391 Lecture 10b decision theory II

What is the main focus of Decision Theory Pt II?

Modeling how decision makers assign utility to alternatives and make choices under uncertainty.

What is an alternative in decision theory?

An available option that a decision maker can choose.

What is a preference order?

An ordered ranking of alternatives according to a decision maker's preferences.

What is a utility function?

A function assigning numerical utility values to alternatives in a way that represents the decision maker's preference order.

What defines a rational actor in the lecture?

An actor who chooses among their highest-ranked alternatives, maximizing utility, with preferences satisfying completeness and transitivity.

What is the axiom of completeness?

Every pair of alternatives must be comparable: one is at least as good as the other, or both are equally good.

What is the axiom of transitivity?

If A is at least as good as B and B is at least as good as C, then A must be at least as good as C.

What is the fundamental requirement for designing a utility function?

It must represent the decision maker's preference order.

What does U(a) > U(b) imply?

The decision maker strictly prefers alternative a to alternative b.

What are the lecture's main guidelines for designing utility functions?

Reflect the preference order, decide whether to use a measurable outcome directly or transform it into utility, and consider easy-to-maximize functions.

Why might a utility function transform a quantifiable outcome instead of using it directly?

Because an outcome's numerical magnitude, such as dollars, may not reflect how much the decision maker values it.

Why are single-peaked utility functions useful?

They have a clear maximum and are often relatively easy to optimize.

What examples of single-peaked utility functions does the lecture mention?

Negative quadratic functions and functions based on absolute value.

What is the one-dimensional policy example?

A legislator chooses among bills located along a policy line from 0 to 1 and has an ideal position on that line.

What does x represent in the one-dimensional policy model?

The position of a proposed bill on the policy line.

What does y represent in the one-dimensional policy model?

The legislator's ideal policy position.

What preferences does the legislator have in the one-dimensional policy example?

The legislator prefers proposals closer to their ideal position and dislikes proposals on either side of it.

What type of utility function is used for the one-dimensional policy model?

A negative quadratic utility function centered on the legislator's ideal position.

What is a representative negative quadratic utility function for the one-dimensional policy model?

U(x) = −(x − y)², where x is the proposal and y is the ideal position.

Where is U(x) = −(x − y)² maximized?

At x = y, where the proposal exactly matches the legislator's ideal position.

Why does a negative quadratic function fit the legislator's preferences?

Utility is highest at the ideal point and decreases as a proposal moves farther away in either direction.

What happens to utility as x moves farther from y in the negative quadratic model?

Utility decreases because the squared distance from the ideal point increases.

How does the lecture extend the policy utility model to multiple dimensions?

By allowing policy positions and ideal points to be vectors rather than single numbers.

What do x and y represent in the multidimensional policy model?

x is a vector of proposed policy positions, and y is a vector of ideal policy positions.

Why might a multidimensional policy model be more realistic?

Because many policy choices involve multiple issues or dimensions rather than a single policy scale.

What is the purpose of using Cobb-Douglas functions for utility?

To represent preferences involving diminishing returns and, in multidimensional cases, tradeoffs between quantities of different goods.

What examples of diminishing returns does the lecture give?

Money or wealth, amounts of food, and studying effort.

What does diminishing returns mean in a utility model?

Additional units of an outcome still increase utility, but each additional unit produces a smaller increase than the previous one.

What is the original Cobb-Douglas production model discussed in the lecture?

A production function involving labor L, capital K, and elasticity constants α and β.

What is a common mathematical form of the Cobb-Douglas production function?

Y = A L^α K^β, where L is labor, K is capital, α and β are elasticity constants, and A is a scaling factor.

How can a one-dimensional Cobb-Douglas-style utility function represent diminishing returns?

By using a power function such as U(x) = x^α with 0 < α < 1, so utility rises at a decreasing rate.

Why does the lecture associate Cobb-Douglas utility with constrained resources?

Because without resource constraints, a utility function that keeps increasing would encourage choosing ever-larger quantities rather than a finite unconstrained optimum.

What is the more common multidimensional Cobb-Douglas utility model used for?

Representing preferences over combinations of goods.

What is a representative two-good Cobb-Douglas utility function?

U(x1,x2) = x1^α × x2^β, where x1 and x2 are quantities of goods and α and β determine their contributions to utility.

What do pi and w represent in the Cobb-Douglas purchasing model?

pi is the unit price of good i, and w is the decision maker's total budget.

What is the purpose of a budget constraint in a purchasing model?

It limits which combinations of goods are affordable.

What is a general budget constraint for two goods?

p1x1 + p2x2 ≤ w, meaning total spending cannot exceed the available budget.

What is the guns-versus-butter model?

A decision model in which a government allocates limited resources between defense spending (guns) and domestic programs (butter).

What do guns and butter represent in the model?

Guns represent defense-related production or spending, while butter represents domestic goods or programs.

What are the alternatives in the lecture's guns-versus-butter model?

Combinations of the number of guns and the number of sticks of butter.

Why is there a tradeoff between guns and butter?

Because GDP and productive resources are finite, so allocating more resources to one category leaves fewer for the other.

What is a Production Possibility Frontier (PPF)?

A curve describing the feasible production tradeoff between two goods, such as guns and butter.

What does the PPF represent in the guns-versus-butter model?

The combinations of guns and butter an economy can produce using its available resources.

What do different shapes of the PPF indicate?

Different economies of scale and opportunity costs associated with producing guns and butter.

What is opportunity cost in the guns-versus-butter model?

The amount of one good that must be given up to produce more of the other.

How are utility functions and the PPF used together?

The PPF describes feasible combinations, while the utility function represents which combinations the decision maker prefers.

What defense-spending percentages are highlighted in the lecture?

Ukraine 37%, Russia 5.9%, Israel 5.3%, and Poland 3.8%, as figures presented in the slides.

Why is defense spending as a percentage of GDP an imperfect measure of national priorities?

It provides only a high-level view and does not fully capture resources, circumstances, or other factors influencing spending.

What two factors does the lecture emphasize when explaining a country's guns-versus-butter decisions?

Available resources and the country's current security outlook.

Why might richer countries be able to obtain more guns and more butter?

They have greater total resources, allowing more production or spending in both categories.

How can uncertainty be incorporated into a decision model?

By assigning probabilities to the possible outcomes of a decision.

How does the lecture represent the possible outcomes of an alternative x?

As a set R = {r1, r2, ..., rn}, with each outcome ri having a probability pi.

What is the expected outcome of choosing an alternative x?

The sum of each possible outcome multiplied by its probability: E[R] = Σ pi × ri.

What is the expected utility of choosing an alternative x?

The sum of the utility of each possible outcome multiplied by its probability: E[U] = Σ pi × U(ri).

What is the difference between expected value and expected utility?

Expected value averages numerical outcomes, while expected utility averages the utilities assigned to those outcomes.

Why is expected value not necessarily a typical outcome?

It is an average across many possible realizations and may not be an outcome that can occur in a single trial.

What are the three classes of risk tolerance discussed in the lecture?

Risk averse, risk neutral, and risk seeking.

What does risk aversion mean?

Preferring a certain outcome over an uncertain outcome with the same expected monetary value.

What does risk neutrality mean?

Being indifferent between a certain outcome and an uncertain outcome with the same expected monetary value.

What does risk seeking mean?

Preferring an uncertain outcome over a certain outcome with the same expected monetary value.

What gamble does the lecture use to illustrate risk tolerance?

Choice A gives $100 for certain; Choice B gives a 50% chance of $200 and a 50% chance of $0.

What is the expected monetary value of both options in the $100-versus-coin-flip example?

Both have an expected value of $100 because the gamble's EV is 0.5($200) + 0.5($0) = $100.

How do risk-averse, risk-neutral, and risk-seeking individuals rank the $100 options?

Risk averse: U(A) > U(B); risk neutral: U(A) = U(B); risk seeking: U(B) > U(A).

What utility-function shapes correspond to the three risk attitudes?

Risk averse: strictly concave; risk neutral: linear; risk seeking: strictly convex.

What does a concave utility function mean for money?

Utility increases at a decreasing rate, so additional money provides progressively smaller gains in utility.

What does a convex utility function mean for money?

Utility increases at an increasing rate, so additional money provides progressively larger gains in utility.

Why does a concave utility function produce risk aversion?

The utility of receiving the expected monetary amount for certain is greater than the expected utility of a lottery with the same monetary expected value.

What examples of risk-averse utility functions does the lecture provide?

Natural-log utility U(x) = ln(x), power utility U(x) = x^a for 0 < a < 1, and an exponential-form utility function.

What does the lecture suggest about risk neutrality in the long run?

It can be advantageous when repeated outcomes average out and the decision maker can absorb losses; the lecture emphasizes that risk neutrality is a privilege.

What reasons does the lecture give for some apparently risk-seeking behavior?

Misestimation of outcome probabilities or manipulation that makes the act of gambling itself rewarding.

What insurance example does the lecture use to demonstrate the value of certainty?

A person has $1,000, faces a 30% chance of paying $500 for car repairs, and evaluates money using natural-log utility.

How is expected utility calculated in the car-repair example?

E[U] = 0.7 ln(1000) + 0.3 ln(500), approximately 6.7.

What equation does the lecture use to find the certain payment equivalent to the car-repair risk?

6.7 = ln(1000 − x), where x is the certain payment that produces the same utility.

What certain payment does the lecture report for the car-repair example?

$187.59, using the lecture's rounded expected-utility calculation.

What is the expected monetary repair cost, and why is the certain payment higher?

The expected repair cost is 0.30 × $500 = $150; the lecture's risk-averse decision maker is willing to pay more to avoid uncertainty.