By John G Michaels; Kenneth H Rosen

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**Example text**

Example 1 Set up a Markov chain to model the following gambling situation. A gambler starts with $2. A coin is flipped; if it comes up heads, the gambler wins $1, and if it comes up tails the gambler loses $1. The gambler will play until having gone broke or having reached a goal of $4. After each play, the observation to be made is how much money the gambler has. Solution: The possible amounts are 0,1,2,3 and 4 dollars. Thus, the state space is S = {O, 1, 2, 3, 4}; the five elements of S describe the status of the gambler's fortune as the game progresses.

BPlC, CP2 B and BPC is equivalent to the hypothesis of Lemma 2. We now have APIB::} APB, APIC::} APC, and BPIC::} BPC. Consideration of the individual preferences B ~ C ~ A and C ~ A ~ B adds BPIA ::} BPA to our list. , the REP is given by R = F(Rl' R2) = Rl (a dictatorship). This proves Lemma 3 since the assumption X/Y implied that the REP is a dictatorship, which is one of the forms of the REP specified in Theorem 2. Our initial choice APlB, BP2 A, and APB was made without loss of generality, hence provides the result for arbitrary X and Y (the reader may repeat the proof with BPA and all the other 2-permutations of A, B, and C) .

R N ). More precisely, if a number e (e stands for "error") is specified, then there is a number ko such that k > ko implies that for each 1 ~ j ~ N, Ip~;) - rj I ~ e, 1 ~ i ~ N . a) Show that if Q is any initial probability distribution, then Qle will be close to the equilibrium distribution for large k. b) Refer to Corollary 1. There, Qle is obtained from the equation Qle = QTle. If we want to compute Qle by first computing Tle with k matrix multiplications and then computing Qle = QT le , how many arithmetic operations (Le.