Puzzle time - three way tie
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Alison, Bonnie, and Clyde run for class president and finish in a three-way tie. To break it, they solicit their fellow students' second choices, but again there is a three-way tie. The election committee is stymied until Alison steps forward and points out that, since the number of voters happens to be odd, they can make two-way decisions. She therefore proposes that the students choose between Bonnie and Clyde, and then the winner would face Alison in a runoff.
Bonnie complains that this is unfair because it gives Alison a better chance to win than either of the other two candidates. Is Bonnie right?
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On the surface of the top, it seems like Bonnie is right.
Like in sports leagues, the "best" team sometimes gets a "free pass" in the first game.
For example, #1 gets the "pass" and #2, and #3 play each other for the right to play #1.
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On the surface of the top, it seems like Bonnie is right.
Like in sports leagues, the "best" team sometimes gets a "free pass" in the first game.
For example, #1 gets the "pass" and #2, and #3 play each other for the right to play #1.
@taiwan_girl said in Puzzle time - three way tie:
On the surface of the top, it seems like Bonnie is right.
Like in sports leagues, the "best" team sometimes gets a "free pass" in the first game.
For example, #1 gets the "pass" and #2, and #3 play each other for the right to play #1.
If they're entirely evenly matched, then maybe this is like rock-paper-scissors. No matter which pair you pit against each other in the first round, the winner loses the next round.
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Horace hinted at the right answer.
SOLUTION: Bonnie is correct — in fact, she has understated the case. Assuming no voters change their minds, Alison will win for sure!
To see this, suppose Alison's supporters mostly prefer Bonnie to Clyde (so that Bonnie would beat Clyde in the proposed two-candidate race). Then Bonnie's supporters must prefer Clyde to Alison, otherwise Clyde would have garnered fewer than 1/3 of the second-place votes; similarly Clyde's supporters prefer Alison to Bonnie. Thus, in this case, Alison will beat Bonnie in the runoff.
If Alison's supporters prefer Clyde to Bonnie, a symmetric argument shows that Alison will beat Clyde in the runoff.
This puzzle serves as a warning: There may be more to some tiebreakers than meets the eye!
