Strategic and Insincere Voting

Real-World Examples of Voting Paradoxes

William H. Riker, Liberalism against Populism (Waveland Press, 1982).

1912 US Presidential race

Popular vote:

42% Wilson

27% Roosevelt

24% Taft

Thanks to the plurality rule, Wilson won (the Electoral College didn’t make a significant difference in this case. Let’s suppose that the voters formed three blocs with the following preference ordering:

42	W, R, T
27	R, T, W
24	T, R, W

Roosevelt is the Condorcet winner. Under approval voting (assuming everyone votes for two candidates), again Roosevelt wins (with 100% of the vote). Under the Hare method, Taft is eliminated, and Roosevelt wins the runoff. On the Coombs rule, Wilson is eliminated, and Roosevelt again wins. Roosevelt is also clearly the Borda winner.

The 1970 New York US Senate race

39	Buckley, Ottinger, Goodel
37	Ottinger, Goodel, Buckley
24	Goodel, Ottinger, Buckley

Buckley won with a plurality. Ottinger would have won the approval ballot, and the Hare and Borda methods. Ottinger is also the Condorcet winner.

Insincere Voting

Let’s suppose that for every preference ordering, there is a unique ballot that will faithfully reflect that ordering, the “faithful” ballot. So, for example, in the case of a single-ballot, plurality election, the “faithful” ballot would be the one that names the voter’s most preferred candidate. In a scoring ballot, if the voter prefers x to y, then the ballot assigns a higher score to x than to y.

On this assumption, a strategic vote would be a situation in which it is rational (given the electoral rules and expectations about others’ voting behavior) for one or more voters to cast an unfaithful or “insincere” ballot.

What’s wrong with insincere voting? Not because there is anything dishonest about casting an insincere ballot. If I assign x a higher score than y, even though I prefer y to x, I am not somehow dishonestly asserting that x is better or preferred. I’m not making any assertion at all. And, by definition, there’s nothing irrational about an insincere ballot. So why is it objectionable?

Simply because if a system is subject to insincere voting, the electorate can face a multiple-equilibria situation – a form of the Stag Hunt game. Consequently, the electorate might find itself trapped at a sub-optimal equilibrium (analogous to “all hunt hare”). We could have a situation in which everyone prefers x to either y or z, and yet y or z wins. How can this happen? Suppose we all believe that no one will actually vote for x. Then, I would be throwing my vote away by voting for x. So, I should case an insincere ballot for either y or z, depending on which of them I prefer to the other.

How can this happen? It can happen pretty frequently in a situation in which there are two or more established parties. Past performance can make all of us virtually certain that one of the established party nominee will win, even if it is common knowledge that we all prefer someone else.

Gibbard-Satterthwaite Theorem

Gibbard (1973).

Where P is a preference ordering, a strategy t is P-dominant for k iff no matter what strategies are fixed for everyone else, strategy t for k produces an outcome at least as high in preference ordering P as does any other. A game form is straightforward if, for every preference ordering P and player k, there is a strategy which is P-dominant for k.

A player k is a dictator for game form g if, for every outcome x, there is a strategy s(x) for k such that g(s) = x whenever s_k = s(x). A game form g is dictatorial if there is a dictator for g.

THEOREM: Every straightforward game form with at least three possible outcomes is dictatorial.

A voting scheme is a game form v such that for some set Z of outcomes, the set S_i of strategies open to each player i is the set of orderings of Z. A voting scheme is manipulable if for some k, for some n-tuple P of orderings of Z, and for some ordering P* of Z, v(P) P* v(P_k/P*).

Manipulability is not a property of a game form alone, it is a property of a voting scheme alone. Voting scheme v is manipulable if for some k and preference n-tuples <P₁…, P_n > and <P*₁…,P*_n>, P_i = P*_i except when i = k, and voter k prefers v(P*₁…, P*_n) to v(P_l, …, P_n).

For, then, given the way the others vote, k prefers the result of expressing preference ordering P*_k to that of expressing P_k.

COROLLARY: Every voting scheme with at least three outcomes is either dictatorial or manipulable.

Satterthwaite 1973

If the voting procedure is not strategy- proof, then there must exist a set of sincere strategies R = (R₁…, R_n) which is not a Nash equilibrium. The voting procedure must also have at least 3 possible outcomes.

Examples of Insincere Voting

Plurality

An insincere vote in a three-way plurality race would be a vote for your second-favorite candidate in place of a vote for your favorite. If I vote for B, when I really prefer A, my vote is (in the technical sense) insincere. It is easy to see that this happens often in plurality systems. Suppose, for example, that there are three candidates, x, y, and z, and seven voters, A-G. Let’s look at the problem from the perspective of G. G prefers the candidates in the order x > y > z. G believes that voters A, B, and C will vote for y, and D, E, and F will vote for z. In the case of a tie, the election will go to H, who is known to prefer z. In this case, the rational strategy for G is to vote for y rather than z. Voting for x is pointless in this situation, while voting for y will break the tie and prevent the worst-case candidate, z, from being elected.

A runoff system just compounds the problem. Let’s suppose that the top two candidates will move on to a runoff. Here are the known preferences:

A, B, C	D, E	F, G	H
z	x	y	x
x	y	z	y
y	z	x	z

Let’s look at it from H’s point of view, and let’s assume for the moment that A-G vote in the first round for their top candidate. If H votes for x, x and z will go on to the runoff, and z (H’s last choice) will beat x. In contrast, if H votes for y in the first round, y and z will go into the runoff, and y will win. For similar reasons, D and E might also vote for y (their second choice) in the first round, in which case y might win a majority in the first round. In any case, the optimal strategy for D, E, and H is not the sincere one (voting for x over y).

In this case, the insincere voting produced a pretty good result (y is the Condorcet winner), but it can also produce a bad one. We saw last time, for example, a case in which plurality does not choose a Condorcet winner.

Borda method

Alf	Betty	Charlie	Doris	Eugene
x	x	x	y	y
y	y	y	x	x
z	z	z	z	z

Using Borda count, we get x = 7, y = 8, and z =15.

Doris wants y to win and knows that x will win if she votes sincerely. So, she changes her vote, moving x to third place, resulting in an x-y tie (7 points each).

Elimination/Runoff

Strategic voting with the Hare method.

Number voters	1st	2nd	3rd	4th
9	w	z	x	y
6	x	y	z	w
2	y	x	z	w
4	y	z	x	w
5	z	x	y	w

Let’s analyze this using sincere ballots. In the first round, z gets the fewest 1^st place votes and is eliminated

Number voters	1st	2nd	3rd
9	w	x	y
6	x	y	w
2	y	x	w
4	y	x	w
5	x	y	w

Now y is eliminated, taking us to the third stage

Number voters	1st	2nd
9	w	x
6	x	w
2	x	w
4	x	w
5	x	w

Now x wins. This example also illustrates a failure of positive monotonicity. If the two players in the first round were to switch their rankings of x and y (moving x up), the result would be that x loses!

Back to insincere voting. Consider the 9 voters who preferred w. They all prefer z to x. If they were to switch their first-choice selection in round 1 from w to x, they can ensure that z survives and w is eliminated instead. It’s easy to see that z will then beat x in round 2, so the group 1 voters will get their second rather than their third choice.

As we shall see in the next post, the only voting method that eliminates insincere voting is the approval method.