From the probabilities that the company uses to move drivers from state to state the next year, we get the following transition matrix:

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1 MAT 121 Solutions to Take-Home Exam 2 Problem 1 Car Insurance a) The 3 states in this Markov Chain correspond to the 3 groups used by the insurance company to classify their drivers: G 0, G 1, and G 2 From the probabilities that the company uses to move drivers from state to state the next year, we get the following transition matrix: P [The transition diagram is drawn based on these transition probabilities: each arrow going from state i to state j is given by the entry p ij ] Note that G 2 is an absorbing state since no G 1 or G 2 driver can move back to G 0 again Also, note that the transition probability to go from G 0 to G 1 is four times greater than the transition probability to go from G 0 to G 2 Finally, check that each row in this matrix is a probability vector whose entries add up to 1 b) Computing the second and third powers of P yield the following matrices: P P Since the 50,000 new drivers have had no accidents, we start with the initial distribution V 0 50, ,

2 The distribution of these drivers in the 3 states of the Markov Chain after 2 and 3 years is then given by and V 2 V 0 P 2 50, ,125 14,500 7,375 V 3 V 0 P 3 50, ,094 15,775 13,131 We therefore conclude that, according to this model, - 28,125 (or 5625%) of the new drivers will still be G 0 drivers after 2 years - 28,906 (or %) of the new drivers will be either G 1 or G 2 drivers after 3 years c) With the grace period we now have: P P Computing the same distributions as before yield the following results: - 29,375 (or 5875%) of the new drivers will still be G 0 drivers after 2 years - 25,906 (or %) of the new drivers will either be G 1 or G 2 drivers after 3 years

3 Problem 2 Confidence Level Here Jack s performance on the court can be modeled with a 2-state Markov Chain, where the first state corresponds to him scoring and the second state corresponds to him missing Based on the given probabilities, the transition matrix is given by P a) If Jack misses a shot, then the initial distribution is given by 0 1 and the probability of him scoring or missing two shots later is given by 0 1P According to this model, we conclude that he has a 31% chance of scoring two shots later after missing a shot b) Let V x 1 x denote the long-term, or equilibrium, distribution of this regular Markov Chain Then x represents Jack s overall shooting percentage (or field goal percentage) We now solve for x in the matrix equation VP V : VP V x x x x 02 1 x x 025x 08 1 x 1 x Solving for x in the first equation then yields: 1 x 075x 02 02x x 055x 02 x 02 x 055x x x ( 444%) Therefore, Jack s shooting percentage is, approximately, 444%

4 Problem 3 Language Using the same procedure as in part b) of Problem 2, we get the following long-term distribution of vowels and consonants, where x and 1 x represent, respectively, the percentage of vowels and consonants in English texts: VP V x x x x x x 088x 0461 x 1 x Solving for x in the first equation then yields: 1 x 012x x x 042x 054 x x x ( 3803%) We thus conclude that approximately 38% of all letters in English texts are vowels Problem 4 Finger Games a You (the row player) and your friend (the column player) have two options in this game, namely to show one finger or two fingers There are then 3 possible outcomes in this game: 1) You both show one finger you win $2 since = 2 is even 2) You both show two fingers you win $4 since = 4 is even 3) You both show a different number of fingers your friend wins $3 since = = 3 is odd Hence we have the following payoff matrix for this game: M By convention, the positive entries of M indicate payoffs (in $) favorable to you

5 We now proceed to solve this game Let A [ p 1 p] be your optimal strategy Then, according to the fundamental principle of game theory, your friend s best counterstrategy has to be pure: either B 1 [1 0] T or B2 [0 1] T, yielding the following two expected values: E1 AMB1 p 1 p p 1 p 2p 3 (1 p) 5p E2 AMB2 p 1 p p 1 p 3p 4 (1 p) 7 p Setting both values equal to each other and solving for p yields E1 E2 5p3 7 p 4 12 p 7 7 p (or p 5833% ) 12 Hence, your optimal mixed strategy is given by A % 4167% The expected value of the game is given by E So you lose! Your friend wins, on average, 8 cents per game b The payoff matrix for this game is now given by: M Using the same procedure as in part a), we get the following two expected values:

6 E1 AMB1 p 1 p p 1 p 4p 9 (1 p) 13p E2 AMB2 p 1 p p 1 p 9 p 16 (1 p) 25 p Setting both values equal to each other and solving for p yields E1 E2 13p 9 25 p 16 38p p (or p 6579% ) 38 Hence, your optimal mixed strategy is given by A % 3421% The expected value of the game is given by E So you lose again! Your friend wins, on average, 45 cents per game c With this last variant of the game, you will now win $2 every time you both show a different number of fingers (since 1 x 2 = 2 is even) and you win $4 when you both show two fingers (since 2 x 2 = 4 is even) The only losing payoff for you now amounts to $1 whenever you both show one finger (since 1 x 1 = 1 is odd) 1 2 The payoff matrix for this new game is then given by M 2 4, which reduces by dominance to the saddle point 2 (also check that row 2 dominates row 1 and column 1 dominates column 2) So you can expect to win $2 each time you play this strictly determined game

7 Problem 5 a) This is a strictly determined game whose saddle point is 0 [check this!] Hence there are no winners since the expected value of the game is 0 b) The expected value of this game is -42, hence the column player wins The column player s optimal strategy is T [check this!] c) This is a strictly determined game whose saddle point is 0 [check this!] Hence there are no winners since the expected value of the game is 0 Bonus Problem: The Prisoner s Dilemma To see what Bonnie should do, think of her as the row player in a game in which she can either confess (row 1) or not confess (row 2) Playing opposite her is Clyde with the same two options: confess (column 1) or not confess (column 2) The payoffs in this game are then the number of years she gets behind bars This yields the following payoff matrix: Since this is a strictly determined game that reduces to the saddle point 2, Bonnie should confess!