I should get email notification of comments I thought (it used to happen...), but must look into that again after realising I missed two. John Lawrence comments on my introduction to lottery-voting, feedback on which is always very welcome (it being my PhD topic)
He plugs his own site, which looks interesting. I haven't had much time to nose around yet, but think I'll be back for a proper look later. I'm not thereby agreeing with or endorsing his views however - some of his criticisms of Arrow seemed (as far as I understand him and Arrow) mistaken. Disagreeing with someone isn't reason to ignore what they might have to say that's of interest though, and he runs his own blog too.
This post is largely a 'reminder to self', but if anyone else is interested they may want to look too. I'd welcome any comments.
Ben,
ReplyDeleteThanks for linking to my website. My criticism of Arrow has to do with his Condition 3, Independence of Irrelevant Alternatives. In "Social Choice and Individual Values," he says what if everyone voted and then one of the candidates dies. Shouldn't you just cross out that guy from everyone's preference list and apply the social choice function again. Thyen he says that should yield the previous social choice with the dead candidate's name crossed out.
Wait a minute. If one candidate dies, your really in the situation of predicting how the voters wopuld have voted given the original alternative set minus the dead candidate. Therefore, you're in the realm of probability, and the best way to predict how the voters would have voted is not to run the same social choice function again on the original set of prefernce lists with the dead candidate's name removed.
#2) Arrow insists that all voters make binary comparisons among all the candidates. Why? Shouldn't the voter be free to vote differently on different candidate sets basing hi9s p[reference list on the entire set and not just binary comparisons.
For example, if you had 4 candidates: A, B, C and Hitler, shouldn't the voter be free to put A, B and C tied for first and Hitler last in his preference list with the rational that anyone would be prefered to Hitler? Then if Hitler dropped out, shouldn't the voter be free to rank A, B and C in some order (not tied)?
The trouble withn the Borda count is that it constrains the voter to rank the candidates without ties. In the above example, a voter would have to vote (for example) ABC(Hitler) which would deny him the possibility of voting anyone except Hitler.
My voting method (if anyone else discovered it, Id be happy to give them all the credit) is a modified Borda count in which there are a number of slots (not necessarily equal to the number of candidates)and voters can put more than one candidate in each slot. Similar to the Borda count the top slot would get a count of N-1 etc. (assuming N slots.
With the computing power available today there's no reason why a more general voting scheme involving the amalgamation of preferences isn't possible.
My final criticism of Arrow is that his Impossibility Theorem doesn't apply if all the voters revote if and when the candidate set changes. Instantaneous revotes should be possible with computing power available today.
Good luck with your PhD.
My email is j.c.lawrence@cox.net.
I should have signed in the last post with my name instead of anonymous. Pardon all the typos. I forgot to proofread.
ReplyDeleteOne final comment. I have another blog than the one you linked to and there I define a social system called Preferensism. The link is http://willblogforfood.typepad.com/will_blog_for_food/2005/12/what_is_prefere.html.
ReplyDeletePreferensism is a combination of social choice and utilitarianism.
@John: "The trouble within the Borda count is that it constrains the voter to rank the candidates without ties. In the above example, a voter would have to vote (for example) ABC(Hitler) which would deny him the possibility of voting anyone except Hitler."
ReplyDeleteIn the textbook by Mas-Colell/Whinston/Green the Borda count is presented in a generalized way with indifference allowed. The tally/per candidate is the average rank of candidates indifferent to that candidate. So with the rankin A>B~C we get T(A)=2 and T(B)=T(C)=0.5.
Sadly, the most exhaustive analysis of the Borda Count (due to D. Saari) uses only strict rankings of individuals. So we don't know much about this generalization.
I think this is a weak problem, but you obviously disagree.