Logic, Game Theory, and Behavioral Economics Lecture Review

Vocabulary-style flashcards covering logic, set theory, game theory equilibria, expected utility, risk aversion, portfolio theory, contract theory, and prospect theory based on Christos A. Ioannou's lecture transcript.

Conditional Statement

A logical statement consisting of a hypothesis or antecedent in the “if” clause and a conclusion or consequent in the “then” clause.

Converse

Given an if-then statement “if p, then q,” the converse is defined as the statement “if q, then p.”

Contrapositive

A related statement to “if p, then q” defined as “If not q, then not p,” which is logically true if the original statement is true.

Tautology

A compound statement in logic that is always true, such as $p \lor [q \Rightarrow \neg p]$.

Theorem

A tautology of the form $[p_1 \land p_2 \land \dots \land p_k] \Rightarrow q$, where the statements $p_1, p_2, \dots, p_k$ are referred to as assumptions.

Modus Ponens

A logical rule of inference stating that from $(p \Rightarrow q) \land p$, one can infer $q$.

Modus Tollens

A logical rule of inference stating that from $(p \Rightarrow q) \land \neg q$, one can infer $\neg p$.

Constructive Dilemma

A logical rule stating that from $(p \Rightarrow q) \land (s \Rightarrow t) \land (p \lor s)$, one can infer $(q \lor t)$.

Universal Quantifier

Denoted by $\forall$, it represents the statement “For all $x$ in $X$, $p(x)$.”

n-fold cross product

Also called the Cartesian product, it is the set of ordered n-tuples from $X_1, X_2, \dots, X_n$, written as $X_1 \times X_2 \times \dots \times X_n$.

Game

A formal representation of a situation in which a number of individuals interact in a setting of strategic interdependence.

Strategic Interdependence

A situation where each individual’s welfare depends not only on her own actions but also on the actions of other individuals.

Action Profile

A combination $a = (a_i, a_{-i})$ consisting of the action chosen by player $i$ and the actions chosen by all other players.

Strictly Dominant Strategy

A strategy $s_i$ is strictly dominant for player $i$ if it uniquely maximizes that player's payoff regardless of the strategies chosen by rivals.

Strictly Dominated Strategy

A strategy for which there exists an alternative strategy that yields a greater payoff for the player regardless of what the other players do.

Nash Equilibrium

A strategy profile where each player’s strategy choice is a best response to the actual strategy choices played by their rivals.

Mixed Nash Equilibrium

An equilibrium where players assign positive probabilities to their strategies such that other players are indifferent over their own strategies.

Rational Preference

A preference relation ($\succsim$) that is both complete (all pairs can be compared) and transitive (ordering is consistent).

Cobb-Douglas utility function

A mathematical utility representation defined as $u(x_1, x_2) = x_1^{\alpha} x_2^{1-\alpha}$ for $\alpha \in (0, 1)$.

St. Petersburg Paradox

A problem proposed by Gabriel Cramer regarding a coin-toss game, used by Daniel Bernoulli to argue that value is based on utility rather than price.

Support

The finite subset of outcomes in a set $X$ to which a probability distribution assigns a positive value.

Reduced Lottery

A simple lottery that generates the same final probability distribution over outcomes as a given compound lottery.

Continuity Axiom (Axiom C)

A preference axiom stating that if $p \succ q \succ r$, there exist probabilities such that mixtures of $p$ and $r$ are ranked relative to $q$, excluding lexicographic preferences.

Independence Axiom (Axiom I)

Also called the Substitution Axiom, it states that preference between two lotteries should be independent of any third lottery they are both mixed with.

Allais’ Paradox

A famous empirical violation of the Independence Axiom described by Maurice Allais in 1953 demonstrating that human choice often contradicts Expected Utility Theory.

Risk Aversion

A characteristic of a decision maker who prefers a certain amount equal to the expected value of a lottery over the lottery itself; equivalent to utility function concavity.

Certainty Equivalent

The guaranteed amount of money, denoted $c(F, u)$, that provides an individual the same utility as a given risky gamble $F(\cdot)$.

Arrow-Pratt coefficient of absolute risk aversion

Defined as $r_A(x) = - \frac{u''(x)}{u'(x)}$; it measures the degree of risk aversion based on the curvature of the Bernoulli utility function.

First-Order Stochastic Dominance

A condition where distribution $F(\cdot)$ yields unambiguously higher returns than $G(\cdot)$ if every expected utility maximizer who values more over less prefers $F$.

Mean-Preserving Spread

A method of creating a riskier distribution from an original one by adding zero-mean noise to the outcomes without changing the overall mean.

Beta (\beta_i)

A measure of non-diversifiable systematic risk of an asset calculated as $\beta_i = \frac{\text{Cov}(R_i, R_m)}{\text{Var}(R_m)}$.

Security Market Line (SML)

A graphical representation of the MEDAF (CAPM) that plots the expected return of an asset relative to its systematic risk (beta).

Efficient Market Hypothesis (EMH)

The hypothesis stating that in an efficient market, prices reflect all available information, making it impossible to systematically beat the market except by chance.

Likelihood Matrix (L)

In information theory, a matrix $L = [q_{m|s}]$ containing the conditional probabilities of receiving message $m$ given state of nature $s$.

Posterior Matrix

A matrix $\Pi = [\pi_{s|m}]$ containing the updated probabilities of state $s$ occurring given the receipt of message $m$.

Value of Information

The expected utility gain from choosing an optimal action after receiving a message compared to the optimal action chosen given only prior beliefs.

Contingent Commodity

A commodity that is specified not only by physical traits, but also by the specific state of nature in which it is made available for consumption.

Arrow Security

A simplified financial asset that pays one unit of value in one specific state and zero in all other states.

Revealing Equilibrium

A Rational Expectations equilibrium where all agents can learn the true state of nature by observing market prices.

Market for Lemons

An adverse selection model by George Akerlof (1970) where asymmetric information regarding quality causes high-quality goods to be withdrawn, potentially collapsing the market.

Signaling

A solution to asymmetric information where the informed agent moves first to reveal their type, such as through education or warranties.

Screening

A solution to asymmetric information where the uninformed agent (principal) moves first to design contracts that force the informed agent (agent) to reveal their type.

Informational Rent

The utility above the reservation level that a 'good type' agent receives in a screening contract to ensure they do not mimic a 'bad type.'

Moral Hazard

A problem of asymmetric information involving hidden actions where the principal cannot observe the agent's actual effort level.

Participation Constraint

A contractual constraint requiring the principal to provide the agent with at least their reservation utility to ensure they accept the contract.

Incentive Constraint

A contractual requirement that induces the agent to choose the action or disclose the information desired by the principal.

Representativeness Heuristic

A cognitive heuristic where probabilities are mistakenly evaluated by the degree to which an event is representative of a stereotypical category.

Availability Heuristic

A bias where the frequency or probability of an event is judged by the ease with which relevant instances come to mind.

Reflection Effect

The behavioral finding that people tend to be risk averse in the positive domain (gains) but risk seeking in the negative domain (losses).

Isolation Effect

The tendency of decision-makers to disregard shared components of alternatives and focus only on the elements that distinguish them.