Lecture IV: Extensive Form Dynamic Games

1
Lecture IV Extensive Form / Dynamic Games

• Recommended Reading
• Dixit Skeath Chapters 3, 6, 9
• Gibbons Chapter 2.4, 2.5 4
• Osborne Chapters 2-4

2
Extensive Form Representation

• Extensive form game has,
• branches
• nodes (decision terminal)
• payoffs
• information sets
• subgames
• Nash Eq concept remains intact
• Generally, solved by working backwards from
  terminal nodes

3, 3
C
j
C
D
0, 4
i
4, 0
D
C
j
D
2, 2
3
Clinton Marbury vs Madison

• Background
• Adams (Fed.) stacks courts before leaving office
• Jefferson (Rep.) withholds some commissions, inc.
  Marburys
• Marbury applies to S.C. for writ of mandamus to
  compel Madison (Jeffersons Sec. of State) to
  deliver commission
• Marshall uses case to strengthen SC by
  establishing judicial review in U.S.
• Argument
• Historians Marshall outwitted Jefferson
  Republicans
• Fails to consider strategic interaction

4
Clinton Marbury vs Madison

• Marshalls Judgement
• Marbury had legal right to commission (hence
  Jefferson acting illegally in withholding it)
• Marbury had right to legal recourse
• Writ of mandamus from SC was not correct remedy
• Marbury sues under Judiciary Act 1789
• Marshall J. Act 1789 contradicts Art III Sec. 2
  of Constitution
• SC declares J Act 1789 unconstitutional invalid
  

5
Clinton Marbury vs Madison
A(1,2)
Deliver commission
B(0,4)
Comply w. SC
J
Ignore SC
J
C(2,0)
Decide for Marbury Uphold J Act 1789
Withhold Commission
D(4,1)
Decide for Madison Uphold J Act 1789
M
Decide for Madison Invalidate J Act 1789
E(3,3)
6
Clinton Marbury vs Madison
A(1,2)
B(0,4)
Deliver commission
Comply w. SC
J
Ignore SC
J
C(2,0)
Withhold Commission
Decide for Marbury Uphold J Act 1789
D(4,1)
Decide for Madison Uphold J Act 1789
M
Decide for Madison Invalidate J Act 1789
E(3,3)
7
Clinton Marbury vs Madison
A(1,2)
B(0,4)
Deliver commission
Comply w. SC
J
Ignore SC
J
C(2,0)
Withhold Commission
Decide for Marbury Uphold J Act 1789
D(4,1)
Decide for Madison Uphold J Act 1789
M
Decide for Madison Invalidate J Act 1789
E(3,3)
8
Eliminating Multiple Equilibria

• Consider modified stag hunt example from last
  class
• Pareto optimality provides compelling reason for
  F, F
• Yet P, P remains a NE

3,2
F
Canada
P
F
0,1
US
F
1,0
P
Canada
P
1,1
9
Backwards Induction Subgame Perfection

• Reasoning from end of tree to beginning, labelled
  backwards induction (or rollback)
• Backwards induction leads to subgame perfect Nash
  equilibrium
• Informally, SGPE is an equilibrium which does not
  allow players to believe (hence to make)
  non-credible threats or promises
• Refines Nash concept, i.e., SGPE ? Nash Nash ?
  SGPE

10
Chain Store Paradox (Selten 1978)

• Chain store has branches in N (finite) towns
• Each town also has a small Mom Pop store
• M P owners will eventually have sufficient
  capital to
• Establish a 2nd store
• Sell, exit the market retire to Florida
• The chain store prefers M P owner exit it can
• Accept the competition, and split the local
  market
• Launch a price war that is costly to both sides
  to drive out M P
• If chain accepts, M P owners prefer to
  establish 2nd store
• Can chain establish reputation for driving out M
  P owners?

11
Chain Store Paradox (Selten 1978)
In Nth town, situation is

• In Nth town, no point in a price war to convince
  the Nth 1 M P owner to exit
• Threat to do so is not credible
• Thus Nth M P owner sets up 2nd store
• Nth-1owner knows Chain has no reason to maintain
  reputation to drive out the Nth M P owner
• Thus Nth-1 M P owner enters
• SGPE is in, accept at each stage

Price War
(0, 0)
Chain
Accept
2nd Store
(2, 2)
M P
Exit
(1, 5)
12
Stackelberg Duopoly Game

• As in Cournot, 2 firms choose optimal quantity of
  good to produce in market where price reflects
  total quantity produced
• In Cournot, firms choose simultaneously in
  Stackelberg, firms choose sequentially
• where q1, q2 0,
• P(Q) a q1 q2 is the market clearing
  price, and
• c gt 0 is the constant cost of production

q1
q2
q1P(Q) c, q2P(Q) c
13
Stackelberg Duopoly Game
q1P(Q) c, q2P(Q) c
q1
q2

• Apply backward induction, starting with firm 2s
  strategy.
• Given any q1, firm 2s quantity should solve,
• s2(q2 (q1)) max q2(a q1 q2 c)
• q2 0
• ? s2(q2 (q1))/?q2 (a q1 2q2 c) 0
• q2 (a q1 c)/2

14
Stackelberg Duopoly Game
q1P(Q) c, q2P(Q) c
q1
q2

• Given q2 (a q1 c)/2, firm 1s optimal
  strategy is,
• s1(q1 (q2)) max q1(a q1 q2 c)
• q1 0
• s1(q1 (q2)) max q1(½a ½q1 ½c)
• q1 0
• ?s1(q1 (q2))/?q1 (½a q1 ½c) 0
• ? q1 (a c)/2, and hence q2 (a c)/4

15
Stackelberg Duopoly Game

• Dynamic games are useful for understanding the
  implications of asymmetric information.
• Consider Stackelberg game where firm 2 does not
  observe firm 1s quantity
• How this affects equilibrium depends crucially on
  firm 2s beliefs about firm 1s production

q1
q2
q1P(Q) c, q2P(Q) c
16
Stackelberg Duopoly Game

• If firm 2 believes that firm 1 has produced its
  Stackelberg equilibrium amount, (a c)/2,
  then firm 2s optimization problem is
• s2(q2 (q1)) max q2(a (a c)/2 q2 c)
• q2 0
• ? s2(q2 (q1))/?q2 (½a 2q2 ½c) 0
• q2 (a c)/4
• This is as before, so it seems as if the
  information asymmetry has no effect.

q1
q2
q1P(Q) c, q2P(Q) c
17
Stackelberg Duopoly Game

• But, firm 2s strategy implies that firm 1
  maximizes w.r.t. (a c)/4
• s1(q2 (q1)) max q1(a (a c)/4 q1 c)
• q1 0
• ? s1(q2 (q1))/?q1 (¾a 2q1 ¾c) 0
• q1 3(a c)/8
• This is a lower payoff for firm 1 than above it
  cannot be subgame perfect
• Firm 1 can does increase production but this
  implies that firm 2s initial beliefs about firm
  1s production cannot be sustained in equilibrium.

q1
q2
q1P(Q) c, q2P(Q) c
18
Asymmetric Information Signalling

• Consider a signalling game
• Player i has private information about their
  type
• Potential Employee Motivated or Lazy
• Politician Honest or Dishonest
• Rival State Peaceful or Aggressive
• Player js incentive is to deal differently with
  different types, e.g., pay motivated employees
  more highly
• But j only knows distribution of types in
  population
• i sends a signal to j ... but how is j to
  interpret that signal?

19
A Signalling Game Example

• Players US Rogue State (RS)
• RS are of two types
• Strong (p .25)
• Weak (p .75)
• US can i) negotiate (costless) or ii) attack
  (costly)
• attacking a strong RS especially costly -10
• attacking a weak RS resolves issue 5
• negotiating delivers 0
• ? US prefers to negotiate with strong RS and
  attack weak RS
• RS can i) sabre rattle (costly) or ii) be quiet
  (costless)
• Sabre rattling is costly -4
• Being attacked is costly -10
• negotiating is costless 0

20
An Example of a Signalling Game (Hawk-Dove)
(-5, 5)
Rogue State
(-9,5)
Quiet (no signal)
Sabre Rattle (signal)
Attack
Attack
Negotiate
Negotiate
Weak p .75
(0, 0)
(-4, 0)
Nature
US
US
Strong p .25
(-14,-10)
(-10, -10)
Attack
Attack
Negotiate
Sabre Rattle (signal)
Negotiate
Rogue State
Quiet (no signal)
(0, 0)
(-4, 0)
21
A Signalling Game Example

• How should US view respond to sabre rattling?
• Sign of strength, so negotiate... but then weak
  RS have incentives to sabre rattle??
• Sign of weakness, so attack (strong RS knows not
  in USs interest to attack it, hence no need to
  sabre rattle)... but then no RS sabre rattles
• US wants a strategy that creates incentives for
  RS to reveal their types via their signals

22
A Signalling Game Example

• How should US view respond to sabre rattling?
• Sign of strength, so negotiate... but then weak
  RS have incentives to sabre rattle??
• POOLING (ON SABRE RATTLING) EQUILIBRIUM
• Sign of weakness, so attack (strong RS knows not
  in USs interest to attack it, hence no need to
  sabre rattle)... but then no RS sabre rattles
• POOLING (ON BEING QUIET) EQUILIBRIUM
• US wants a strategy that creates incentives for
  RS to reveal their types via their signals
• SEPARATING EQUILIBRIUM

23
Solving a Signalling Game

• Backward induction not possible because USs
  later actions hinge on beliefs updated by RSs
  earlier actions
• Receivers beliefs are crucial must be a)
  specified, b) updated if possible, and c) in
  equilibrium.
• Plan
• Fix Receivers beliefs
• Have Sender signal
• Use Bayes Rule to update Receivers beliefs if
  possible
• Determine Receivers best response given updated
  beliefs
• Check if Sender has incentive to alter signal
  given Receivers actions
• If a-e lead to self-enforcing situation, we have
  a Perfect Bayesian Equilibrium (PBE)

24
Bayes Rule

• A rule for updating conditional probabilities
• Let E F be 2 correlated events, i.e., knowing
  that E occurs tells you something about whether F
  is more or less likely to co-occur.
• e.g., what is the probability that the
  Conservatives win a majority (E) given that they
  have 3 lead in polls (F)?
• Pr(EF) Pr(E F)/Pr(F)

25
Is there a Separating Equilibrium?

• Note 2 possible separating equilibria
• Strong RS sabre rattle weak RS stay quiet
• Strong RS stay quiet weak sabre rattle.
• Fix US beliefs
• ?(ti SSabre Rattle) ?(ti WQuiet) 0
• ?(ti SQuiet) ?(ti WSabre Rattle) 1
• Strong RS Quiet, Weak RS Sabre Rattle
• Beliefs remain intact after signal (consistent w.
  Bayes Rule)

26
An Example of a Signalling Game
Quiet leads US to think its at this node
Sabre rattling leads US to think its at this node
(-5, 5)
Rogue State
(-9,5)
Quiet (no signal)
Sabre Rattle (signal)
Attack
Attack
Negotiate
Negotiate
Weak p .75
(0, 0)
(-4, 0)
Nature
US
US
Strong p .25
(-14,-10)
(-10, -10)
Attack
Attack
Negotiate
Sabre Rattle (signal)
Negotiate
Rogue State
Quiet (no signal)
(0, 0)
(-4, 0)
27
Is there a Separating Equilibrium?

• US best response if Quiet ? Negotiate if Sabre
  Rattle ? Attack
• i) Given US strategy, would a strong RS remain
  quiet?
• Yes Quiet delivers 0, Sabre Rattling delivers
  -14.
• ii) Given US strategy, would a weak RS continue
  sabre rattling?
• No Sabre Rattling delivers -9, Quiet delivers
  0.
• This is not a self-enforcing situation Weak RS
  changes strategy US beliefs cannot then be
  sustained.

28
Is there a Pooling on Quiet Equilibrium?

• If W S RS remain quiet, US cannot update on
  following beliefs
• µ(ti WQuiet) p .75
• µ(ti SQuiet) 1-p .25
• Given above beliefs, the US recoups
• EUUS(A, Quiet) (5 ? .75) (-10 ? .25) 3.75
  2.5 1.25
• EUUS(N, Quiet) (0 ? .75) (0 ? .25) 0
• USs best response to Quiet is A

29
Is there a Pooling on Quiet Equilibrium?

• Well be more rigorous here define USs
  off-equilibrium path beliefs, i.e., to a Sabre
  Rattle message
• µ(ti WSabre) ? ? 0, 1
• EUUS(A, Sabre) µ(ti WSabre) ? 5 1-µ(ti
  SSabre) ? -10
• 5 ? ? (1-?) ? -10 15 ? -10
• EUUS(N, Sabre) µ(ti WSabre) ? 0 1-µ(ti
  SSabre) ? 0 0
• ? USs best response to Sabre Rattle N if ?
  2/3, A if ? gt 2/3

30
An Example of a Signalling Game
USs off-equilibrium path beliefs cover these
situations
(-5, 5)
Rogue State
(-9,5)
Quiet (no signal)
Sabre Rattle (signal)
Attack
Attack
Negotiate
Negotiate
Weak p .75
(0, 0)
(-4, 0)
Nature
US
US
Strong p .25
(-14,-10)
(-10, -10)
Attack
Attack
Negotiate
Sabre Rattle (signal)
Negotiate
Rogue State
Quiet (no signal)
(0, 0)
(-4, 0)
31
Is there a Pooling on Quiet Equilibrium?

• Strong RS Quiet, Weak RS Quiet
• If all types quiet, no updating is possible, so
  US beliefs thus US strategy remains in place
• Do strong RS remain quiet given US strategy?
• Yes, if ? gt 2/3 Sabre Rattling would still lead
  to an attack and would be costly in itself
• Do weak RS remain quiet given US strategy?
• Yes, if ? gt 2/3, for same reason
• But if ? 2/3 then sabre rattling induces
  negotiation by US, and all types switch, and
  beliefs not sustainable in equilibrium