More Advanced Game Theory

1
More Advanced Game Theory Applications

• Ian Larkin Evan Rawley
• MBA 299 Strategy
• April 21st, 2006

2
Agenda for today

• Hand back case write-ups
• More game theory

3
Grading philosophy and approach
Philosophy
Approach

• 1st pass to establish independent grade
• 2nd pass to ensure rank order is right
• 3rd read for anyone on the margin

Grades matter I take them seriously My strong
presumption is that you will write very
intelligent papers Grading is more lenient on
mid-term work than on the final Final typically
makes up a lot of the variability in grades
4
Grades

• 20 Outstanding lt10
• 19 Very strong analysis no major
  flaws 10-20
• 18 Very good analysis w/clear thesis some
  problems 10-20
• 17 Good analysis overall at least one major
  issue 25-35
• 16 Some good analysis, but at least 2 major
  problems 15-25
• lt15 A few good points, but problems tend to
  dominate lt10
• Very good performance overall
• The only time a letter grade will be assigned is
  for your final course grade

5
Thinking ahead to the final

• WE NEED MORE OF
• Establishing a clear thesis and approach, and
  building an organization/ framework which
  supports this thesis
• Work on quant. Analysis
• More of it going deeper
• Better justifications for assumptions
• Better support even if non-quantitative in nature
• Deeper thinking about strategic dynamics
• Consistent logic
• Clarity around predictions

• WE NEED LESS OF
• Mechanical/exhaustive application of standard
  frameworks which dont add value to case in
  question
• Over-summarization of case facts
• Bullet pointed lists
• Approaching it as building a business plan
  rather than analyzing a case
• Exhibits without quantitative analysis (save
  these for the boardroom)

6
Agenda for today

• Hand back case write-ups
• More game theory

7
More game theory

• Game theory warm up
• Backwards induction
• Oligopoly games
• Repeated games
• Signaling games

8
The Chicken Game wheres the Nash equilibrium?
B
Stay the course
Swerve
-999,-999
50, -50
Stay the course Swerve
A
-20, -20
-50, 50
9
The Chicken Game another example of a
coordination game
B
Stay the course
Swerve
-999,-999
50, -50
Stay the course Swerve
A
-20, -20
-50, 50
10
Questions

• What type of player do you think is going to win
  this game? What does game theory say about whos
  going to win this game? Does this result say
  anything about real-life behavior (business or
  otherwise)?
• Assuming youre not insane, whats the best move
  when facing games like this?

11
More game theory

• Game theory warm up
• Backwards induction
• Oligopoly games
• Repeated games
• Signaling games

12
Sequential games

• In sequential games, players move in a
  pre-determined order, and can observe moves of
  other players that happened before they move
• This type of game is useful in developing
  predictions in situations where one firm moves
  first and others follow
• Firms with a dominant player (e.g., AB/Bud
  advertising)
• Capacity decisions (e.g., Nutraweet)
• Patent games (e.g., Pharmaceuticals)

13
Capacity Expansion and Entry is one relevant
example

• An established manufacturer is facing possible
  competition from a rival
• The established retailer can try to stave off
  entry by engaging in a costly capacity expansion,
  which increases supply and lowers price charged
  to customers
• Rival can observe whether incumbent expands
  capacity or not before deciding on entry

14
Incumbent decides to expand or not, then rival
decides whether to enter
In
1,1

R
3,2
Expand
Out
I
2,4
In
R
Do not expand
4,2
Out
15
Game is solved using Backward Induction

• Look to the end of the game tree and prune back
  (similar to working backwards through a decision
  tree as in 201A)
• Rationality assumption implies that players
  choose the best strategy at each node
• Theres no incomplete information in this game,
  so theres no uncertainty in the prediction

16
What will rival do?
In
1,1

R
3,2
Expand
Out
I
2,4
In
R
Do not expand
4,2
Out
17
Rivals Choice

In
1,1
R
Expand
3,2
Out
I
2,4
In
R
Do not expand
4,2
Out
18
What will incumbent do?

In
1,1
R
Expand
3,2
Out
I
2,4
In
R
Do not expand
4,2
Out
19
Incumbents Choice
In
1,1

R
Expand
3,2
Out
I
2,4
In
R
Do not expand
4,2
Out
20
Equilibrium Prediction

• The prediction from this model is that the
  incumbent will expand capacity and this will
  effectively forestall entry
• Notice that even in absence of actual entry, the
  potential competition from the rival eats into
  the incumbents profits.
• By thinking dynamically, game theory allows a
  refinement of the typical economics monopoly
  prediction of MRMC

21
Is Flexibility an Advantage?

• Preceding game assumed rival could move at the
  last moment, after seeing incumbents decision
• Suppose that the rival is less flexible in its
  management practices.
• It must commit to enter or not before the
  capacity expansion decision of the incumbent.
• How does this affect the outcome of the game?

22
Game Tree Rival moves first

Expand
1,1
I
4,2
In
Not expand
R
2,3
Expand
I
Out
2,4
Not expand
23
Backwards Induction Rival moves first

Expand
1,1
I
4,2
In
Not expand
R
2,3
Expand
I
Out
2,4
Not expand
24
Equilibrium Prediction

• The absence of flexibility on the part of the
  rival improves its outcome relative to the case
  where it retained flexibility.
• This game has a first-mover advantage
• Sometimes flexibility to commit is more
  important than flexibility to wait and see
• Is it always true in sequential move games that
  there is a first-mover advantage?

25
Defining the Rules Properly is Critical An
Example of What Can Go Wrong
6,6,6,6,6
Join
1B 2U 3T 4S 5E
5
Join
Abstain
4
Join
4,4,4,4,0
Abstain
3
Join
2,2,2,0,0
Abstain
2
Buy
0,0,0,0,0
Abstain
1
-10,0,0,0,0
Dont buy
What do you think happened?
0,0,0,0,0
26
BACKWARDS INDUCTION (I)Monks Cerecloth
1

• What are the BI outcome when a4?
• R (6,8)

L
R
2
l
r

• What is the BI outcome when a 8?
• R (6,8)

3 2
a -6
6 8
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BACKWARDS INDUCTION (II)Shoved Environment
1

• What is the BI outcome?
• L1, r1 (2,2)

L1
R1
2
r1
1
2 2
2
L2
R2
l2
r2
3 1
2
1 3
3 1
l3
r3
4 6
2 7
28
More game theory

• Game theory warm up
• Backwards induction
• Oligopoly games
• Repeated games
• Signaling games

29
COURNOT DUOPOLY WITH IDENTICAL FIRMSSet-up

• Two identical firms (firm 1 and firm 2) producing
  widgets
• Firms choose their quantities (q) simultaneously
• Profit for each firm (i) is given by the standard
  function
• p i Pqi Ci(qi)
• For simplicity assume no fixed costs Ci(qi)
  cqi
• p i Pqi ciqi
• Note that total supply is Q q1 q2
• Assume demand is linear Qa-P
• P(Q) a Q (inverse demand)
• Assume c lt a

P
c
Q
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COURNOT DUOPOLY WITH IDENTICAL FIRMSSolution

• p i(q1,q2)P(Q)qi - cqi
• qiP(qiqj)-c
• qia-(qiqj)-c
• Recall Nash EQM gt max p i given js best play
  (and vice versa)
• How do we find the maximum of a function? Take
  derivatives!
• F.O.C. with respect qi for firm i (assuming
  qjlta-c) is . . .
• a 2qi qj c 0
• qi ½(a qj- c)
• Solving the pair of equations by substitution
• q1 ½ (a q2 c) ½ (a ½(a q1 c) c)
• (3/2)q1 ½ (a c)
• q1(a-c)/3 q2

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COURNOT DUOPOLY WITH IDENTICAL FIRMSIntuition

• Remember that the monopoly outcome is
• qm(a-c)/2
• pm (a-c)2/4
• The optimal outcome for the two firms would be to
  divide the market at the monopoly output level
• For example qiqj ½ qm
• But each firm has a strong incentive to deviate
  at this qm
• Check ½ qm is not firm 2s best response to ½
  qm by firm 1

32
COURNOT EQUILIBRIUM WITH Ngt2Homogeneous
Consumers and Firms
Solution pi (q1,q2 . . . qn) qiP(Q)-c
qia-bQ-c Recall Nash EQM gt max profit for i
given all other players best play So F.O.C. for
qi, assuming qjlta-c qi1/2(a-c)/b
?qj Solving the n equations q1q2 . .
.qn(a-c)/(n1)b
j?i
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COURNOT DUOPOLY (N2)Homogeneous Consumers,
Firms Have Different Costs
Solution pi (q1,q2) qiP(qiqj)-ci
qia-(qiqj)-ci So F.O.C. for qi, assuming
qjlta-c qi1/2(a-qj-ci) Solving the pair of
equations qi2/3a - 2/3ci 1/3cj qj2/3a - 2/3cj
1/3ci
34
STACKELBERG DUOPOLOYGame Set-up

• Firm 1 moves first and chooses q10
• Firm 2 observes q1 and chooses q20
• Payoffs are ?(qi,qj)qiP(Q)-c
• P(Q) a - Q
• Q q1 q2
• Solve by backwards induction

35
STACKELBERG DUOPOLOYEquilibrium Outcomes

• Firm 2
• max ?2(q1,q2) max q2a-q1-q2-c
• q20 q20
• q2R2(q1) a - q1 - c/2 (assuming q1 lt a -c)
• Firm 1
• anticipates Firm 2s move
• max ?1(q1,R2(q1)) max q1a-q1-R2(q1)-c
• q10 max q1a-q1-c/2
• q1 a-c/2 and R2(q1) q2 a-c/4

36
STACKELBERG DUOPOLOYAnalysis (I)

• Recall the Cornout duopoly outcome was a-c/3
  for both firms so total output is higher in the
  Stackelberg game . . . and therefore prices are
  lower
• Cornout total output 2a-c/3
• Stackelberg total output 3a-c/4
• Firm 1 is better off in the Stackelberg game but
  Firm 2 is worse off
• Firm 1 could have chosen the Cornout output level
  but did not therefore Firm 1 must be better off
• Firm 2 produces less at lower prices so must be
  worse off

37
STACKELBERG DUOPOLOYAnalysis (II)

• Observe the role of information
• Firm 1 knows that Firm 2 will optimize its
  output based on what Firm 1 does
• Firm 2 knows that Firm 1 knows that Firm 2 will
  optimize its output based on what Firm 1 does
• Order matters too
• If Firm 1 went after Firm 2 and Firm 2 chose
  a-c/4, Firm 1s optimal output would be
  3a-c/8 . . . so Firm 2s choice would not be an
  equilibrium outcome . . . and we would end up
  back at the Cournot output level

38
STACKELBERG DUOPOLOYIntuition

• Firm 1 is the dominant firm, in fact it acts like
  a monopolist, while Firm 2 takes the scraps off
  the table (acting as a monopolist in the
  remaining market)
• Firm 1 is the first mover in a market while Firm
  2 is the late-comer
• Note the difference in equilibrium in a
  sequential move game versus a simultaneous move
  game
• Do you observe this effect in the CSG game when
  you are entering a market already held by an
  incumbent?

39
More game theory

• Game theory warm up
• Backwards induction
• Oligopoly games
• Repeated games
• Signaling games

40
What about repeated games?

• What happens when we play the price war game over
  and over again?

B
Fight Accommodate
0,0
4,-3
Fight Accommodate
A
1,1
-3,4
41
Equilibrium in repeated play

• Consider the strategy If you fought last round
  I will fight forever . . . If you accommodated in
  the last round I will accommodate until you fight
• Assume no discounting for simplicity
• 40000 . . . . 4
• 11111 . . . . n

42
Thinking about price wars

• Why do price wars stop?
• PV (nice payoffs) gt PV (bitter competition)
• Why do price wars start?
• How do you credibly signal commitment to fight
  forever?
• What happens if its not an infinite game? Would
  you ever cooperate?

43
DISCOUNTING IN INFINITELY REPEATED GAMES
Discounting in infinitely repeated games Assume
discount factor is ? 1/1r lt 1 S ?xp xt
xt1 xt2 . . . ?S ? xp xt1 xt2
xt3 . . . S - ?S (1-?)S xt S xt/(1-?)
infinity
pt
infinity
pt
44
THE FOLK THEOREM
As long as gamma is big enough (players are
sufficiently patient) any stage game outcome with
payoffs in excess of the stage game Nash
Equilibrium outcome can be supported This is a
nice result because it tells us we can support
cooperation with trigger strategies in
long-period repeated games where players are
patient as long as neither player knows when the
game will end Doesnt rely exclusively on doom
triggers but punishments must be big Problem
An embarrassment of riches
45
More game theory

• Game theory warm up
• Backwards induction
• Oligopoly games
• Repeated games
• Signaling games

46
Signaling game set-up

• Imagine there are two types of people in the
  world, but that type is private information know
  only to individual
• people who good at business but not at art
  (business people)
• people who are very talented at art but not
  business (artists)
• Employers want to very business people not
  artists
• Businesses pay very well such that artists would
  like to have business jobs
• How should business find business people?

47
How do business find business people?

• One solution is to ask people, are you a
  business person or an artist?
• What will business people say?
• What will artists say?
• Is there a signal business people can send?
• What if wearing a suit signals that one is a
  business person? What will artists do?
• Is there a credible signal business people can
  send that businesses will believe?

48
Credible signals

• A signal is credible if it is costly enough such
  that artists will not want to invest in signaling
• One potential credible signal is going to
  business school
• Interestingly the signal works even if business
  school does not affect business peoples
  productivity

49
Signaling game set-up

• ½ the people in the world are business people and
  ½ are artists
• Business people are worth 5 to businesses, while
  artists are worth 4
• There are only enough business jobs for ½ the
  people in the world
• Firms pay 3 to anyone they hire, regardless of
  type (since this is unobservable)
• Business school is free however, it costs 1 of
  effort from business people and 2 of effort from
  artists (artists dislike business school)
• School does not change the productive capacity of
  the workers

50
Signaling game
Business people
3,2
2,2
H
H
B School
No B School
50
0,0
-1,0
N
N
Nature
H
H
3,1
1,1
50
B School
No B School
N
N
0,0
-2,0
Artists
51
Equilibrium

• Business people go to business school, artists do
  not and firms only hire business school graduates
• This is the only equilibrium in this game, no one
  can do better by changing their strategy given
  what other players do
• Note that if everyone goes to school the expected
  value for artists is ½ (etc. etc.)
• Note the role of business school in this game
• Business people dont learn anything useful in
  business school in this set-up
• However business school is still a socially
  useful institution since it allows business
  people to send credible signals to potential
  employers

52
A few comments

• A signal is only valuable if it is credible
• Credible signals must be costly to send
• This game is actually more complicated than what
  Ive laid out here . . . If you want more take
  John Morgans game theory class

53
Next week!

• More game theory! (Just joking Evan and Ian (as
  well as Jonathan and Rui) are economists and like
  game theory but thats all we have time for take
  John Morgans class if youre interested in
  learning more)
• Final preparation (I) Work on building theses,
  prioritizing analysis and deciding on
  organization/structure, based on a real-world
  case example in the news