# [EM] [CES #4445] Re: Looking at Condorcet

Juho Laatu juho4880 at yahoo.co.uk
Wed Feb 8 10:25:05 PST 2012

```On 8.2.2012, at 7.33, robert bristow-johnson wrote:

> On 2/7/12 6:30 PM, Juho Laatu wrote:
>> On 7.2.2012, at 5.31, robert bristow-johnson wrote:
>>
>>> how can Clay build a proof where he claims that "it's a proven mathematical fact that the Condorcet winner is not necessarily the option whom the electorate prefers"?  if he is making a utilitarian argument, he needs to define how the individual metrics of utility are define and that's just guessing.
>> Yes, I think Clay assumes that we know how the "aggregate utility" of a society is to be counted. There could be many opinions on how to define "aggregate utility" or "electorate preference", and also opinions that it can not be defined.
>>
>> It is actually not necessary to talk about those general concepts. It is enough to agree what the targets of the election are. Maybe Clay should tell explicitly that in this particular election that he considers the maximal sum of individual (sincere or given strategic) utilities to be the target.
>
> so he's seeking to maximize a measure of utility that is the sum of individual utilities and, again, i see no mathematical expression of the individual utility to sum up.  how does Clay maximize this sum of undefined quantities?

Yes, this is a problem. Clay's next explanation might be that people should normalize their ratings. (We would be back to one-vote-one-person then, but further away from the "utility sum" idea.)

>
> as best as i can tell, we only know what this quantity of individual utility is for a simple two-choice election.  assuming all of the voters are of equal weight, if the candidate some voter has voted for is subsequently elected, the utility to that voter is 1.  if the other candidate is elected, the utility to that same voter is 0.

Yes, we have now normalized the utilities to 0 and 1. We assume equal weight but not equal (sincere) utility difference between the two candidates (for each voter).

>
> but when there is a multi-candidate race, this is much more poorly defined.  say there are 3 candidates, if the candidate that some voter votes for is elected, the measure of utility (to that voter) is 1.  if the candidate that this voter ranked last is elected, the utility to that voter is 0.  but what about that voter's 2nd choice?  it depends who it is and who it is to the voter.  if we were to always assume that the utility is 1/2, then it seems like the kind of assumption Borda makes.  but the voter's 1st and 2nd choice could be very close to each other, or the 2nd choice could be a piece of crap just a little better than the last choice.  we don't know.  so how do you put together an argument that "it's a proven mathematical fact that the Condorcet winner is not necessarily the option whom the electorate prefers" when you just don't know whom the electorate prefers because you don't know the utility metrics for each voter?
>
> if you answer, "we ask the voters what the utility measure is with a Score ballot", then my response is: "how do you know that this is accurate?  that the voter even knows or that the voter isn't lying on his ballot to try to bury his 2nd choice or to compromise and forsake his favorite candidate?"  there are so many assumptions made here, it's like we're pulling numbers out of our butts.
>
> hardly constitutes anything approximating "a proven mathematical fact".

Agreed.

>
>
>>  And then he could continue to say that Condorcet is not designed to meet this target.
>
> how does he know when this target is not even operationally defined.

Condorcet targets are quite clear, so a badly defined method must differ ;-).

>
>>  Condorcet may however perform quite well as a method that approxmates that target in a highly competitive environment.
>>
>> For some other election the target could be to let the majority decide, or to maximize the worst outcome to any individual voter. Clay's target (corrected to refer to the sum of preferences based target of the election, not to the ambiguous electorate preference) may thus be valid for some elections but not all. (Also Range could be used to approximate majority decisions or Condorcet criterion, but only approximate.)
>>
>>> now, with the simple two-candidate or two-choice election that is (remember all those conditions i attached?) Governmental with reasonably high stakes, Competitive, and  Equality of franchise, you *do* have a reasonable assumption of what the individual metric of utility is for a voter.  if the candidate that some voter supports is elected, the utility to that voter is 1.  if the other candidate is elected, the utility to that voter is 0.  (it could be any two numbers as long as the utility of electing my candidate exceeds the utility of not electing who i voted for.  it's a linear and monotonic mapping that changes nothing.)  all voters have equal franchise, which means that the utility of each voter has equal weight in combining into an overall utility for the electorate.  that simply means that the maximum utility is obtained by electing the candidate who had the most votes which, because there are only two candidates, is also the majority candidate.
>> I wouldn't say that "the maximum utility is obtained" because that is a too much general utility oriented term. I'd say that "the maximum utility to the society, as agreed, is obtained". Or maybe "the most reasonable practical result is obtained" (based on the conditions that you gave). I thus want to see also your conditions as one possible agreed way to define the (in this case maybe only sensible) targets for the election.
>
> what other conditions could be agreed on?  Two-candidate is a given, High stakes and Competitive are pretty hard to agree to change, they are just there.  if you want to consider a variance to Equal franchise, then whose ballots are going to be attenuated?  will you be able to get those voters to agree to have their ballots each count less than your ballot?
>
> this is soooo fundamental.  all i want to do is get people to agree that when there are only two choices, that the candidate with the most votes wins, which is simple enough.  if you *don't* agree with this, what are the conditions you are envisioning for when election to office is awarded to the candidate with the fewest votes?  sometimes when considering a simplified case like this, you have to ask yourself about the contra-indication.  either you award the election to the candidate with the greater number of voters or you award the election to the candidate with the fewer number of votes.  i am astonished that anyone can see this in any more nuanced manner.  how would you *ever* award election to the less-supported candidate?

Yes, these assumptions are quite fundamental. We might have some other approaches like random ballot, but I assume that you had some extra requirements/targets that took care of this.

>
>>> if Clay or any others are disputing that electing the majority candidate (as opposed to electing the minority candidate) does not maximize the utility, can you please spell out the model and the assumptions you are making to get to your conclusion?
>> I think he made his assumptions / definition of the general utility of the society
>
> and what are they?   general utility of the society is equal to the sum of the individual utilities, so how are the individual utilities defined?  i don't see an answer there and i don't see how there *can* be an answer without making a lot of assumptions.  and then if you do that, i don't see much confidence in the answer arrived at.

Maybe one could claim that it is possible to agree verbally what the utility scale is (e.g. 2 = "quite bad"), and one could build a reasonably sensible model (including assumptions on summable utilities) for non-competitive elecions. But that model would not work well in competitive elections.

>
>>  and then assumed that this can be set as an universal target also for all single-winner elections. I wouldn't generalize that approach that much. For example majority oriented elections are a common practice in most societies. So we have at least two fundamentally different approaches to defining the targets of an election. For competitive environments I find your approach to be a very sensible approach. You can either assume that majority rule is what you want, or that majority rule is what you must satisfy with in a competitive environment.
>
> if it's not the majority that rule, what's the alternative?

I'm not aware of any good alternatives to majority rule in competitive two-candidate elections (with some extra assumptions that rule out random ballot etc.).

Juho

>
> once we can settle this simple issue, i'll move on to "why Condorcet".
>
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