Question

7. Guessing what box. Consider a game as in Examples 1 and 2, where I pick one of the three boxes, then you guess which box I picked after seeing the color of a ball drawn at random from the box. Then you learn whether your guess was right or wrong. Suppose we play the game over and over, replacing the ball drawn and mixing up the balls between plays. Your objective is to guess the box correctly as often as possible. Section 1.5. Bayes' Rule 55 a) Suppose you know that I pick a box each time at random (probability 1/3 for each box). And suppose you adopt the strategy of guessing the box with highest posterior probability given the observed color, as described in Example 1, in case the observed color is white. About what proportion of the time do you expect to be right over the long run? b) Could you do any better by another guessing strategy? Explain. c) Suppose you use guessing strategy found in a), but I was in fact randomizing the choice of the box each time, with probabilities (1/2,1/4, 1/4) instead of (1/3,1/3, 1/3). Now how would your strategy perform over the long run? d) Suppose you knew I was either randomizing with probabilities (1/3,1/3, 1/3) or with probabilities (1/2,1/4, 1/4). How could you learn which I was doing? How should you respond, and how would your response perform over the long run?

(Continuation of Exercise 7, due to David Blackwell.) The question now arises: What randomizing strategy should I use to make it as hard as possible for you to guess correctly? Consider what happens if I use the (f:J, f:J, strategy, and answer the following questions:

a) What box should you guess if you see a black ball? b) What box should you guess if you see a white balJ? c) What is your overall chance of winning?

You should conclude that with this strategy, your chance of winning is at most f: J, no matter what you do. Moreover, you have a strategy which guarantees you this chance of winning, no matter what randomization I use. It is the following:
If black, guess 1 with probability 2 with probability and 3 with probability O. If white, guess 1 with probability 0, 2 with probability H, and 3 with probability K

d) Check that using this strategy, you win with probability f: J, no matter what box I picked.

According to the above analysis, I can limit your chance of winning to f:J by a good choice of strategy, and you can guarantee that chance of winning by a good choice of strategy. The fraction f:J is called the value of the above game, where it is understood that the payoff to you is 1 for guessing correctly, 0 otherwise. Optimal strategies of the type discussed above and a resulting value can be defined for a large class of games between two players called zero-sum games. For further discussion consult books on game theory.

Answer 1

What randomizing strategy should I use to make it as hard as possible for you to guess correctly? Consider what happens if I use the (f:J, f:J, strategy, and answer the following questions:

a) What box should you guess if you see a black ball? b) What box should you guess if you see a white balJ? c) What is your overall chance of winning?

You should conclude that with this strategy, your chance of winning is at most f: J, no matter what you do. Moreover, you have a strategy which guarantees you this chance of winning, no matter what randomization I use. It is the following:
If black, guess 1 with probability 2 with probability and 3 with probability O. If white, guess 1 with probability 0, 2 with probability H, and 3 with probability K

d) Check that using this strategy, you win with probability f: J, no matter what box I picked.

According to the above analysis, I can limit your chance of winning to f:J by a good choice of strategy, and you can guarantee that chance of winning by a good choice of strategy. The fraction f:J is called the value of the above game, where it is understood that the payoff to you is 1 for guessing correctly, 0 otherwise. Optimal strategies of the type discussed above and a resulting value can be defined for a large class of games between two players called zero-sum games. For further discussion consult books on game theory.