Game Theory II

Question 1

Story of the Bach or Stravinsky (/ battle of the sex) game

Answer 1

• Two people want to get together for an evening of entertainment, but they have no means of communication
• They can either go to the ballet or the fight

Question 2

Possible actions of the Bach or Stravinsky game

Answer 2

• Player 1 prefers going to a boxing fight
• Player 2 prefers going to the ballet
• But they prefer being together over being alone

Question 3

Bach or Stravinsky game pay offs

Answer 3

• Each get 0 utility if they are not together and 2 if they do what they most prefer
• 1 utility if they do something together but the action is not his or her most preferred action

Question 4

What is a sequential game?

Answer 4

• Players take turns in making choices
• Previous choices are known to players - still complete information
• Game represented as a tree (also called extensive form)

Question 5

What does a subgame perfect equilibrium do to a Nash equilibrium?

Answer 5

Subgame perfect equilibrium refines the concept of Nash equilibrium to account for:
- Fact that not all Nash equilibria are sensible in extensive form games
- Players maximize their rewards based off what they can still obtain, not what they could have obtained

Question 6

How can we find the equilibrium in the escalation game?

Answer 6

Backward induction
- We start at the end and move backwards towards the beginning

Question 7

What are subgame perfect equilibria?

Answer 7

• A subgame perfect equilibrium is a complete and contingent plan of action
• It is important to note that all subgame perfect equilibria are Nash equilibria
• A profile of strategies in an extensive form game is a Subgame Perfect Nash Equilibrium (SPNE) if it induces a Nash equilibrium in every subgame

Question 8

When can a mixed strategy exist in a strategic game?

Answer 8

A mixed strategy exists in a strategic game, when the player does not choose one definite action, but rather, chooses according to a probability distribution over his actions

Question 9

Story of matching pennies game

Answer 9

• You and a friend simultaneously reveal a penny
• If both pennies show heads or both show tails, your friend must pay you $1
• If one penny shows heads and the other shows tails, you must pay your friend $1

Question 10

What is the issue with the matching pennies game?

Answer 10

• The matching pennies game is a zero-sum game
• No clear PSNE: there is always an incentive to change

Question 11

How did Nash contribute to the solution of the matching pennies game (games with no clear PSNE)

Answer 11

If the game is finite (finite number of players and finite number of strategies) and there are no Pure Strategy Nash equilibrium, then there must be a Mixed Strategy Nash Equilibrium
- It is possible to find a randomized mutual strategy that satisfies Nash Equilibrium requirements

Question 12

What is a mixed strategy?

Answer 12

• A mixed strategy is a probability distribution over two or more pure strategies
• In MSNE, each player’s probability distribution makes all other player’s indifferent between their pure strategies (only true for 2x2 games)
  – That is the utility of playing strategy 1 must equal the utility of playing strategy 2

Question 13

Critiques of mixed strategies?

Answer 13

Randomization, central in mixed choices, lacks behavioral support
- Seldom do people make their choices following a lottery

Question 14

How does one calculate Mixed Strategy Nash Equilibrium (MSNE)?

Answer 14

• Find the probability of each outcome to occur in equilibrium
• For each outcome, multiply the probability by a particular player’s payoff