Thursday, April 25, 2013

Dilemma 19: Predictor’s Paradox

Predictor’s Paradox

This paradox is a failed attempt to resolve Dilemma. It teaches us that not even a confrontation with a Superior Being can make certain people behave themselves.
In the Predictor’s Paradox, you (an ordinary mortal) are shown a pair of boxes. Box A is open; $1 can be seen within. Box B is shut; it contains either $100 or $0. The other player claims to be a Superior Being who can predict your actions. “If you choose to take both boxes”, says the Being, “then you’ll discover that I’ve punished you by putting nothing in box B; but if you have faith in me and take only box B, then you’ll find my reward of $100 there.”
Let us assume that previous experience has shown that the Being can apparently make good on its claim of being able to predict your actions; what should you do? Here’s the game matrix:

payoff for mortal   

            |  rewards    punishes
takes box B |   100    |     0     |
mortal                -|----------|-----------|
takes both  |   101    |     1     |

Meanwhile, what’s in it for the Being? Let us suppose, for the sake of symmetry, that the Being’s game matrix is as follows:

payoff for Being

            |  rewards    punishes
takes box B |   100    |    101    |
mortal                -|----------|-----------|
takes both  |    0     |     1     |

Presumably the Being values your faith one hundred times more than the material profit of punishing you. This way you and the Being are in a Dilemma game, where win = 101, truce = 100, draw = 1 and lose = 0.
Your best move depends on how well the Superior Being can foretell your actions. If the Being can correctly predict your actions with probability exceeding (in this case) 50.5 %, then the expected value of taking one box exceeds that of taking both. In this case, reasoning by expected value favors leaving the $1 alone; yet taking the $1 would still be a dominant strategy!
Two different lines of argument yield two opposite recommendations. How are we to decide? That is the question. The Predictor’s Paradox represents a dispute between the principles of Dominance and Expectation. It shows that not even a confrontation with a Superior Being can make Iron-rule players behave themselves! That $1 sitting there, just begging to be taken... how can they resist such a devilish temptation? If there  really were a Superior Being, then this little test would ensure the self-defeat of all habitual exploiters, whilst humbler folk win riches!

Actually I’m satisfied neither with blind faith, nor with exploitation. Blind faith in the Being is a Gold Rule strategy, and as such is vulnerable if the Being is an Iron Rule player. Similarly the attempt to exploit the Being is an Iron Rule strategy. The optimum long run strategy is the Silver Rule:
Do Unto Others As They Have Done Unto You.
This principle of cosmic justice is so powerful that even a Superior Being must meet us there on equal terms.
Therefore, if I were ever to confront a Superior Being in this fashion, I would form the intention to take either $1 or $100, but not $0 nor $101. After all, why should I ignore the $1 bill if I get nothing otherwise? And conversely, why should I try to cheat a Superior Being of $1 if it already gave me $100?
Thus I, a mere mortal ruled by greed and fear, propose to make myself the equal of a Superior Being! I leave it to you, dear reader, to judge the soundness of my thinking; but note that within this mental context, the Being has every reason to give me $100.

Some may object that there are no Superior Beings in evidence with whom to play this game; to this I reply that the Silver Rule is so powerful that it enables a mere mortal to make a passable imitation of a Superior Being, provided that the shadow of the future is long enough.

For let us iterate this game with open bounds, with replay probability 99/100, so the expected number of plays is 100. Rescale the payoffs accordingly, at 1/100th of the payoffs noted above; i.e. 1 cent in box A, $0 or $1 in box B. In this Predictor’s Tournament, I shall play the Superior Being’s role; my strategy will be tit-for-tat. If you take both boxes on a given round, then on the next round I’ll leave box B empty; and if you take only box B, then on the next round I’ll put $1 in box B. (In a sense, the Superior Being that I emulate is Reciprocity itself!)
If you are rational, and if play is long enough, then we will attain truce; you will always take only box B, and you will always find $1 in it. Though I can’t predict you, I do remember you; so in the long run it will be in your interest to act as if I could predict you. Continuity is the key; if the “shadow of the future” is long enough, then my memory, like the Superior Being’s prophesy, will enforce social harmony. My hindsight equals Reciprocity’s foresight.
A shadow is haunting Earth; the shadow of the future. Will we be or will we not be? That is the question. When the future’s shadow is short, not even a Superior Being can deter the wicked from maximizing profit; but when the shadow of the future extends, then you and I can be like unto Superior Beings, and peace breaks out! Blessed be the shadow of the future!


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