From the safety of your street to the quality of the air you breathe- Voting is an important way of having your say on the issues you care about. BY Voting you are also deciding who will take decisions on issues affecting your everyday life History shows that democracies in danger of losing their freedom register frightening low voter turnouts. In thriving democracies, people vote in large numbers and the people's voice remains supreme. Support the candidates and ballot measures that can help your community, state, and even the nation for the greater good.
Make your voice heard in these elections. Our communities are made up of friends, loved ones, neighbors, and children. Make the decision to vote for yourself and those around you.
Make sure your voice is heard — vote! September 10, Here are just a few reasons why you should get registered and vote: 1. Elections have consequences. Not voting is giving up your voice. Formal studies of Liquid Democracy have focused on: the possibility of delegation cycles and the relationship with the theory of judgement aggregation Christoff and Grossi ; the rationality of delegating votes Bloembergen, Grossi and Lackner ; the potential problems that arise when many voters delegate votes to only a few voters Kang et al.
This section introduced different methods for making a group decision. One striking fact about the voting methods discussed in this section is that they can identify different winners given the same collection of ballots.
This raises an important question: How should we compare the different voting methods? Can we argue that some voting methods are better than others? There are a number of different criteria that can be used to compare and contrast different voting methods:. In this section, I introduce and discuss a number of voting paradoxes — i.
Consult Saari and Nurmi for penetrating analyses that explain the underlying mathematics behind the different voting paradoxes. A very common assumption is that a rational preference ordering must be transitive i.
Many authors argue that voters with cyclic preference orderings have inconsistent opinions about the candidates and should be ignored by a voting method in particular, Condorcet forcefully argued this point. A key observation of Condorcet which has become known as the Condorcet Paradox is that the majority ordering may have cycles even when all the voters submit rankings of the alternatives.
This means that there is no Condorcet winner. This simple, but fundamental observation has been extensively studied Gehrlein ; Schwartz The Condorcet Paradox shows that there may not always be a Condorcet winner in an election. However, one natural requirement for a voting method is that if there is a Condorcet winner, then that candidate should be elected. Voting methods that satisfy this property are called Condorcet consistent.
Many of the methods introduced above are not Condorcet consistent. I already presented an example showing that plurality rule is not Condorcet consistent in fact, plurality rule may even elect the Condorcet loser. The example from Section 1 shows that Borda Count is not Condorcet consistent.
Consider the following voting situation with 81 voters and three candidates from Condorcet Using the Borda rule, we have:. To simplify the calculation, assume that candidates ranked last receive 0 points i. But, of course, it is counterintuitive to give more points for being ranked second than for being ranked first. Peter Fishburn generalized this example as follows:. Theorem Fishburn So, no scoring rule is Condorcet consistent, but what about other methods?
A number of voting methods were devised specifically to guarantee that a Condorcet winner will be elected, if one exists. The examples below give a flavor of different types of Condorcet consistent methods. See Brams and Fishburn, , and Fishburn, , for more examples and a discussion of Condorcet consistent methods.
If there is a Condorcet winner, then that candidate wins the election. Otherwise, all candidates tie for the win. If there is a Condorcet winner, then that candidate is the winner. Otherwise, use Borda Count to determine the winners.
Calculate the Borda score for each candidate. The candidates with a Borda score below the average of the Borda scores are eliminated.
The Borda scores of the candidates are re-calculated and the process continues until there is only one candidate remaining. See Niou, , for a discussion of this voting method. The winners are the smallest set of candidates that are not beaten in a one-on-one election by any candidate outside the set Schwartz The candidate s with the fewest swaps is are declared the winner s.
The last method was proposed by Charles Dodgson better known by the pseudonym Lewis Carroll. Interestingly, this is an example of a procedure in which it is computationally difficult to compute the winner that is, the problem of calculating the winner is NP-complete. See Bartholdi et al. These voting methods and the other Condorcet consistent methods guarantee that a Condorcet winner, if one exists, will be elected.
But, should a Condorcet winner be elected? Many people argue that there is something amiss with a voting method that does not always elect a Condorcet winner if one exists. The idea is that a Condorcet winner best reflects the overall group opinion and is stable in the sense that it will defeat any challenger in a one-on-one contest using Majority Rule.
The most persuasive argument that the Condorcet winner should not always be elected comes from the work of Donald Saari , The majority ordering is.
Groups 1 and 2 constitute majority cycles with the voters evenly distributed among the three possible rankings. Such profiles are called Condorcet components. These profiles form a perfect symmetry among the rankings. Balinski and Laraki , pgs. Balinski and Laraki , pg. See the discussion of the multiple districts paradox in Section 3. A voting method is monotonic provided that receiving more support from the voters is always better for a candidate. There are different ways to make this idea precise see Fishburn, , Sanver and Zwicker, , and Felsenthal and Tideman, It is easy to see that Plurality Rule is monotonic in this sense: The more voters that rank a candidate first, the better chance the candidate has to win.
Surprisingly, there are voting methods that do not satisfy this natural property. The most well-known example is Plurality with Runoff. Consider the two scenarios below.
Note that the only difference between the them is the ranking of the fourth group of voters. See Felsenthal and Nurmi for further discussion of voting methods that are not monotonic. In this section, I discuss two related paradoxes that involve changes to the population of voters. Voting methods that do not satisfy this version of monotonicity are said to be susceptible to the no-show paradox Fishburn and Brams Suppose that there are 3 candidates and 11 voters with the following rankings:.
Suppose that 2 voters in the first group do not show up to the election:. Plurality with Runoff is not the only voting method that is susceptible to the no-show paradox. The Coombs Rule, Hare Rule and Majority Judgement using the tie-breaking mechanism from Balinski and Laraki are all susceptible to the no-show paradox.
It turns out that always electing a Condorcet winner, if one exists, makes a voting method susceptible to the above failure of monotonicity. Theorem Moulin If there are four or more candidates, then every Condorcet consistent voting method is susceptible to the no-show paradox.
See Perez , Campbell and Kelly , Jimeno et al. Multiple Districts Paradox : Suppose that a population is divided into districts. If a candidate wins each of the districts, one would expect that candidate to win the election over the entire population of voters assuming that the two districts divide the set of voters into disjoint sets.
This is certainly true for Plurality Rule: If a candidate is ranked first by the most voters in each of the districts, then that candidate will also be ranked first by a the most voters over the entire population.
Interestingly, this is not true for all voting methods Fishburn and Brams The example below illustrates the paradox for Coombs Rule. District 1 : There are a total of 10 voters in this district. District 2 : There are a total of 5 voters in this district. There are 15 total voters in the combined districts. None of the candidates are ranked first by 8 or more of the voters. The other voting methods that are susceptible to the multiple-districts paradox include Plurality with Runoff, The Hare Rule, and Majority Judgement.
Note that these methods are also susceptible to the no-show paradox. As is the case with the no-show paradox, every Condorcet consistent voting method is susceptible to the multiple districts paradox see Zwicker, , Proposition 2. I sketch the proof of this from Zwicker pg.
Consider the following two districts. In district 1, there are no Condorcet winners. See Saari , Section 4. The paradox discussed in this section, first introduced by Brams, Kilgour and Zwicker , has a somewhat different structure from the paradoxes discussed above. Voters are taking part in a referendum , where they are asked their opinion directly about various propositions cf.
Suppose that there are 13 voters who cast the following votes for the three propositions so voters can cast one of eight possible votes :. When the votes are tallied for each proposition separately, the outcome is N for each proposition N wins 7—6 for all three propositions. Putting this information together, this means that NNN is the outcome of this election. However, there is no support for this outcome in this population of voters.
This raises an important question about what outcome reflects the group opinion: Viewing each proposition separately, there is clear support for N on each proposition; however, there is no support for the entire package of N for all propositions.
Brams et al. See Scarsini , Lacy and Niou , Xia et al. However, a majority of the voters voters 1, 2 and 3 do not support the majority outcome on a majority of the issues note that voter 1 does not support the majority outcome on issues 2 and 3; voter 2 does not support the majority outcome on issues 1 and 3; and voter 3 does not support the majority outcome on issues 1 and 2!
The issue is more interesting when the voters do not vote directly on the issues, but on candidates that take positions on the different issues. In the discussion above, I have assumed that voters select ballots sincerely. That is, the voters are simply trying to communicate their opinions about the candidates under the constraints of the chosen voting method.
However, in many contexts, it makes sense to assume that voters choose strategically. One need only look to recent U. The most often cited example is the U. A detailed overview of the literature on strategic voting is beyond the scope of this article see Taylor and Section 3.
I will explain the main issues, focusing on specific voting rules. There are two general types of manipulation that can be studied in the context of voting. The first is manipulation by a moderator or outside party that has the authority to set the agenda or select the voting method that will be used. So, the outcome of an election is not manipulated from within by unhappy voters, but, rather, it is controlled by an outside authority figure.
To illustrate this type of control, consider a population with three voters whose rankings of four candidates are given in the table below:. A second type of manipulation focuses on how the voters themselves can manipulate the outcome of an election by misrepresenting their preferences.
Consider the following two election scenarios with 7 voters and 3 candidates:. The only difference between the two election scenarios is that the third voter changed the ranking of the bottom three candidates. This is an instance of a general result known as the Gibbard-Satterthwaite Theorem Gibbard ; Satterthwaite : Under natural assumptions, there is no voting method that guarantees that voters will choose their ballots sincerely for a precise statement of this theorem see Theorem 3.
Much of the literature on voting theory and, more generally, social choice theory is focused on so-called axiomatic characterization results. The main goal is to characterize different voting methods in terms of abstract principles of collective decision making. Consult List and Gaertner for introductions to the vast literature on axiomatic characterizations in social choice theory.
In this article, I focus on a few key axioms and results and how they relate to the voting methods and paradoxes discussed above. I start with three core principles.
Anonymity : The names of the voters do not matter: If two voters swap their ballots, then the outcome of the election is unaffected. Neutrality : The names of the candidates, or alternatives, do not matter: If two candidates are exchanged in every ballot, then the outcome of the election changes accordingly. In other words, no profile of ballots can be ignored by a voting method. One way to make this precise is to require that voting methods are total functions on the set of all profiles recall that a profile is a sequence of ballots, one from each voter.
Other properties are intended to rule out some of the paradoxes and anomalies discussed above. In section 4. The next principle rules out such situations:. These are natural properties to impose on any voting method. A surprising consequence of these properties is that they rule out another natural property that one may want to impose: Say that a voting method is resolute if the method always selects one winner i.
First, consider the situation when there are exactly 3 candidates in this case, we do not need to assume Unanimity. Notice that this last election scenario can be generated by permuting the voters in the first election scenario to generate the last election scenario from the first election scenario, move the first group of voters to the 2nd position, the 2nd group of voters to the 3rd position and the 3rd group of voters to the first position.
That is, there are no Resolute voting methods that satisfy Universal Domain, Anonymity, Neutrality, and Unanimity for 3 or more candidates note that I have assumed that the number of voters is a multiple of 3, see Moulin for the full proof. Section 3. There are many ways to state properties that require a voting method to be monotonic. The following strong version called Positive Responsiveness in the literature is used to characterize majority rule when there are only two candidates:.
I can now state our first characterization result. Theorem May A voting method for choosing between two candidates satisfies Neutrality, Anonymity, Unanimity and Positive Responsiveness if and only if the method is majority rule. See May for a precise statement of this theorem and Asan and Sanver , Maskin , and Woeginger for alternative characterizations of majority rule.
When there are only two alternatives, the definition of a ballot can be simplified since a ranking of two alternatives boils down to selecting the alternative that is ranked first. The above characterizations of Majority Rule work in a more general setting since they also allow voters to abstain which is ambiguous between not voting and being indifferent between the alternatives.
A natural question is whether there are May-style characterization theorems for more than two alternatives. A crucial issue is that rankings of more than two alternatives are much more informative than selecting an alternative or abstaining.
They also show that a minor modification of the axioms characterize Approval Voting when voters are allowed to select more than one alternative. Note that focusing on voting methods that limit the information required from the voters to selecting one or more of the alternatives hides all the interesting phenomena discussed in the previous sections, such as the existence of a Condorcet paradox. This is a very strong property that has been extensively criticized see Gaertner, , for pointers to the relevant literature, and Cato, , for a discussion of generalizations of this property.
A striking example of a voting method that does not satisfy Independence of Irrelevant Alternatives is Borda Count. Consider the following two election scenarios:. In Section 3. An example of a method that is not susceptible to the multiple districts paradox is Plurality Rule: If a candidate receives the most first place votes in two different districts, then that candidate must receive the most first place votes in the combined the districts. More generally, no scoring rule is susceptible to the multiple districts paradox.
This property is called reinforcement:. The reinforcement property explicitly rules out the multiple-districts paradox so, candidates that win all sub-elections are guaranteed to win the full election. In order to characterize all scoring rules, one additional technical property is needed:. Theorem Young See Merlin and Chebotarev and Smais for surveys of other characterizations of scoring rules.
Additional axioms single out Borda Count among all scoring methods Young ; Gardenfors ; Nitzan and Rubinstein For example, it is often remarked that Borda Count and all scoring rules can be easily manipulated by the voters. Saari , Section 5. Note that the reinforcement property refers to the behavior of a voting method on different populations of voters. To make this precise, the formal definition of a voting method must allow for domains that include profiles i.
A variable domain voting method assigns a non-empty set of voters to each anonymous profile—i. Of course, this builds in the property of Anonymity into the definition of a voting method. For this reason, Young does not need to state Anonymity as a characterizing property of scoring rules. In order to characterize the voting methods from Section 2. There are also local and state elections to consider.
While presidential or other national elections usually get a significant voter turnout, local elections are typically decided by a much smaller group of voters. A Portland State University study found that fewer than 15 percent of eligible voters were turning out to vote for mayors, council members, and other local offices.
Low turnout means that important local issues are determined by a limited group of voters, making a single vote even more statistically meaningful. You may not be able to walk into a voting booth, but there are things you can do to get involved:. Participating in elections is one of the key freedoms of American life. Many people in countries around the world do not have the same freedom, nor did many Americans in centuries past.
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