Math 112

What can go wrong if independence doesn't hold?

(Theorem) If a *voting method violates the property of independence*, then there necessarily exist *elections within that voting method that are manipulable*. If a voting method fails independence, then there exist elections within it in which some voter v has a lot of power— even if they don't know it. Indeed, as (4.4) and (4.5) made clear, v may well be better off submitting a ballot that differs from their "genuine" ballot (even if they don't know it)— and this is precisely game theory. Having said that, the Theorem doesn't tell us which elections, in particular, can be "gamed," how likely they are to occur, which lucky voter it is who can alter the outcome all by themselves, or what their strategy for doing so 42 should be. (Indeed, go back to the 1998 Minnesota election: it's impossible that one voter could have changed the outcome by altering their ballot, even though plurality voting, as we saw, violates independence.) However, what our Theorem lacks in detail, it more than makes up for by its generality — *it applies to any preferential ballot voting method that fails independence!*

The property of the independence

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dictatorship

A dictatorship is a voting method in which one voter — the designated "dictator" — determines the winner of any election all by themselves; i.e., in the final ranking, one candidate, say A, will be ahead of another candidate, say B, if and only if the dictator wanted it that way! Only the dictator's ballot matters.

Condorcet winner

An option that beats all other options in a series of pair-wise contests When a candidate beats all other candidates head-to-head, that candidate is known as the Condorcet winner, named after the Marquis de Condorcet. (We shall discuss this at length later.) Question: if a candidate is the least favorite of a majority of voters, must they also lose head-to-head contests with all other candidates? I have to check. its definitely more likely theyll lose.

what is wrong with instant runoff voting

Can violate the Condorcet criterion -

Plurality

Candidate or party with the most votes cast in an election (not necessarily more than half). We can have the absurd situation wherein a candidate who is literally the least favorite of a majority wins the election outright. *However, it's not possible, in a plurality election, to increase a candidate's chances of winning by strategically not voting for them.* For strategic manipulation, if there are three or more candidates, they can switch their vote to ensure their favorite wins. all other candidates combined must have less than 50%. No matter what, no other candidate can have as many votes as the majority candidate. Hence, any candidate with a majority must necessarily be the plurality winner, so the plurality method satisfies the majority criterion. Satisfies majority criterion: a majority candidate has the most first-place votes. • Satisfies monotonicity: if in a reelection the votes change only to favor the previous winner, there can only be more first-place votes for the candidate that already had most of the first-place votes. • Violates the Condorcet criterion: in Election 1, B is a Condorcet candidate yet loses the election by plurality.

Vote splitting

Candidates similar to B hurt B's chances in the election.

Monotonicity

Ensures that time stamp values always increase (?)

Majority Criterion

If a candidate receives more than 50% of the votes, they should win. The majority criterion states that if a candidate has a majority of first choice votes, then that candidate should be the winner of the election. If no majority candidate exists, then the majority criterion does not apply. There are many reasons—mathematical, political, sociological—why we'd like to see the majority criterion satisfied, but one particularly good reason, is to prevent nuclear war!

Theorem. If a voting method violates the property of independence, then there necessarily exist elections within that voting method that are manipulable. whats a consequence to this

If a voting method fails independence, then there exist elections within it in which some voter v has a lot of power — even if they don't know it. V may well be better off submitting a ballot that differs from their "genuine" ballot (even if they don't know it) — and this is precisely game theory. However, what our Theorem lacks in detail it more than makes up for by its generality — it applies to any preferential ballot voting method that fails independence!

We now commence our study of independence by asking: what's the worst that can happen if independence does not hold?

If a voting method violates the property of independence, then there necessarily exist elections within that voting method that are manipulable. for proof, there exist two elections and two candidates A and B such that in one election A > B, while in the other election B > A—and that the voters who preferred A to B were exactly the same in both elections.

Another useful consequence of the anti-symmetry property is:

Lemma. At every round of an instant runoff Borda (IRB) election, the sum of all the candidates' total margin of victory is 0. Stare at any margin of victory matrix: for every number in it, its negative is also there (on the "other side" of the diagonal of 0's). Now, since adding up all the margins of victory is tantamount to adding up all the numbers in the matrix, and since every such number has its negative, then that means that everything cancels out at the end of the day, leaving you with 0.

The only criterion that doesn't satisfy Borda count

Majority criterion

The Borda count is susceptible to the game theory --

One candidate (in this case B) introducing "clone" candidates in such a way as to hurt the chances of other candidates (in this case A). More general than vote splitting.

Manipulable in voting methods

One voter who could have ensured that their personal preference between A and B is reflected in the final outcome simply by ordering their ballot in a certain way — and this assuming that all other voters left their ballots unchanged. The last bit in this definition is absolutely crucial: if a bunch of voters other than v go back and change their ballots, too, then the election completely changes — it's now a different election — so maybe v loses their magical abil- ity to affect the outcome -- Definition (final form). The outcome of an election is manipulable if there exists one voter, v, with the following property: if A [v prefers] B but B > A, then v can alter their ballot so as to make A > B. -- example of manipulable voting on pg. 37 It makes no mention of any particular voting method, but rather to the outcome of a particular election within a given voting method. Never forget that the only ballot being altered here is that of v. Our definition says nothing about how v could have changed the out- come — it simply asserts that it is possible for v to do this. No strategy is offered for what v should do in order to "get what they want." Nor does it say that if v likes A 2nd best and B 3rd best, then v can ensure that A comes in 2nd place and B in 3rd place in the election; i.e., it doesn't claim that the final ranking between A and B will be a mirror image of v's ranking between A and B.

IRV (Instant Runoff Voting) or ranked-choice voting (RCV)

Voters rank all candidates, from their favorite to their least favorite. The candidate who receives the fewest 1st place votes is then eliminated from everyone's ballot, and all votes are redistributed accordingly; e.g., if a voter had the least favorite in 1st place on their ballot, then after eliminating this candidate their 2nd favorite now effectively moves to 1st place on their ballot, etc. This process continues until there is one candidate left, who is then declared the winner. Observe that there is only one round of actual voting here (hence "instant"). Observe also that in the final round, when there are only two candidates remaining, the winner will necessarily have won a majority of votes compared to the other candidate. Instant runoff voting (IRV), also called plurality with elimination, is a modification of the plurality method that attempts to address insincere voting. in IRV, voting is done with preferential ballots - whoever has the fewest 1st place votes is eliminated first, then any votes for that candidate is redistributed to the voters next choice. This continues until a choice has a majority (over 50%). To employ game theory with IRV, one of the voting groups can split off and, with enough people, vote in a way that ensures the candidate they like the least comes last in terms of first-place votes. here in IRV, we have the (greater?) absurdity that a candidate can lose the election solely by virtue of gaining more votes! This is arguably the biggest flaw with IRV. Satisfies majority criterion: a majority candidate wins in the first round. • Violates the Condorcet criterion: in Election 6, D is the winner by this method, but B is a Condorcet candidate. • Violates monotonicity: in Election 7, C is the winner by this method, but if in a reelection the two voters in the last column switch their votes and move C ahead of A, the winner of the reelection is B.

Satifying Fairness Criteria

When we can show that a voting method is guaranteed to fulfill a given fairness criterion in any possible election, we say that the method satisfies the criterion. If a voting method does not satisfy a given fairness criterion, even if it fails in only one election, then we say that the method violates the given criterion. As soon as we find one example of a violation, we know that the method in question violates the criterion we're looking at. Instead, to show that a voting method satisfies a fairness criterion, we must construct an argument explaining why the criterion will be satisfied in any possible election using that method. Remember that if no majority candidate exists, then the criterion does not apply: it is neither satisfied nor violated.

When you want to disprove that a voting method satisfies independence

Whenever you encounter a voting method that violates independence, you should immediately think: "there exist two elections and two candidates A and B such that A > B in one election, B > A in the other election — but the voters who preferred A to B were exactly the same in both elections." This statement is precisely the negation of the definition of independence.

Definition of Independence

[pg. 33] if there are ever two elections — with the same voters and the same candidates — such that the voters who preferred A to B were exactly the same in both elections, -- you can never have A finish ahead of B in one election but B ahead of A in the other --

Borda Count Method

a numerical score is given to each candidate, as follows: if an election has, say, ten candidates, then a voter will assign 10 points to their favorite, 9 points to their 2nd favorite, 8 points to their 3rd favorite, and so on, with 1 point given to their least favorite candidate. Satisfies monotonicity: if in a reelection the votes change only to favor the previous winner, there can only be more points for the candidate that already had the most points. • Violates the Condorcet criterion. • Violates majority criterion.

clone invariant.

if a voting method is such that the introduction of clone candidates do not "significantly" change the outcome of the election.

In runoff elections (whether instant or multiple rounds), it is impossible.... [but it is possible for..]

impossible for the least favorite of a majority to win the election (why?), but it is possible to improve your candidate's chances of winning by not voting for them.

plurality voting

in plurality voting, we can have the absurd situation wherein a candidate who is literally the least favorite of a majority wins the election outright. However, it's not possible, in a plurality election, to increase a candidate's chances of winning by strategically not voting for them (why?).

why the property of independence is also flawed

it can't distinguish between an "honest" ballot and a "strategically manipulated" ballot... pg. 35 i. Game theory: the C crowd knows their candidate can't win, so some of them choose B simply to make sure that A doesn't win. ii. No game theory: maybe in this parallel universe, some of the C crowd just decided that they prefer B over C after all. The problem is that there is a world of difference between the "honesty" of the C voters in i. and ii.—but because their ballots are identical, independence can't tell the difference.

Runoff elections (whether instant or multiple round)

it is impossible for the least favorite of a majority to win the election (why?), but it is possible to improve your candidate's chances of winning by not voting for them.

What doesn't and does satisfy independence

neither plurality voting nor IRV satisfies independence. . . . A dictatorship satisfies independence (why?). . . . a plurality election with only two candidates satisfies independence (why?). In neither of these elections is a voter ever incentivized to manipulate their ballot, to submit a ballot that is anything other than their true preference (why?). majority voting in a two-candidate election is the very ideal of democracy — there is no controversy as to how to vote or how to decide the winner.