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". > > -- > > r b-j r...@audioimagination.com > > "Imagination is more important than knowledge." > > > > > ---- > Election-Methods mailing list - see http://electorama.com/em for list info ---- Election-Methods mailing list - see http://electorama.com/em for list info
