The Polchinski paradox provides a succinct representation of the complications surrounding time travel at a macroscopic scale. By proposing a simple scenario - a billiard ball journeying through a wormhole and emerging in the past, it sidesteps intricate questions about quantum superposition, free will, and paradoxical situations. In this scenario, the future version of the ball collides with its past version, redirecting its path and preventing it from ever entering the wormhole.
This fascinating paradox was put forward by Joe Polchinski to Kip Thorne in 1990, and Thorne extensively discusses it in his 1994 book, "Black Holes and Time Warps". The paradox also sparked the interest of Fernando Echeverria and Gunnar Klinkhammer, who explored the physics underlying the possible trajectories and collisions of the ball.
Their analysis revealed numerous scenarios where the future version of the ball either knocks the past version into the wormhole or fails to hit it with enough force to stop it from entering. Thorne and Klinkhammer later applied quantum mechanics to these scenarios, arriving at the conclusion that Polchinski's setup can yield different outcomes, but the likelihood of a non-paradoxical outcome is significantly high. This aligns with the Novikov self-consistency principle, which insists that the probability of a non-paradoxical outcome must be 100%.
The paradox resides not in the physical laws, but in the semantics of the proposition. Our language allows us to treat the two occurrences of the ball being hit at 3:30 as separate events, but the universe does not differentiate between them. The dilemma lies in our inability to simultaneously depict a collision and the absence of one.
Turning to real-world instances of Parrondo's Paradox, it's often considered quite mundane. Imagine two activities - work (performed at the office) and leisure (enjoyed at the park). You can only participate in either activity if your friends are present at the respective locations. Your severe social anxiety means that your discomfort from being alone significantly outweighs the joy derived 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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