(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.
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.
(Continuation of Exercise 7, due to David Blackwell.) The question now arises: What randomizing strategy should...
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