### Artificial IntelligenceAIMA Exercises

Economists often make use of an exponential utility function for money: $U(x) = -e^{-x/R}$, where $R$ is a positive constant representing an individual’s risk tolerance. Risk tolerance reflects how likely an individual is to accept a lottery with a particular expected monetary value (EMV) versus some certain payoff. As $R$ (which is measured in the same units as $x$) becomes larger, the individual becomes less risk-averse.
1. Assume Mary has an exponential utility function with $R = \$500$. Mary is given the choice between receiving$\$500$ with certainty (probability 1) or participating in a lottery which has a 60% probability of winning \$5000 and a 40% probability of winning nothing. Assuming Marry acts rationally, which option would she choose? Show how you derived your answer. 2. Consider the choice between receiving$\$100$ with certainty (probability 1) or participating in a lottery which has a 50% probability of winning $\$500$and a 50% probability of winning nothing. Approximate the value of R (to 3 significant digits) in an exponential utility function that would cause an individual to be indifferent to these two alternatives. (You might find it helpful to write a short program to help you solve this problem.) Economists often make use of an exponential utility function for money:$U(x) = -e^{-x/R}$, where$R$is a positive constant representing an individual’s risk tolerance. Risk tolerance reflects how likely an individual is to accept a lottery with a particular expected monetary value (EMV) versus some certain payoff. As$R$(which is measured in the same units as$x$) becomes larger, the individual becomes less risk-averse. 1. Assume Mary has an exponential utility function with$R = \$500$. Mary is given the choice between receiving $\$500$with certainty (probability 1) or participating in a lottery which has a 60% probability of winning \$5000 and a 40% probability of winning nothing. Assuming Marry acts rationally, which option would she choose? Show how you derived your answer.
2. Consider the choice between receiving $\$100$with certainty (probability 1) or participating in a lottery which has a 50% probability of winning$\$500$ and a 50% probability of winning nothing. Approximate the value of R (to 3 significant digits) in an exponential utility function that would cause an individual to be indifferent to these two alternatives. (You might find it helpful to write a short program to help you solve this problem.)

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