Burr dilemma

The Burr dilemma or chicken dilemma is a kind of strategic voting that may affect approval or score voting. The term was used in The Journal of Politics (2007) by Jack H. Nagel, who named it after Aaron Burr, who initially tied with Thomas Jefferson for Electoral College votes in the United States presidential election of 1800. According to Nagel, the electoral tie resulted from "a strategic tension built into approval voting, which forces two leaders in appealing to the same voters to play a game of Chicken."

Scenario
In a Burr scenario, a group of voters prefer two candidates (traditionally called Jefferson and Burr) from the same political party or faction (traditionally called the Republicans or Democratic-Republicans, not to be confused with the Republican Party). These candidates face a unified opposition (the supporters of Adams). This creates a dilemma:
 * If a Republican voter supports both Jefferson and Burr, they are effectively casting no votes because they do not make a distinction between the top two candidates. If most Republicans support both, the election is also a near-tie, with the outcome being determined essentially by chance.
 * On the other hand, if too many electors vote only for Jefferson or only for Burr, Adams will be elected.

History
The 1800 United States presidential election was conducted using a voting-rule similar to approval voting, though not quite identical. Each member of the Electoral College was allowed to vote for two candidates, with all votes being counted anonymously and simultaneously. The candidate with the most votes would be president, and the one with the second-most would be vice-president.

The Democratic-Republicans held a majority in the Electoral College that year, with 73 electors versus only 65 Federalists. Most of these electors were instructed to vote for both Jefferson and Burr, with the intention of securing both the Presidency and the Vice-Presidency for their party.

Ultimately the electors fell into the first trap, with all 73 Republicans (terrified by a possible Adams victory) supporting both Jefferson and Burr. The resulting tie nearly caused a constitutional crisis when the tiebreaking mechanism deadlocked as well.

Score voting
Score voting lessens the Burr dilemma when some voters are either honest or willing to cooperate. In the real Burr election, honest electors might assign a score of 10/10 to Jefferson, 9/10 to Burr, and 0/10 to Adams; or they might all agree ahead of time to use this strategy. This strategy gives Jefferson 73 points and Burr 65.7 points.

However, score voting systems can behave similarly to approval if voters act in a way that is individually strategic, suggesting score voting can only be a partial solution.

Ranked Condorcet voting
Paired tournaments (i.e. Condorcet methods) elect candidates who win a pairwise majority vote. In the race above, Jefferson would win against Adams 73-65, while Jefferson would likely defeat Burr in a one-on-one election, allowing Jefferson to safely secure the presidency.

However, Condorcet methods have their own advantages and disadvantages; for example, they can suffer from spoiler effects, which do not occur in honest approval and score voting.

STAR voting
STAR voting is a compromise between score and the ranked Condorcet methods, which still maintains a "mostly score-based" system. Candidates give each candidate a score, and the top-two candidates face a runoff, where the candidate preferred by a majority wins.