persons have a right to vote. In capitalism corporations are set up with the rights of individuals. In preferensism only individual persons would have the rights of individuals.

Therefore, there would be no stock market in preferensism since there would be no privately owned companies. In capitalism even publicly traded companies are privately owned in the sense that they are not owned by all of the citizens in general but only by that group of citizens who have bought stock in that company. A business enterprise would be set up in preferensism according to the demand by the citizens for the products which that enterprise would produce. Therefore, supply would equal demand. Several competing enterprises could be set up just to keep everyone honest and to distribute the work geographically. If only one enterprise were set up, there wouldn't be any checks or balances on that one business. Several competing businesses could keep things in check. But they wouldn't be competing price wise only in terms of efficiency and quality.

So relationships among producers, consumers, workers and property owners would all be regulated in preferensism so that no one citizen had any more power over the process than any other. In capitalism lone individuals are no match for large corporations.

In preferensism the law of supply and demand would be operative but would be contained or held in check due to the fact that aggregate consumer demand would be calculated from citizens' preference lists and work assignments and business start-ups and contracts would be such as to satisfy that demand. An excess beyond consumer demand would not be produced while production insofar as is possible would equal demand. Therefore, the aggregate amount of work and consumption of resources would be minimized since nothing in excess would be produced. No one would have an advantage in the economic process in the sense that they would be able to corner a market or dictate prices. Likewise, no business would have to fear bankruptcy since the businesses as entities in themselves would be assembled and disassembled according to societal needs.

The law of supply and demand would be in balance. There would be no unemployment since everyone would have equal access to the job market. The work week would be minimized and tailored to individual wants since work would tend to be spread around among as many people as possible. There would be no gluts in the marketplace since nothing would be produced that wasn't already earmarked for consumption. There would be no overproduction and no shortages. Natural resources would be conserved since there would be no wastage or wastage would be minimized.

Not everyone would have an equal outcome. Some would be richer than others, but the gross inequality that exists in capitalism would tend to be mitigated due to the fact that instant billionaires due to IPOs would not be produced.


Posted by jclawrence at 4:31 PM PST
Updated: Monday, February 20, 2006 8:09 PM PST
Friday, January 20, 2006
Issues for a Social Choice Based Political-Economic System (2): Welfare
Mood:  chillin'
Now Playing: Miles Davis
Topic: Social Choice
Unlike socialism or communism which claim to right the wrongs of capitalism, unequal distribution of wealth being among them, a social choice based system or Preferensism would not necessarily be a panacea for the ills of society such as poverty, ecological destruction or unequal distribution of wealth. Basically, since it's an extension of democracy, it would reflect the wishes of the people. If the populace wanted to collectively help poor people or those who are disadvantaged, they could do so. If they didn't want to, they could vote their preferences to do that.

The same goes for the environment. If people wanted to protect the environment, they could vote to do that and vice versa. The only qualification would be that Preferensism is not necessarily a majoritarian system. In other words it caters to minorities. So a minority could vote to help poor or disadvantaged people, and some help would be provided. Also the amount of help would be a function of the aggregate of all citizens' preference lists. Therefore, it is likely that those who want to work an extra amount in return for generating funds to help the poor or funds to ameliorate the environment could do so. How many people list these concerns how high on their list of preferences would determine the funds that would be allocated to those areas.

So it wouldn't necessarily be an up or down vote: either to help poor people or not. Chances are that funds would be mostly allocated out of the paychecks of those who have those concerns as high priorities, but little or no funds would be allocated out of the paychecks of those who don't have those concerns as high priorities.

Therefore, liberals or those who favor government help for the poor and disadvantaged could have it their way and vote to help the poor and disadvantaged. Conservatives or those who are not in favor of government helping the poor and disadvantaged could vote not to help them. The social choice function or social decision function would seek to maximize the utility of the entire population. Not everyone, of course, would get their top preference, but the system would attempt to give everyone as high a preference as possible.

There would be a variation over the entire population in terms of how much each individual would pay depending on his or her preferences. So everyone would not be "taxed" the same but only according to their preferences.

"From each according to his preferences; to each according to her preferences" subject to the maximization of utility with each person being treated the same in the process characterizes Preferensism.

Posted by jclawrence at 4:37 PM PST
Updated: Friday, January 20, 2006 4:40 PM PST

Newer | Latest | Older