Dr Smith identified a class of voting systems where each voter is asked to send back a vector (one number per candidate). The vectors are summed. The candidate whose corresponding number is largest wins.
Many of the voting systems others have discussed are of this category. They each restrict what kind of vector get sent back. Assume, there are three candidates, B, N and G
In conventional voting, each voter simply sends back a one for a single candidate. That is each person has to vote for either B, N and G. thus, we may have the votes
Then, there is approval voting. Here, each vector is still limited to zeros and ones. However, we allow the voter to enter as many one's as they care to. In the above election, we might have.
Then, there is the Borda voting, where each person gives a rank ordering. And we count a first place as one more than a second-place vote. In a three-way election, we will have the numbers zero, one and two with two going to our first choice.;
Range voting gives everybody the most expressiveness. Everyone can put any number between zero and one. (Actually, one can set up range voting with any defined range, say zero to ten like in the Olympics.) Question for the reader--why don't we allow to the voter to put any number without restricting them to a range.
There are other voting systems that one can use. However, they have the disadvantage as the algorithm has to store all the votes. The above types of systems which Dr. Smith calls COAF total, can just work with the totals. I wrote about some of them earlier. Some of them take exponential computer time just to find out who won. Also, each voting machine has to send all the votes to the main office so it can find out the winner. COAF systems are better, the voting machine can send the totals for each candidate to the central machine. That would add the subtotals to find out who won.
Range voting gets out of a lot of the paradoxes and problems in voting theorems. Both Arrow and Gibbard assumed that voters have to give a ranking. In range-voting, each voter provides real numbers.
I have seen several descriptions of Arrow's famous impossibility theorem. I like Dr. Smith's explanation the best (which he attributes to Dr. Fishburn). Each voter gives a ranking of all the voters. My job as a programmer is to write a program that takes the set of ranks and generates a ranking of the candidates. (Of course, in an election for a single senator or governor, we only need to know who won, we don't need to know who the first loser is.) I can write the program anyway I want, but I have to obey the following rules. Arrow also assumed that I have a finite number of voters. (Of course, I question whether the thing might go away with a huge number of voters and) where the probability of a manipulation goes down with the cube of the number of voters.)
In any event, I cannot write an algorithm that fulfills all of these conditions.
- There is a finite number of voters--obviously. in an election with three (or more) candidates.
- If all voters agree than one candidate is better than the other will that candidate come out ahead
- Let there be two sets of V voters each and we run the algorithm in (two parallel universities). Only one voter differs between the two universes. However, they both rank candidates B more than G but rearrange their choices for other candiates. The algorithm should both rank B more than G (or the other way around). This is that the final result should not be affected by how voters behave on irrelevant alternatives. Thus if B wins, if everyone rearranged their choices ranked beledow B, it shouldn't change the fact that B has won. (I have seen some argue that this is not an important criteria.)
- We also don't allow me to write a dictator solution. That is, I can't just copy one person's choices and ignore everyone else's choices.
Dr. Smith programmed what I would consider the obvious simulation. Some voters are honest. That is they simply send in their utility. Others try to game the system. Dr. Smith calls them rational. They are simply the people who will vote the lesser of two evils. That is those who would prefer N to win but who look at the polls would vote for B or G.
He generated random sets of preferences for candidates. Then he simulates each of them and see how preferred the candidates for each voting system. The honest voters vote their preference. The dishonest (or "rational") voters look at the polling data and vote the way that they think will give them the best possible candidate. (Dr. Smith also includes some nice results for the probability that one's vote will affect the outcome for various models.)
Dr. Smith tried 144 scenarios and in all of them range voting was the one that selected a candidate that made people, on average, the happiest. (I should add that Dr. Smith's papers on voting system are full of wonderful details and arguments and are a joy to read. I urge all to go to his home page and read his political science papers--he also writes papers on a wide variety of topics unrelated to politics.)