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Game theory in the popular press.

Schrödinger's Games

For quantum prisoners, there may be no dilemma

Scientific American
January 2000
text is a cache of http://www.sciam.com/article.cfm?articleID=00013742-4C8B-1C76-9B81809EC588EF21

The FBI has seized your computers and brought you in for questioning. They know about you and your colleague's plan to plunder the U.S. financial system with an ingenious new computer virus. Fortunately for you, the evidence is securely encrypted. But here's their offer: if you tell them the password, so they can access the evidence, they'll throw the book at Alice, your co-conspirator, to set an example, and hire you as a computer security expert. Of course, you can bet they have Alice in for questioning, too. If you both keep your traps shut, they'll have to let you go. If you both rat, you'll both do some time, but with some chance of parole, eventually.

This is the classic prisoner's dilemma: whatever Alice chooses, your best option, as a self-interested perp, is to rat on her. Unfortunately, the same logic will make her rat on you, the turncoat, and you'll both end up doing time instead of going free.

Is there any way out? Ethical considerations aside, no. The mathematics is watertight. The logic does change if you will be "playing" prisoner's dilemma many times indefinitely. Then the most profitable strategy seems to be "tit-for-tat": don't defect against a partner unless they have previously defected against you. But that's of no use in one round of the game for monumental stakes.

It turns out, as shown in a paper in the October 11 Physical Review Letters, that there can be a way out if the situation is ruled by quantum mechanics. The essence of quantum mechanics, and how it can help, is embodied by another hapless inmate: Schrödinger's cat. In the infamous thought experiment the cat becomes a superposition of alive and dead inside its infernal box, and only when a measurement is made--someone opening the box and looking in--does it become wholly alive or dead.

Similarly, one can conceive of a superposition of defecting and not defecting. Physicist Martin Wilkens of the University of Potsdam in Germany and his co-workers show how to extend the prisoner's dilemma to a theoretical quantum system. Each prisoner encodes his choice (some superposition of defect and cooperate) onto a simple quantum element inside a device. The device combines the two elements, performs a measurement and announces a definite choice (either defect or cooperate) for each prisoner.

When the device is configured to take the most advantage of another quantum effect--when it "maximally entangles" the two choices--the dilemma vanishes: among the new quantum choices available is one that will let each prisoner reap the benefit of keeping quiet. Neither player has a motive for deviating from this preferred quantum strategy; doing so would lower his or her expected payoff. The entanglement links the announced choices, so that you and Alice can cooperate without risk (or temptation) of unilateral defection.

Another quantum variant of game theory was studied by David A. Meyer of the University of California at San Diego: a game called "penny flipover." A coin inside a box begins heads-up, and the players (called Picard and Q, by Meyer) take turns flipping it over, or not, without seeing which side is up; first Q, then P, and finally Q again. If the coin finishes heads up, Q wins. With a classical penny, each player does best by flipping at random, winning half the time. But if Q can exploit quantum superpositions, he can win every time. First he puts the coin into an equal superposition of heads and tails. This state is unchanged even if P flips the coin. On his final turn, Q returns the superposition back to purely heads

The importance of such results is not for avoiding the clutches of quantum G-men or con men. Instead they provide an instance of how quantum principles alter information processing. Furthermore, the coin flipper is a prototype error-correction system--negating the effects of random "errors" introduced by player P's move.


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