Monday, April 29, 2013

Dilemma 21: The Unexpected Departure

The Unexpected Departure

The Axelrod upper equilibrium requires certain conditions. One of them is that the expected number of plays be great enough; another is that the play not end at too definite at time. If it does, then a “backwards induction paradox” destroys the Axelrod truce, no matter how long the tournament.
Consider the following scene:
Curly is about to play with Moe in a dilemma tournament sceduled to last exactly 100 rounds. Curly, a Silver Rule player, is optimistic that he can convince Moe (an Iron Rule player) that it’ll be in his own best interest to cooperate.
But Moe said, “What about the 100th round? Won’t that be the last one?”
Curly said, “Yes.”
“There won’t be any after the 100th?”
“Yes,” said Curly.
Moe asked, “So in the very last play, what’s to keep me from defecting?”
“‘Cause I’ll defect the next...” Curly said, then slapped himself on the face. “Alright, nothing will stop you from defecting on the 100th play.”
“So you might as well defect too, right?” Moe said, smiling.
“I guess so,” Curly said reluctantly. “On the 100th play.”
Moe continued, “And what about the 99th play? What’s to keep me from defecting then?”
“‘Cause I’ll defect the next...” Curly said, then slapped himself on the face. “But I’ll defect on the 100th play anyhow.”
“That’s right,” Moe said, smiling.
“So nothing’s keeping you from defecting on the 99th play.”
“That’s right,” Moe said, smiling.
“So I should defect on the 99th play also,” said Curly.
“That’s right,” Moe said. “Now, what about the 98th play?”
And so they continued! Moe whittled down Curly’s proposed truce, one play at a time, starting from the end. By the time the conversation was over, Moe had convinced Curly that the only logical course was for them to defect from each other 100 times, drawing the tournament. And so they did; yet when Curly played with Larry (a Gold Rule player) they cooperated 100 times, for a truce!
Thus we deduce, by mathematical induction, that the prospect of abruptly terminated play, even if in the far future, poisons the relationship at its inception. That is the “backwards induction paradox”.
In dilemma play, cooperation requires continuity to the end. Departure should not be at an expected time lest that light the backwards-induction fuse; departure should be unannounced, at an unexpected time.
We need an unexpected departure; but this yields a paradox. Consider this following story about an Unexpected Exam:

Once upon a time a professor told his students, “Sometime next week I will give you an exam; and that exam will be at an unexpected time. Right up until the moment I give you the exam, you will have no way to deduce when it will happen, or even if it will happen. It will be an Unexpected Exam.”
One of the professor’s students objected, “But then the exam couldn’t happen on Friday; for by then it would be expected!”
The professor said, “True.”
The student continued, “So Friday’s ruled out.”
Another student said, “But if Thursday’s the last possible day for an Unexpected Exam, then it’s ruled out too; for by Thursday the Thursday exam will be expected!”
The professor said, “True.”
And so on; by such steps the students concluded that the Unexpected Exam can’t happen on Friday, Thursday, Wednesday, Tuesday, or Monday; so it can’t happen at all!
“So you don’t expect it?” said the professor.
His students said, “No!”
The professor smiled...
The next Wednesday, he handed out an exam, to the students’ surprise.

That’s the Paradox of the Unexpected Exam. This also is a backwards induction paradox; but this time it is a strangely false result rather than a strangely undesirable result. This match suggests the following fable.

The same professor visited the Dean; he said, “I will depart this school sometime during the next month. To ensure cordial relations between us until that time, my departure will take place on an unexpected day.”
The Dean retorted, “You couldn’t leave on the 31st, for by then your Unexpected Departure would be expected.”
The professor agreed.
The Dean added, “Having ruled out the 31st, the 30th is also ruled out; for it would be expected.”
The professor agreed to that too.
And so the conversation continued; and in the end the Dean concluded, “Your Unexpected Departure can’t happen on any day. Therefore I don’t expect it.” The professor agreed.
On the seventeenth day of the month the professor departed, to the Dean’s astonishment.

This Paradox of the Unexpected Departure is just what the doctor ordered; for here the failure of backwards induction (so puzzling to the reason) is precisely what is needed to defend the Axelrod equilibrium from its backwards induction proof!
Above I insisted that dilemma tournaments use “open bounding”; that is, replay only if a random device permits it. This ensures an Unexpected Departure; play will be finite, but there will be no definite last play during which the Iron player is safe from the danger of Silver retaliation.

The conclusion then is clear; let none of your social relationships end too definitely; let there be some possibility that you might encounter that person again, soon. (And conversely, when you must leave, slip away quietly!)

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