1. Finitely repeated games

We ‘repeat’ a normal form game $$T$$ times. Strategies need to be defined to take in to account the entire history of the game.

What did we do?

• We played an iterated prisoner’s dilemma tournament.

What do I need to be able to do?

• Define a repeated game.
• State and prove the theorem of a sequence of stage Nash profiles.
• Draw possibly utility outcomes.
• Obtain subgame perfect Nash equilibria that are not a sequence of state Nash equilibria.
2. Infinitely repeated games

We now ‘repeat’ a normal form game an infinite amount of times. To deal with this we use discounting.

What did we do?

• We played an infinite iterated prisoner’s dilemma tournament by introducing a probability of the game carrying on.

What do I need to be able to do?

• Calculate expected utilities with discounting.
• Prove particular instances where a given strategy is better than another.
• Prove the Folk theorem.
3. Population Games and Evolutionary stable strategies

We move on to something different and consider populations and how a particular strategy evolves in a population.

What did we do?

• We played a brief game with cards in class.

What do I need to be able to do?

• Define a population game.
• State and prove the theorem for necessity of stability.
• Define a post entry population $$\chi_{\epsilon}$$ and an ESS.
• Obtain ESS in a game against the field.
4. Nash equilibrium and Evolutionary stable strategies

We consider a particular type of game against the field: member of the population are randomly matched and play a normal form game.

What did we do?

• Nothing really.

What do I need to be able to do?

• Define a pairwise contest game.
• Find ESS in a pairwise contest game from first principles.
• State prove and use the theorem relating an ESS to a Nash equilibrium of a normal form game.
5. Random events and incomplete information

Here we look at extensive form games with random events.

What did we do?

• 3 volunteers played a version of matching pennies with some randomness involved (someone won a melon)

What do I need to be able to do?

• Obtain the normal form representation of a game with incomplete information.
• Describe basic utility theory.
• Solve the principal agent game.
6. Stochastic games

Games where strategies not only define the utilities but the next game.

What did we do?

• We played a modified version of the Prisoner’s dilemma in the form of a stochastic game (this was a 4 team round robin tournament)

What do I need to be able to do?

• Define a stochastic/Markov game, a Markov strategy.
• State the requirement for Nash equilibrium.
• Obtain Nash equilibrium for stochastic games.
7. Matching games

We look at the stable marriage problem and the Gale Shapley algorithm.

What did we do?

• We matched 3 volunteers to 3 of my toys (Donatello, Zoe and my Tech Deck).

What do I need to be able to do?

• Define a matching game.
• State and apply the Gale-Shapley algorithm.
• State and prove the theorem guaranteeing a unique matching as output of the Gale Shapley algorithm.
• State and prove the theorem of reviewer sub optimality.
8. Cooperative games

Sharing resources when considering contribution of coalitions.

What did we do?

• We played basketball (kind of).

What do I need to be able to do?

• Define a characteristic value game.
• Define monotone and superadditive games.
• Define Efficiency, Null player, Symmetry and Additivity (see exercises - you could be asked to prove that the Shapley value has these properties).
• Define and be able to calculate the Shapley value.
9. Routing games

Games that can be used to model behaviour in system affected by congestion.

What did we do?

• Nothing fun at all…

What do I need to be able to do?

• Define a routing game, set of paths and feasible paths.
• Define and obtain (from first principles) Optimal flows.
• Define and obtain (from first principles) Nash flows.
• Define the potential function.
• Define marginal costs.
• State and use the theorem connecting the Nash flow to the optimal flow.
• State and use the theorem connecting optimal flows to Nash flows.