Example 1

EXAMPLE 1 Manipulating the Borda Count with Four Candidates and Two Voters

Suppose there are two voters and four candidates, and suppose the true preferences of the voters are reflected in the following ballots:

Voter 1	Voter 2
$A$	$B$
$B$	$C$
$C$	$A$
$D$	$D$

Using the Borda count with point values 3, 2, 1, 0 (or by counting the number of occurrences of other candidates below the one in question, as described in Section 9.3), we see that the Borda scores of the four candidates are as follows:

The Borda score of $A$ is 4.
The Borda score of $B$ is 5.
The Borda score of $C$ is 3.
The Borda score of $D$ is 0.

Thus, Candidate $B$ wins this election. Voter 1, however, would have preferred to see Candidate $A$ —her top choice, according to her true preferences—win this election rather than Candidate $B$ , her second choice.

Assume that Voter 1 had known that Voter 2 planned to submit the ballot that he cast above. Could Voter 1 have secured a victory for Candidate $A$ by submitting a disingenuous ballot?

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The answer here, as we'll show, turns out to be "yes." The intuition is fairly transparent: Voter 1 wants to pretend that $B$ is not her second choice, but her last choice. Let's see if this is enough to bring about the desired switch in winner from $B$ to $A$ . The new ballots and Borda scores are as follows:

Voter 1	Voter 2
$A$	$B$
$C$	$C$
$D$	$A$
$B$	$D$

The Borda score of $A$ is 4.
The Borda score of $B$ is 3.
The Borda score of $C$ is 4.
The Borda score of $D$ is 1.

Close, but not quite what we wanted: Candidates $A$ and $C$ now tie for the win, and we wanted the winner to be just Candidate $A$ . But a moment's inspection reveals that Voter 1 can achieve this if, in addition to plunging Candidate $B$ to the bottom of her ballot, she also flip-flops $C$ and $D$ . That is, the desired ballots (and Borda scores) that yield Candidate $A$ as the sole winner are as follows:

Voter 1	Voter 2
$A$	$B$
$D$	$C$
$C$	$A$
$B$	$D$

The Borda score of $A$ is 4.
The Borda score of $B$ is 3.
The Borda score of $C$ is 3.
The Borda score of $D$ is 2.

Thus, Voter 1 can change her ballot and—with Voter 2 making no change at all— cause the election outcome to go from $B$ to $A$ . Moreover—and this is very important— Voter 1 prefers $A$ to $B$ ! The reason we know that Voter 1 prefers $A$ to $B$ is that we are assuming the original ballots represented the voters′ true preferences, and Voter 1 ranked $A$ over $B$ on her original ballot.