Game Theory

Game theory is a branch of mathematics and economics that provides a systematic framework for studying outcomes that depend on the choices and decisions of multiple players. Its objective is to predict outcomes and design mechanisms that can steer players toward desired outcomes.

This introductory course covers the theory and application of non-cooperative games. It discusses two important formulations of games: static games, which are one-shot games, and sequential-move games, along with the solution concepts used to analyze their outcomes. One-shot games can be used to analyze scenarios such as the Prisoner's Dilemma, markets in which firms compete with one another, and electoral competition. Sequential-move games are more representative of real-world situations, which are typically sequential in nature. Examples include war-like scenarios, markets in which one firm moves before others, and decision-making situations in which an individual must repeatedly choose whether to proceed with a particular course of action.

 The course applies game-theoretic concepts to such real-world scenarios and also introduces the theory of mechanism design, which builds on the foundations of game theory. An example of a mechanism is an auction, which is widely used in practice to allocate goods and services efficiently. The course explores mechanism design through relevant and practical real-world-inspired examples.

The course is a formal study of the theory of games. The course introduces two fundamental game-theoretic frameworks: static games and sequential-move games. Static games are one-shot type games, where the players can move just once and simultaneously. The course discusses several such games inspired from real-life scenarios such as the Prisonner’s dilemma, hunting game, market settings such as the Cournot model and electoral competition. The course introduces a very common notion of equilibrium or outcome of a game called the Nash Equilibrium (NE) to analyze the outcomes of games. The course then presents an analysis of the outcomes of several of the one-shot games by using the concept of NE.

In the next part, the course discusses the extension of the Nash equilibrium concept to sequential-move games. In sequential-move games, the moves of the players must be sequentially rational from a player’s point of view. This leads to an equilibrium called Sub-game Perfect Equilibrium (SPE). The course discusses several examples where a sequential move-game is applicable such as a Stackleberg game, dictatorship game, centipede game, market examples, electoral competition, etc., and their outcomes by applying the concept of SPE. The course also discusses specific forms of sequential-move games such as repeated games. These are games where a one-shot type of game is played repeatedly over a period. A specific example, which is the repeated game version of the prisoners’ dilemma is discussed in-depth, and various common-sense strategies like trigger strategies, tit-for-tat, etc. that the players could deploy are analyzed in detail.

In the next part, the course discusses scenarios where the players have imperfect information about the game. These are scenarios where the agent may not have complete information of its state and therefore can only move, or act based on the chance or probabilities of its underlying state. Games of such type in the one-shot setting are called the Bayesian games. The course discusses the Bayesian version of several one-shot games such as the Cournot market model and their outcomes. The course also discusses such scenarios in the sequential move games and presents examples.

In the final part, the course discusses the theory of mechanism design, and an important class of mechanisms called auctions. Auctions are mechanisms that can be used for allocating goods and services in an efficient manner. Auctions are used for example to determine which service provider is most fit to transmit over a spectrum bandwidth. The course discusses several auction types such as sealed first price and sealed second price and discusses their outcomes by applying the game-theoretic solution concepts. The course then discusses real world applications such as spectrum auctions.

By the end of this course, each student will have had the opportunity to:

1. Understand the various game-theoretic settings.
2. Apply the game-theoretic solution concepts to analyze game-type scenarios.

By the end of this course, each student will have had the opportunity to:

1. Describe/explain the various game formulations | Know/Knowledge Outcome  
2. Describe/explain the various game-theoretic solution concepts | Know/Knowledge Outcome
3. Demonstrate solving game-theoretic problems | Comprehend Outcome 

• Introduction to Game Theory, Martin J Osborne.

• Games of Strategy, Avinash Dixit, Susan Skeath and David Reiley.
• A Course in Game Theory, Martin J Osborn and Ariel Rubinstein.