Game theory and IO Flashcards
What is a payoff?
The possible expected utilities that each player could receive, which each depends on the choices of all the players
What are the timing options in a game? [3]
- Simultaneous / sequential choices
- One shot / repeated games
- Finite / infinite
Define the Nash equilibrium
The set of strategies, such that each strategy is the best response to the other player’s action.
- At the Nash equilibrium no player wants to change their strategy given what the other players are doing.*
- Games can have multiple Nash equilibria. Any Nash equlibrium might not be Pareto efficient.*
What is a pure strategy?
A plan of action, a complete specification of what the player chooses to do whenever that player has to make a choice.
What is a mixed strategy?
One that allows randomisation: you can select probabilities on which pure strategy will be followed.
If we add probabilities to the game we create a mixed strategy equilibrium.
What is a dominant strategy?
A best response to every possible strategy profile for the other players.
- Whatever the other player does, the best response is always the dominant strategy - e**.g. in the Prisonner’s Dilemma, betrayal is the dominant strategy.*
- Most games don’t have a dominant strategy - e.g. in the Battle of the Sexes, there is no dominant strategy but two Nash equilibria - it’s a coordination game.*
What is the extensive-form game?
The game tree - illustrates actions available and the sequence in which they occur
What do we mean by perfect information?
Everyone knows the same amount - all agents have all the relevant information with which to make a decision
(Imperfect = some people know more than others)
What do we mean by complete information?
At every stage in the game, every player knows what has been played by the other players, and the payoffs/strategies available to them.
What is the normal-form game?
Matrix representation - players move simultaneously, or at least don’t observe the other player’s move before making their own.
What is Cournot competition?
Firms set quantities simultaneously, and the market determines prices so that the market clears
What is the Cournot reaction function?
πi(qi,qj) = p(qi+qj)*qi - c*qi
qj = quantity of other firm, taken as given
Firm i assumes qj and chooses qi to maximise profits
Solve the two reaction functions simultaneously to find the Nash equilibrium (gives each firm’s quantity choice as the best response to the other firm)
What is Stackelberg competition?
Same as Cournot but with sequential moves - e.g. firm j sets their quantity first
What is Bertrand competition?
Firms simultaneously set prices and the market (customers) chooses quantities at the prices set.
What is the consequence of Bertrand competition?
Perfect competition: customers will buy from the lowest-priced firm, so the best response for each firm is to set output where P=MC
The market cannot have a higher equilibrium price, because there is always the incentive to undercut and take the whole market (unless firms collude)
What is a subgame perfect equilibrium and how is it identified?
A strategy profile is a subgame perfect equilibrium if it represents a Nash equilibrium of every subgame of the original game.
Identified through backwards induction: if the players played any smaller game within the larger game and their behavior could achieve a Nash equilibrium of that smaller game, then that behavior is a subgame perfect equilibrium of the larger game.
What does subgame perfect equilibrium rule out?
Incredible threats, which could have existed in a Nash equilibrium.
What is the equation for collusion in a Cournot duopoly?
qm = qi + qj
maxqiqj {πi(qi, qj) + πj(qi, qj)} = max πm = max {πim + πjm}
Why is collusion not possible in a finite game?
Because the eventual outcome in the final game will be betrayal - this unravels back through all of the earlier subgames.
Finite game = one shot game
What are the factors that determine whether collusion successfully occurs? [4]
- Number of firms
- Firm size
- History of the market
- Punishment mechanisms
What is a cartel?
A formal (explicit) agreement among competing firms. Found to decrease the welfare of consumers.
What is a (grim) trigger strategy?
Play X as long as your opponent doesn’t play Y, in which case play Z.
Grim trigger: once Z is initiated, continue playing Z for ever.
What happens with a collusion in an infinite game?
It can be sustained. Each firm maximizes the discounted sum of all its future profits.
δ = 1/1+r = the possibility that, with some probability, the game ends after the current period = patience of players
When would someone betray collusion in an infinite game?
When the discounted future profits from colluding are less than the profits from the betrayal:
δ >/= (πd - πm/2) / (πd - πc)
δ > one-period deviation payoff / payoff after that
The more patient you are (larger δ), or the lower the interest rate, the more you resist the temptation to deviate.
What is the committment problem?
The SPNE where the firms collude is not renegotiation proof: after defection, the firms are supposed to play Cournot-Nash forever, but both would be better off if they renegotiated and started to collude again.
What is a minmax punishment?
The worst that one player can do to the other, given the other is responding optimally
What is the Folk Theorem?
Any feasible payoff pair, which gives each player at least her minmax punishment, can be supported as a Nash equilibrium of an infinitely repeated game if the discount factor (δ) is sufficiently close to one.
