I am in my office in Cambridge, but I have to catch a flight...

I am in my office in Cambridge, but I have to catch a flight from Heathrow this afternoon. I must decide whether to go to Heathrow by coach, which comes relatively cheap at $£ 40$, or buy a train ticket for $£ 70$. If I take the coach I might get stuck in an intense traffic jam and miss my flight. I would then have to buy a new ticket for $£ 100$. According to the latest statistics, the traffic jam on the M25 to Heathrow is intense one day in three. (a) Should I travel by train or coach? (b) This description of my decision problem overlooks a number of features that might be relevant. Which? (a) You are in Las Vegas. The probability of winning a jackpot of $$\$ 350,000$$ is one in a million. How much should you, who find no reason to reject the principle of maximising expected utility, be prepared to pay to enter this gamble? Your utility of money is $u(x)=\ln (x+1)$. (b) This time the probability of winning the jackpot of $$\$ 350,000$$ is one in a thousand. How much should you, who find no reason to reject the principle of maximising expected utility, be prepared to pay to enter this gamble? Your utility of money is $u(x)=\ln (x+1)$. (c) Why is the difference between the amount you are willing to pay in (a) and (b) so small? (d) Why did we assume that your utility function is $u(x)=\ln (x+1)$, rather than just $u(x)=\ln (x)$ ?

I am in my office in Cambridge, but I have to catch a flight from Heathrow this afternoon. I must decide whether to go to Heathrow by coach, which comes relatively cheap at $£ 40$, or buy a train ticket for $£ 70$. If I take the coach I might get stuck in an intense traffic jam and miss my flight. I would then have to buy a new ticket for $£ 100$. According to the latest statistics, the traffic jam on the M25 to Heathrow is intense one day in three. (a) Should I travel by train or coach? (b) This description of my decision problem overlooks a number of features that might be relevant. Which?
(a) You are in Las Vegas. The probability of winning a jackpot of $$\$ 350,000$$ is one in a million. How much should you, who find no reason to reject the principle of maximising expected utility, be prepared to pay to enter this gamble? Your utility of money is $u(x)=\ln (x+1)$.
(b) This time the probability of winning the jackpot of $$\$ 350,000$$ is one in a thousand. How much should you, who find no reason to reject the principle of maximising expected utility, be prepared to pay to enter this gamble? Your utility of money is $u(x)=\ln (x+1)$.
(c) Why is the difference between the amount you are willing to pay in (a) and (b) so small?
(d) Why did we assume that your utility function is $u(x)=\ln (x+1)$, rather than just $u(x)=\ln (x)$ ?

Key Concepts

Utility Function Specification

The specification of a utility function is essential in capturing an individual's preferences and attitudes towards risk. The choice between functions like u(x) = ln(x+1) and u(x) = ln(x) is important because it determines the practical domain of the function (e.g., avoiding undefined values at zero) and reflects subtle differences in behavior near small amounts of wealth. This choice can have significant implications for decision analysis, particularly when potential outcomes approach lower bounds.

Model Assumptions and Omitted Factors

When formulating a decision problem, it is crucial to account for all relevant variables that might affect the outcome. Omitted factors, such as time buffers, opportunity costs, personal preferences, and external influences like weather or operational delays, can significantly alter the decision landscape. Recognizing these omitted variables ensures a more robust and realistic analysis of the risks and benefits involved.

Expected Utility Theory

Expected utility theory is a framework for making rational decisions under uncertainty. It involves calculating the utility of each possible outcome and then weighting these utilities by their probabilities to obtain an overall measure of the desirability of different choices. This approach is used to compare alternatives that involve various risks and rewards, such as travel options or gambling scenarios.

Risk Aversion and Diminishing Marginal Utility

Risk aversion describes the tendency of individuals to prefer a certain outcome over a risky one with the same expected monetary value. This behavior is typically modeled by concave utility functions, such as the logarithmic function, which capture the concept of diminishing marginal utility of wealth. As wealth increases, the additional satisfaction gained from an extra unit of money decreases, influencing decision making under uncertainty.

Decision Making Under Uncertainty

Decision making under uncertainty involves selecting among alternatives when the outcomes are not guaranteed, and probabilities are associated with different events. In this context, key factors include the variability of outcomes, potential losses or gains, and the probabilities of these events occurring. This concept is fundamental when evaluating choices like transport methods where delays or additional costs may be probabilistically determined.

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