Question 15.57

25. On an overcast morning, deciding whether to carry your umbrella can be viewed as a game between yourself and nature as follows:

Weather
Rain No rain
You Carry umbrella Stay dry Lug umbrella
Leave it home Get wet Hands free

Let’s assume that you are willing to assign the following numerical payoffs to these outcomes, and that you are also willing to make decisions on the basis of expected values (i.e., average payoffs):

662

  1. If the weather forecast says there is a 50% chance of rain, should you carry your umbrella or not? What if you believe there is a 75% chance of rain?
  2. If you are conservative and wish to protect against the worst case, what pure strategy should you pick?
  3. If you are rather paranoid and believe that nature will pick an optimal strategy in this two-person zero-sum game, what strategy should you choose?
  4. Another approach to this decision problem is to assign payoffs to represent what your regret will be after you know nature’s decision. In this case, each such payoff is the best payoff you could have received under that state of nature, minus the corresponding payoff in the previous table.
Weather
Rain No rain
You Carry umbrella
Leave it home

What strategy should you select if you wish to minimize your maximum possible regret?

25.

(a) 50% chance of rain: leave umbrella; 75% chance of rain: carry umbrella

(b) Carry umbrella

(c) Saddlepoint at"carry umbrella"and"rain,"giving value −2

(d) Leave umbrella

A-37