1. Static Games of Complete Information: Subgame Perfection
APEC 8205: Applied Game Theory

2. Objectives
Why is Nash not enough in dynamic games of complete information?
Subgame Perfect Refinement to Nash Equilibrium
What is a subgame?
Definition of Subgame Perfect Equilibrium
Application of Subgame Perfect Equilibrium
Stackelberg Duopoly
Stackelberg Rent Seeking
Empirical Weaknesses of Subgame Perfection

3. Why is Nash not enough in dynamic games of complete information?
Player 1
Player 1’s Strategies:
Do Not
Enter
Enter
{Do Not Enter, Enter}
Player 2
Player 2’s Strategies:
(70,60)
High
Output
Low
Output
{If Enter, High Output;
If Enter, Low Output}
(60,10)
(90,50)
(Player 1’s Payoff, Player 2’s Payoff)
How will/should this game be played?

4. Normal Form Game & Nash Equilibria
Player 1
1(L| E=1) = 90L + 60 (1 - L) = 30L + 60
1(L| E=0) = 70L + 70 (1 - L) = 70
1(L| E=1) >/=/< 1(L| E=0) for L >/=/< 1/3
Player 2
2(E| L=1) = 50E + 60 (1 - E) = E
2(E| L=0) = 10E + 60 (1 - E) = E
2(E| L=1) ≥ 2(E| L=0)
Nash Equilibria:
{(1,0),(1,0)}, {(0,1),(0,1)}, & {(0,1),(1/3 ≥ L > 0,1- L)}
There are an infinite
number of Nash
equilibria!

5. Can we do better than this?
Consider the Nash equilibrium {(0,1),(0,1)}!
Player 1
If Player 1 unilaterally deviates by Entering,
Player 2 will choose a High Output according
its equilibrium strategy.
Do Not
Enter
Enter
Player 2
However, executing such a strategy means
that Player 2 will receive 10 instead of
50 from choosing a Low Output.
(70,60)
High
Output
Low
Output
Is this sensible?
(60,10)
(90,50)
If not, why should Player 1 ever choose
Do Not Enter?
(Player 1’s Payoff, Player 2’s Payoff)
This equilibrium is sustained by what
is called an incredible threat!
If we rule out such threats as unreasonable, we are left with a unique equilibrium: {(1,0),(1,0)}!

6. How can we rule out incredible threats?
We can rule out incredible threats by solving
the game backward!
Player 1
Do Not
Enter
Since Player 2 moves last, we will start by
asking the question, what is Player 2’s
best response assuming Player 1 Enters?
Enter
Player 2
(70,60)
High
Output
Low
Output
Low Output
Since Low Output is a best response to
Enter, lets eliminate High Output as an
option for Player 2 given Enter?
(60,10)
(90,50)
Now that High Output is out of the
picture, what is Player 1’s best response?
(Player 1’s Payoff, Player 2’s Payoff)
Enter
Therefore, we are left with the
unique equilibrium: {(1,0),(1,0)}!

7. Subgame Definition A subgame in an extensive form game:
begins at a decision node n that is a singleton information set,
includes all the decision and terminal nodes following n in the game tree (but no nodes that do not follow n), and
does not cut any information sets (i.e. if a decision node n’ follows n in the game tree, then all other nodes in the information set that contains n’ must also follow n and must be included in the subgame).
A subgame is a piece of a larger game that can be solved without considering the rest of the game!

8. Example Subgames How many subgames are there in this game? 1 + 1 + 1
Player 1
Player 2
Player 3
Player 4
Player 4
How many subgames are there in this game? 1
+ 1
+ 1
= 3

9. Example Subgames How many subgames are there in this game? 1 = 1
Player 1
Player 2
Player 3
Player 4
How many subgames are there in this game? 1
= 1

10. Subgame Perfect Equilibrium
A Nash equilibrium is subgame perfect if the players’ strategies constitute a Nash equilibrium in every subgame (Selten, 1965).

11. Application: Stackelberg Duopoly
Who are the players?
Two firms denoted by i = 1, 2.
Who can do what when?
Firm 1 chooses output.
After Firm 1 chooses output, Firm 2 chooses output.
Who knows what when?
Firm 1 does not know Firm 2’s output when choosing.
Firm 2 knows Firm 1’s output when choosing. .
How are firms rewarded based on what they do?
gi(qi, qj) = (a – qi – qj)qi – cqi for i ≠ j.
Question: What is a strategy for each firm?
Firm 1: q1 ≥ 0
Firm 2: q2(q1) ≥ 0 for all possible q1.

12. What is the subgame perfect equilibrium?
Firm 2 has the last move knowing Firm 1’s output, so lets start here!
Firm 2’s optimization problem is then:
FOC for interior: a – 2q2 – q1 – c = 0
SOC: –2 < 0 is satisfied
Solve for q2:
This is Firm 2’s Nash equilibrium strategy for the subgames starting after Firm 1 has chosen its output.
Note that there are an infinite number of these subgames.

13. Now we know Firm 2’s best response, lets solve for Firms 1 taking into account this information?
Firm 1’s optimization problem is:
FOC for interior:
SOC:
But: , and
So, the SOC is satisfied and

14. And the subgame perfect equilibrium is?
Stackelberg:
Strategies
Outputs
Total Output
Profits
Cournot:
Strategies
Outputs
Total Output
Profits

15. Implications Regarding the Impacts of Better Information for Firm 2
Output
Firm 1’s Increases
Firm 2’s Decreases
Total Increases
Profit
Total Profit Decreases
Even though Firm 2 has better information
to make its choice, it is worse off!

16. Is this the only Nash equilibrium strategies?
No!
How about ?
This is a Nash equilibrium strategy!
It is not subgame perfect, because q2 = a – c is not a best response to any q1 ≥ 0.
There are in fact an infinite number a Nash equilibria for this game.

17. Application: Stackelberg Rent Seeking
Who are the players?
Two firms denoted by i = 1, 2 competing for a lucrative contract worth Vi.
Who can do what when?
Firm 1 chooses effort (x1) for preparing its proposal.
After Firm 1 chooses effort, Firm 2 chooses effort (x2).
Who knows what when?
Firm 1 does not know Firm 2’s effort when choosing.
Firm 2 knows Firm 1’s effort when choosing. .
How are firms rewarded based on what they do?
gi(xi, xj) = Vi xi / (xj + xj) – xi for i ≠ j.
Question: What is a strategy for each firm?
Firm 1: x1 ≥ 0
Firm 2: x2(x1) ≥ 0 for all possible x1.

18. What is the subgame perfect equilibrium?
Firm 2 has the last move knowing Firm 1’s effort, so again lets start here!
Firm 2’s optimization problem is:
FOC for interior:
SOC:
Solve for x2:

19. Now we know Firm 2’s best response, lets solve for Firms 1 taking into account this information?
Firm 1’s optimization problem is:
FOC for interior:
SOC satisfied:
Solve for x1:
Which implies:

20. Is Firm 2 better or worse off from knowing Firm 1’s effort?
Stackelberg
Strategies
Rent Dissipation
Payoffs
Cournot
Strategies
Rent Dissipation
Payoffs

21. Implications Firm 1 better off with Firm 2 knowing its effort.
Firm 2 may be better or worse off knowing Firm 1’s effort:
Better off if V2 > V1 > 0
Worse off if 2V2 > V1 > V2
So we can get results contrary to the duopoly model. Having more information is not always bad!
Why?
Information Effect: Beneficial to Second Mover
Timing Effect: Detrimental to Second Mover
In the Duopoly model with linear demand the timing effect always dominates.
In the Rent Seeking model, the information effect can dominate.

22. Predictive Weaknesses of Subgame Perfection
Incredible Threats That Are Actually Credible
Pareto Improving Non-Equilibrium Play

23. Incredible Threats That Are Actually Credible
Player 1
Player 1
Enter
Do Not
(70,60)
Player 2
High
Low
(60,48)
(90,50)
Do Not
Enter
Player 2
(70,60)
High
Low
12%/0%
32%/12.5%
36%/75%
(60,10)
(90,50)
0%/25%
88%/75%
Subgame Perfection Does Pretty Well!
Subgame Perfection Does Pretty Poorly!
(Player 1’s Payoff, Player 2’s Payoff)
Percentage of Observed Play

24. Pareto Improving Non-Equilibrium Play
(0.40,0.10)
(0.20,0.80)
(1.60,0.40)
(0.80,3.20)
(6.40,1.60)
Player 1 2
Take
Pass
5%/0%
7%/50%
36%/25%
37%/0%
15%/25%
Player
(0.40,0.10)
(0.20,0.80)
(1.60,0.40)
(0.80,3.20)
(6.40,1.60) 1 2
(3.20,12.80)
(25.60,6.40)
Take
Pass 1% 6%
20%
38%
25% 5% 4%
(Player 1’s Payoff, Player 2’s Payoff)
Percentage of Observed Play