Horse race. Consider a horse race with four horses. Assume that each horse pays

Question

Horse race. Consider a horse race with four horses. Assume that each horse pays 4-for-1 if it wins. Let the probabilities of wining of the horses be {1/2, 1/4, 1/8, 1/8}. If you start with $100 and bet optimally to maximize your long-term growth rate. Let b = (b_1, b_2, b_3, b_4), sigma b_1 = 1, be the amount invested on each of the horses. a) What is the expected log wealth W(b) on each race? b) What are your optimal bets b* on each race? c) Approximate how much money would you have after 20 races with this strategy? d) Show that the wealth achieved in repeated races should grow to infinity like 2^nW with probability 1. e) 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.

Explanation / Answer

b) The optimal betting stratergy is propotional betting that is , dividing the investment in proportion to the probabilities of each horse winning. Thus the bets on each horse should be ( 50% , 25% , 12.5%, 12.5%) and the growth rate achieved by this stratergy is log 4 - H(p) = log 4 - H (1/2 , 1/4 , 1/8 , 1/8) = 2 - 1.75 = 0.25

c) After 20 races with this stratergy the wealth is approximately 2nW = 25 = 32 and hence the wealth would grow approximately 32 folds over 20 races.

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