Sunday, February 7, 2021

What is Polchinski’s paradox?

The Polchinski paradox is an elegantly simple summary of the problems with time travel at a macroscopic level.  It neatly avoids questions of free will and grandfathers and baby Hitlers and quantum superposition by presenting an extremely simple scenario: A billiard ball traveling through empty space enters one mouth of a wormhole and exits the other mouth 30 minutes in the past; the future-ball proceeds on a collision course with past-ball, altering past-ball’s trajectory so that it never enters the wormhole at all.

Joe Polchinski proposed the scenario to Kip Thorne in 1990.  Thorne discusses it at length in his book Black Holes and Time Warps (1994), as well as efforts to resolve the paradox by Fernando Echeverria and Gunnar Klinkhammer.

Given that the ball will pass through the wormhole and hit itself, Echeverria and Klinkhammer examined the physics of the possible courses/collisions of the ball.  They found many scenarios where the future-ball either knocks past-ball into the wormhole (figure a) or doesn’t hit past-ball hard enough to prevent it from entering the wormhole (figure b).  Thorne and Klinkhammer later evaluated these various scenarios with quantum mechanics; they concluded that, although Polchinski’s setup may lead to different outcomes each time, the probability of a non-paradoxical outcome is quite high.  This conforms to the Novikov self-consistency principle, which asserts that the probability must be 100%.

The contradiction lies not in the laws of physics but the semantics of the proposal: A ball was not hit at 3:30, then it was hit at 3:30.  Our language allows us to consider these two 3:30’s as separate states in a narrative, but the universe doesn’t.  Even the diagram at the top struggles with this–it can show the collision and non-collision paths of the ball, but it cannot simultaneously depict the collision and the absence of a collision.

What are some real-life examples of Parrondo's Paradox?

As far as I know, Parrondo's Paradox is actually pretty boring. Let's say you have two activities: work, which you do at the office, and leisure, which you do at the park. Let's say you can only do work or leisure if your friends are at the venue in question. And your crippling social anxiety means that your disutility from loneliness is many times greater than your utility from work or leisure.

Two bad strategies would be:

Go to the park every day

Go to the office every day

But what if you alternate these two losing strategies, switching to the "Office" strategy every Monday and the "Park" strategy every Saturday? Presto, Parrondo! You've found a way to alternate two losing strategies in order to create a single winning strategy.

This also seems to work with, e.g., a strategy of only exhaling or only inhaling. Alternate between them every couple seconds, and you have a viable strategy called breathing.

The mathematical illustrations of Parrondo's Paradox usually involve games where one payout is based on the player's capital. To construct the kind of Parrondo's Paradox that won't irritate mathematicians, you need two payoff functions like:

You are a cab driver in a city with nice neighborhoods and rough neighborhoods. In nice neighborhoods, there's lots of competition, so it's hard to get a fare. In rough neighborhoods, it's easy to get a fare, but you can get mugged. Muggers will take half of whatever money you have. Your bank requires you to have at least $20 to make a deposit, and you get $10 per fare. Your strategies are:

1. Find a fare in a nice neighorhood.

2. Find a fare in a rough neighborhood, at the risk of getting robbed.

If you just do #1, you can take a long time. If you just do #2, your return may go like this: $10, $5, $15, $7.50, $17.50 ... But if you do #2, then drive like hell and switch to #1, then bank your money and do #2, you may have the optimal strategy. (Like all cabbies, you do not know fear, and laugh in the face of death.)

This example is somewhat close to a possible real-world example, where the less money you have, the more risky strategies pay off. It might work for games where reputation is at stake, too. But Parrondo's Paradox requires such an elaborate, weird setup that it's hard to find real examples.

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