Question

consider a lottery that pays $ 2 if n consecutive heads turn up in (n+1) tosses if a fair coin( i.e the sequence of coin flips ends with the first tail).If you have a logarithmic utility function U(W)=In W What is the utility of the expected payoff? what is the expected utility of the payoff.

Answer 1

Solution:

As per the given situation, the only situation that is satisfied to win the lottery is: Tails (T) in first toss and Heads (H) in all remaining n tosses, that is

T, H, H, H, H,...H

With a fair coin, probability of tails = probability of heads = (1/2)

The probability of this occurring (with a fair coin) is:

(1/2)*(1/2)*(1/2)*(1/2)*...*(1/2) (n+1) times

= (1/2)ⁿ⁺¹ = 1/2ⁿ⁺¹

The probability of loosing the lottery (that is earning nothing) is then 1 - (1/2)ⁿ⁺¹

So, expected payoff = probability of winning*amount won + probability of losing*0

E.P = (1/2ⁿ⁺¹)*2 + (1 - 1/2ⁿ⁺¹)*0 = 1/2ⁿ

So, utility of expected payoff, U(EP) = ln (1/2ⁿ) = -n*ln 2

Expected utility of payoff EU(P) = probability of winning*utility from winning amount + probability of not winning*utility from it

EU(P) = (1/2ⁿ⁺¹)*(ln 2) + (1 - 1/2ⁿ⁺¹)*(ln 0)

EU(P) = 0.693/2ⁿ⁺¹ + (-∞) = -∞