Syllabus

Notes

Lecture 2

• At the core of the course is the concept of a rational agent.

• In this lecture, we will discuss a core theorem in economics, that will justify expected utility maximization, the main tool/assumption we’ll use throughout the course

Outcomes and Lotteries

• Let be a set of outcomes

• A lottery is a distribution over outcomes

  • We write and for the set of lotteries on

Ordinal Preferences

• A preferences relation indicates the relative preference of an agent for outcomes and lotteries

• We write:

  • if the agent weakly prefers to
  • if the agent strictly prefers to
  • if the agent is indifferent between and
• These extend to lotteries as well if the agent weakly prefers lottery to

Cardinal Preferences

• A utility function is a function

• It represents a preference relation if

• A utility function expresses not only which of the outcomes is preferred, but by how much.

Ordinal vs Cardinal

• At first glance, cardinal utilities seem much stronger than ordinal utilities, but under some mild conditions, von Neumann and Morgenstern proved that ordinal utilities imply cardinal utilities

Axioms: Completeness and Transitivity

• Axiom 1 (completeness): For every pair of lotteries and :

• Axiom 2 (transitivity): For any lotteries :

• In other words, the preference relation is a weak order

  • This is important to prevent money pumps
• Axiom 3 (independence): For any lotteries and every :

• Axiom 3 (continuity): If for lotteries then there exists an such that

The von Neumann-Morgenstern Theorem [1947]

• A preference relation over lotteries satisfies completeness, transitivity, independence, and continuity iff there is a utility function over outcomes such that, for all lotteries and :

• Here, for

(Forward Direction)

• The proof is constructive

  • We will build a utility function (and then check that the expected utility reproduces )
• Lemma 1 (independence preserves indifference): If , then for every lottery and every :

• Proof: The cases and are immediate, so assume . Since , we have both and . Applying independence to each gives both mixture inequalities, which together are exactly indifference.
• Lemma 2 (independence preserves strict preference): If , then for every lottery and every :

• Proof: Suppose toward a contradiction that . Independence runs in both directions, so this would give , contradicting . Completeness implies that the mixture preference is strict.
• Lemma 3 (simultaneous substitution): Suppose for , and let be weights such that . Then:

• Proof: Apply Lemma 1 one component at a time, replacing by inside the mixture, then by , and so on. Every replacement preserves indifference, and transitivity chains the intermediate steps into one.

Step 1: A best and a worst outcome

• Because is finite and is complete and transitive, the outcomes can be ranked, so there are outcomes with for every

  • Call a best outcome and a worst outcome. Assume for now (we handle at the end)

• For define the reference gamble

  • As increases, we go from the worst to the best outcome

• Lemma 4 (more probability on the best outcome is better): For all :

• Proof: The endpoints and are immediate, since is and is . Let . Applying Lemma 2 to gives .
• Now take , and set . Since , Lemma 2 gives . The LHS is just , and substituting on the RHS simplifies it to . So implies ; reversing the roles covers , and is the same lottery.

Step 2: Assign a utility number to every outcome

• Fix an outcome . Since is best and is worst,
• By continuity, there exists such that
• Define
• Uniqueness: Suppose two weights both work, and for . Transitivity implies . But by Lemma 4, if then , and if then — a contradiction. So
• Therefore, for every outcome there is a unique such that

with and

Step 3: Reduce an arbitrary lottery to a best-worst lottery

• Let . Then by definition, for weights with
• By equation (1), for every we have
• Using Lemma 3:

• Distributing and using :

• Define . Then

Step 4: Expected utility represents preferences

• Let be two arbitrary lotteries
• and
• By transitivity,
• By Lemma 4,
• So , i.e.

where preferences are expressed as expected utilities

Corner case ()

• For every , since is best and is worst,
• Since , transitivity gives , so , so , so , for every
• So for every lottery :
• Thus for every represents these preferences

What we’ve shown: given a preference relation over lotteries satisfying Completeness, Transitivity, Independence, and Continuity, there is a utility function over outcomes such that for all lotteries .

• It remains to show the converse: if there is a utility function that represents a preference relation, then the preferences satisfy the axioms (homework!)

Impact

• The von Neumann–Morgenstern theorem is one of the most (if not the most) influential theorems in Economics

Realistic?

• Are the axioms a realistic description of how people actually choose? Allais (1953) suggests not so much

• Scenario 1:

  • gives $1 million for sure
  • gives $1 million w.p. 0.89, $5 million w.p. 0.1, and $0 w.p. 0.01
  • Most people choose

• Scenario 2:

  • gives $1 million w.p. 0.11 and $0 w.p. 0.89
  • gives $5 million w.p. 0.1 and $0 w.p. 0.9
  • Most people choose
• But means , while means the opposite
• Question (homework?): which axiom is violated?