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Sony vs Microsoft
• Assignment 1: Given the information in Exhibits 1, 2, 3
– Draw AVC and ATC
– Calculate Sony’s profit over variable cost when
Ps=Pm=399
– Assuming demand curves are linear, calculate ownprice elasticities of demand at price 399
– Calculate Sony’s and Microsoft’s profits for each
combination of prices.
• Would you predict that Sony and/or Microsoft will
want to reduce console price by $100?
Who are the players?
Costumers
Competitors
(substitutes)
Company
Competitors
(complements)
Game Theory
• Strategic interdependence
– Each player’s best choice depends on what
he expects other players to do
– Outcome depends con choices by all players
• Rational players
– Maximise their payoffs
• players have to reason strategically
Elements of a game
• Set of players I
• Set of strategies for each player Si
• Outcomes (s1, s2,…sI)
• Payoffs ui (s1, s2,…sI)
• Rules (timing and information)
Suppliers
1
At the bar
https://www.youtube.com/watch?v=2d_dtTZQyUM EN
• The situation …. It is a game
• Players: boys
• Strategies: "go for the blonde" or "go for a
brunette“
• Rules: Each boy has to decide what to do
without knowing what the others will do.
Types of games and solution concepts
timing\ info
Complete
Incomplete
Static
Nash Equilibrium
Bayesian NE
Dynamic
Prisoners’ dilemma
Sony vs Microsoft
S\ M
$ 399
(in normal form)
$ 299
$ 399
1032.5
960
$ 299
1186.8
637
767.3
978.8
Subgame Perfect Perfect Bayesian
1\2
1012.5
NC
920
C
NC
2
C
2
0 3
3 0
1 1
2
definitions
Nash equilibrium
• Dominant strategy: maximises player’s payoff
regardless of strategies chosen by the others.
– In the PD C is the dominant strategy
• Dominated strategy: there is always another
choice taht gives higher payoffs it is never a
good choice
– In the PD NC is dominated by C
Nash equilibrium: an example
R\C
L
C
• A strategy profile is a Nash equilibrium if
each player’s strategy is a best response
to the strategies chosen by other players
•  in equilibrium palyers no player can
change his trategy and do better (noregret)
Nash equilibrium: an example
• (B, C) is a NE
• To find a NE, write the BR functions
R
– For Row player
T
4
0
2 2
2
3
M
1
2
1 0
4
0
B
2
1
3 3
2
2
• BRR(L)=T
• BRR(C)=B
• BRR(R)=M
– For Column player
• BRC(T)=R
• BRC(M)=L
• BRC(B)=C
3
Nash equilibrium
Nash equilibrium: an example
R\C
L
C
R
T
4
0
2 2
2
3
M
1
2
1 0
4
0
B
2
1
3 3
2
2
Problems
• There may be multiple NE
• An equilibrium may not exist
•
•
•
•
•
Define the best reply function: si=Bi(s-i)
I= 2 ….
A NE strategy profile is s*=B(s*)
∗
∗
i.e. 𝜋𝑖 (𝑠𝑖∗ , 𝑠−𝑖
) ≥ 𝜋𝑖 (𝑠𝑖′ , 𝑠−𝑖
)
To find a NE
– Write BR functions
– Find a fixed point, i.e. Solve simultaneously
Multiple NE: a coordination game
R\C
L
R
T
3 3
0 0
B
0 0
1 1
4
Multiple NE: the «stag hunt»
Stag
Rabbit
Stag
3 3
0 2
Rabbit
2 0
1 1
Matching pennies
The Battle of the Sexes
C
S
C
2 3
0 0
S
1 1
3 2
Mixed strategies
H
T
H
1 -1
-1 1
T
-1 1
1 -1
• Players randomise, i.e. choose
probability distributions over pure
strategies
• Players must be indifferent between pure
strategies: expected payoff from strategies
must be equal
• Matching pennies
• Harsanyi’s interpretation of MS as
uncertainty over opponent behaviour
5
Mixed strategy equilibrium: an example
• Matching pennies
• For player 1: prob (H)= p1; prob(T)=1-p1
• For player 2: prob (H)= p2; prob(T)=1-p2
• Randomization makes sense if players are indifferent
between H and T
• EU(H)=EU(T) ….  p1=
Nash equilibrium: existence
• Nash’s theorem: Any finite game has at
least an equilibrium, possibly in mixed
strategies (Nash, 1950)
– Proof by a fixed-point theorem
p2=1/2
Quantity competition: an example
(CW p. 246)
Cournot Competition
• Assumptions:
– n=2
– strategy: output
– Homogeneous product
– Firms choose simultaneously
– one shot
• In what follows assume,
– AC1=AC2=c
– p=a-bQ
6
Nash-Cournot Equilibrium
Graphically
• Find best response functions
– Firm 1 considers residual demand (total
demand - q2 ) and maximises profits,
which requires MR1=MC1
– Analogously for firm 2
MC
q1*
Nash-Cournot Equilibrium
Firm 1 Best Response function
q1*(q2)
q2
qc
qM
qM
N
qc
qM
qc
• A pair of strategies
which are mutually best
response
• Graphically, it is where
best response functions
intersect
• Analitically, one has to
solve the system of
equations given by the
best response functions
7
Price competition
Exercises
P=a-bQ
Q=q1+q2
MCi=cqi
• In a wide variety of markets firms compete in
prices
–
–
–
–
ch. 8 CW
Internet access
Restaurants
Consultants
Financial services
• With monopoly setting price or quantity first
makes no difference
• In oligopoly it matters a great deal
– nature of price competition is much more aggressive the
quantity competition
33
Price competition
an example: Microsoft and Netscape
• Product: browser (Netscape Navigator and
Internet Explorer) were good substitutes
• Navigator, since 1995; Explorer since
1996.
• Price of N = $49; Price of E = $0
• Microsoft signed agreements with PC
producers to incentivate use of E
• Netscape in 1998 N free and open source
Bertrand Competition
• Assumptions:
– n=2
– strategy: price
– Homogeneous product
– No capacity constraints
– Firms choose simultaneously
– one shot
• In what follows assume,
– AC1=AC2=c
– p=a-bQ
8
Bertrand Competition
Demand
Bertrand Competition
Best Response functions
• No capacity constraints 

0 se p1  p2

D1  p1    D p1  se p1  p2
1
 D p1  se p1  p2
2
Bertrand’s paradox
• Nash equilibrium p1=p2=c
– (no firm wants to deviate)
• “Proof”:
– p1 p2 is not an equilibrium (firm setting lowe
price wants to deviate)
– p1=p2=p’>c is not an equilibrium (both firms want
to deviate)
• Result depends on assumptions:
 p M se p2  p M

p1*  p2    p2   se c  p2  p M
c se p  c
2

Price competition with
differentiated products
• With differentiated products a firm setting
a price higher than rival does not loose all
costumers
• Demand for each firm depends on prices
set by both firms
• Firms have market power ( P>MC)
– No capacity constraints
– Homogeneous product
– One shot game
9
An example of product
differentiation
Exercise
Coke and Pepsi are similar but not identical. As a result, the
lower priced product does not win the entire market.
• q1=130-1,5p1+0,5p2
• q2=130-1,5p2+0,5p1
Econometric estimation gives:
QC = 63.42 - 3.98PC + 2.25PP
• MC1=MC2=10
MCC = $4.96
QP = 49.52 - 5.48PP + 1.40PC
MCP = $3.96
40
Price competition with
capacity constraints
• For the p = c equilibrium to arise, both firms need enough
capacity to fill all demand at p = c
– But when p = c they each get only half the market
– So, at the p = c equilibrium, there is huge excess
capacity
• Firms are unlikely to choose sufficient capacity to serve the
whole market when price equals marginal cost
• since they get only a fraction in equilibrium
– so capacity of each firm is less than needed to serve the
whole market
– but then there is no incentive to cut price to marginal cost
Price competition with
capacity constraints
• With capacity constraints a firm setting a price
higher than rival does not loose all costumers
• Two-stage game: first firm invest in capacity
and the compete over prices
• Kreps and Sheinkman (1983) show that the
solution to this two-stage game is Cournot
equilibrium
10
Price competition in
repeated interactions
• If firms are in the market for many periods
they may be able to reach a tacit
agreement (collusion) to fix prices higher
than NE.
• In non-cooperative games this agreement
must be self-enforcing (i.e. in the
players’interest)
Strategies vs Actions
• Actions are the choices available to a
player when it is his turn to move.
• Strategies are complete plan of actions;
tell what to do in every possible
circumstance (if ...then ... else)
Dynamic Games
Game tree: sequential games
• The Battle of the Sexes
S
3 2
C
1 1
W
S
M
S
C
W
0 0
C
2 3
Another example: entry deterrence
E
E
NE
I
A
100 150
W
-10 0
0 450
11
Backward Induction
Capacity Expansion
• Entry deterrence
– Price war is not credible
• Procede backward
– Incumbent must choose between Accept entry
(payoff=20) or Price War (payoff=-10)
– Correctly anticipating this decision E enters
• The credible equilibrium is: [E; A/E]
• I invests in capacity at a cost equal 200.
Capacity is only used if production is high.
E
E
NE
I
A
100 -50
W
-10 0
0 250
Subgame perfect equilibrium
Evidence on predatory expansion
• Some anecdotal evidence
• Alcoa
– evidence that consistently expanded capacity in advance of
demand
• Safeway in Edmonton
– evidence that it aggressively expanded store locations in
response to potential entry
• DuPont in titanium oxide
– rapidly expanded capacity in response to to changes in
rivals’ costs
– market share grew from 34% to 46%
• Subgame: starting from any decision
node, it include all subsequent choices
• A strategy profile is a subgame perfect
Nash equilibrium (SPNE) if the strategies
are a Nash equilibrium in every subgame.
• In a finite game the SPNE can be found by
backward induction (example 9.4 in CW)
52
12
Subgame perfect equilibrium
An example (9.4 in CW)
Sequential quantity competition:
Stackelberg
• Firms choose outputs sequentially
– leader sets output first, and visibly
– follower then sets output
• The firm moving first has a leadership
advantage
– can anticipate the follower’s actions
– can therefore manipulate the follower
• For this to work the leader must be able to
commit to its choice of output
• Strategic commitment has value
56
Sequential price competition
• Firms choose prices sequentially
• Price competition gives a second mover
advantage.
Repeated games
• In many economic situations strategic interaction
is repeaded over time.
• A repeated game is a dynamic game in which
the same basic game is repeated many times.
The basic game can be
– In normal form (simultaneous move) :
G(Si,ui)iI e.g PD
– In extensive form (sequential moves): game
tree e.g. entry-deterrence
• Players remeber history of play
13
Repeated games
• A SPNE is a set of strategies optimal in
every subgame.
• In a repeated game a subgame starts
each period.
• SPNE adds sequential rationality to NE,
thus refining NE to rule out non-credible
threats.
The collusion dilemma in Cournot model
The collusion dilemma
• Recognising that strategic interdependence may
lead to inefficient outcome, players may try to
reach an agreement to maximise joint profits
(COLLUSION).
• Collusion must be self-enforcing …
• To mantain collusive agreements collusive
strategies must contemplate Detection of
deviation and Punishment of deviators
The collusion dilemma in Bertrand model
1\2
Collude
Deviate
1\2
Collude
Deviate
Collude
1800 1800
1350 2025
Collude
1800 1800
0 3600
Deviate
2025
1600 1600
Deviate
1350
3600
0
0 0
14
The collusion dilemma in the PD
Finitely Repeated PD
• The unique SPNE is (D, D …, D)
1\2
Collude
Deviate
Collude
2
2
0 3
Deviate
3 0
1 1
– ….
• If history does not influence the strategies,
then repeating the game does not change
the equilibrium.
Infinitely Repeated PD
Infinitely repeated PD
• Trigger strategy:
– Play the collusive strategy in each period as long
as all the others have done so in the past
– If anyone has ever deviated from collusion, play
the punishment strategy for ever.
• If the opponent is playing the trigger strategy (i.e. is
choosing Coll)
– By choosing Coll., the stream of discounted payoffs is
2
2  2  2 2  ... 
1 
– By choosing Dev., the stream of discounted payoffs is
– In the example, if δ>1/2 then efficient
outcome can be sustained by the trigger
strategy (play Coll. as long as the other has
done so, play Dev. otherwise)
3     2  ...  3 
– Thus if

1 
2

1
 3
cioè  
1 
1 
2
– The trigger strategy is a SPNE
15
Folk Theorem
Folk Theorem
• Let a* be a static equilibrium of the stage
game with payoffs ui .
• For any feasible payoff v, with vi > ui for all
i ∈ I, there exists some δ < 1, such that for all
δ > δ, there exists a subgame perfect
equilibrium of G ∞(δ) with payoffs v
•
•
•
•
•
The Great Salt Duopoly
The evolution of cooperation
(CW case study 10.4)
http://www.cultureofdoubt.net/download/docs_cod/evolution%20of%20cooperation,%20axelrod.pdf
Two firms: BS (55%) and WP (45%)
Excess capacity: 25% (BS)and 35% (WP)
Theory prediction: ….
Observed behaviour: parallel pricing
Rees: Firm behaviour is consistent with
collusion supported by carrot-and-stick
strategies
• Under what conditions will cooperation emerge in a
world of egoists without central authority?
• To find a good strategy to play in the PD Axelrod
invited experts in game theory to submit programs
for a computer PD tournament.
• The winner was tit for tat (reciprocity, provocability,
forgiveness, simplicity)
• Small number interacting for extended periods
(indefinite)
• Altruism nor trust are needed
16
Experiments
• In lab experiments, there is more
cooperation in prisoners’ dilemma games
than predicted by theory.
• More interestingly, cooperation increases
as the game is repeated, even if there is
only finite rounds of repetition.
• Why?
– Altruism
– Fairness
– revenge
17