1. What is the fundamental distinction between "risk" and "uncertainty" in decision theory?

Risk refers to situations where the probabilities of various outcomes are known and can be objectively determined. Uncertainty, on the other hand, describes situations where the probabilities of outcomes are unknown or cannot be objectively determined. This means that for uncertain situations, it's difficult to assign numerical likelihoods to potential events.

2. How is a situation involving "risk" typically represented in decision theory?

A situation involving risk is typically represented as a lottery (L). A lottery is defined as a set of pairs (p1: x1, p2: x2, ..., pn: xn), where each outcome x is associated with a known probability p. This representation clearly indicates the likelihood of each potential result.

3. How is a situation involving "uncertainty" typically represented in decision theory?

A situation involving uncertainty is typically represented as an act (A). An act is defined as (s1: x1, s2: x2, ..., sn: xn), where outcomes x are associated with states of the world (s) rather than known probabilities. This implies that the specific outcome depends on which state of the world materializes, and the probabilities of these states are unknown.

4. Define and explain how to calculate the Expected Value (EV) of a lottery.

The Expected Value (EV) of a lottery is the average outcome one would expect if the lottery were played many times. It is calculated as the sum of each outcome multiplied by its probability: EV(L) = p1x1 + p2x2 + ... + pnxn. EV provides a measure of the central tendency of the lottery's potential results.

5. Why is Expected Value (EV) often insufficient for understanding how people make decisions under risk?

Expected Value alone often fails to capture how people actually make decisions, especially in situations involving large sums or varying personal preferences. It assumes a linear valuation of money, which doesn't account for an individual's subjective valuation of outcomes or their attitude towards risk. The St. Petersburg Paradox perfectly illustrates this limitation.

6. Define and explain how to calculate the Expected Utility (EU) of a lottery.

Expected Utility (EU) accounts for an individual's subjective valuation of outcomes. It is calculated as the sum of the utility of each outcome multiplied by its probability: EU(L) = p1u(x1) + p2u(x2) + ... + pnu(xn), where u(x) is the utility function that assigns a subjective value to each outcome x. This approach better reflects individual preferences than EV.

7. What is meant by the statement that the utility function in Expected Utility Theory is "cardinal"?

When the utility function in Expected Utility Theory is described as "cardinal," it means that the differences in utility values are meaningful, not just the order. This implies that not only can we say one outcome is preferred over another, but we can also quantify the strength of that preference. This allows for mathematical operations like summing utilities multiplied by probabilities.

8. Briefly describe the St. Petersburg Paradox and its significance.

The St. Petersburg Paradox describes a game with an infinite Expected Value, yet most people are only willing to pay a small, finite amount to play. This paradox highlights the limitations of Expected Value as a sole predictor of human decision-making under risk. It demonstrates that people do not value additional wealth linearly, especially at higher amounts.

9. How does Expected Utility Theory resolve the St. Petersburg Paradox?

Expected Utility Theory resolves the St. Petersburg Paradox by incorporating a utility function that reflects diminishing marginal utility of wealth. For example, using a logarithmic utility function like u(x) = ln(x) makes the Expected Utility of the game finite. This aligns more closely with the small amounts people are willing to pay, as people value additional wealth less as their total wealth increases.

10. Define a "risk-averse" individual in the context of Expected Utility Theory.

A risk-averse individual is someone who prefers the Expected Value of a lottery for sure over the lottery itself. This means they would rather have a guaranteed amount equal to the lottery's average outcome than take the chance on the lottery. Mathematically, this is represented as (100%: EV(L)) is preferred to L, indicating a preference for certainty over potential variability.

11. Define a "risk-neutral" individual in the context of Expected Utility Theory.

A risk-neutral individual is someone who is indifferent between the Expected Value of a lottery for sure and the lottery itself. This means they consider a guaranteed amount equal to the lottery's average outcome to be equivalent in value to playing the lottery. Mathematically, this is represented as (100%: EV(L)) is equivalent to L, indicating no particular preference for or aversion to risk.

12. Define a "risk-seeking" individual in the context of Expected Utility Theory.

A risk-seeking individual is someone who prefers a lottery over its Expected Value for sure. This means they would rather take the chance on the lottery than accept a guaranteed amount equal to its average outcome. Mathematically, this is represented as (100%: EV(L)) is less preferred than L, indicating a preference for potential variability and higher upside, even if it comes with higher risk.

13. What is the "Certainty Equivalent" (CE) of a lottery?

The Certainty Equivalent (CE) of a lottery is the guaranteed amount of money that an individual considers equivalent in value to the lottery itself. It represents the sure amount of money that would make the individual indifferent between receiving that amount for certain and playing the lottery. The CE helps quantify an individual's risk attitude.

14. How does the Certainty Equivalent (CE) relate to the Expected Value (EV) for a risk-averse individual?

For a risk-averse individual, the Certainty Equivalent (CE) is less than the Expected Value (EV) of the lottery (CE(L) < EV(L)). This is because a risk-averse person is willing to accept a lower sure amount to avoid the risk associated with the lottery. They value certainty over the potential for a higher, but uncertain, outcome.

15. How does the Certainty Equivalent (CE) relate to the Expected Value (EV) for a risk-neutral individual?

For a risk-neutral individual, the Certainty Equivalent (CE) is equal to the Expected Value (EV) of the lottery (CE(L) = EV(L)). This reflects their indifference between a sure amount and a risky lottery with the same average outcome. They do not place a premium on avoiding or seeking risk.

16. How does the Certainty Equivalent (CE) relate to the Expected Value (EV) for a risk-seeking individual?

For a risk-seeking individual, the Certainty Equivalent (CE) is greater than the Expected Value (EV) of the lottery (CE(L) > EV(L)). This means they require a higher sure amount to forgo the thrill or potential upside of the lottery. They are willing to accept a lower expected value in exchange for the chance of a larger gain.

17. Describe the shape of the utility function for a risk-averse individual.

A risk-averse individual has a concave utility function. This shape implies that the marginal utility of wealth decreases as wealth increases, meaning each additional unit of wealth provides less satisfaction than the previous one. This diminishing marginal utility is what drives their preference for certainty over risk.

18. Describe the shape of the utility function for a risk-neutral individual.

A risk-neutral individual has a linear utility function. This shape implies a constant marginal utility of wealth, meaning each additional unit of wealth provides the same amount of satisfaction, regardless of their current wealth level. This constant marginal utility leads to indifference between a sure amount and a risky lottery with the same expected value.

19. Describe the shape of the utility function for a risk-seeking individual.

A risk-seeking individual has a convex utility function. This shape implies that the marginal utility of wealth increases as wealth increases, meaning each additional unit of wealth provides more satisfaction than the previous one. This increasing marginal utility drives their preference for risk and potential higher gains, as the thrill of larger rewards outweighs the risk.

20. What is the "Asian Disease problem" and what key concept does it illustrate?

The Asian Disease problem is a classic experiment that illustrates the impact of "framing" on decision-making. It shows how people's preferences for risky choices can reverse depending on whether the outcomes are presented as gains (lives saved) or losses (lives lost), even when the underlying probabilities and outcomes are objectively identical. This highlights a violation of Expected Utility Theory.

21. How do people typically respond to the Asian Disease problem when outcomes are framed as "gains"?

When outcomes in the Asian Disease problem are framed as "gains" (e.g., lives saved), most people tend to be risk-averse. For instance, they prefer a program that saves a smaller number of lives for sure (Program A: saves 200 people) over a program with a higher potential to save more lives but also a risk of saving none (Program B: 1/3 probability of saving 600, 2/3 of saving none). This demonstrates a preference for certainty when dealing with positive outcomes.

22. How do people typically respond to the Asian Disease problem when outcomes are framed as "losses"?

When outcomes in the Asian Disease problem are framed as "losses" (e.g., lives lost), most people tend to be risk-seeking. For instance, they prefer a program with a higher risk of more deaths but also a chance of no deaths (Program D: 1/3 probability of nobody dying, 2/3 of 600 dying), over a program that guarantees a smaller number of deaths (Program C: 400 people dying for sure). This demonstrates a preference for taking risks to avoid certain losses.

23. Why does the Asian Disease problem pose a challenge to Expected Utility Theory?

The Asian Disease problem challenges Expected Utility Theory because it demonstrates an inconsistency in preferences based purely on how a problem is presented (framing), rather than on the objective probabilities and outcomes. EU Theory assumes rational agents make consistent choices regardless of framing, which is violated when preferences shift between risk-aversion for gains and risk-seeking for losses, even when the underlying scenarios are mathematically equivalent.

24. What is "Prospect Theory" and how does it differ from Expected Utility Theory?

Prospect Theory is a descriptive theory of decision-making under risk that aims to explain observed violations of Expected Utility Theory. Unlike EU Theory, which assumes rational agents and objective probabilities, Prospect Theory incorporates psychological insights. It suggests that individuals evaluate outcomes as gains or losses relative to a reference point and exhibit different risk attitudes for gains versus losses, making it a more realistic model of human behavior.

25. Explain the concept of "reference points" in Prospect Theory.

In Prospect Theory, "reference points" are baseline levels against which outcomes are evaluated as either gains or losses. Instead of evaluating absolute wealth, individuals perceive outcomes relative to their current state or some other relevant benchmark. This means the same objective outcome can be perceived differently depending on the chosen reference point, influencing whether it's seen as a positive or negative change.