L7 - Game Theory: Repeated Games

Question 1

What is a Nash Equilibrium?

Answer 1

A Nash equilibrium is a combination of strategies where no player has any incentive to change his strategy, given the strategies of the other players

Question 2

Why do we get Pareto inefficient outcomes in the Prisoner Dilemma?

Answer 2

• The outcome is Pareto inefficient and rather disappointing. Some might argue that it is also unrealistic as one might expect the players to cooperate. Is the theory missing anything?
• One important element that we have ignored so far is the possibility that the players interact more than once.
• In many situations, interactions are not one-shot and instead, they are repeated and ongoing.
• Examples: labour relations, firms, governments, central banks, political parties, terrorist groups, neighbours, friends, relationships, sports teams…

Question 3

What is a repeated game?

Answer 3

• A repeated game is a one-shot game or stage game that is played successively T>1 times.
• Keeping assumptions of complete information, rationality and common knwoledge
• Further assumption - players can recall the full history of the game – past actions are common knowledge.
• Remember that a strategy is a complete contingent plan, describing which action a player will choose in all potential situations.
• Here, in this context, strategies can be very complex. For example, a strategy could have to describe actions for all possible game histories

Question 4

Considering the example below, how can repeated interaction allow cooperation to be sustained as part of equilibrium where T = 2?

Answer 4

• More precisely, as this game is no longer a normal form game, and is now a dynamic game, we need to consider SPNE. Can cooperation exist in a SPNE?
• Remember:
  
  • A SPNE demands that each subgame is played in NE.
  • (Roughly) A subgame is defined as a section of a game, starting at a node (where there is no uncertainty) and ending at a terminal node.

There is still no co-operation!

Question 5

How do we solve games that allow for Finite Repetitions, T

Answer 5

• In ever stage players will still defect and their will be no cooperation

Question 6

How can the Grim Trigger strategy be used to solve games with infinite repetitions?

Answer 6

• Higher discount factor means that you value a pound tomorrow in a similar way that I value a pound today, That means that if I’m sufficiently patient I’m okay and will cooperate
  
  • Going to put more weight on getting 3 forever than getting a short term gain of 5 and only receiving 2 from then on

Question 7

Note on Discounting math formulas?

Answer 7

Question 8

How do we solve Indefinite repetition games

Answer 8

Question 9

What are the steps for solving a Cournot duopoly in an infinitely repeating continuous game?

Answer 9

• Equilibrium in this sense is whether both firms would reduce their quantity below what they would normally do in a way that actually enhances their profits
• maximise joint payoffs

Question 10

The solution to the Cournot Duopoly continuous game with infinite repetitions?

Answer 10

1. Step 1: Have the same BR response as the Cournot one-shot game, and just need to calculate the profits using that BR quantity
2. Step 2: What would the BR quantity be for a monopoly? if quantity is 0.5 then if both firms were acting as if they were the monopoly firm combined they would both produce 1/4
3. Step 3: So what are profits when you deviate?
   
   1. Firm J will produce the half monopoly quantity, well we then need to find at what level of production maximises my profits given their quantity

Companies that therefore compete regularly could restrict output and increase their profits if they are patient enough (energy companies or OPEC boosted their global output to help economies as they recover)