An optimal stopping paradox

by Yuxi_Liu 1 min read12th Nov 201910 comments

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Consider an optimal stopping problem: a company at each time step grows by some constant, and has a certain probability of shutting down. You decide when to sell the company.

Since the math is cleaner in continuous time, we consider the continuous time. Then the company has a linearly increasing value βt, and an exponentially decaying survival curve e^(-αt).

Another framing of the paradox: Schrodinger wants to make a new record for the longest surviving cat, so he put a cat in the box with an atom that might decay and kill the cat, and waits. When should he open the box?

Since at each moment in time, you face the exact same problem (linearly increasing reward, α-exponentially decaying survival rate), if you decide to wait at t=0, you would decide to wait forever, and thus receive no reward.

There are several possible replies to this paradox, none of which is satisfactory to me:

  1. "This looks like St. Petersburg Paradox.". No, because at time t=0, the expectation is β/α^2. In fact, the payoff can grow faster than βt, such as like t^3, and it would still have finite expectation.
  2. Claim that expectation maximization decision theory is flawed. This doesn't stop the procrastination. As long as your decision is purely based on the future, and your rational decision process is constant in time, you either immediately sell the company or never sell the company.
  3. Try some kind of discounting, like exponential discounting. This doesn't stop the procrastination., since at any time, selling the company gives you 0 extra expected reward, and waiting gives you some positive extra expected reward, no matter how much you discount the future.
  4. Claim that there should be a finite lifetime. You can't wait forever. If there is a finite lifetime, then the same decision analysis would tell you to procrastinate until the very end. This effectively is procrastinating forever. It does not converge to a reasonable finite waiting time as your lifetime goes to infinity.
  5. Claim that one should stick to past decisions even when they don't make sense from a purely future-looking decision theory. Such decision theory seems to be just sweeping time-inconsistency under the rug, and I'm sure would suffer from serious paradoxes of their own.
  6. Claim that there is no paradox, and procrastination is really the rational action. I'd not claim a strategy that guarantees 0 reward to be rational.

Option 5 seems at least to have some meaning to it. Sticking to it would mean that, for example, one would at t=0 decide to choose T to maximize βT e^(-αT), then at t=T really sell the company, even though it's irrational, conditional on the company still alive at t=T.

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