Bringing together what others have said, I propose a solution in three steps:
Adopt a mixed strategy where, for each day, you open the box on that day with probability p. The expected utility of this strategy is the sum of (p (1-p)^n 2^n), for n=0… which diverges for any p in the half-open interval (0,0.5]. In other words, you get infinite EU as long as p is in (0,0.5]. This is paradoxical, because it means a strategy with a 0.5 risk of ending up with only 1 utilon is as good as any other.
Extend the range of our utility function to a number system with different infinities, where a faster-growing series has greater value than a slower-growing series, even if they both grow without bound. Now the EU of the mixed strategy continues to grow as p approaches 0, bringing us back to the original problem: The smaller p is, the better, but there is no smallest positive real number.
Realize that physical agents can only choose between a finite number of strategies (because we only have a finite number of possible mind states). So, in practice, there is always a smallest p: the smallest p we can implement in reality.
So that’s it. Build a random number generator with as many bits of precision as possible. Run it every day until it outputs 0. Then open the box. This strategy improves on the OP because it yields infinite expected payout, and is intuitively appealing because it also has a very high median payout, with a very small probability of a low payout. Also, it doesn’t require precommitment, which seems more mathematically elegant because it’s a time-symmetric strategy for a time-symmetric game.
Bringing together what others have said, I propose a solution in three steps:
Adopt a mixed strategy where, for each day, you open the box on that day with probability p. The expected utility of this strategy is the sum of (p (1-p)^n 2^n), for n=0… which diverges for any p in the half-open interval (0,0.5]. In other words, you get infinite EU as long as p is in (0,0.5]. This is paradoxical, because it means a strategy with a 0.5 risk of ending up with only 1 utilon is as good as any other.
Extend the range of our utility function to a number system with different infinities, where a faster-growing series has greater value than a slower-growing series, even if they both grow without bound. Now the EU of the mixed strategy continues to grow as p approaches 0, bringing us back to the original problem: The smaller p is, the better, but there is no smallest positive real number.
Realize that physical agents can only choose between a finite number of strategies (because we only have a finite number of possible mind states). So, in practice, there is always a smallest p: the smallest p we can implement in reality.
So that’s it. Build a random number generator with as many bits of precision as possible. Run it every day until it outputs 0. Then open the box. This strategy improves on the OP because it yields infinite expected payout, and is intuitively appealing because it also has a very high median payout, with a very small probability of a low payout. Also, it doesn’t require precommitment, which seems more mathematically elegant because it’s a time-symmetric strategy for a time-symmetric game.