What should you do when all you can say is: "I know nothing. Nothing!"?
I'm reading Michael Schwarz's very interesting "Decision Making under Extreme Uncertainty", with its fascinating result that "invariance restrictions alone are sufficient to pin down the agent’s choices in some decision problems":
Suppose an agent... has no information relevant for estimating the variable. In this case her actions in decision problems where payoff is contingent on different dimensional variables are the same, i.e., in this case the name of a variable is merely an uninformative label. (Effectively, provided that an agent has never heard of either tugric or dugric her strategy for selecting a guess from an interval [1,4] is the same regardless if she is guessing the value of a ”tugric” or the length of a “dugric”.) This imposes a sever restriction on an agent’s choices. For instance, if an agent is asked to “guess” exchange rate between currencies A and B conditional on the rate being between a and b, a “guess” of (a + b)/2 is not “reasonable” because, if this decision problem is reformulated in terms of exchange rate between B and A the range becomes 1/b to 1/a and the “guess” in the mirror decision problem (1/b+1/a)/2 is not a reciprocal of the guess in the original problem. A remarkable property of invariant decision problems is that a strategy in the “image game” must be reciprocal of the strategy in the original game.
Surprisingly, in an information vacuum the invariance consideration along are sufficient to uniquely pin done the strategy of an agent in some decision problems. We showed that if the payoff relevant range in an invariant decision problem is given by [a, b], then an agent’s strategy in such a decision problem is approximated by the geometric mean given by √ab. Combining the results of this section with expected utility axioms one can show that the prior of an agent in an information vacuum corresponds to the Jeffreys’ prior 1/x...
I find myself wondering if there isn't a connection between Schwarz's idea of "invariance" and the "Grass Is Greener" Switching-Envelopes Problem, but I'm not smart enough to see clearly why I have a hunch that there is a connection.