Chooser -- let's say that's you --- plays a game wherein you choose whether to open both boxes A and B or just B. For whatever odd reason -- Predictor is an omniscient diety, Predictor is running a very repeatable simulation of the Chooser's mind, time travel, the Predictor has repeatedly run the Chooser through this experiment before, etc. -- the Predictor already knows with very high probability what box(es) the Chooser will select. The first step in the game is that the Predictor makes his prediction. Then a trustworthy third party puts $1,000 into box A, and $1,000,000 into box B if the Predictor has predicted B only, otherwise nothing into box B. After these two steps are completed the Chooser selects which box(es) to open.
Here's the payoff matrix for Chooser:
|Predicted choice||Actual choice||Payoff|
|A and B||A and B||$1,000|
|A and B||B only||$0|
|B only||A and B||$1,001,000|
|B only||B only||$1,000,000|
What is Chooser's best strategy? According to game theory, A and B beats B-only by $1,000 whether the Predictor predicts A and B or just B. Therefore Chooser should choose to open both A and B.
But another strategy for Chooser is the following: The Chooser, knowing that the Predictor has very good information about his forthcoming choice and has mechanically acted on it, and thus is almost surely correct, eliminates those choices where the Predictor is wrong and chooses B-only over A and B because of the remaining two choices it has a higher payoff.
By following 2-player game theory, the expected value of Chooser's winnings is only a bit over $1,000, while by taking into account the high accuracy of the Predictor's prediction and maximizing expected value the expected value is a bit over $1,000,000.
The paradox dissappears if you stop assuming a game with two players. The Predictor does not have any stake in the outcome and is not a player. Indeed, we are told that Predictor will make no free choice at all, but will simply mechanically predict what Chooser will do. It's the Chooser's free will in the face of a mechanical but oddly well-informed Predictor. Chooser should not be misled by the payoff matrix into assuming this is a 2-player game, but should instead choose the best of the two choices of significant probability.
Enrico Fermi asked, if there are extraterrestrial civilizations, why haven't we seen them? No alien artifacts large or small on earth, nor anything visibly unnatural on the many millions of square kilometers of billion-year-old (or more) surface we've observed elsewhere in our solar system, nor any visible megastructures in our galaxy. Under Darwinian evolution life and civilization tends to spread to use as much energy and matter as it can. New volcanic islands and areas where life has been destroyed by volcanoes are quickly colonized by a very observable spread of plants that soon soak up a significant fraction of the incoming sunlight. Human civilizations have similarly in the blink of astronomical time spread all over our planet, leaving a number of highly visible artifacts such as the Great Wall of China, a dazzling display of lights on the night side of our planet, and in our atmosphere increased carbon dioxide and a potpouri of odd chemicals. Photosynthetic life has given our planet a bizarre oxygen atmosphere and turned our continents significantly darker. Even if an alien society somehow turned radically un-Darwinian and thus remained obsessively small and hidden, it would take just one "crazy hermit" to appear once in hundreds of millions of years to take off and replicate his crazy Darwinian verion of that civilization across the galaxy, building visible structures all over the galaxy to efficiently use stellar energy and recycle volatile elements.
The average star in our galaxy is about 10 billion years old; if it takes 5 billion years for life to appear and evolve into a civilization and 200 million years for the their descendants to spread across our galaxy (which only requires travel at a small fraction of the speed of light), the average civilization that has already emerged in our galaxy should have spread across it 2.3 billion years ago.
If they exist, or existed, evidence of this existence should be all over the galaxy, just as with the existence of life and civilization on earth. But we see no evidence that any advanced civilization has ever expanded into our solar system or any where else in our galaxy. Even worse, we see no evidence of artificiality in other galaxies. Artificial entities would severely change what we observe in the cosmos, but instead the millions of galaxies we've observed seem to look quite natural. This can be easily tested by studying data from our current telescopes. Unless we discover a significant proportion of galaxies that are oddly dim in the optical but bright in the infrared (or perhaps, if the alien machines are extremely clever and very miserly, in the microwave), indicating artificially efficient use of stellar energy, and oddly bright in the heavy element spectra and dim in the volatile elements spectra, indicating artifically efficient enclosure of the volatile elements (for efficient volatile recycling), we must conclude that the odds of finding a civilization in any given galaxy, and thus of another civilization in our own galaxy, are remote. Fermi's paradox simply proves its major assumption wrong: there are no little green men in our galaxy. Sorry to pop that good old sense of wonder.
It would be fun to listen in on a civilization from a distant galaxy, if we ever find one and if we ever figure out how to detect such faint signals. Imagine the Wikipedia of a billion-year-old civilization!
Perhaps less easily resolved, because one of its assumptions is the vague and subjective idea of "intent", is the paradox of Gregory Kavka's toxin: you get $1 million put in your bank account at 9 AM if at 7 AM you intend to drink an extremely painful but not otherwise harmful toxin at 11 AM. The toxin basically causes you to live in sheer hell for 24 hours. You get to keep the million dollars whether you drink the toxin or not. Can you intend to do something that when the time comes would not be rational to do? Kavka says you can't -- that there is no way to win the $1 million. Turn off your alarm and sleep in.
My analysis is that the only ways to win the $1 million are through credible commitment or self-delusion. Thus the Wikipedia entry cites election-year political promises that would actually be too expensive to implement as an example of Kavka's toxin. Of course the political party only fools others (and perhaps themselves) into believing its intent. Standard election procedures, in which campaign promises are not legally binding, prevent credible commitment, so serious intent only could arise through self-delusion.