# Module7 - Module 7 Lecture 1 Information Theory and...

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Module 7, Lecture 1 Information Theory and Gambling G.L. Heileman Module 7, Lecture 1

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Background – Contest Consider a contest (e.g., a race, lottery, beauty contest, etc.) involving m contestants (e.g, horses, people, dogs, etc.) in which one and only one participant can win a given instance of the contest (i.e., no ties). Let p i , i = 1 , . . . , m , be the probability that contestant i wins a given contest. Let o i , i = 1 , . . . , m , be the payoﬀ (i.e., bookie odds) a gambler receives for a \$1 bet on contestant i if contestant i wins the contest. G.L. Heileman Module 7, Lecture 1
Background – Odds There are two ways to think about odds: 1 a -for-1 odds – A \$1 bet paid before the contest (to bookie) yields an \$ a payout after the contest (to gambler) if selected contestant wins, and a \$0 payout if selected contestant loses. 2 a -to-1 odds – A \$1 bet is placed before the contest without payment. The gambler pays \$1 after the contest (to bookie) if selected contestant loses, and receives \$ a (to gambler) after the contest if the selected contestant wins. Notes: In terms of total payout. A bet at ( a - 1)-to-1 odds is the same as a bet at a -for-1 odds — in the latter case the bookie is already holding \$1 from the gambler. It only makes sense to consider odds where a 1. (Why would you bet if this were not the case?) G.L. Heileman Module 7, Lecture 1

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Background – Wealth Relative Assume a gambler starts with a ﬁxed amount of money, and all of the money a gambler holds is bet on each contest, with b i the fraction bet on contestant i , i.e., b i 0, and m i =1 b i = 1. After the contest, if contestant i won, then the gambler’s wealth is b i o i . Let us call S ( i ) = b i o i the wealth relative , and note that since the b i ’s are normalized to 1, it is the ratio of the gambler’s ﬁnal wealth (after one contest) to initial wealth. The wealth relative is an RV (that the gambler would obviously like to maximize) – actually it’s a function of the RV X i , which denotes the winner of contest i , i.e.: S ( X i ) = b ( X i ) o ( X i ) . Next we’ll consider the more interesting case of repeated contests. G.L. Heileman Module 7, Lecture 1
Background – Wealth Relative Theorem Let X , X 1 , . . . , X n be iid RVs with X p ( x ) , where X i denotes the winner of contest i. Then the gambler’s wealth grows exponentially at a rate n E log S ( X ) for n suﬃciently large. Proof: The gambler wealth after n contests is S n n Y i =1 S ( X i ) (a product since gambler is “reinvesting” money) log S n = n X i =1 log S ( X i ) 1 n log S n = 1 n n X i =1 log S ( X i ) p -→ E log S ( X ) by the WLLN (and by nothing that X 1 , . . . , X n iid log( X 1 ), . . . ,log( X n ) iid). Thus, S n p 2 n E log S ( X ) . G.L. Heileman Module 7, Lecture 1

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Background – Doubling Rate Since the base of the exponent in the previous proof is 2, E log S ( X ) is referred to as the doubling rate , and denoted W ( b , p ), where b = b 1 , . . . , b m and p = p 1 , . . . , p m .
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Module7 - Module 7 Lecture 1 Information Theory and...

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