376B/Stat EE 376B Information Theory Prof. T. Cover
Handout #24 Thursday, June 1, 2006 Due Tuesday, June 6, 2006
Homework Set #9 1. Horse race. Three horses run a race. A gambler offers 3-for-1 odds on each horse. These are fair odds under the assumption that all horses are equally likely to win the race. The true win probabilities are known to be p = (p1 , p2 , p3 ) = 1 1 1 . , , 2 4 4 (1)
Let b = (b1 , b2 , b3 ), bi 0, bi = 1, be the amount invested on each of the horses. The expected log wealth is thus
3
W (b) =
i=1
pi log 3bi .
(2)
(a) Maximize this over b to find b and W . Thus the wealth achieved in repeated horse races should grow to infinity like 2nW with probability 1. (b) Show that if instead we put all of our money on horse 1, the most likely winner, we will eventually go broke with probability 1.
2. Dutch Consider book. a horse race with m = 2 horses, X = 1, 2 1 1 P = , 2 2 odds (for one) = 10, 30 bets = b, 1 - b. The odds are superfair. (a) There is a bet b that guarantees the same payoff regardless of which horse wins. Such a bet is called a Dutch book. Find this b and the associated wealth factor S(X). (b) What is the maximum growth rate of the wealth for the optimal choice of b? Compare it to the growth rate for the Dutch book. 1
3. Investment growth rate. Let X=
(1, a), (1, 1/a),
with probability 1/2 with probability 1/2 ,
where a > 1. This vector X represents a stock market vector of cash vs. a hot stock. Let W (b, F ) = E log bt X and W = max W (b, F )
b
be the growth rate. (a) Find the log optimal portfolio b . (b) Find the growth rate W . (c) Find the asymptotic behavior of
n
Sn =
i=1
bt Xi
for all b.
2