ECN 416 - Game Theory Week 1: Vocabulary Flashcards

Key concepts and terminology introduced in Week 1 of ECN 416 covering the basics of sets, functions, randomness, and decision-making under uncertainty.

Game theory

The study of decision-making and responding to incentives when others are making decisions; not prescriptive or descriptive, but a tool to understand strategic interactions.

Decision-making under uncertainty

Choice by an agent when payoffs depend on an uncertain state θ ∈ Θ; modeled with utilities and maximizing expected utility.

State θ

A realization of uncertainty that affects payoffs; θ takes values in Θ.

Utility function (u)

A function u: Si → R that assigns a numeric payoff to a strategy si for a single agent.

Utility function under uncertainty

An extension u: Si × Θ → R where payoff depends on both the strategy and the state of the world.

Expected utility (Eu)

The probability-weighted average of payoffs across states; Eu(si,θ) = Σj pj u(si, θ_j).

Rational agents maximize expected utility

Among available strategies, choose si that maximizes Eu(si,θ) given the (probabilistic) states.

Pure strategies

Deterministic strategies; a function from information to a single action.

Mixed strategies

Randomized strategies; a probability distribution over pure strategies.

Function (f: A → B)

A mapping from input set A (domain) to output set B (codomain); outputs are f(a).

Domain

The input set A of a function f.

Codomain

The output set B that a function f can map into.

Random map (π: A → ∆B)

A stochastic mapping from A to B that yields a distribution over B for each a ∈ A; ∆B is the set of distributions over B.

∆B

The set of all probability distributions over B.

Correspondence (F: A ⇒ B)

A multi-valued function; for each a ∈ A, F(a) ⊆ B rather than a single value.

Multivalued function

A generalization of a function that assigns a set of outputs to each input.

Cartesian product

A1 × A2 × … × At, the set of tuples (a1,…,at) with ai ∈ Ai.

Set

A collection of objects; sets are denoted by capital letters.

Element

An individual object that belongs to a set.

Subset

S′ ⊆ S means S′ is a subset of S.

Strict subset

S′ ⊂ S means S′ is a subset of S and S ≠ S′.

Finite set

A set with a finite number of elements.

Countable infinity

A set that can be put in a one-to-one correspondence with the natural numbers (e.g., N = {1,2,3,…}).

Uncountable infinity

A set that cannot be put into a one-to-one correspondence with the natural numbers (e.g., real numbers).

States Θ

The collection of possible uncertain states influencing payoffs.

Probabilities (p_j)

The probabilities associated with states θj, with Σj p_j = 1.