## Exercises

1» (Simple wheel strategy) Consider a strategy of the form ()',0, 0) for the investment wheel Show that the overall factor multiplying your money after n steps is likely to be (1 -j- 2y)"/2(l — y)"'1 Find the value of y that maximizes this factor

2,. (How to play the state lottery) In a certain state lottery, people select eight numbers in advance of a random drawing of six numbers. If someone's selections include the six drawn, they receive a large prize, but this prize is shared with other winners Victor has discovered that some numbers are "unpopular" in that they are rarely chosen by lottery players He has computed that by selecting these numbers he has one chance in a million of winning \$10 million for a \$1 lottery ticket He has odds of 10 to 1 in his favor Victor 's current wealth is \$100,000, and he wants to maximize the expected logarithm of wealth.

(a) Should Victor buy a lottery ticket?

(b) Victor knows that he can buy a fraction of a ticket by forming a pool with friends What fraction of a ticket would be optimal?

3, (Easy policy) Show that |) is the optimal policy for Example 15 2

4. (A general betting wheel o) Consider a wheel with n sectors If the wheel pointer lands on sector i, the payoff obtained is rf for every unit bet on that sector T he chance of landing on sector / is , / — 1,2, , n Let a,- be the fraction of one's capital bet on sector i We require < 1 and a,- > 0 for / = 1, 2, ., //.

(a) Show that the optimal growth strategy is obtained by solving

(b) Assuming that cti > 0 for all / = I, 2, , n, show that the optimal values must satisfy Pkfk sp Pi

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