MATH 181 D2: A MATHEMATICAL WORLD

W5: Manipulability & Weighted Systems Objectives: SWBAT

Manipulate a given election to produce the desired winner under any manipulable voting system. r Recognize which voting systems are manipulable. r Describe a weighted voting system using the quota and the voting weights r Determine which players are dictators or dummy voters r

Recall

Arrow’s Impossibility Theorem: With three or more candidates and any number of voters, there does not exist–and there never will exist–a voting system that always produces a winner, satisfies the Pareto condition and IIA, and is not a dictatorship.

So how badly do the suggested voting systems do? We are going to start by looking more closely at the manipulability of the system.

Group Activity: Worksheet W5.1

Def: A voting system is said to be manipulable if there exist two sequences of preference list ballot and a voter (call the voter Jane) such that

1. Neither election results in a tie. 2. The only ballot change is by Jane. 3. Jane prefers–assuming that her ballot in the first election represents her true preferences–the outcome of the second election to that of the first election.

We see this illustrated in the example from the worksheet. Note: Neither , in the case of two candidates, nor Condorcet’s method, in the case of three or more candidates are manipulable. What about the ? We saw earlier that with four candidates and 2 voters it was manipulable, what about with 3 candidates? Group Activity: Worksheet W5.2

1 Example: Suppose the candidates are A, B, and C, and that you prefer A to B, but B is the election winner using the Borda count. We want to manipulate the ballot so that A emerges as the winner. Case 1: Your sincere ballot is A over B over C. No ballot change on your part can increase A’s Borda score, and you can decrease B’s Borda score by no more than 1. Thus, at best, you can make a unilateral change that results in A and B having the same Borda score, whereas successful manipulation on your part requires that A have a strictly higher Borda score than B after your ballot change. Case 2: Your sincere ballot is C over A over B. No ballot change on your part can decrease B’s Borda score, and you can increase A’s Borda score by no more than 1. Thus, at best, you can make a unilateral change that results in A and B having the same Borda score, whereas successful manipulation on your part requires that A have a strictly higher Borda score than B after your ballot change. Case 3: Your sincere ballot is A over C over B. No ballot change on your part can increase A’s Borda score or decrease B’s Borda score. Thus, after your ballot change, B will still have a higher Borda score than A, so your attempt at manipulation has failed in this case as well.

Theorem: With exactly three candidates, the Borda count cannot be manipulated. With four or more candidates (and two or more voters), the Borda count can be manipulated.

What about the other systems? Group Activity: Worksheet W5.3

As we can see, sequential pairwise voting is not only manipulable by rearranging the agenda, as we saw last class, but it can also be manipulated by a single voter. We’ve been looking so far at voting systems that treat all the voters equally, but this isn’t always the case. For example in a corporation shareholders are asked to vote on motions presented by the corporation’s board of directors, they are allotted one vote per share that they own, so if two shareholders own different numbers of shares, they are not treated equally as voters. Systems in which the voters have varying number of votes are called weighted voting systems.

A weighted voting system is described by specifying the voting weights w1, w2, . . . , wn of the players (P1,P2,...,Pn), and the quota, q. The following notation is a shorthand way of making these specifications: [q : w1, w2, . . . , wn]

If the sum of the voting weights of voters who favor a motion is greater than or equal to the quota, then “yes” wins. If the total voting weight of voters favoring the motion is less than the quota, then “no” wins. It must be greater than half the total weight of all the voters, to avoid situations where contradictory motions can pass, and it cannot be greater than the total weight, or no motion would ever pass. The U.S. functions a as a weighted voting system when electing the president. The voters are the states. 2 Examples: 1. [51 : 40, 60] - two voters, with voting weights 40 and 60, and the quota is 51. This system has a dictator. This means that a motion will pass if and only if the dictator is in favor, and it doesn’t matter how the other participants vote. Most voting systems we consider will not have a dictator, if there is one, his or her voting weight must be equal to or more than the quota. 2. [8 : 5, 3, 1] - three voters, with voting weights 5, 3, and 1. The weight-1 voter is called a dummy voter because he will never have the opportunity to cast a deciding vote. A motion will pass only if it has the support of the weight-5 and the weight-3 voters, and then the additional 1 vote is not needed.

3