VL02: Game Theory

Question 1

Name all the assumption needed for a Nash equilibrium

Answer 1

• Static game (all players move simultaneously)
• One-stage game
• Complete information (each player knows his own and the payoff function of the other players)
• Perfect information (all players know the actions so far)
• Rational players (player will play the most profitable strategy given the strategy of the other players)
• Non-cooperative game (each player chooses the most profitable strategy for himself)
• Pure strategies (each strategy has a probability of 1 to be played)

Question 2

What is the definition of a Nash equilibrium?

Answer 2

In a Nash equilibrium no player has an incentive to deviate unilaterally from his chosen strategy.

Question 3

Is it possible to have several Nash equilibriums?

Answer 3

Yes

Question 4

What non-cooperative market games do you know?

Answer 4

Static games

• Cournot game (game in quantities)
• Bertrand game (game in prices)

Dynamic (multi-stage) game
- Stackelberg (leader - follower) game

No game: perfect competition (no reaction to other players)
- Cost-minimization problem (optimization)

Question 5

What market does a Cournot game describe?

Answer 5

• Duopoly (two players)

- Oligopoly

Question 6

Describe the Cournot Game

Answer 6

Suppliers decide simultaneously on the quantities to offer, given their competitors offers and the known payoff functions of all the players.

Question 7

Describe the Cournot-Nash equilibrium

Answer 7

No firm has an incentive to deviate because each obtains its maximum profit given the strategy of the other firm.

Question 8

Name the assumptions of the Stackelberg game

Answer 8

• Complete Information (of all payoff functions)
• Perfect information (of all previous actions)
• Rationality of players

Question 9

Describe the Stackelberg Game

Answer 9

Leader follower market game: sequential moves in two stages

1) Leader decides on his optimal quantity knowing how the followers will react to it.
2) After observing the quantity supplied by the leader, the followers decide on their optimal quantities to offer on the market

Question 10

True or false?

Stackelberg equilibrium quantity < Cournot equilibrium quantity
(aggregated quantities)

Compare the quantities of the leader and the follower to the Cournot equilibrium quantities.

Answer 10

False!

Stackelberg equilibrium quantity > Cournot equilibrium quantity

Stackelberg leader benefits from knowing that the followers will take his decision into their optimization and can set higher quantities than as a Cournot player. The follower set a lower quantity than he would in the Cournot Game.

Question 11

How many Nash equlilibriums are possible?

Nash equilibrium:
Cournot game:
Stackelberg game:

Answer 11

Nash equilibrium: More than one possible

Cournot game: Unique Nash equilibrium

Stackelberg game: More than one possible

Question 12

True or false?

In a Cournot game players take their effect on the demand level into account when choosing their quantities.

Answer 12

True!

Question 13

What conditions do you need for calculations?

Perfect competition

Answer 13

q1 = q2 (if same cost functions)

P = MC (Price = Marginal Costs)

Question 14

What conditions do you need for calculations?

Duopoly - Cournot competition

Answer 14

• max q1 Pi1(q1,q2) and max q2 Pi2(q1,q2)
  (Maximize the profit of company i depending on the quantity of the other company, that will give you the reactionfunctions)
• AFTER the maximization insert von reactionfunction into the other one

Question 15

What conditions do you need for calculations?

Duopoly - Stackelberg-Game

Answer 15

• Backwards induction

1) max q2 Pi(q1,q2)
(Follower, result will be the reactionfunction R2(q1))

2) max q1 Pi(q1,R2(q1))
(Leader, result will be the solution for q1)

Question 16

What conditions do you need for calculations?

Duopoly - Joint profit maximization

Answer 16

1) max Q Pi(q1) + Pi(q2)
(result is the optimal Q)

2) q1 = q2 = Q/2 (identical cost functions)

Question 17

What conditions do you need for calculations?

Monopolist

Answer 17

• max q Pi(q) = max q P(q)*q-c(q)

- cost function stays the same for the syndicate, just change quantity variable from q to Q

Question 18

What conditions do you need for calculations?

Oligopol - Cournot game

Answer 18

• max q1 Pi1(q1, q2, … , qn)
  (result: reaction function R1(q2, … , qn))
• once reaction function is found: q1 = q2 = … = qn
  (if cost functions are identical)