Friday, May 31, 2013

On Troikas 15: Agenda Manipulation and Chairman’s Paradox

Agenda Manipulation and Chairman’s Paradox

One fine day Moe decided to seize absolute power. To this end he rigged an election. His nefarious scheme succeeded, but it didn’t do him any good.
“Boys,” he said after a particularly confusing wrangle, “we’re getting too many tie votes. I say we need a chairman to cast tie-breaking votes!”
“That’s a good idea!” Curly enthused. “But who should be the chairman?”
“Who but me?” said Moe. “Wasn’t I the one clever enough to think of the idea?”
“But I’m not so sure I want you for Chairman,” Larry ventured.
“Why then, let’s put it to a vote,” said Moe.
“That’s fair,” said Curly. “I nominate myself!”
“And I nominate myself,” Larry added.
After much bargaining, discussion, and fisticuffs, they settled on these preferences:
Moe:    Curly   <          Larry   <          Moe
Larry:  Moe     <          Curly   <          Larry  
Curly:  Larry   <          Moe     <          Curly

    <                  <                     2/3 majority each
                          Curly             >                 Moe

“This isn’t getting us anywhere,” Curly complained.
“Why not try a run-off election?” Moe suggested.
“You’d be the last person I’d vote for!” Larry said.
“Okay,” said Moe, “then why don’t you and Curly face off?”
“That sounds fair,” said Curly; and so the first round of voting was between Larry and Curly.
Larry won the first round, thanks to Moe’s and Larry’s vote. But then Larry faced Moe, who won with Moe’s and Curly’s votes:
   Larry         Curly
            \           /
  Larry             Moe
\           /

Had Larry been the last one considered, then the elections would have been like this:

    Moe          Curly
\           /
 Curly              Larry
\           /
And had Curly been the last one considered, then the elections would have been like this:

    Larry           Moe
\           /
  Moe               Curly
\           /
Therefore, in an election like this, the last one to be considered wins. That’s Agenda Manipulation; how Moe become Chairman of the Stooges!

“Well then!” Moe said, eagerly rubbing his hands. “Let’s decide a few things, shall we? Eh, boys?” Larry and Curly looked up. They had been discussing something together. Moe continued, “Now let’s try ranking fairness, power, and logic. Which is best? Curly?”
“Fairness is best,” said Curly.
“How about you, Larry?” Moe said gleefully. He was expecting an answer of ‘Logic’, so he could vote ‘Power’ and invoke a tie.
But Larry said, “I agree with Curly.”
More sophisticated voting! For Larry was going along with his second-favorite choice, to keep Moe from exercising the chairman’s power.
“Oh,” said Moe, crestfallen. “All right then, which one is your least favorite? Larry?”
“Power is worst,” said Larry.
“How about you, Curly?” Moe was expecting an answer of ‘Logic’, so that he could vote ‘Fairness’ and invoke a tie.
But Curly said, “I agree with Larry.”
Once again, a sophisticated vote! Curly went along with Larry’s choice, to keep Moe from using the chairman’s power.
Moe said grimly, “I see. And is Logic in the middle?”
“That’s what you believe,” Curly said.
“So that’s how we’ll vote,” Larry added.

Actual preferences:
Moe:    Fairness   <     Logic      <       Power
Larry:  Power     <       Fairness   <     Logic  
Curly:  Logic      <       Power     <       Fairness

Preferences according to Sophisticated Vote:
Power  <          Logic   <          Fairness

“But that’s the exact opposite of what I want!” Moe yelled.
“That’s because you’re chairman,” Larry explained.
Curly added, “Now we have a reason to gang up on you!”
“Power corrupts, and mathematical power corrupts mathematically,” Larry explained.
“It’s the Peter Principle,” Curly confided. “You’ve just risen to your Level of Incompetence!”
“That is what the troika is for,” Larry explained.
Curly chirped, “It’s a king trap!”
Moe hollered, “And I’m the Stooge who fell for it!”
Curly chuckled: nyuck-nyuck-nyuck!

No comments:

Post a Comment