EC941 - Game Theory

1

EC941 - Game Theory

Prof. Francesco SquintaniEmail:

[email protected]

Lecture 5

2

Structure of the Lecture

Definition of Extensive-Form Games

Subgame Perfection and Backward Induction

Applications: Stackelberg Duopoly, Harris-Vickers Race.

3

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.

4

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.

5

Let us model the entry game in strategic form.

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.

Fight Share In

Out

-1,-1

0, 2 0, 2

1, 1

6

The extensive form underlines that the incumbent choice takes place only if the entrant enters,

and after entry.

1

O I

F S0, 2

-1,-1 1, 1

2

Extensive Form Representation

Starting from the end, player 2 prefers S to F, because 1 > -1.

Carry the value [1, 1] to

player 2 decision node.

1

O I

F S0, 2

-1,-1 1, 1

2 [1, 1]

Backward Induction Solution

7

8

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).

1

O I

F S0, 2

-1,-1 1, 1

2 [1, 1]

[1, 1]

9

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.

10

1

O I

F S0, 2

-1,-1 1, -2

2

1

O I

F S0, 2

-1,-1 1, 1

2

2B N


11

1

O I

F S0, 2

-1,-1 1, -2

2

1

O I

F S0, 2

-1,-1 1, 1

2

2B N


12

General DefinitionsDefinition 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.

13

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.

14

Strategies and OutcomesDefinition 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.

15

For example, in the game below,

the strategies of player 1 are {AA, AD, DA, DD},

and the strategies of player 2 are {a, d}.

11 2

1,12,2 0,0

3,3

D Dd

A Aa

16

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.

17

The strategic form representation is:

The Nash Equilibrium are

(AA, a) and any mix

between DA and DD,

with player 2 playing d.

a d DD

DA

2, 2

2, 2 2, 2

2, 2

AD

AA

0, 0

3, 3 1, 1

1, 1

18

In terms of outcomes in the extensive form, they correspond to:

The backward-induction solution is (AA, a).

11 2

1,12,2 0,0

3,3

D Dd

A Aa[3,3][3,3][3,3]

19

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.

20

SubgamesDefinition 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.

21

Example:

These are the 3 subgames of the game:1. The whole game2. The game starting with player 2’s

decision3. The final decision of player 1.

11 2

1,12,2 0,0

3,3

D Dd

A Aa

22

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.

Subgame Perfect Equilibrium

23

Solving for the subgame-perfect equilibrium

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.

11 2

1,12,2 0,0

3,3

D Dd

A Aa

24

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.

11 2

1,12,2 0,0

3,3

D Dd

A Aa

25

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.

26

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.

27

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.


28

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).


29

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).

30

The SPE outcome of the Stackelberg game is: q1

S = (a − c)/2, q2S = (a − c − q1

* )/2 = (a − c)/4.

Firm 1’s profit is q1S(P(q1

S+q2S) − c) = (a

− c)2/8, firm 2’s profit is q1

S(P(q1S+q2

S) − c) = (a − c)2/16. Recall that in the unique Nash equilibrium

of Cournot’s (simultaneous-move) game, q1

C = 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.

31

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.

32

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.

33

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.


34

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.

35

1 2 3 4 Firm k wins game Gk(1, 1): whoever moves

first takes one step towards the finish line and wins.

3

2

1 f

k1

k2


36

1 2 3 4 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).

4

3

2

1 f f

f

k1

k2

37

1 2 3 4 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).

4

3

2

1 f f

f f

k1

k2

38

1 2 3 4 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 games, firm 1 wins. Hence 2 takes 0 steps.

4

3

2

1 f f

f f

1

1

k1

k2 Firm 1 wins games G1(1, 3) and G1(1, 4) with 1 step.

1 2 3 4 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.

4

3

2

1 f f

f f

k1

k2

1 1

1 1

39

40

1 2 3 4 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.

4

3

2

1 f f

f f

2 2

k1

k2

2 2

1 1

1 1

41

1 2 3 4 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.

4

3

2

1 f f

f f

2 2

k1

k2

2 2

1 1

1 1

f f

f f

42

1 2 3 4 5 6 7 8

Hence, the matrix can then be completed inductively.

5

4

3

2

1 f f

f f

2 2k1

k2

2 2

1 1

1 1

f f

f f

2 2

2 2

2 2

2 2

2 2

2 2

2 2

2 2

1 1 1 1 f f 2 2

43

Summary of the Lecture

Definition of Extensive-Form Games

Subgame Perfection and Backward Induction

Applications: Stackelberg Duopoly, Harris-Vickers Race.

44

Preview of the Next Lecture

Extensive-Form Games with Imperfect Information

Spence Signalling Game

Crawford and Sobel Cheap Talk

Documents

EC941 - Game Theory