Game Theory: History & Applications

Compute the mixed strategy Nash Equilibrium in 2x2 games using the principle of indifference.

State Nash's Existence Theorem and explain its significance for the analysis of finite games.

Represent sequential games in extensive form using game trees, nodes, and edges.

Define perfect and imperfect information in extensive-form games using the concept of information sets.

Model a simultaneous-move game (e.g., Rock-Paper-Scissors) in extensive form using an information set.

Define Subgame Perfect Nash Equilibrium (SPNE) and apply backward induction to find it in finite games of perfect information.

Analyze the Ultimatum Game and the Centipede Game to understand the power and limitations of backward induction.

Model Cournot (quantity) competition as a simultaneous game and derive the Nash Equilibrium.

Model Stackelberg (quantity) competition as a sequential game and compare its outcome to the Cournot model.

Distinguish between finitely and infinitely repeated games and their different strategic possibilities.

State the Folk Theorem and explain how it allows for cooperative outcomes in infinitely repeated games with discounting.

Analyze the strategic dynamics of the iterated Prisoner's Dilemma, referencing the results of Axelrod's tournaments.

Define a Bayesian game of incomplete information, including player types and prior beliefs.

Define and compute a Bayesian Nash Equilibrium in a simple static game of incomplete information (e.g., a variant of the Prisoner's Dilemma).

Define signaling games as a class of dynamic Bayesian games with asymmetric information.

Analyze a canonical signaling game (e.g., Spence's job-market model) to distinguish between separating and pooling equilibria.

Define a cooperative (coalitional) game and represent it using a characteristic function.

Define the core of a cooperative game as a set of stable outcomes and determine if it is non-empty for a given game.

Define and compute the Shapley value for a given coalitional game.

Apply the Shapley value to a practical cost allocation problem.

Apply the Shapley value to measure voting power in a weighted voting system.

Define mechanism design as 'reverse game theory' and describe its core components (agents, types, social choice function).

Define direct mechanisms and incentive compatibility, and state the Revelation Principle.

Describe the Vickrey-Clarke-Groves (VCG) mechanism and demonstrate its strategy-proofness property.

Describe the rules of English, Dutch, sealed-bid first-price, and Vickrey (sealed-bid second-price) auctions.

Analyze why truthful bidding is a dominant strategy in Vickrey auctions.

Derive the equilibrium bidding function in a sealed-bid first-price auction with independent private values.

State the Revenue Equivalence Theorem and explain its implications for auction design.

Define and calculate the Price of Anarchy in a network routing game, illustrating with Braess's paradox.

Describe the computational complexity of finding a Nash equilibrium and the significance of the PPAD complexity class.

Analyze a multi-agent reinforcement learning (MARL) system from a game-theoretic perspective, identifying players, strategies, and equilibria.

Model a simple principal-agent problem in LLM alignment using the framework of Bayesian games.

Analyze the strategic incentives in a plurality voting system and explain the potential for tactical voting (e.g., Duverger's Law).