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# Game Theory

## Game theory - a master course

The course introduces a person to non-cooperative and cooperative game theory. The students will receive a broad overview of the branches of game theory, and subsequently dive into non-cooperative solution concepts and efficiency measures. We consider several models and important classes of such games.

The second part of the course deals with cooperative solution concepts and classes of games

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The required preliminaries consist of calculus, probability and combinatorics.

Additionally, this course assumes a good understanding of mathematics, notions such as definition and proof, basic set theory, linearity, sufficiency, necessity, characterisations, etc. Understanding the mathematical way of thinking is necessary.

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Topics:

1:

Game theory areas (non-cooperative, cooperative, MD (auctions, etc.), epistemic GT, evolutionary GT, logic in GT, etc.)

Utility theory (Von-Neumann Morgenstern) and rationality assumption

Normative approach here (rather than descriptive)

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Non-cooperative game theory:

Normal form games

Extensive games

Succinct representations (polymatrix games, graphical games, etc.)

(In)Complete and (im)perfect information assumptions

(In)finite games

2:

Normal (= strategic)-form games:

(Pure) Nash equilibrium

Examples (of non-existence too)

Efficiency (prices if anarchy and stability (PoA, PoS))

Strongly/weakly dominant strategies

(1st and 2nd price) auctions example

Mixed strategies, mixed extension, and MNE (Mixed Nash Equilibrium)

Strongly/weakly dominated strategies

Elimination + order-dependent for weak (and independent for strong - later)

The influence of elimination on NE

3:

(Exact) potential games

Equivalence to congestion games

4:

Zero-sum games

(maxmin, minmax, value, exchangeability of NE strategies)

5:

Mixed extension

Mixed NE

(Finite existence)

6:

Properties of mixed NE, general (mixed dominance), symmetric games, constant-sum, potential

Finding mixed NE (general alg. and examples)

Rationalizability + order independence of elimination

7:

Social welfare

Prices of anarchy and stability

Examples (coordination, routing, etc.)

8:

Correlated and coarse correlated equilibrium

Strong Nash equilibrium

Evolutionary equilibrium and evolutionary games

9:

Extensive games

A winning strategy and proof techniques

Zermelo's algorithm

Examples: chess, checkers, chomp

10:

SPE (Subgame-Prefect Equilibria)

Existence

11:

(In)finitely repeated games (prisoner's dilemma, etc.)

Folk theorems

12:

Cooperative games:

Non-transferable and transferable utility

General properties

Transferable utility:

Simple games

13:

Core

Bondareva-Shapley characterisation theorem

14:

The Shapley value and its axiomatic characterisation

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Material:

The necessary material consists of the slides, lectures, tutorials and home works.

The additional reading consists of:

A Course in Game Theory by Martin J. Osborne and Ariel Rubinstein, 1994, besides the definition of extensive form games

Game Theory by Michael Maschler, Eilon Solan and Shmuel Zamir, 2013

An Introduction to Game Theory by Martin J. Osborne, 2004, besides the definition of extensive form games

Game Theory: A Multi-Leveled Approach by Hans Peters, 2008

Game Theory And Mechanism Design by Y. Narahari, 2014

Algorithmic game theory, edited by Noam Nisan, Tim Roughgarden, Eva Tardos and Vijay V. Vazirani

A site to acquire a deeper understanding:

plato.stanford.edu

Concrete topics from their creators:

Non-Cooperative Games by John F. Nash, 1951 - about mixed Nash equilibrium

Potential Games by Dov Monderer and Lloyd S. Shapley, 1994 - about potential games