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.