In the system of \"approval voting\", a citizen may vote for as many candidates
ID: 1191753 • Letter: I
Question
In the system of "approval voting", a citizen may vote for as many candidates as she wishes. If there are two candidates, say A and B, for example, a citizen may vote for neither candidate, for A, for B, or for both A and B. As before, the candidate who obtains the most votes wins. Show that any action that includes a vote for a citizen's least preferred candidate is weakly dominated, as is any action that does not include a vote for her most preferred candidate. More difficult: show that if there are k candidates, then for a citizen who prefers candidate 1 to candidate 2 to ... to candidate k, the action that consists of votes for candidates 1 and A: - 1 is not weakly dominated.Explanation / Answer
Approval Voting has been adopted by a number of scientific and technical organizations, such as the American Mathematical Society, and the America Statistical Association to elect officers and make other important decisions. Also, approval voting is used to elect the Secretary-General of the United Nations and to elect new members into the National Academy of Sciences. It was proposed independently by several policital analyst in the 1970's and has many advantagous qualities. It works in the following way:
Note that, the system accepts as input only those preference orders in which the symbol > appears exactly once. So the only preference orders allowed are those that have one or more candidates tied for first place, followed by all of the remaining candidates, who are tied for last place
Approval voting is clearly not a dictatorship and also preserves Citizen Sovereignty. Howevery, to see that it is monotone, just consider a voter changing there preference order (for candidate A) to go from dissapproval to approval, this can only increase the candidates ranking in the societal preference order. Hence, approval voting is satisfies monotonicity.
We assume that people have a need to make statements, and construct a model in which this need is the sole determinant of voting behavior. In this model, an individual selects a ballot that makes as close a statement as possible to her ideal point, where abstaining from voting is a possible (null) statement. We show that in such a model, a political system that adopts approval voting may be expected to enjoy a significantly higher rate of participation in elections than a comparable system with plurality rule.
For approval voting, it turns out that this is indeed the case for several interesting scenarios: – If all voters are optimistic or pessimistic (not shown). – If there are at most three candidates and one of these hold: Voter preferences are governed by our “reasonable axioms”. A rational chair is breaking the ties (not shown). – If uniform tie-breaking is used and the voters are expected-utility maximisers.
Every voting rule for three or more candidates must be either dictatorial or manipulable. Let 'k' be a finite set of candidates and let P the set of all linear orders over k. A voting rule for n voters is a function f : P n k, selecting a single winner given the (reported) voter preferences. A voting rule is dictatorial if the winner is always the top candidate of a particular voter (the dictator). A voting rule is manipulable if there are situations where a (single) voter can force a preferred outcome by misreporting his preferences.
We can try to axiomatise the range of possible choices for we wish to admit. One very reasonable option is this: • is reflexive and transitive. • (DOM) A B if #A = #B and there exists a surjective mapping f : A B such that a f(a) for all a A. • (ADD) A B if A B and a b for all a A and all b B A. • (REM) A B if B A and a b for all a A B and all b B.