DECISION THEORY

Module 1: Introduction to Decision Theory

What is the core claim of methodological individualism?

The starting point of any analysis of social interactions is always the individual — not groups or institutions.

How does methodological individualism explain what a firm is?

A firm is a network of contractual relationships between the individuals involved — employees, owners, customers (stakeholders).

How is market analysis approached through methodological individualism?

By identifying and studying the behavior of individual market participants — suppliers and consumers.

How does methodological individualism apply to the legal system?

The legal system is analysed by examining how laws affect and shape individuals' behavior and choices.

What are the three characteristics of Homo Economicus?

Stable needs (preferences), rational actions, and self-interest.

What does it mean that the individual has "stable needs"?

Their preferences do not change randomly — their wants remain consistent over time.

What does "rational actions" mean in the Homo Economicus model?

The individual makes logical decisions aimed at best satisfying their needs and preferences.

What is scarcity in the context of Homo Economicus?

The individual cannot satisfy all needs at once because resources are limited, forcing them to make choices.

What are "alternatives" in Decision Theory?

The measures or actions available to an individual to satisfy their needs — limited by real-world restrictions

Why are the individual's alternatives limited?

Because restrictions exist in the real world that prevent every possible action from being available.

What is the opposite of methodological individualism, and how does it differ?

Holism — it analyses groups or societies as wholes, rather than as the sum of individual actions and decisions.

What is the full definition of Homo Economicus?

An individual who, in a scarcity situation, chooses the alternative that maximises their satisfaction of needs.

What is the Rational-Choice Approach?

The assumption that individuals always pick the option that best maximises their satisfaction given their available alternatives and restrictions.

What is the difference between normative and descriptive decision theory?

Normative is about how decisions should ideally be made. Descriptive is about how decisions are actually made in real life.

What is the difference between cooperative and non-cooperative game theory?

Cooperative — players can team up and make binding agreements. Non-cooperative — every player acts alone in their own interest.

What is game theory?

The study of decisions involving multiple people, where each person's choice affects the others.

What is the main difference between decision theory and game theory in this classification?

Decision theory = one person deciding alone with no interaction. Game theory = multiple people whose decisions affect each other.

What is the difference between deciding under certainty vs. under risk and uncertainty?

You know exactly what will happen. Risk & uncertainty = the outcome depends on things you cannot fully predict or control.

What is the difference between game theory with and without binding contracts?

Without binding contracts = promises can be broken. With binding contracts = everyone is legally locked in and must keep their word.

What is the action space (A)

The full set of all alternatives or choices available to the decision-maker.

What is the state space (Θ)?

The set of all possible and relevant states of the world that could affect the outcome of a decision.

What is the outcome space (X)?

The set of all possible outcomes that could result from any combination of action and world state.

What does the outcome function g(a, θ) do?

It takes an action a and a state of the world θ, and assigns them a unique outcome x. Written as x = g(a, θ).

What does the outcome matrix show?

A table where every row is an action, every column is a state of the world, and every cell shows the outcome of that action-state combination.

How do you read a specific cell in the outcome matrix, for example x(i,j)?

t is the outcome you get when you choose action i and the world turns out to be in state j.

What is the relationship between the outcome matrix and the outcome function g(a, θ)?

The outcome matrix is just the outcome function written out as a table — each cell shows the result of g(a, θ) for every possible combination of action and state.

What are decisions under certainty?

Situations where every action leads to a completely known and fixed outcome — no randomness or surprise at all.

What are decisions under risk?

Situations where outcomes are not known in advance, but probabilities can be assigned to each possible outcome.

What are decisions under uncertainty?

Situations where outcomes are unknown and no probabilities can be assigned — you are completely in the dark.

What is the key difference between risk and uncertainty?

Under risk you know the probabilities. Under uncertainty you do not even know the probabilities — you cannot put any numbers on the chances at all.

What is a binary relation?

A rule that describes which pairs of items from a set are connected. Formally it is a subset of the Cartesian product X × X.

What does X × X mean?

The Cartesian product — the set of every possible pair of items that can be made from set X.

What does the notation x1Rx2 mean?

That x1 is related to x2 under relation R. It is shorthand for writing (x1, x2) ∈ R.

Give two simple examples of binary relations on real numbers.

"=" (equals) and "≤" (less than or equal to) — both describe a relationship between two numbers at a time.

What does it mean for a relation to be reflexive?

Every item is related to itself — for all x, xRx is true.

What does it mean for a relation to be complete?

Any two items can always be compared — for any x and y, either xRy or yRx (or both) must hold.

What does it mean for a relation to be symmetric?

If x is related to y, then y is also related to x — the connection goes both ways.

What does it mean for a relation to be transitive?

If x relates to y, and y relates to z, then x must also relate to z — the relation chains together.

What is an equivalence relation?

A binary relation that is reflexive, symmetric, and transitive all at the same time. Example: "=" on numbers.

What is a preference relation?

A binary relation that summarises and orders a decision-maker's preferences over outcomes, written with the symbol ≽.

What does x ≽ y mean?

"x is weakly preferred over y" — meaning x is at least as good as y in the decision-maker's eyes.

When is a preference relation called weak-order rational?

When it is both complete (you can always compare any two options) and transitive (your preferences chain together logically).

What does strict preference x ≻ y mean?

You genuinely prefer x over y. x ≽ y is true but y ≽ x is NOT true — the preference only goes one way.

What does indifference x ~ y mean?

You do not care which option you get — x ≽ y AND y ≽ x both hold, meaning both options feel equally good.

What is the difference between x ≽ y and x ≻ y?

x ≽ y means x is at least as good as y (could be equal). x ≻ y means x is strictly better — y is definitely not as good as x.

What is a utility function?

A function u: X → ℝ that assigns a real number to each outcome so that x ≽ y if and only if u(x) ≥ u(y).

What does it mean that utility is an ordinal concept?

Only the ranking matters, not the size of the gaps between numbers. If u(x) > u(y), all we know is x is preferred over y — not by how much.

What is a monotone transformation of a utility function?

Any rescaling that keeps the same order — bigger stays bigger. It always represents the exact same preferences as the original utility function.

Why does utility have no cardinal meaning?

Because only the order of the numbers matters. The actual size of the gap between utility scores is meaningless — doubling or adding to all scores changes nothing about the preferences.

What does the invariance theorem say about utility functions?

Any strictly increasing rescaling of a utility function represents exactly the same preferences — only the order of scores matters, not the actual numbers.

What makes a transformation of a utility function valid?

It must be strictly increasing — meaning bigger inputs always produce bigger outputs, so the ranking order is preserved.

What does the representability theorem say?

If outcomes are finite or countably infinite, having complete and transitive preferences is equivalent to having a utility function that represents them — rational preferences and utility functions always go together.

What are the two equivalent conditions in the representability theorem?

(1) ≽ is complete and transitive. (2) There exists a utility function u: X → ℝ that represents ≽. Both mean exactly the same thing.

What does u'(x) > 0 mean for a utility function?

The function is always increasing — more of something always gives at least a little more utility.

What does u''(x) < 0 mean for a utility function?

The function is concave — it is still increasing but getting flatter. Each extra unit adds less happiness than the one before.

What is diminishing marginal utility?

The more you already have of something, the less each extra unit adds to your happiness. Captured by a concave utility function.

In the graph, why is the step up in u(x) smaller for each equal step along x?

Because the utility function is concave — equal increases in x produce smaller and smaller increases in happiness as x gets larger.

What is the Maximin Rule?

A decision rule that looks at the worst possible outcome for each action and picks the action whose worst outcome is the best. Maximise the minimum.

What are the steps to apply the Maximin Rule?

For each action find the worst possible outcome. Then pick the action where that worst outcome is the highest.

Why does the Maximin Rule not require probabilities?

Because it only looks at outcomes, not how likely each state is. It works even under pure uncertainty where no probabilities can be assigned.

What kind of attitude does the Maximin Rule reflect?

A pessimistic attitude — it always assumes the worst possible state of the world will happen, whatever you choose.

What is the Minimax Regret Rule?

A decision rule that picks the action whose maximum possible regret across all states is the smallest. Minimise the worst case regret.

How is regret calculated in the Minimax Regret Rule?

Regret = the best possible utility in that state minus the utility you actually got from your chosen action. It measures how much you missed out on.

What are the three steps of the Minimax Regret Rule?

Step 1 — calculate regret for every action in every state. Step 2 — find the worst regret for each action. Step 3 — pick the action with the smallest worst regret.

What do the Maximin Rule and Minimax Regret Rule have in common?

Neither requires probabilities — both work purely by comparing outcomes, so both apply under uncertainty as well as certainty.

What is the μ-Principle?

A decision rule that picks the action with the highest expected value — the probability-weighted average outcome. Requires known probabilities.

What is expected value?

The average outcome you would get if you repeated a decision many times. Calculated by weighting each outcome by its probability and summing them up.

How do you calculate expected value for a discrete distribution?

Multiply each possible outcome by its probability and add them all up. μ = Σ xₖ · pₖ

How do you calculate expected value for a continuous distribution?

Use an integral — μ = ∫ x · f(x) dx — where f(x) is the probability density function telling you how likely each value is.

What is the key difference between the μ-Principle and the Maximin/Minimax Regret rules?

The μ-Principle requires known probabilities so it only works under risk. Maximin and Minimax Regret need no probabilities so they work under uncertainty too.

What is the main advantage of the μ-Principle?

It is easy to apply — just calculate the expected value for each action and pick the highest one.

What is the main disadvantage of the μ-Principle regarding information?

It reduces the entire probability distribution to just one number (the average), losing all information about spread, risk, and extreme outcomes.

Why does the μ-Principle fail to account for risk?

Two actions with the same expected value can have completely different risk levels — one safe, one a gamble — but the μ-Principle treats them as identical.

Why does the μ-Principle neglect subjective meaning?

It gives the same answer to everyone regardless of personal attitudes toward risk — ignoring the fact that different people feel very differently about the same options.

What is the St. Petersburg Paradox?

A coin flip game where you double your prize for every heads before tails. Its expected value is infinite, but real people only pay a small amount to play — contradicting the μ-Principle.

Why is the expected value of the St. Petersburg game infinite?

Each round contributes (1/2)^k × 2^k = 1 to the expected value. Since there are infinitely many rounds, the total is 1+1+1... = infinity.

What does the St. Petersburg Paradox prove about the μ-Principle?

That maximising expected value alone cannot explain real human decisions — people care about risk and personal feelings, not just the mathematical average.

What does the μ-σ Principle add compared to the plain μ-Principle?

It adds variance (σ²) as a second criterion — so you consider both the average outcome AND how risky/spread out the outcomes are.

What is variance in simple terms?

How spread out the possible outcomes are around the average. High variance = outcomes jump around wildly = risky. Low variance = outcomes stay close to the average = safe.

What is standard deviation and why do we use it instead of variance?

Standard deviation = square root of variance. We use it because it is in the same units as the outcomes, making it easier to interpret.

What is the main weakness of the μ-σ Principle?

It still only uses two numbers (μ and σ), throwing away the rest of the information in the distribution. It can also violate state-by-state dominance.

What does it mean to violate state-by-state dominance?

Recommending an option that is actually worse than another option in every single possible state of the world — a clear logical error.

What does the preference function Φ(μ, σ) = μ − σ² do?

It combines expected value and variance into one score. Higher average = better. Higher variance = worse. Pick the action with the highest score.

In the example, why does a1 beat a2 despite a2 having a higher average?

a1 has zero variance (σ² = 0) so its score is 5. a2 has variance of 4 so its score is only 7 − 4 = 3. The risk penalty makes a1 the winner.

What would the plain μ-Principle have chosen in this example, and why is that a problem?

What would the plain μ-Principle have chosen in this example, and why is that a problem? Back: It would have chosen a2 because μ = 7 > 5. But this ignores the extra risk of a2 — showing why the μ-Principle alone is not enough.

What is a risky alternative?

An option where multiple outcomes are possible but the actual outcome is uncertain at the time of the decision — like picking a mystery box.

Why can risky alternatives be called "lotteries"?

Because each alternative has known probabilities over possible outcomes — just like a lottery ticket with known prize probabilities.

What is the key assumption about probabilities in this section?

The probabilities for all possible outcomes of each alternative are known — placing us in the world of decisions under risk, not uncertainty.

What is a simple lottery?

A list pairing every possible outcome with its probability. Written as L = (x₁, p₁; ...; xₙ, pₙ). Every probability ≥ 0 and all probabilities sum to 1.

What are the two rules every simple lottery must satisfy?

Every probability must be non-negative (pₙ ≥ 0) and all probabilities must sum to exactly 1 (Σpₙ = 1).

Why must all probabilities in a lottery sum to 1?

Because something must always happen — the total chance of all possible outcomes together must equal 100%.

What is a compound lottery?

A lottery where the outcomes are themselves simple lotteries rather than fixed prizes — a lottery over lotteries.

What is a real life example of a compound lottery?

A game show where spinning a wheel puts you into one of several different prize games — you first win a lottery ticket, then play that lottery.

What are the two probability rules for a compound lottery?

Every probability αₖ must be non-negative (αₖ ≥ 0) and all probabilities must sum to 1 (Σαₖ = 1) — same rules as a simple lottery.

What is the key difference between a simple lottery and a compound lottery?

Simple lottery — prizes are fixed outcomes. Compound lottery — prizes are other lotteries. You win a game to play another game.

What is a reduced lottery?

A simple lottery that collapses a compound lottery into one layer, giving the same final probabilities over outcomes.

How do you calculate the probability of outcome n in a reduced lottery?

pₙ = Σ αₖ · pₙᵏ — multiply each sub-lottery's probability αₖ by its own probability of outcome n, then sum across all sub-lotteries.

Why do the probabilities of a reduced lottery still sum to 1?

Because something must always happen — no matter which sub-lottery you land in, you always get an outcome. The total stays exactly 1.

Why does only the reduced lottery matter to the decision-maker?

Because only the final probabilities over outcomes matter — not the process or number of stages that produced them.

When are two lotteries considered equivalent?

When they have the same reduced form — meaning identical final probabilities over all outcomes, regardless of how they are structured.