Game Theory
Section 5: Subgame-Perfect Eqm
Agenda
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•
•
•
•
Main ideas
Key terms
Extensive Games, Backward Induction
Subgame Perfect Nash Equilibrium
Baron and Ferejohn: Bargaining in Legislatures
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The strategic setting
Two-round, 3 candidate closed rule game
Two-round, n-candidate closed rule game
Infinite-round, 3 candidate closed rule game
Open-rule games
Endogenizing the rules
Key Terms
• Sequential Rationality: Players should be rational at
every decision-making opportunity (information set)
• Backward Induction: Process of analyzing a game from
back to front, eliminating actions which are dominated
given the terminal nodes that would be reached
– Most useful in extensive-form games of complete information
– If there are no ties in payoffs, backward induction completely
solves the game, finding the single rational strategy profile
• Subgame: A node x initiates a subgame if neither x nor
its successors are in an information set that contains
nodes that are not successors of x. A subgame is the tree
structure defined by such a node x and its successors
– What is a proper subgame? Let’s do an-class exercise
How Many Subgames Do You See?
Challenger
ø
Acquiesce
Challenger
Out
In
Incumbent
ø
Fight
Acquiesce
Challenger
Out
In
Incumbent
In
Incumbent
Acquiesce
ø
Fight
Fight
Slack off
Incumbent
Challenger
Spend
Save
1
Out
In
ø
Out
2
Fight
Acquiesce
1
Spend
1
Save
Hit Back
Back out
Extensive Games w/ Perfect Info
An Extensive Form Game is Defined by Four Components
• Players
• Terminal History: a set of sequences (chronologically
ordered actions, histories), describing a complete game
• Player Function: the function that assigns a player to
every sequence
• Preferences for the Players: preferences over the set of
terminal histories
– Any preference over terminal histories may be translated
directly into preferences over outcomes, (and vice-versa?*)
for example: the “Entry” Game...
*See Osborne, p. 155
Example: The Entry Game
An Extensive Form Game is Defined by Four Components
•
•
•
•
Players: an incumbent and a challenger
Terminal History: (In, Acquiesce), (In, Fight), and (Out)
Player Function: P(ø) = challenger , P(In) = incumbent
Preferences for the Players:
– Challenger’s preferences represented by payoff function u1
• u1(In, Acquiesce) = 2, u1(Out) = 1, u1(In, Fight) = 0
– Incumbent’s preferences represented by payoff function u2
• u2(In, Acquiesce) = 1, u2(Out) = 2, u2(In, Fight) = 0
Acquiesce
The Entry Game
Payoff is defined as
(Challenger, Incumbent)
Challenger
Out
In
ø
(1, 2)
(2, 1)
Incumbent
Fight
What’s the solution?
(use backward induction)
(0, 0)
When Backward Induction Fails
A Variant of the Entry Game (Empty Threat Game)
•
•
•
•
Players: an incumbent and a challenger
Terminal History: (In, Acquiesce), (In, Fight), and (Out)
Player Function: P(ø) = challenger , P(In) = incumbent
Preferences for the Players:
– Challenger’s preferences represented by payoff function u1
• u1(In, Acquiesce) = 2, u1(Out) = 1, u1(In, Fight) = 0
– Incumbent’s preferences represented by payoff function u2
• u2(In, Acquiesce) = 1, u2(Out) = 2, u2(In, Fight) = 1
Acquiesce
The Entry Game
Payoff is defined as
(Challenger, Incumbent)
Challenger
Out
In
ø
(1, 2)
(2, 1)
Incumbent
Fight
What’s the solution?
(just try to use backward induction...)
(0, 1)
Subgame Perfect Nash Eqm
To be a SPNE, the Strategy Must Be Credible Along the Eqm Path
• Subgame Perfect Nash Equilibrium: A strategy profile is
called a SPNE if it specifies a Nash equilibrium in every
subgame of the original game
– Equilibrium concept incorporating idea of sequential rationality
– If any particular subgame is reached, then we can expect the
players to follow through with the prescription of the strategy
– In this sense, SPE is a robust solution concept (robust to errors)
• Procedure
– Examine the matrices corresponding to all of the subgames
– Locate Nash equilibria
• For infinite games, examine subgames toward the end of the extensive
form (hoping that these subgames have unique Nash equilibria
• Go backwards, embedding the equilibrium outcomes in larger subgames
SPNE: Robusto!
To be a SPNE, the Strategy Must Be Credible Along the Eqm Path
Acquiesce
Challenger
Out
In
ø
(2, 1)
Incumbent
Nash Equlibria
Fight
(0, 0)
(Out, Fight), (In, Acquiesce)
(1, 2)
• In a strategic game, NE rationale is that in steady state, each player’s experience
playing the game leads her belief about the other players’ actions to be correct.
• In a sequential game, this rationale does not apply, because (for example) a
challenger who always chooses Out never observes the incumbent’s action after
the history In.
– Nash Equilibrium of an extensive game: a slightly perturned steady state in which, on
rare occasions, non-equilibrium actions are taken and the perturbations allow each
player eventually to observe every other player’s action after every history. Given
such perturbations, every player eventually learns the others entire strategies.
– Interpreting (Out, Fight) as such a perturbed steady state: On rare occasions when the
challenger enters, the subsequent behavior of the incumbent to fight is not a steady
state in the remainder of the game; if the challenger enters, the incumbent is better of
acquiescing than fighting. The NE is not a robust steady state of the extensive game.
Still confused about SPNE?
More ways of saying the same thing yet again, slightly differently
Subgame Perfect Nash Equilibrium
– Each player’s strategy must be optimal for every history after
which it is the player’s turn to move, not only at the start of the
game, as in the definition of Nash Equilibrium
• NE may not be optimal in some subgames, but a NE is optimal in any
subgame reached when players follow their strategies (by definition)
– A subgame perfect NE generates a NE in every subgame
• So every player’s strategy is optimal, given other players strategies,
throughout the entire game
– Significance of SPNE versus NE:
• Requirement of SPNE is that each player’s strategy be optimal after
histories that do not occur if players follow their strategies,
– Like the history In when the challenger’s action is Out at the start of the
entry game.
When Backward Induction Fails
And SPNE is There to Pick up the Pieces
• The incumbent is indifferent
– If challenger is In, incumbent is indifferent
– If challenger is Out, incumbent is indifferent
• Challenger prefers In when incumbent chooses Acquiesce
• Challenger prefers Out when incumbent chooses Fight
• Two SPNE: (In, Acquiesce), (Out, Fight)
– Both correspond to a steady-state outcome
• The SPE (In, Acquiesce) is a perfectly reasonabe (risky?) steady state
– If you had played the game 100s of times against opponents drawn from the same population, and
your opponent had always chosen Acquiesce, you could reasonably expect your next opponent to
choose Acquiesce, and thus you could optimally choose In
Acquiesce
The Entry Game
Payoff is defined as
(Challenger, Incumbent)
Challenger
Out
In
ø
(1, 2)
(2, 1)
Incumbent
Fight
What’s the solution?
(use SPNE, not backward induction...)
(0, 1)
Ok--So What is SPNE Again?
SPNE is an extension of backward induction
– Corresponds to backward induction in finite games of perfect
information when there is a single best action for the player who
moves at the start of each subgame
– What happens in a game in which at the start of some subgames
more than one action is optimal?
• We trace back separately the implications for behavior in the
longer subgames of every combination of optimal actions in
the shorter subgames
– Don’t always have perfect info; sometimes it’s simultaneous...
2
1
Left
Right
ø
(2, 2)
2
Down
(3, 1)
A
Forward
1
C (2, -2)
1
D (-2, 2)
C (-2, 2)
B
2
Left
Right
ø
(2, 2)
D (2, -2)
2 Forward
Down
(3, 1)
(0, 0)
Baron & Ferejohn: Bargaining in Legislatures
Key Concepts
• Model: legislature dividing up a budget, with
legislatures trying to serve their own districts
– Lots of structure imposed on this model
– Procedures and rules of order are specified in detail
• Equilibria: stable, self-enforcing terminal history
– Legislative outcomes reflect the institutional structure of agenda
formation and voting mechanism, as well as time-preference
• Issues under investigation
– Does equilibrium reflect majoritarian nature of the voting rule?
– Does equilibrium allow benefits to distributed universally?
– When the choice of institutions (open versus closed rules) is made
endogenous, what determines which institution is selected?
– How does time-preference affect the rules and distributive outcome?
Discounting and Time-Preference: 0   
1
Discount factor : the cost of the passage of time
– When  = 1, time has no cost
• $100 today (time t = 1) is equal to $100 tomorrow (t = 2),
from the perspective of someone today.
– When  = 0, time is as costly as it can be: “do or die”
– When  = 0.5, the value of an agreement tomorrow is
worth half the value of the same agreement today
– This is a way of getting at impatience, or the notion
that “time is money”
• In this paper, they attribute it to reelection concerns to
distribute benefits sooner, as well as the probability that
lawmaker will be in office in the next period.
Pass
X1
V
Fail
1
1
Baron & Ferejohn
X1
V
Fail R
2X2 V
Fail
3
Closed-Rule Game
3 Lawmakers, 2 Rounds
Pass
Pass
V
Pass
X3
Fail
Pass
X1
V
Fail
1
R
2
X2
V
Fail R
ø
2X2 V
Members can’t make binding
commitments. An equilibrium
strategy must be self-enforcing in
the sense that the member would
wish to execute it at each point at
which action can be taken.
Therefore, the equilibrium is
required to be subgame perfect.
Fail
3
Pass
Pass
V
Pass
X3
Fail
Pass
X1
V
Fail
1
3
X3
V Fail R
2X2 V
3
Pass
Pass
Fail
V
Pass
X3
Fail
Equilibrium Given 3 Lawmakers, 2 Rounds
Begin backward induction (n = 3 lawmakers)
• Final (2nd) round, each has a 1/3 (or 1/n) chance
– Proposer must offer rivals more than BATNA* (0)
– Proposer offers rivals 0, takes it all
• 1/3 chance of getting it all, 2/3 chance of getting 0
– Expected value is (1/3)*(1) + (2/3)*(zero) = 1/3 or (1/n)
• 1st round, each has 1/3 chance of being proposer
– Must offer 1/2 the others more than their BATNAs
• From perspective of round 1, BATNA is *(1/3) or ( /n)
• Offers /3 to one lawmaker, and proposes 1 - (/3) for himself
» This is better than the expected value of round two ( /3) 
– Motion carries, because majority is no worse off than BATNA
*Best Alternative to a Negotiated Agreement
Equilibrium With n Lawmakers, 2 Rounds
Same as Before…
• 1/3 chance to get it all, 2/3 chance of getting 0
– Expected value is (1/n)*(1) + (2/3)*(0) = (1/n)
• 1st round, each has 1/n chance of being proposer
– Must offer 1/2 the others more their BATNAs
• From perspective of round 1, BATNA is /n
• Offers /n to 0.5*(n-1) lawmakers
• Proposes that he should get to keep 1 - [(1/2)(n-1)*( /n)]
– Motion carries, because majority prefers this to BATNAs
– As n → ∞: the proposer gets to keep 1- [(1/2)* )
• Recall that when n = 3, proposer kept 1- [(1/3)* )
– Depending on n, proposer keeps between 1/2 and 1/3 the benefits
Baron & Ferejohn
Pass
Infinite Round Closed-Rule Game
3Given
Lawmakers
a minimum level of
1
X1
impatience, there are infinite
subgame perfect equilibria
sustainable by the threat
of future punishment.
All equilibria punish those who
would deviate by giving them
a payoff of zero. Members
expect the punishment
R
2
X2
to be enforced, since
ø
those who fail to punish
are themselves punished.
If people are really impatient,
this is equivalent to saying
they care very little for what
will happen in the future;
impatient people can’t be
3
X3
threatened by future
punishment.
Even so, there is an argument that these equilibria are
unconvincing & not robust. Ex-post, people may be
indifferent about punishing; or, ignorant of the history.
X1
V
Fail
1
V
Fail R
2X2 V
Pass
R
Fail
3
Pass
R
V
Pass
X3
R
Fail
Pass
X1
V
Fail
1
V
Fail R
2X2 V
Pass
R
Fail
3
Pass
R
V
Pass
X3
R
Fail
Pass
X1
V
Fail
1
V Fail R
2X2 V
3
Pass
Pass
R
R
Fail
V
Pass
X3
R
Fail
Closed Rule and Stationary Equilibria
Collapsing infinite-sessions into a one-shot game
• What are stationary equilibria?
– Formally: An equilibria is stationary if the
continuation values for each structurallyequivalent subgame are the same
• Continuation value: value of going to next subgame (vj)
• Structurally-equivalent subgame: When subgames are
identical in every way but their prior history
Member recognized in round 1 chooses (x21, x31)
Feasible allocation set (A): x21 + x31  1 and x21, x31 
0.
Member 1 maximizes: 1 - (x21 + x31) s.t. A and the
requirement that the proposal passes by a majority
1
Closed Rule and Stationary Equilibria
Solution to the Infinite Round Game
Member 1 maximizes: 1 - (x21 + x31) s.t. A and the
requirement that proposal passes by a majority
To pass, xj1  vj, for lawmaker j in the majority
Recall that prob (being in majority) = (n-1)/n
Continuation values vj are:
vj = payoff (i is proposer)*(probability i is chosen) +
payoff (i is in majority)*(prob i is in majority) +
payoff (i gets zero)*(probability i gets zero).
vj = [1 - ( v(n-1)/2)])*(1/n) + [(vj)*0.5*(n-1)/n] + 0.
vj* = 1/n.
Proposer claims all that would go to excluded lawmakers.
With discounting, the proposer also benefits from the
impatience of members in his winning coalition:
(see p. 1193 on the bottom)
Open Rule Bargaining
How does this game differ?
(1) Chance selects a proposer
(2) M1 makes offers xi1 to one or more others (i)
(3) Chance recognizes another member, M2
(4) M2 moves the question to a vote, or amends
If M2 amends, and the next round begins at step 3
If M2 moves, there’s a vote. If x1 is defeated, return to step 1.
If M2 moves and x1 passes, payoffs are received.
So… is there a (universalistic) equilibrium
where all other members receive an offer?
xi1 =  [1-(n-1) x1], xi1 =  /(1 + 2), (proposer) = 1- (n -1)* xi1
Open Rule Bargaining
What are the Distributive Comparative Statics?
Proposer must buy off a majority/supermajority) of others.
• The more lawmakers he buys, the higher the likelihood
that his offer will be approved.
– All else equal, the more lawmakers there are, the fewer the proposer will
chose to buy off
– Low discount factors mean that lawmakers deeply discount the future, and
are relatively unwilling to risk the chance someone will propose an
amendment and take up extra time
• Legislature is certain to complete its task in round 1 if and only if the offer in round 1 is
universalistic
• This will happen only if discount factors are low enough
– Distribution is more equal under an open rule than under a closed rule
– Open rule reduces the power of the recognition, relative to a closed rule
• Yet, the one recognized member will never do worse than anyone else
– And, recognition advantage increasing in membership impatience
Endogenizing the Rules
What are the Distributive Comparative Statics?
• Unless there is no impatience, ex-ante*, legislature
prefers a closed rule to a simple open rule
– The sum of the values of a closed-rule game is always at least as much as
that of an open-rule game
– This is because the closed rule finishes the game in one round, and most of
the open-rule equilibria involve the chance of multiple rounds
• Every time it goes to the next round, value is lost forever via discounting
• Nevertheless, an open rule does have its benefits
– More equal ex-post distribution
– More compatible with democratic theory, int hat it allows greater
opportunity to members than does a closed rule
By ex-ante here I mean prior to the very first round, before any one legislator is selected as proposer.