cd a b The following game is proposed to you. A fair coin will be tossed...

 

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cd a b The following game is proposed to you. A fair coin will be tossed repeatedly until a tail appears (the first tail will end the game). The pot starts at 1 and is doubled every time a head appears. You win whatever is in the pot when the game ends. Thus, you win 1 if a tail appears on the first toss, 2 if on the second, 4 if on the third, 8 if on the fourth, etc. In short, you win 2*-1 if the coin is tossed k times until the first tail appears. A reasonable way to analyse the value of this game is by calculating the certainty equivalent of the game's random payoff. Calculate the expected utility and the certainty equivalent of the game in the following situations: risk-neutral utility function: U(x) = x (hint: note that the probability of the first tail appearing in the k-th toss is ); power utility function: U(x) = bx with b = -1; c logarithmic utility function: U(x) = log(x) (hint: use the formula -1 (k-1)/2 = 1). The three parts carry, respectively, 40%, 30%, and 30% of the marks. cd a b The following game is proposed to you. A fair coin will be tossed repeatedly until a tail appears (the first tail will end the game). The pot starts at 1 and is doubled every time a head appears. You win whatever is in the pot when the game ends. Thus, you win 1 if a tail appears on the first toss, 2 if on the second, 4 if on the third, 8 if on the fourth, etc. In short, you win 2*-1 if the coin is tossed k times until the first tail appears. A reasonable way to analyse the value of this game is by calculating the certainty equivalent of the game's random payoff. Calculate the expected utility and the certainty equivalent of the game in the following situations: risk-neutral utility function: U(x) = x (hint: note that the probability of the first tail appearing in the k-th toss is ); power utility function: U(x) = bx with b = -1; c logarithmic utility function: U(x) = log(x) (hint: use the formula -1 (k-1)/2 = 1). The three parts carry, respectively, 40%, 30%, and 30% of the marks.