Game Theory Flashcards
What is Game Theory?
Study of how People Behave in Strategic Situations
What is a Strategic Situation?
Where each Decision-maker must Consider how others will respond to its actions
What are the 5 components of a Game?
Players Strategies Payoffs Timing An Equilibrium concept
What are the Players?
Parties who make Decisions
What are Strategies?
Range of decisions the Players can Make
What are Payoffs?
Each Player is Motivated by what they will Gain from the Outcome
What is Timing?
Who can do what, when?
What is an Equilibrium Concept?
What kind of Outcome will be looked for
What is a Nash Equilibrium?
Where no Player can change their Equilibrium Strategy + Receive a Higher Payoff
- No force acting for change- No Player wants to Deviate from their Equilibrium
- No Player regrets Strategy Decision
Under a Nash equilibrium, where can there ALWAYS be Equilibrium?
Dominant Strategies Equilibrium
What are the 3 main reasons we Observe Cooperation?
- Different Games
- Repeat Play of Games
- Lack of Information
Why might Different Games lead to Cooperation?
Game may not be a Prisoner’s Dilemma style
How can Repeat Play of Games lead to Cooperation?
More games allows Players to learn about each other + develop Trust
-Allows to threaten punishment for cheating
How can Lack of Information lead to Cooperation?
Players may not know Numbers in Payoff Matrix
–> More Inclined to Cooperate
What does Cournot Quantity Competition model?
Models Strategic Interactions between few firms using Rev./Cost/Demand Curves
What are the Total Costs of Firms?
TCi = cqi i = different firms
What are the Marginal Costs of each firm?
MC1 = MC2 = c
What are the 3 assumptions of firms?
Each produce Homogenous Goods
Produce quantities qi
Firms are Symmetrical
if there are 2 firms, what is the Profit Function of firm 1?
Prof. 1 = P(Q)q1 - cq1
Derive the FOC of the Profit Function. given that Market Demand = a - bQ
Q = q1 + q2 ==> P = a - b(q1 + q2)
Prof. 1 = P(Q)q1 - cq1
==> [a - b(q1 + q2)]q1 - cq1
FOC (dProf/dq1): a - b(q1 + q2) - b(1 + dq2/dq1)q1 - c = 0
What doe dq2/dq1 show?
How Firm 1 thinks Firm 2’s choice of q2 will respond to Firm 1’s choice of q1
What is the Cournot Conjecture + what is it assumed to be + what does it mean?
dq2/dq1 = 0
Firm 1 assumes Firm 2 will NOT adjust q2 in response to Firm 1’s choice of q1
–> Each firm chooses Output assuming others will NOT change their Output
Derive Firm 1’s Reaction Function from the FOC, given Cournot’s Conjecture
FOC: a - b(q1 + q2) - bq1 - c = 0
q1 = (a - bq2 - c)/2b ==> (a-c)/2b - q2/2
What does the Reaction Function show?
Shows Best Value of q1 in response to a given q2
If a Increases, what happens to q1?
Increased a = Increased Demand
q1 Increases
If c Increases, what happens to q1?
Increased c = Increased MC
q1 Decreases
If q2 Increases, what happens to q1?
Q = q1 + q2
q1 Decreases to keep Price high
If b Increases, what happens to q1?
Increased b = Demand becomes Steeper- Increased P from Lower q1
q1 Decreases
What would Firm 2’s Reaction Function be and why?
q2 = (a - bq1 - c)/2b
Due to Symmetry
Where would the Nash equilibrium be for the 2 firms?
Where the 2 firms’ Reaction Functions Intersect
How do you solve for the Cournot Nash equilibrium?
Solve Simultaneous Equations of Reaction Functions
solve by Symmetry or substitution
q1nc = q2nc
Given the firms’ Reaction Function to be (a - bq - c)/2b, what would be q1nc and q2nc- the Cournot Nash equilibrium?
q1nc = (a - bq2 - c)/2b = (a - bq1 - c)/2b = q2nc
==> q1nc = (a-c)/3b = q2nc by Symmetry
How would an Increase in Demand (a) affect the Reaction Functions?
Both Reaction Functions would shift to the Right by Increase in a
Increased Demand–> Output more Profitable–> Increased Output
How would a Decrease in Firm 1’s MC (c) affect the Reaction Functions?
Firm 1’s Reaction Function shifts to the Right by Decrease in c
Firm 1 now more Cost Competitive than Firm 2–> Gains larger Market Share
For N Symmetric firms, what is the Profit Function and its FOC?
Profit i = P(Q)qi - cqi
FOC: P + qi(dP/dQ)(dQ/dqi) - c = 0
For N Symmetric firms, what is MR and MC?
MR = P + qi(dP/dQ)(dQ/dqi) MC = cqi
For N Symmetric firms, what is the Cournot Conjecture?
dQ/dqi = 1
By Symmetry, for N Symmetric firms, what is qi?
qi = Q/N
For N Symmetric firms, using the Cournot Conjecture + qi, what is the equation for MR = MC?
P + (Q/N)(dP/dQ) = c
==> P [1 + (1/N) (Q/P) (dP/dQ)] = c
=> P [1 + (1/Ne)] = c
e = PeD
Given P [1 + (1/Ne)] = c, what happens to the Outcome as N tends to 1?
Monopoly Outcome- P > c
Given P [1 + (1/Ne)] = c, what happens to the Outcome as N tends to Infinity?
Perfectly Competitive Outcome- P = c
Given P [1 + (1/Ne)] = c, what happens to the Outcome as e tends to Infinity?
Perfectly Competitive Outcome- P = c
Given P [1 + (1/Ne)] = c, what happens to the Outcome as N Increases or e Decreases (towards Negative Infinity)?
P tends to c
What is Stackelberg Quantity Leadership?
One Firm sets Output before others + can stick to it
- Firm can Increase Profits as it can Influence Follower’s Output Decision
- -> Leader can always choose Cournot NE if it wanted to
What is Bertrand Price competition?
Firms Simultaneously choose PRICE not Output
- Firms can now Lose ALL Market Share- even if only 2 firms
Explain Bertrand Price competition
Assume Duopoly: Homogenous Output, MC1 = MC2 = c, No Fixed Costs
If Firm 1 chooses P1 > c => Firm 2 can choose P2 where c < P2 < P1 - Firm 2 gains All Market Share + Profit
In Bertrand Price competition, why can’t P2, where c < P2 < P1, be a Nash equilibrium?
Firm 1 would want to Deviate + set Price where c < P1 < P2
Where is the only Nash equilibrium under Bertrand Price competition?
c = P1BN = P2BN
Outcome is Efficient- same as Perfectly Competitive outcome
