Belief
A player's assessment of the strategies of other players in a game. The blank a player holds about another players specific strategy X is represented by θ.
θ2(X)
How likely player 1 thinks it is that player 2 will choose the pure strategy X.
θ-1(A, B)
In a three player game, the belief of player 1 about the probability that player 2 chooses A while player 3 chooses B.
Strategic uncertainty
A situation a player may face in a game where they are uncertain about what the other player(s) will do.
Risk Aversion
When players who face strategic uncertainty elect to choose less risky strategies. Demonstrated by the concavity of the utility function. A more concave utility function means they are more likely to choose less risky options.
The extensive form
A way to represent a game using a set of decision trees. Allows for display of sequential moves.
Matrix form game disadvantage
Only allows for display of simultaneous moves.
Decision Node
The point where a player makes a decision
Branches
Represent actions players can choose
Information Set
A set that, for a particular player, establishes all the possible moves that could have taken place in the game so far, given what that player has observed.
Dotted Line in Game Tree
Represents that a given player does not know what has transpired beforehand at a given set of connected nodes.
Strategy
A complete contingent plan for a player in the game
Complete contingent plan
A full specification of a player's behavior. Describes the actions that the player would take at each of his possible decision points.
Strategy Set
Denoted by Si a set comprising each of player i's possible strategies.
si ϵ Si
A particular strategy in Si
Subgame
A game consisting of the decisions that are left to be made after an initial decision, or set of decisions, has been made.Split information sets (IE: dotted line connected nodes) cannot be these.
Split information sets
Decision nodes connected by dotted lines. Cannot be subgames.
Sequential Rationality
When players choose optimally at every information set, even information sets which are not reached in equilibrium.
Subgame-perfect Nash equilibrium (SPE)
A strategy profile which results in Nash equilibrium at every subgame in an extensive form game.
Backward Induction
A method of finding SPE, where you start at the furthest subgames in the tree and you find Nash Equilibria at each. Continue to work backwards until the beginning of the tree is reached.
Stackelberg duopoly
The cournot duopoly, but firms choose quantities to sell sequentially. (IE: the second firm's strategy is a function of the first firm's strategy) (2 firms, identical products, market price decreases with total quantity, constant marginal cost)
What do Cournot duopolist's do wrong?
They produce too much relative to the profit maximizing amount because each ignores the externality that it imposes on the other firm.
Stage game
A game which is repeated. Every time it is repeated, a new subgame begins.
history of play
What happened in previous games when playing multiple rounds
Strategies in repeated games
A player's strategy states what they will do in every period t > 1 as a function of what both they and the other player(s) have done in the previous rounds. Also at t=1.
How do you solve for the SPEs in finitely repeated extensive form games?
Use backwards induction, starting at the last repetition and moving backwards towards the start of the set of repetitions.
Infinitely repeated games requirements for analysis
After every period: There is a constant probability that the game ends.Players discount future payoffs Payoffs after each period discounted by multiples of a factor d ϵ (0,1)
Trigger Strategy
A strategy by which in the first period, players play cooperatively, and then continue to play cooperatively unless the other player(s) defect. If the other players defect, play a Nash Equilibrium strategy from the stage game as punishment.
What do trigger strategies form?
An SPE where players discount the future by a factor d close enough to 1. This causes the threat of future punishment to matter enough to players.
Factors affecting the likelihood of collusion
More likely when:There is a smaller number of firms in the marketDeviations are easier to detectProducts are homogeneous / commodity marketsThe firms interact frequentlyManagers are able to communicate with each other