Lecture 6
Extensive Games with Perfect Information
Jiangtao Li
Singapore Management University
Econ 206
Outline
Subgame Perfect Nash Equilibrium (SPNE)
Stackelberg game
Ultimatum game
Review of Lecture 1 - Lecture 4 (if time permits)
Q&A session (if time permits)
Subgame perfect Nash equilibrium
• Action profiles: (In,Acquiesce), (In,Fight), (Out,Acquiesce),
(Out,Fight)
Subgame perfect Nash equilibrium
• NE: (In,Acquiesce), (Out,Fight)
• Drawback of NE: Non-credible threat!
• NE is not the appropriate solution concept for extensive form games.
Subgame perfect Nash equilibrium
• A subgame is a game represented by a subtree starting from any
node in the orginal tree.
• A subgame perfect NE is a strategy profile with the property that in
any subgame, every player’s strategy is a best response against
opponents’ strategies.
• Every SPNE is a Nash Equilibrium since the original game is a
subgame.
Subgame perfect Nash equilibrium
• SPNE: a strategy profile which induces a NE in every subgame
• SPNE: (In,Acquiesce)
More examples
• SPNE: (C,EH)
Class exercise 1
Class exercise 2
Subgame perfect Nash Equilibrium
First, SPNE must be a NE of the game.
Second, SPNE must induce a NE in every subgame.
Subgame perfect Nash Equilibrium
Backwards induction:
It proceeds by examining the last point at which a decision is to be made
and then identifying what action would be most optimal at that moment.
Using this information, one can then determine what to do at the
second-to-last time of decision. This process continues backwards until
one has determined the best action for every possible situation at every
point in time.
A magical tree!
In period 1, there is one dollar. Tom decides whether to take it or not. If
Tom takes it, the game ends. If Tom does not take it, the game
continues to period 2.
In period 2, there are two dollars. Jerry decides whether to take it or not.
If Jerry takes it, the game ends. If Jerry does not take it, the game
continues to period 3.
In period 3, there are three dollars. Tom decides whether to take it or
not. If Tom takes it, the game ends. If Tom does not take it, the game
continues to period 4.
In period 4, there are four dollars. Jerry decides whether to take it or not.
If Jerry takes it, the game ends. If Jerry does not take it, the game
continues to period 5.
In period 5, there are five dollars. Tom decides whether to take it or not.
If Tom takes it, the game ends. If Tom does not take it, the five dollars
go to Jerry, and the game ends.
A magical tree!
What if instead of 5 periods, there are 100 periods?
Stackelberg game
• Players: {1, 2} — 1 is the leader, 2 the follower
• Player 1’s strategies: q1 ∈ R+
• Player 2’s strategies: q2 : R+ → R+ , q2 (q1 )
• In this extensive game, we must write down the quantity chosen by
the second player as a function of the quantity chosen by player 1.
Stackelberg game
• Payoffs:
P(Q) =
α−Q
0
if Q ≤ α
if Q > α
u1 (q1 , q2 (·)) = P(q1 + q2 (q1 ))q1 − cq1
u2 (q1 , q2 (·)) = P(q1 + q2 (q1 ))q2 (q1 ) − cq2 (q1 )
• α>c >0
Stackelberg game
• Period 2: B2 (q1 ) =
{ 12 (α − c − q1 )}
{0}
if q1 ≤ α − c
if q1 > α − c
• Period 1: Player1’s maximization problem
1
(α − q1 − { α−c−q
})q1 − cq1
2
u1 (q1 , q2 (·)) =
(α − q1 − 0)q1 − cq1 ≤ 0
if q1 ≤ α − c
if q1 > α − c
Stackelberg game
• The optimal q1 should be in the first range: q1 ≤ α − c
1
• Compute the FOC for (α − q1 − { α−c−q
})q1 − cq1 with respect to
2
q1
• FOC: α − 2q1 −
• SPNE: q1∗ =
q2∗ (q1 ) =
α−c
2
+ q1 − c = 0.
α−c
2 ,
α−c−q1
2
if q1 ≤ α − c and 0 if q1 > α − c
• SPNE outcome: q1∗ =
α−c
2 ,
q2∗ (q1∗ ) =
α−c−q1∗
2
• SPNE and SPNE outcome are different.
• The first-mover advantage
=
α−c
4
The ultimatum game (6.1.1)
The ultimatum game
Discrete offers
Discrete offers
Now play this game with your neighbour!
The ultimatum game (6.1.1): continuous offers
• Period 1: Player 1 chooses x ∈ [0, c]
• Period 2: Player 2 chooses Accept or Reject
Interpretation
• Player 1 chooses an offer x.
• If player 2 accepts x, then player 1 receives c − x. However, if player
2 rejects, both players do not receive anything at all.
Backward induction: Case 1 (player 2 accepts the offer x = 0)
• SPNE: Player 1 offers 0, Player 2 accepts ALL offers.
Backward induction: Case 2 (player 2 rejects the offer x = 0)
• What is player 1’s best response?
• In other words, what is the smallest positive number?
Backward induction: Case 2
• If offers are discrete, there would be a best response strategy for
player 1
• However, if offers are continuous, there would be no best response
for player 1, since there is no smallest offer that makes player 2
accept.
• There is no SPNE in this case.
• In this game, there is a unique SPNE which comes from Case 1.
What happens when we run this game with real-life people?
Werner Guth, Rolf Schmittberger, and Bernd Schwarze (1982), An
experimental analysis of ultimatum bargaining, Journal of Economic
Behavior & Organization 3 (4), 367 - 388
Mid-term examination
• Time: 9 - 11am on 24 Feb 2024 (Saturday).
• Online via eLearn’s Quiz function.
I
Requires Respondus LockDown Browser + Webcam
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The exam is only available for a certain amount of time.
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Do not be late.
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If you are late, you may not be able to participate.
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There is no make-up exam.
• If you are new to this:
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See Respondus LockDown Browser - Online Exam Instructions.
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Do try the Mock Test (see pdf) to ensure things run smoothly.
• I asked IT to assist with any tech issues during the exam.
• I will forward to you the contact once available.
Mid-term examination
• The exam covers materials up to mixed strategy NE.
• Open book exam (printed materials only, internet not allowed).
• This paper consists of 3 questions.
• You need to type your answers in the boxes provided. As this may
take time (especially for equations), the questions are made easier in
terms of typing equations.
• Calculators may be used.
• Show your work, unless explicitly mentioned otherwise.
• If you have a question during the exam, use your phone to send a
private telegram message to me. I will join the telegram group on
23 Feb 2024. You can use your phone only for this purpose. The
webcam will record your activities.
• PS: There is no lecture in Week 7.
Additional consultation hours (for Week 7 only)
(1) 3:30 - 5:30pm, 19 Feb (my regular consultation hour, in my office).
(2) 3:15pm - 5:15pm, 22 Feb (in my office).
(2) 9 - 11am, 21 Feb (via zoom)
(3) 9 - 11am, 23 Feb (via zoom)
via zoom click here: Consultation with Jiangtao Li
Review
• Strict dominance, weak dominance (Prisoners’ dilemma)
• Pareto dominance
• Best response =⇒ Nash equilibrium
• Cournot competition, voting, auction
• Mixed strategy NE (Matching pennies)
• Iterated elimination of strictly dominated strategies
Question 2 in More practice questions
Question 2 in More practice questions