EC941 - Game Theory
Lecture 5
Prof. Francesco Squintani
Email: f.squintani@warwick.ac.uk
1
Structure of the Lecture

Definition of Extensive-Form Games

Subgame Perfection and Backward Induction

Applications: Stackelberg Duopoly, Harris-Vickers Race.
2
Extensive Form Games

The strategic form representation is appropriate to
describe simultaneous move games.

In order to account for the sequence of moves in a
game, we introduce the extensive form representation.

This will allow to refine the predictions of the Nash
Equilibrium concept, and formulate more precise and
reasonable predictions.
3
An Entry Game

A potential entrant chooses whether to enter a market
controlled by a monopolist.

If the entrant enters the market, the monopolist can
either start a price war, or share the market with the
entrant.

Intuitively, the incumbent cannot credibly commit to a
price war the entrant, and hence cannot deter entry.
4
Let us model the entry game in strategic form.
Fight
In
Out
Share
-1,-1
1, 1
0, 2
0, 2
The game has two Nash Equilibria: (In, Share) and
(Out, Fight). Only the first one is “reasonable”.
The second one contains a non-credible threat.
5
Extensive Form Representation
1
The extensive form underlines that
the incumbent choice takes place
I
only if the entrant enters,
and after entry.
2
O
0, 2
F
-1,-1
S
1, 1
6
Backward Induction Solution
1
O
I
2
[1, 1]
Starting from the end,
player 2 prefers S to F,
because 1 > -1.
Carry the value [1, 1] to
player 2 decision node.
0, 2
F
-1,-1
S
1, 1
7
1
[1, 1]
O
I
2
[1, 1]
0, 2
F
-1,-1
S
1, 1
Given that player 2 plays
S if player plays I, player
1 chooses I over O,
because 1 > 0.
Carry the value [1, 1] to
player 1 decision node.
This is the BI value of
the game.
The BI solution is (I, S).
8
The Value of Commitment




Before a potential competitor chooses whether to enter
a market or not, the incumbent may or may not “burn
bridges”, i.e. make an investment that reduces her payoff
if sharing the market with the competitor.
By burning bridges, she successfully deters entry.
The competitor correctly anticipates that if entering the
market, she will be fought by the incumbent.
As a result, the competitor chooses not to enter the
market.
9
Extensive Form Representation
2
B
N
1
O
1
I
O
I
2
2
0, 2
0, 2
F
-1,-1
S
1, -2
F
-1,-1
S
1, 1
10
Backward Induction Solution
2
B
N
1
O
1
I
O
I
2
2
0, 2
0, 2
F
-1,-1
S
1, -2
F
-1,-1
S
1, 1
11
General Definitions
Definition An extensive game with perfect information G is:
 a set of players I,
 a set of sequences E (terminal histories) with the property
that no sequence is a proper subhistory of any other
sequence,
 a function P (the player function) that assigns a player to
every sequence that is a proper subhistory of some
terminal history,
 for each player i, preferences ui over the set of terminal
histories.
12
For example, in the entry game,




Players: The challenger and the incumbent.
Terminal histories: (In, Share), (In, Fight), and Out.
Player function: P(∅) = Challenger, P(In) = Incumbent.
Preferences The challenger’s preferences are given by the
payoff function u1 with u1(I, S) = 1, u1(O) = 0, u1(I, F) = -1.
The incumbent’s preferences are given by payoff function
u2 with u2(O) = 2, u2(I, S) = 1, u2(I, F) = -1.
13
Strategies and Outcomes
Definition A strategy si of player i in an extensive game
with perfect information is a function that assigns to each
history h after which it is player i’s turn to move (i.e.,
P(h) = i) an action in A(h), the set of actions available after h.

A player’s strategy specifies the action the player chooses
for every history after which it is her turn to move.
14

For example, in the game below,
1 A
2
D
d
2,2
1,1
a
1
A
3,3
D
0,0
the strategies of player 1 are {AA, AD, DA, DD},
and the strategies of player 2 are {a, d}.
15

Each strategy profile s uniquely determines a terminal
history e that is reached through the strategy s.

Because each terminal history e is associated with
payoffs u(e), from each extensive form game
G = (I, E, P, u), we obtain a normal form game
G=(I, S, u), by assigning the utility u(e) to the strategy
profiles s identifying terminal node e.
16

The strategic form representation is:
a
DD
DA
AD
AA
d
2, 2
2, 2
2, 2
2, 2
0, 0
1, 1
3, 3
1, 1
The Nash Equilibrium are
(AA, a) and any mix
between DA and DD,
with player 2 playing d.
17

In terms of outcomes in the extensive form, they
correspond to:
1 A
[3,3]

2 a
[3,3]
D
d
2,2
1,1
1 A
[3,3]
3,3
D
0,0
The backward-induction solution is (AA, a).
18
Theorem In any extensive form game of perfect information, the
backward-induction solution is a Nash Equilibrium.
Proposition In any extensive form game of perfect information in
which there are no ties in payoffs, the backward-induction solution is
unique.
Proposition There exists Nash Equilibria of extensive form games
of perfect information that do not correspond to a backward induction
solution.
19
Subgames
Definition Let Γ be an extensive game with perfect
information, with player function P. For any nonterminal
history h of Γ, the subgame Γ(h) following the history h is the
following extensive form game:
 Players The players in Γ.
 Terminal histories The set of all sequences h’ of actions
such that (h, h’ ) is a terminal history of Γ.
 Player function The player P(h, h’ ) is assigned to each
proper subhistory h’ of a terminal history.
 Preferences Each player prefers h’ to h’’ if and only if she
prefers (h, h’ ) to (h, h’’ ) in Γ.
20
Example:
1 A
2
D
d
2,2
1,1
a
1
A
3,3
D
0,0
These are the 3 subgames of the game:
1. The whole game
2. The game starting with player 2’s decision
3. The final decision of player 1.
21
Subgame Perfect Equilibrium
Definition The strategy profile s∗ is a subgame perfect
equilibrium if it induces a Nash equilibrium in every
subgame.
Theorem The strategy profile s∗ is a subgame perfect
equilibrium if, for every player i, every history h after which
it is player i’s turn to move (i.e. P(h) = i), and every strategy ri
of player i, the terminal history Oh(s∗) generated by s∗ after
the history h yields utility to player i no less than the terminal
history Oh (ri, s∗−i) generated by the strategy profile (ri, s∗−i)
where player i plays ri while every other player j plays s∗j.
22

Solving for the subgame-perfect equilibrium
1 A


2
D
d
2,2
1,1
a
1
A
3,3
D
0,0
The unique Nash equilibrium of the smallest subgame is A.
The middle subgame has 2 Nash equilibria (D,d) and (A,a),
but only (A, a) induces a N.E. in the smaller subgame.
23
1 A

2
D
d
2,2
1,1
a
1
A
3,3
D
0,0
The game has 3 (pure-strategy) Nash equilibria (DD, d),
(DA, d) and (AA, a), but only (AA, a) induces a Nash
Equilibrium in every subgame.
24

In games of perfect information, the subgame perfect
equilibrium coincides with the backward-induction
solution.

But subgame-perfect equilibrium is a more general
concept, defined also for extensive form games without
perfect information.
25
Application: Stackelberg duopoly





There are two firms in the market.
Firm i’ s cost of producing qi units of the good is
Ci (qi); the price at which output is sold when the total
output is Q is P(Q).
Each firm’s strategy is the output, as in Cournot model.
But the firms make their decisions sequentially, rather
than simultaneously.
One firm chooses its output, then the other firm does
so, knowing the output chosen by the first firm.
26
Extensive Form Representation






Players: The two firms.
Terminal histories: The set of all sequences (q1, q2) of
outputs for the firms (where qi, the output of firm i, is a
nonnegative number).
Player function: P(∅) = 1 and P(q1) = 2 for all q1.
Preferences: The payoff of firm i to the terminal history
(q1, q2) is its profit qi P(q1 + q2) − Ci(qi), for i = 1, 2.
A strategy of firm 1 is an output choice q1.
A strategy of firm 2 is a function that associates an
output q2 to each possible output q1 of firm 1.
27
Backward Induction Solution

For any q1 output of firm 1, we find the output b2(q1) of
firm 2 that maximize its profit q2 P(q1 + q2) – C2 (q2).
In any subgame perfect equilibrium, firm 2’s strategy is b2.

We then find the output q1 of firm 1 that maximize its
profit, given the strategy b2(q1) of firm 2.
Firm 1’s output in a subgame perfect equilibrium is the
value q1 that maximizes q1P(q1 + b2(q1)) − C1 (q1).
28





Suppose that Ci(qi) = cqi for i = 1, 2, and P(Q) = a −
Q if Q ≤ a, P(Q) = 0 if Q > a, where c > 0 and c < a.
We know that firm 2’s best response to output q1 of
firm 1 is b2(q1) = (a − c − q1)/2 if q1 ≤ a − c,
b2(q1) = 0 if q1 > a − c.
In a subgame perfect equilibrium of Stackelberg’s game
firm 2’s strategy is this function b2 and firm 1’s strategy
q1* maximizes q1 (a − c − (q1 + (a − c − q1)/2))
= q1 (a − c − q1)/2.
The first order condition yields q1* =(a − c)/2.
The unique subgame perfect equilibrium is (q1*, b2).
29




The SPE outcome of the Stackelberg game is:
q1S = (a − c)/2, q2S = (a − c − q1* )/2 = (a − c)/4.
Firm 1’s profit is q1S(P(q1S+q2S) − c) = (a − c)2/8,
firm 2’s profit is q1S(P(q1S+q2S) − c) = (a − c)2/16.
Recall that in the unique Nash equilibrium of Cournot’s
(simultaneous-move) game, q1C = q2C = (a − c)/3, so
that each firm’s profit is (a − c)2/9.
First-Mover Advantage: The first firm to move
produces more output and obtains more profit than if
firms move simultaneously, the second firm produces
less output and obtains less profit.
30
Application: Harris-Vickers Race

Two firms compete to develop new technologies.

How does competition affect the pace of activity?

How does a leading advantage translates into a final
outcome?

We model this race as an extensive game where players
alternately choose how many “steps” to take towards a
finish line.
31

Player i = 1, 2 is initially ki > 0 steps from the finish line.

On each of her turns, a player can either take 0 steps (at
zero cost), 1 step, at a cost of c(1), or 2 steps, at a cost of
c(2) > 2c(1).

The first player to reach the finish line wins a prize,
worth vi > 0 to player i; the losing player’s payoff is 0.

If, on successive turns, neither player takes any step, the
game ends and neither player obtains the prize.
32
Extensive Form Representation

The game G1(k1, k2) is defined as follows.

Players: The two parties.

Player function: P(∅) = 1, P(x1) = 2 for all x1, P(x1, y1) = 1
for all (x1, y1), P(x1, y1, x2) = 2 for all (x1, y1, x2), etc.
For all t, xt is the number of steps taken by player 1 on
her tth turn, and yt are the steps taken by 2 on tth turn.
33

Terminal histories: The set of sequences of the form
(x1, y1, x2, y2, . . . , xT) or (x1, y1, x2, y2, . . . , yT) for some
integer T, where xt and yt are 0, 1, or 2, there are never
two successive 0’s except possibly at the end of a
sequence, and either x1+…+xT = k1, y1+…+yT < k2
(1 wins the race), or x1+…+xT < k1 and
y1+…+yT = k2 (2 wins the race).

Preferences: For an end history in which player i loses, her
payoff is minus the sum of the costs of all her moves;
for an end history in which she wins it is vi minus the
sum of these costs.
34
k2
Backward Induction Solution
3
2
1
f
1

2
3
4
k1
Firm k wins game Gk(1, 1): whoever moves first takes
one step towards the finish line and wins.
35
4
k2
3
2
f
1
f
1


f
2
3
4
k1
Firm 1 wins game G1(1, 2) by taking 1 step, and wins game
G1(2, 1) by taking 2 steps. (If she took 1 only step, the next
round would be equivalent to game G2(1, 1), where 2 wins.)
Similarly, firm 2 wins games G2(1, 2) and G2(2, 1).
36
4
k2
3

2
f
f
1
f
f
1
2
3
4
k1
Firm 1 wins game G1(2, 2) by taking 2 steps. (If she took 1
step, the next round would be equivalent to game G2(1, 2),
where 2 wins.) Similarly, firm 2 wins game G2(2, 2).
37
4


k2
1
Firm 1 wins games G1(1, 3)
and G1(1, 4) with 1 step.
3
1
2
f
f
1
f
f
1
2
3
4
k1
In games G2(1, 3) and G2(1, 4), firm 2 loses.
With 1 step, the next round is G1(1, 2) and G1(1, 3), with 2
steps, the next round is G1(1, 1) and G1(1, 2): in all such
38
games, firm 1 wins. Hence 2 takes 0 steps.
k2
1
1
3
1
1
2
f
f
1
f
f
1
2
4


3
4
k1
Firm 1 wins games G1(2, 3) and G1(2, 4): with 1 step, the
next round is G2(1, 3) and G2(1, 4), where firm 2 loses.
Firm 2 loses in G2(2, 3) and G2(2, 4): with j > 0 steps, the
next round are G1(2, 3-j) and G1(2, 4-j), where 1 wins. 39
4

k2
1
1
3
1
1
2
f
f
2
2
1
f
f
2
2
1
2
3
4
k1
Analogously to the previous steps, firm 2 wins games
Gk(3, 1), Gk(4, 1), Gk(3, 2) and Gk(4, 2) for k=1, 2.
40
4
k2
1
1
f
f
3
1
1
f
f
2
f
f
2
2
1
f
f
2
2
1



2
3
4
k1
Consider games Gk(3, 3), Gk(3, 4), Gk(4, 3) and Gk(4, 4).
Reaching “regions” where one player wins for sure is the
same as reaching the finish line.
So, player k wins all these games.
41
k2
5
1
1
1
1
f
f
2
2
4
1
1
f
f
2
2
2
2
3
1
1
f
f
2
2
2
2
2
f
f
2
2
2
2
2
2
1
f
f
2
2
2
2
2
2
k1
1

2
3
4
5
6
7
8
Hence, the matrix can then be completed inductively.
42
Summary of the Lecture

Definition of Extensive-Form Games

Subgame Perfection and Backward Induction

Applications: Stackelberg Duopoly, Harris-Vickers Race.
43
Preview of the Next Lecture

Extensive-Form Games with Imperfect Information

Spence Signalling Game

Crawford and Sobel Cheap Talk
44