Talk:Gibbard–Satterthwaite theorem

message: The Gibbard-Satterthwaite theorem, as written, is about linear orders mapping to linear orders.  The result holds for weak orders mapping to linear orders as well (ties in the ballots don't change things).  I explained this briefly in the entry, but I think we should have a reference for this.  I cited an overview work of Taylor, but if someone else has a reference to an older paper stating this (ideally for the first time), please replace my reference.  It almost makes it look like Taylor came up with the idea as I wrote it.
If anyone would like to look into this but has trouble getting started, I'd be happy to give you the list of references from the Taylor paper.  I'm looking through that and a number of other sources, but it could take some time. CRGreathouse (t | c) 05:17, 28 August 2006 (UTC)
thread_title:  Weak vs. linear orders 

message: It is important to note that the theorem only holds for deterministic systems. This is easiest to see if the following well-known non-deterministic counter-example ("Random Ballot") is considered: Each voter marks her favourite candidate on a ballot, then one of the ballots is drawn at random to determine the winner. This system is neither dictatorial, all candidates can possibly win, and it is always strategically optimal to honestly mark your favourite candidate. User:heitzig-j 04 Nov 2008  —Preceding undated comment was added at 03:28, 5 November 2008 (UTC)
replies: text: I agree and added non-random as a qualifier. Otherwise the result might be interpreted too widely. --Xeeron (talk) 12:27, 2 March 2010 (UTC)
thread_title:  Hidden assumption: Determinism 

message: The word 'dictatorial' seems to have a specific, technical meaning in this context- one I'm not familiar with. Could someone knowledgeable put it in the article? -Toptomcat (talk) 07:45, 8 November 2008 (UTC)
replies: text: I provided a link to Arrow's theorem. Kiefer.Wolfowitz (talk) 13:10, 21 February 2010 (UTC)
thread_title: 'Dictatorial'

message: Here's a little theorem with most of the same force as Gibbard-Satterthwaite but expressed in terms of manipulating the set of nominees rather than misrepresenting a voter's preferences.  In the context of public elections, where the set of nominated alternatives isn't fixed by nature and many voters are strategically unsophisticated, this may be more relevant than Gibbard-Satterthwaite's framework (which assumes a fixed set of alternatives).
:Axiom: Any ordering of alternatives is an admissible vote. (Universal Domain)
:Axiom: One of the alternatives must be elected. (Prime Directive)
:Theorem:  Given any voting method that reduces to majority rule when only two alternatives are ranked, both of the following hold:
:1. There is at least one collection of votes such that deleting a losing alternative from everyone's vote can cause another losing alternative to become the winner.
:2. There is at least one collection of votes such that inserting an additional nominee into everyone's vote can cause a losing alternative to become the winner.
:Proof:  Suppose the nominees are {x,y,z}.  Suppose about a third of the voters vote "x over y over z," about a third vote "y over z over x" and the rest vote "z over x over y."  Without loss of generality, assume x is the winner. (We could swap the alternatives' labels if necessary to make "x" the label of the winner.)  Suppose y, a losing alternative, is deleted from every vote.  Only two alternatives remain, so majority rule applies, and since about 2/3 vote z over x, the choice from {x,z} is z. That proves the first part of the theorem.  Now restore y to the votes; this changes the winner back from z to x, which proves the second part of the theorem.
Significance:  When many voters are not strategically sophisticated or will have trouble coordinating their voting strategies with other voters (such as agreeing with other voters on which compromise to vote for or rank on top), the outcome can often be manipulated during the nomination phase by adding or deleting alternatives.  This raises important questions: How are the nominees nominated, and by whom, and which (non-dictatorial, non-imposing, usually deterministic) voting method is least manipulable by strategic nomination? (Hint: take a look at the "Maximize Affirmed Majorities" voting method, which finds the order of finish that minimizes the size of the largest unaffirmed majority, in the minlexmax sense. Definition: A majority who ranked x over y is called "affirmed" if the order of finish ranks x over y.)
Example 1:  Due to the spoiling problem given plurality rule, many potential candidates choose not to run and each party nominates at most one candidate per office.  This kind of manipulation is not necessarily more harmful to society than nominating more alternatives would be (given plurality rule) but the lack of competition can cause many problems: The best candidates might not be nominated; parties (and their supporters) will regret their choice of nominee if negative information about him/her comes to light before election day; voters must accept a nominee's undesired policies along with those desired; issues don't get settled since a small swing of votes can cause many large changes of policies; accountability is undermined since "single issue" minorities can add together to form a winning majority coalition; and accountability on most issues is undermined since most issues (e.g., corrupt government payoffs to rich special interests) are not of paramount importance to voters and thus do not sway elections.
Example 2:  (Egregious problem with Borda voting method)  Suppose 66% of the voters prefer X over Y.  With most voting methods, X would beat Y by a landslide.  However, in many contexts it is easy to find an additional alternative Y2 that is very similar to Y, so the minority who prefer Y over X would prefer Y2 over X too, and also slightly inferior to Y, so that nearly all voters would prefer Y over Y2.  Given the Borda method, Y would win if the inferior clone is also nominated (assuming the voters' vote sincere preferences): X would receive 2 points from 66% and zero points from 34%, while Y would receive 2 points from 34% plus 1 point from 66%. The greater the number of inferior clones nominated, the larger the landslide that can be overcome.  If we assume a few supporters of X are strategically sophisticated too, inferior clones of X would also be nominated. (Sauce for the goose is sauce for the gander.)  This means public elections would turn into farces given the Borda method.
(Note: Gibbard-Satterthwaite is more general since it also covers voting methods that do not reduce to majority rule when there are only two alternatives.  A theorem similar to the above and as general as Gibbard-Satterthwaite can be proved, but its proof is more complex than the proof below, just as the proof of Gibbard-Satterthwaite is more complex.)
(Note 2: Voting methods that don't accept orders of preference from the voters, such as Approval voting or Range Voting, can also be shown to be manipulable by strategic nomination, given reasonable assumptions about how voters vote. For Approval, assume a significant number of voters will approve their most preferred nominee and disapprove their least preferred nominee. For Range Voting, assume a significant number of voters will vote the maximum score for their most preferred nominee and the minimum score for their least preferred nominee. These assumptions are reasonable because these strategies are optimal--other strategies waste at least part of one's voting weight--and they're obvious or easy to learn, like the strategy of not wasting one's vote on a candidate who appears to have no chance to win is obvious when using the "vote for one; plurality rule" method.  Note that if all voters vote optimal strategy when there are only two nominees, then Approval, Range Voting and most other methods reduce to majority rule when there are only two nominees, and if most voters vote optimal strategy, then a slight alteration of the proof shows that Approval, Range Voting and many other methods are manipulable by strategic nomination too. Also, even if different assumptions are made about voters' behavior, it may still be easy to demonstrate strategic nomination.)
(Note 3: A theorem like Arrow's can be fitted to this framework too, unsurprisingly. For instance, Independence of Irrelevant Alternatives becomes Choice Consistency: If x is chosen and y is not chosen when the set of nominees is {x,y}, then y must not be chosen from any set of nominees that contains x. For completeness, here's a simpler yet powerful Arrow-like theorem: Every voting method that always chooses at least one winner, satisfies Universal Domain and reduces to majority rule when only two alternatives are ranked fails Choice Consistency.)
SEppley (talk) 22:53, 27 August 2011 (UTC)
thread_title:  Similar theorem about "strategic nomination" 

message: The article says Arrow's theorem is similar to Gibbard-Satterthwaite except it is for social ordering functions instead of social choice functions. That's not the only difference, and in my opinion it's not the most significant difference, since Arrow's theorem can be rewritten for social choice functions (as a number of writers have done). SEppley (talk) 01:49, 28 August 2011 (UTC)
replies: text: Sen and others have thought that this was an important distinction. Please improve the article using high quality references, such as a reliable source explaining your  point.  Kiefer.Wolfowitz 12:30, 28 August 2011 (UTC)

text: @SEppley do you have a link to these authors? –Maximum Limelihood Estimator 20:26, 21 April 2024 (UTC)
thread_title:  The article's comment about Arrow's theorem is misleading. 

message: At least. The same statement can be made of psychosis (schizophrenia, irrationality): psychotic voter will not vote according to his preferences, even if following his own (? ...) preferences leads to his own well being while the contrary does not (crazy). DJB  — Preceding unsigned comment added by 4.35.92.11 (talk) 19:26, 30 May 2016 (UTC)
thread_title:  Tactical voting is a misnomer