Lecture Notes 17: Voting
Voting Rules
Binary voting methods involve choice over two alternatives. In the majority rule, the winner is whichever choice receives more votes.
When there are more than two alternatives, things get complicated. Pairwise voting consists of repeated binary votes. There are two common pairwise voting variations.
• Condorcet voting: The winner is the candidate who can win a pairwise election against every other candidate. Unfortunately, a Condorcet winner may not always exist.
• Copeland Index: The winner is the candidate who can accumulate the most wins in a series of pairwise contests. An example is the first round of the World Cup.
Plurative voting methods select among more than two alternatives simultaneously. Here are some common plurative voting rules.
• Plurality Rule: Each voter is asked to select his most preferred candidate. The candidate receiving the most votes wins.
• Antiplurality Rule: Each voter is asked to select his least preferred candidate. The candidate receiving the fewest votes wins.
• Borda count: Each voter ranks all candidates. For example, if there are three candidates, the voter’s top choice receives three points, the second choice receives two points and the last choice receives one point. The candidate who receives the most points wins.
• Approval voting: Each voter selects whichever candidates are acceptable (as many or few as he wants). The candidate receiving the most votes wins.
Mixed voting methods combine different voting rules. Here are a few common ones.
• Majority Runoff: Each voter selects one candidate. If one candidate secures an outright majority, he wins. If no candidate secures a majority, then the top two go on to a binary vote. An example is French Presidential elections.
vote-getter among these options is eliminated. This continues until there are only two candidates, at which point the majority wins. This is how the city for the Olympics is chosen.
• Instant Runoff: Voters rank their choices. If no candidate gets a majority of #1 votes, then the candidate with the fewest number of #1 votes is eliminated. All voters who selected the eliminated candidate as their #1 choice have their votes automatically transferred to their #2 choice. This continues, with candidates successively eliminated, until there is a candidate who secures a majority of the votes. This method is becoming increasingly popular for local elections in the US. Proponents argue that it encourages diversity. It allows a voter to vote for his preferred candidate without worrying that he will “waste his vote” on a non-major candidate. If the voter’s preferred candidate is eliminated, his vote is not “wasted”, but automatically switched to his next-most preferred candidate.
Condorcet Paradox
There are many voting methods that can be used to choose a winner, especially when there are a large number of alternatives. There is no perfect method, and many have weaknesses. We will go through some examples before finishing the unit with some general results.
Consider an election with three candidates and three voters. The candidates are 𝐴𝐴, 𝐵𝐵 and 𝐶𝐶. The voters’ preferences are given below.
Voter 1: 𝐴𝐴 ≻ 𝐵𝐵 ≻ 𝐶𝐶 Voter 2: 𝐵𝐵 ≻ 𝐶𝐶 ≻ 𝐴𝐴 Voter 3: 𝐶𝐶 ≻ 𝐴𝐴 ≻ 𝐵𝐵
Suppose we want to choose a winner using pairwise voting.
• In an election between 𝐴𝐴 and 𝐵𝐵, the winner is 𝐴𝐴 (with support of voters 1 and 3).
• In an election between 𝐵𝐵 and 𝐶𝐶, the winner is 𝐵𝐵 (with support of voters 1 and 2).
• In an election between 𝐴𝐴 and 𝐶𝐶, the winner is 𝐶𝐶 (with support of voters 2 and 3).
First, there is no Condorcet winner. None of the candidates can win a pairwise election against both alternatives.
Agenda Paradox
Suppose in the example above that we want to set up the rules for 𝐴𝐴 to win. Consider the following voting rule.
• First, vote pairwise on 𝐵𝐵 and 𝐶𝐶.
• Second, vote pairwise on 𝐴𝐴 and the winner of the first election.
If voters represent their preferences honestly, then 𝐵𝐵 wins the first election. 𝐵𝐵 then goes up against 𝐴𝐴, and 𝐴𝐴 wins.
But suppose that we wanted to set up the rules for 𝐵𝐵 to win. We could use the following voting procedure.
• First, vote pairwise on 𝐴𝐴 and 𝐶𝐶.
• Second, vote pairwise on 𝐵𝐵 and the winner of the first election.
Again, assuming that voters represent their preferences honestly, 𝐶𝐶 will win the first election. 𝐶𝐶 will then go up against 𝐵𝐵, who will win the second election.
A similar setup could make 𝐶𝐶 the winner. Putting all of this together, any candidate can win depending upon the voting rule chosen! This problem is known as the agenda paradox: setting up the rules differently can result in a different outcome, even if voters’ preferences don’t change.
Example: Agenda Paradox in Court
Consider a panel of three judges who are deciding on the fate of an accused criminal. There are three possible outcomes. The defendant can be set free, sent to jail, or put to death. The preferences of the three judges are given below.
Judge 1: Death ≻ Jail ≻ Free Judge 2: Jail ≻ Free ≻ Death Judge 3: Free ≻ Death ≻ Jail
Consider three different voting rules.
• Roman system: The judges start with the most serious penalty, working down, with the defendant set free if no penalty is approved.
Use backwards induction. If the death penalty is not approved initially, then the defendant will be sent to jail, since judges 1 and 2 prefer jail to setting the defendant free. Given this, the death penalty is approved initially since judges 1 and 3 prefer the death penalty to jail. The Roman system puts the defendant to death.
• Mandatory sentencing system: The judges first set the sentence for a guilty defendant, and then rule on whether the defendant is guilty.
Use backwards induction. If the court decides on a jail sentence in the first stage, then the defendant will be found guilty since judges 1 and 2 prefer a jail sentence to setting the defendant free. But if the court decides on the death penalty in the first stage, then the defendant will be found not guilty since judges 2 and 3 prefer setting the defendant free to the death penalty. Knowing this, judges 1 and 2 set a jail sentence in the first stage, as they prefer that to setting the defendant free (the outcome if the death penalty is chosen). The mandatory sentencing system sends the defendant to jail.
This is the agenda paradox again. Even with the same judges and the same preferences, we can actually get any possible outcome simply by using different decision rules.
Example: Violation of Pareto Principle
There are three legislators who are voting on filling a vacancy to the Supreme Court. The candidates are 𝐴𝐴, 𝐵𝐵 and 𝐶𝐶, but the legislators could also leave the seat vacant. The preferences of the legislators are as follows.
Legislator 1: 𝐶𝐶 ≻ vacant ≻ 𝐴𝐴 ≻ 𝐵𝐵 Legislator 2: 𝐵𝐵 ≻ 𝐶𝐶 ≻ vacant ≻ 𝐴𝐴 Legislator 3: vacant ≻ 𝐴𝐴 ≻ 𝐵𝐵 ≻ 𝐶𝐶
The voting is as follows. There is a yes/no vote on 𝐴𝐴. If 𝐴𝐴 does not attain a majority, then there is a yes/no vote on 𝐵𝐵. If 𝐵𝐵 does not attain a majority, then there is a yes/ no vote on 𝐶𝐶. If 𝐶𝐶 does not attain a majority, then the seat is left vacant.
We can find the outcome by reasoning backwards.
• If the last stage is reached, 𝐶𝐶 wins (since legislators 1 and 2 prefer 𝐶𝐶 to a vacant seat).
• Knowing this, 𝐴𝐴 is approved in the first vote (since legislators 1 and 3 prefer 𝐴𝐴 to 𝐵𝐵, and they realize that 𝐵𝐵 will win if the next stage is reached).
Using the voting rule specified, the outcome is that 𝐴𝐴 wins in the first election. But this is a strange outcome since all three voters unanimously prefer a vacant seat to a victory by 𝐴𝐴. This voting rule violates the Pareto Principle, which states that 𝑋𝑋 should be chosen over 𝑌𝑌 if voters unanimously prefer 𝑋𝑋 to 𝑌𝑌.
You might think that a solution here is some kind of collusion beforehand, but that requires trust. If the legislators collude to reject all candidates, so that the seat remains vacant, legislators 1 and 2 might violate their agreement in the last election since they prefer 𝐶𝐶 to a vacant seat. Knowing this, legislator 3 might not go along with the deal at all.
Strategic Voting
The examples from the previous section assume sincere voting. In other words, we assume that all voters represent their preferences honestly. But, for some elections, there might be an incentive for voters to use strategic voting, where voters vote for alternatives that are not their preferred alternatives in order to manipulate the overall outcome in their favor.
Let us reconsider the example from earlier, where three voters choose from among three candidates. Their preferences are as follows.
Voter 1: 𝐴𝐴 ≻ 𝐵𝐵 ≻ 𝐶𝐶 Voter 2: 𝐵𝐵 ≻ 𝐶𝐶 ≻ 𝐴𝐴 Voter 3: 𝐶𝐶 ≻ 𝐴𝐴 ≻ 𝐵𝐵
Suppose that the chairperson establishes the following voting rule.
• First, vote pairwise on 𝐵𝐵 and 𝐶𝐶.
• Second, vote pairwise on 𝐴𝐴 and the winner of the first election.
When voters represent their preference honestly, 𝐵𝐵 beats 𝐶𝐶 in the first election, with the support of voters 1 and 2. Then 𝐴𝐴 beats 𝐵𝐵 in the second election with the support of voters 1 and 3.
ultimately wins in the second election. But if she misrepresents her preferences and votes in the first election for 𝐶𝐶 instead of 𝐵𝐵, then 𝐶𝐶 wins the first election and 𝐶𝐶 wins the second election. This is actually a better ultimate outcome for voter 2.
Notice that this strategic voting involves a coalition of voters 2 and 3 who unite together to support candidate 𝐶𝐶 in both elections in order to defeat 𝐴𝐴. This is a common feature of strategic voting.
But the chairperson might set up the voting rule as follows.
• First, vote pairwise on 𝐴𝐴 and 𝐶𝐶.
• Second, vote pairwise on 𝐵𝐵 and the winner of the first election.
Under sincere voting, 𝐵𝐵 would win. Voters 2 and 3 would support 𝐶𝐶 in the first election, and then voters 1 and 2 would support 𝐵𝐵 over 𝐶𝐶 in the second election.
But if the voters vote strategically, then voter 3 might realize that he is better off to misrepresent his preferences and support 𝐴𝐴 in the first election. If he did that, 𝐴𝐴 would win against 𝐵𝐵 in the second election, with the support of voters 1 and 3. Ultimately, this is a better outcome for voter 3 since he prefers for 𝐴𝐴 to win than for 𝐵𝐵 to win, which was the outcome when he represented his preferences honestly in the first election.
Examples of Strategic Voting
Plurality elections with many candidates are rife with strategic voting.
• Many people do not vote their top choice in plurality elections because they do not want to “throw away their vote.” If you prefer a minor candidate, but he has very little chance of winning, you might instead to use your vote for a major candidate to avoid wasting it.1
• In a plurality election with two major candidates that are closely matched, a spoiler can enter to divert votes from one of them. In the 2000 US Presidential election, conservative George Bush and liberal Al Gore were very close in the polls. Ralph Nader ran as an extreme liberal, even more liberal than Gore. He was hoping to extract concessions from the Gore campaign by threatening to take votes away from him. Nader ended up costing Al Gore the election. Ironically, Nader’s efforts ultimately helped to elect George Bush!
• There are different ways to allocate legislative seats. The US and the UK have district systems, where one representative is chosen from each geographical region by plurality.
1 In 1992, a plurality (40%) of American voters said that they would have voted for Ross Perot “if they thought he
France and Italy have proportional systems, where seats in the legislature are allocated to parties in proportion to the total number of votes they receive nationwide. Duverger’s Law in political science asserts that two-party systems tend to arise in countries with district voting, whereas multi-party systems tend to arise in countries with proportional voting. Using the idea of strategic voting, we can see why. Voters in a district system might be afraid to vote for minor parties that have no chance of winning their districts, and strategically vote for their most preferred major candidate instead. But, in proportional systems, even minor parties can get some fraction of legislative seats, so it’s not throwing away your vote to support them. Thus, a larger number of small parties can arise in a proportional system.
Here are some other examples of strategic voting.
• In multi-stage voting, you might support something ridiculous in the first election in order to increase the chance of a preferred outcome in the second election. For example, in US presidential elections, we first hold a primary to select one candidate for each party, and then the two primary winners square off in a general election. A Democrat might think about voting in the Republican primary and supporting the worst Republican in order to increase the chance that the Democrat will ultimately win the general election.
• Legislators often front-load budgets with frivolous items at first, knowing that when the crunch arrives, important projects must be financed and the budget passed at the end. This can contribute to excessive spending and waste.
General Results on Voting and Social Choice
If society is going to use a voting rule or social choice mechanism to select from among alternatives, there are some features that we would like the mechanism to display.
1. Complete – The system can select among all possible alternatives.
2. Transitive – If the system chooses 𝐴𝐴 over 𝐵𝐵 and chooses 𝐵𝐵 over 𝐶𝐶, then the system should also choose 𝐴𝐴 over 𝐶𝐶.
3. Pareto Property – If all voters prefer 𝐴𝐴 to 𝐵𝐵, then the system should not choose 𝐵𝐵. 4. Non-Dictatorial – No single voter should determine the group’s choice.
5. Independence of Irrelevant Alternatives (IIA) – No change in the set of candidates should impact the ranking of other candidates.
The following is a central result in social choice theory.
• Arrow’s Impossibility Theorem: There is no voting system that satisfies all of (1) – (5).
In other words Arrow’s Impossibility Theorem states that any voting system will violate one of these five properties. Another way to think about this is that the only voting system that satisfies (1) – (3) and that satisfies IIA is a dictator.
For example, Condorcet’s Paradox shows us that pairwise majority voting does not always satisfy transitivity. However, under the special case of single-peaked preferences, where alternatives are ordered on some continuum and voters have a single peak as far as which alternative they prefer, the Condorcet Paradox goes away.
In addition to voting outcomes that seem not to represent social preferences very well, economists are also interested in the susceptibility of voting systems to strategic manipulation. The following result is fundamental
• Gibbard-Satterthwaite Theorem: If there are three or more alternatives, then the only
voting system not subject to strategic manipulation is a dictatorship.
Alas, we can’t really do anything about strategic manipulation and misrepresentation of preferences. The Gibbard-Satterthwaite Theorem says that every voting system, other than a dictatorship, is subject to strategic manipulation.
• Median Voter Theorem: In a two-candidate election, both candidates will adopt a position that is the same as the median voter’s position.
Note that the median voter is the voter whose position is such that half the voters lie to the left and half to the right.
The reason for the result is that both candidates staking out the median position is the Nash Equilibrium. If the candidates positioned themselves anywhere else, then moving towards the middle would only increase the candidate’s vote total. Consider a candidate who was located towards the left end. He could move towards the middle to pick up some moderate votes, but the ones to his left would still prefer to vote for him.
Indeed, the usual complaint in two-party systems is that voters don’t have all that much of a choice. The two parties end up with positions that are fairly similar to each other. This is exactly the result of the median voter theorem.
Problems
1. A group of 50 residents are attending a town meeting. They must choose one of three proposals for dealing with garbage. Proposal 1 is that the town collect garbage and tax residents. Proposal 2 is that the town hire a private garbage collector who bills residents individually. Proposal 3 is that residents be responsible for their own garbage. There are three types of voters, whose preferences are as follows:
Type I (20 voters): 1 ≻ 2 ≻ 3 Type II (15 voters): 2 ≻ 3 ≻ 1 Type III (15 voters): 3 ≻ 1 ≻ 2
a. Using a plurality system, which proposal wins?
b. Using a Borda count system (with 3 points for the top-ranked proposal, 2 points for the second-ranked proposal and 1 point for the last place proposal), which proposal wins if the voters vote honestly?
c. Describe how Type II voters and Type III voters can vote strategically to improve their outcome under a Borda count. Under this strategic voting, how many points does each proposal get and which proposal wins?
2. There are three candidates running for governor: Bustamante (B), Schwarzenegger (S) and McClintock (M). There are 3 types of voters whose preferences are as follows. Suppose that 40% of voters are liberal, 30% are moderate and 30% are conservative.
Liberal: 𝐵𝐵 ≻ 𝑆𝑆 ≻ 𝑀𝑀 Moderate: 𝑆𝑆 ≻ 𝐵𝐵 ≻ 𝑀𝑀 Conservative: 𝑀𝑀 ≻ 𝑆𝑆 ≻ 𝐵𝐵
a. Describe the way in which a plurality election is subject to strategic manipulation. b. Does the election have a Condorcet winner?
3. Two senators are trying to decide which policy to implement out of the set {𝑋𝑋, 𝑌𝑌, 𝑍𝑍}. Senator Michael’s preferences are 𝑋𝑋 ≻ 𝑌𝑌 ≻ 𝑍𝑍, while Senator George’s preferences are 𝑍𝑍 ≻ 𝑌𝑌 ≻ 𝑋𝑋. The rules require the senators to decide as follows. First, Michael eliminates one of the three policies from consideration. Then George eliminates one of the remaining two policies from consideration. Whichever policy remains is implemented.
Describe the equilibrium of this game. Which policy is ultimately implemented?
4. Consider an election with four candidates. McCain (M), Romney (R), Huckabee (H) and Paul (P). There are seven types of voters, whose preferences are represented below, along with the percentage of each type of voter.
Type I (16%): 𝑀𝑀 ≻ 𝑅𝑅 ≻ 𝐻𝐻 ≻ 𝑃𝑃 Type II (28%): 𝑅𝑅 ≻ 𝑀𝑀 ≻ 𝐻𝐻 ≻ 𝑃𝑃 Type III (13%): 𝑅𝑅 ≻ 𝐻𝐻 ≻ 𝑀𝑀 ≻ 𝑃𝑃 Type IV (21%): 𝐻𝐻 ≻ 𝑅𝑅 ≻ 𝑀𝑀 ≻ 𝑃𝑃 Type V (12%): 𝐻𝐻 ≻ 𝑀𝑀 ≻ 𝑅𝑅 ≻ 𝑃𝑃 Type VI (6%): 𝑃𝑃 ≻ 𝑅𝑅 ≻ 𝐻𝐻 ≻ 𝑀𝑀 Type VII (4%): 𝑃𝑃 ≻ 𝐻𝐻 ≻ 𝑅𝑅 ≻ 𝑀𝑀
The voting proceeds as follows. Voters each choose their preferred candidate. If no candidate wins more than 50%, then the candidate with the fewest votes is dropped and everyone votes again. This continues until one candidate wins a majority.
In the first round, voters represented their preferences honestly and Romney was the top vote-getter with 41%. Paul was dropped.
a. What would the results of the second round have been if voters had honestly represented their preferences for the remaining three candidates?
b. In reality, Huckabee won 52% in the second round, with Romney winning 47% and McCain winning 1%. Given the preferences of McCain voters, why might this have happened? It might be important to note that other states held similar elections on later dates and that McCain and Romney were widely viewed as the only two candidates with any serious chance of winning nationally.
5. In instant-runoff voting (IRV), voters first rank all candidates. If a single candidate has a majority of the top votes, this candidate wins. If no single candidate has a majority of the top votes, then the candidate with the lowest number of top votes is eliminated. Any voter who chose this candidate has his vote automatically redistributed to his next highest-ranked choice that is still on the ballot. This continues until a single candidate has a majority of the votes. Consider an IRV election with 3 potential candidates: Jack (J), Kate (K) and Locke (L). There are 5 voters, whose preferences are as follows.
Voter 1: 𝐽𝐽 ≻ 𝐾𝐾 ≻ 𝐿𝐿 Voter 2: 𝐽𝐽 ≻ 𝐾𝐾 ≻ 𝐿𝐿 Voter 3: 𝐾𝐾 ≻ 𝐿𝐿 ≻ 𝐽𝐽 Voter 4: 𝐿𝐿 ≻ 𝐾𝐾 ≻ 𝐽𝐽 Voter 5: 𝐿𝐿 ≻ 𝐽𝐽 ≻ 𝐾𝐾
a. Which candidate would win the IRV election if the voters represented their preferences honestly?
