Term 2 Week 8: Repeated Games and Coordination Failure Flashcards
What is a finitely repeated games, and some examples (1,4)
-In finitely repeated games, players will play the same ‘stage game’ for a set number of periods
Examples of repeated prisoners dilemma’s include:
-Competition between two firms in a market (compete vs collude)
-Overfishing (tragedy of the commons)
-Climate change
-International tax competition
What is the grim trigger strategy (3)
-The Grim trigger strategy aims to threaten to the opponent defecting by defecting, hence limiting their maximum achievable payoff
-The strategy is that ‘I will cooperate but if you ever defect i’ll do the same forever’
-With this strategy, a player might offer an incentive to cooperate today by threatening punishment tomorrow, the punishment being the removal of the opportunity to cooperate in the future
How does cooperation in the grim trigger unravel in a finite game (2,3,1)
-Suppose we play a game for 12 rounds, where (C,C) = 1,1, (C,D) = -1, 2, (D, C) = 2, -1 and (D,D) = 0,0
-If both players cooperate they get 12 overall, but by defecting they get 2 initially than 0 after
-Using backward induction to test if cooperation is a SPE, there is no incentive to cooperate in the final period, as no punishment can occur
-But why then would they cooperate in period 11 if they know they get punishment in period 12 anyway
-Thus, cooperation completely unravels
-The backwards induction SPE is only always defecting
How does data view cooperation in a finitely repeating prisoners dilemma (2)
-Data backs up the unravelling argument, that defection is more common towards the end of the game
-However, the SPE prediction of completely unravelling does not materialise
How can we discount future payoffs in an infinitely repeating game (1,2)
-Say that players discount future payoffs with discount factor δ < 1
-δ could be a measure of patience
-δ could be the probability of 2 players meeting again next period
How can we model the grim trigger strategy in an infinitely repeating game (2,2,2,2)
-Play C at t = 0, play C at t>0 if neither player has played D at any time t’ < t
-If either player has played D at any t’ < t, then play D
-πj = 1 + δ + δ2 + …
-πj = Σδt = 1/1-δ
-The payoff from cheating is 2 + 0 + … = 2
-So the best response is to also cooperate if 1/1-δ ≥ 2, or δ ≥ 1/2
-The grim trigger strategy can hence sustain cooperation as a NE for a sufficiently high δ
-It is also possible to show the grim trigger is an SPE (outcome can become self-enforceable)
What are the issues with the grim trigger strategy (3)
-Strict punishment (no way back to cooperation)
-Renegotiation proof (it is not in your interest to get stuck in the punishment phase)
-This is not the only cooperative strategy
What are some examples of different strategies in infinitely repeated games (4)
-Grim
-Tit-for-tat (copy what was played last period)
-Joss (TFT and randomly defect 10% of the time)
-Tester (start with D, then play C twice, if D is seen go TFT, if not then play D onwards
Why did Tit for Tat win Axelrod’s tournaments (3)
-Niceness (never first to defect)
-Retaliation (respond to defection quickly)
-Forgiveness (allowed cooperation again after a defection)
When is the stag hunt game applied (2,1)
-The stag hunt game is often applied to situations where there is an outcome which is pareto superior to all others, but cannot be achieved unilaterally
-Uncertainty is created due to a trade-off between mutual benefit and individual risk
-This is different from the prisoner’s dilemma, where cooperation doesn’t give the highest individual payoff
What is coordination failure (2)
-Coordination failure is a situation where multiple parties fail to coordinate on a mutually beneficial outcome
-They can end up with an outcome that no one wants, and there exists another pareto superior outcome
What does coordination failure motivate (2)
-Coordination failure motivates collective action and intervention
-Game theory assumes people act independently, but here they benefit from acting jointly
How can we represent the bad vs good equilibrium in a game between protesting and staying at home (1,2,2)
-Draw payoff matrix, where (P,P) = (25, 25), (H,P) = (0,-50), (P,H) = (-50, 0), (H,H) = 0,0
-Now draw our payoff diagram, with payoff on the y axis, fraction of the population who process on the x axis
-Then, stay at Home = payoff 0, protest is a straight line from (0, -50) to (1, 25)
-We can then see there is a bad NE (H far left) and a good NE (P far right)
-We are more likely to end up in the bad RE, due to staying at home being risk dominant (higher payoff when x = 1/2)
How can we diagramatically represent the pareto superior area (2)
-Have πi on the x axis, and πj on the y axis
-Plot a point, anywhere in the area above and to the right of it is pareto superior (nothing to the left or below at all)
How can we assess the likelihood of coordination failure and how easy it is to escape in different types of games (2,2,2)
Stag Hunt:
-Coordination failure is likely to occur, depending on the payoffs
-It requires joint action to escape from the suboptimal outcome, but the new outcome is stable once reached
Pure coordination:
-Coordination failure is possible in the short come
-Escaping the suboptimal outcome requires unilateral action, as one player best responds, but the new outcome is stable
Prisoner’s dilemma:
-Coordination failure is likely to occur
-Escaping the suboptimal outcome requires joint action and some way to enforce the new outcome
What are institutions and different types (1,3)
-Institutions are a broad term describing a collectively accepted system of formal and informal rules, which regulate how people interact
These include:
-Formal laws
-Conventions (expectations on how people act)
-And social norms/sanctions (expectations on how people should act)
How can acting in accordance with institutions resolve coordination failures (2)
-Common knowledge about what to expect of others can prevent miscoordination
-The possibility of sanctions provides an incentive to stick to explicit formal agreements and societal norms
What happens when you repeat the prisoners dilemma (1)
-It turns into a stag hunt
What are the best responses in a repeated prisoners dilemma (2)
-If the other player is cooperating it is a best response to cooperate for a high δ (TFT)
-However, if the other player is playing always defect then always defect becomes the BR
How may different societies arise in repeated games (2)
-Different societies may arise where different equilibria become the norm
-Expectations can become entrenched, preventing movement to better norms
How can we implement a social sanction into a game + example (3,2)
-Social sanctions such as ostracism or other punishment can prevent cheating in PD. But these require repeated interactions, and a high delta
-Beyond that, people may have social preferences
-One simple approach is to introduce an extra cost of ‘K’ to playing defect if the other cooperates (k is the psychological cost of guilt associated with playing the cheating strategy)
-Create a 2x2 game, where (C,C) = (3,3), (C, D) = (0, 5-K), (D,C) = (5-K, 0), (D,D) = 1,1
-For k > 2, we make C,C a nash equilibrium, even in a one period interaction
How do institutions regulate social interaction (2)
-Solidifying and coordinating expectations
-Modifying the payoffs of the game to achieve some other outcome
What is the folk theorem (2)
-The folk theorem is an important result in game theory, showing that one-shot games with unique equilibria have a multitude of other equilibria when repeated
-The cooperative outcome is far from guaranteed, even if players sufficiently value their future payoffs
How can we draw a convex combination when one player randomises and one player cooperates (2,5)
-Outcome (C,C) has payoff (1,1)
-Outcome (C,D) has payoff (-1, 2)
-Suppose player I always picks c, but player j picks c with probability a
-Draw a payoff diagram, with πi on the x axis, and πj on the y axis
-Draw 2 points, C,C at (1,1) and C,D at (-1, 2)
-Draw a line between the two, this being the convex combination with a(1,1) + (1-a)(-1, 2)
-The higher a is, the closer you are to point C,C
How can we draw a convex hull (2,4)
-Let C,C = (1,1), C,D = (-1,2), D,D = (0,0)
-If both players randomise they achieve a convex combination of case 1 and 2
-Draw a payoff diagram, with πi on the x axis, and πj on the y axis
-Draw 4 points, C,C at (1,1), C,D at (-1, 2), D,C at (2, -1) and D,D = (0,0)
-Connect these 4 points, and anything within it is the ‘convex hull’
-These are all the possible average payoffs which can be achieved through some strategy in a repeated prisoner’s dilemma (2,4)
