# Game Theory

### Prof. S Modica

**2018 Students please click here (December 28)**### What we have done this year (2018)

We have done very little theory (proof of the expected utility theorem, characterization of the structure of zero-sum games, existence of subgame perfect equilibria, and little more); we have worked on finding solutions to non-trivial decision problems and equilibria in non-trivial games. In strategic form games we have worked on pure strategy equilibria, mixed equilibria, correlated equilibria and Bayes-Nash equilibria. In extensive form games we have done Nash and subgame-perfect equilibria, and we have only seen a couple of examples of imperfect information games (without going into the subleties of subgame-perfect refinements). In cooperative games we have computed the Core, Shapley value and the Nucleolus, with particular attention to the airport game and the bankruptcy game. We have finished off with a brief introduction to repeated games and the idea of cooperation opportunities in long run relationships. Thank you for attending the course!S Modica

### Syllabus (a superset of what we do each year)

- Expected Utility Theory
- Games in strategic form
- Nash equilibrium, pure and mixed strategies
- Zero-sum games
- Correlated equilibrium
- Bayes-Nash equilibrium
- Extensive form games, Perfect Information
- Subgame Perfection
- Repeated Prisoner's dilemma
- Repeated Games Folk Theorems
- Extensive form games, Imperfect Information
- Nash, Perfect Bayesian and Sequential Equilibrium (theory and examples)
- Cooperative Games: Core, Shapley, Nucleolus. Some theory and some examples

### Class material

The coach problem (from Dixit-Nalebuff*Thinking Strategically*)

Osborne-Rubinstein Proposition 22.2 (zero-sum games)

Mixed extension and Osborne-Rubinstein Lemma 33.2 (and equation 32.2)

Mixed extension of zero-sum games

The Rock-Paper-Scissors game

The bus game

A two-by-two simple game (hiring, from Gibbons)

A couple of exercises on normal form games by Joel Watson

The penalty game

Li Calzi exercise n.4.8 p.51 (last bit)

Battleships (based on Binmore's

*Fun and Games*)

A voting game

Examples of correlated equilibria

Clarifications on correlated equilibrium

Battle of Sexes with Uncertainty (or: Yes, girls are definitely smarter than boys)

Definition of game tree gives equivalent definition of extensive form game

A bank deposit game

Three-way duels (or: the games of your life, Dixit-Nalebuff)

Fairness (Fehr-Schmidt-QJE1999)

Matching pennies with spying (Blackwell-Girshick)

Perfect Bayesian Equilibrium from MWG (ignore the word "weak")

Example of sequential equilibrium

More examples of PBE and sequential equilibria with solutions

Preliminaries on discountijng flows

Long Run Cooperation in oligopoly

Bankruptcy Solutions (Peyton Young in

*Equity*)

Some basic points and examples about Core and Shapley Value

Shapley and Nucleolus in the bankruptcy game

Shapley and Nucleolus in the airport (or elevator) game

The heritage game

Nucleolus of the 4-player weighted majority game

### More notes on games and decisions

Pareto Optima of the classical Tragedy of the Commons### Texts

- Peter Wakker,
*Prospect Theory,*Cambridge University Press 2010 -
Martin J Osborne, Ariel Rubinstein,
*A Course in Game Theory*, MIT Press 1994 - Martin J Osborne,
*An introduction to game theory*, Oxford University Press 2003 - Marco Li Calzi,
*Teoria dei Giochi*, Etas Libri 1995 (esercizi)