Halfway handout

The whole idea behind game theory:
What happens to me depends on what I do but also on what everyone else does.
What did we do?
 We played the 2/3rds of the average game.
What do I need to be able to do?
 Define an extensive form game and represent a game in extensive form;
 Define an information set.

Normal Form Games:
Also called bimatrix games. This is often used to represent situations in which players make decisions at the same time (and have a finite strategy set).
What did we do?
 We played some matching pennies games (I blogged about the results, links are in the discussion boards).
What do I need to be able to do?
 Define a normal form game;
 Calculate expected utility with mixed strategies;
 Plot expected utilities against/with pure strategies.

Dominance:
The idea of being able to ‘remove’ strategies that will never be played.
What did we do?
 Nothing too exciting.
What do I need to be able to do?
 Define dominated strategies (strict and weak);
 Define Common knowledge of rationality;
 Predict rational behaviour using iterated removal of dominated strategies;
 Recognise games that cannot be approached using this tactic.

Best responses:
The idea of doing what ‘I’ should do if I know what ‘you’ would do.
What did we do?
 We played a game similar to Golden balls in class. (I blogged about it, link in the discussion board)
What do I need to be able to do?
 Identify best responses in pure strategies (underline stuff);
 Identify best responses against mixed strategies;
 Define the sets \(UD_i\) and \(B_i\);
 State and prove certain cases of theorems connecting these two sets.

Nash equilibria in pure strategies:
Here we are officially defining what we meant by ‘predicted rational behaviour’. In essence: ‘not having a reason to change what you are doing’.
What did we do?
 Nothing too exciting.
What do I need to be able to do?
 Give definition of Nash Equilibria in pure strategies;
 Continue to use best responses to identify these;
 Calculate Nash equilibria in pure strategies when strategy sets are continuous.

Nash equilibria in mixed strategies:
Congratulations we’re now doing real game theory.
What did we do?
 We played a Rock, Paper, Scissor, Lizard, Spock game in class (I blogged about it, link on a discussion board).
 The weak after seeing this we played against a random number generator (I blogged about that too, link in the same place).
What do I need to be able to do?
 Define Nash equilibrium (completely, ie in mixed strategies);
 Identify Nash equilibrium using best responses against mixed strategies;
 State and prove the Equality of Payoffs theorem;
 Apply the equality of Payoffs theorem;
 State and prove specific cases of Nash’s theorem.

Extensive form games (again):
We go back to the tree things and get a bit more precise.
What did we do?
 We played the centipede game but I’ll come back to that when we talk about subgames.
What do I need to be able to do?
 Recognize valid extensive form games;
 Define sequential rationality;
 Define backwards induction;
 Prove the theorem of existence of Nash equilibrium in games of perfect information;
 Solve extensive form games using backwards induction;

Subgame perfection:
We connect Normal form games and extensive form games and define a stronger type of equilibrium condition.
What did we do?
 We played the centipede game and analysed the equilibrium strategies: everyone should take AT ALL STAGES.
What do I need to be able to do?
 Write down the normal form representation of an extensive form game;
 Define a subgame;
 Define subgame perfect equilibria;
 Obtain subgame perfect equilibria in games.