Seeking help / thought process guidance on the following interview problem, which seems centred on game theory
Setup: there’s a number X which we can measure once with error following N(0, 1). We can choose whether to receive $X (negative means lose money)
a) When do we choose to receive X?
b) Now another agent can also measure X with iid error, and choose to receive X before you (if they do so we can no longer receive X). What’s the strategy now?
c) Back to single-player. Now after measuring, X moves up/down by one every second with even probability. We cannot observe again but know each second which direction X moves. What’s the strategy?
d) Same as c) except the other agent comes back in and has the same information/setup (iid measure, can see X’s moves, can collect before us each second). What’s the best strategy now?
For a), I was thinking we should take the observed value as the best estimate and accept if it is positive. For b), I guess if the other player doesn’t accept when we see a positive value, that makes it more likely the true value is lower?
For (b), suppose we observe y_1 = X + e_1, and the other agent observes y_2 = X + e_2.
Then I guess we want to compute E(X | y2 < 0) = y_1 - E(e_1 | y_1 - e_1 + e_2 < 0)?
I tried bashing it out, but this seems to require some convolution integral, or likely that I'm overcomplicating it...
