UC Berkeley
Fall 2007
Economics 201A - Section 4
Marina Halac
1
What we learnt this week
• Basics: relationship between extensive form and normal form, behavioral strategies
• Classes of games: dynamic imperfect-information games
• Solution concepts: Nash equilibrium
(ii) even when his own action prevents h from being reached.
2
− A pure strategy,
Problems
si : Hi → A, specifies an actioin si (h) ∈ Ch for player i at each
h ∈ Hi . si (·) ∈ Si .
Problem 1: Splitting 100 dollars
• There are two ways of randomization.
Consider a bargaining situation in which two individuals are considering undertaking a business
− Awill
mixed
strategy
∈ ∆(S
over pure
σi ∈agree
Σi . on how to split the 100
venture that
earn
themσi 100
dollars
in profit,
but strategies.
they must
i ) randomizes
dollars. Bargaining
works as follows: The two individuals each make a demand simultaneously. If
− A behavior strategy bi : h ∈ Hi 7→ bi (h) ∈ ∆(Ch ) randomizes over available actions at
their demands sum to more than 100 dollars, then they fail to agree, and each gets nothing. If
each information set of player i. bi ∈ Bi
their demands sum to 100 dollars or less, they do the project, each gets his demand, and the rest
− When
studying
extensiveutility
form games
more convenient
useThe
behavior
goes to charity.
Each
individual’s
fromdirectly,
gettingitxisdollars
is u(x) =tox.
individuals do not
strategies.
value charity.
Example
4.2. player’s
(Strategies
in Extensive
Form
Game) Consider the following extensive
(i) What
are each
strictly
dominated
strategies?
formare
game:
(ii) What
each player’s weakly dominated 1strategies?
(iii) What are the pure-strategy Nash equilibria
of this
U
D game?
2
w
2
y
x
Problem 2: Quick questions
z
(i) Consider an extensive-form game where each player has N decision nodes. How many information sets must each player have for the game to be one of perfect information?
The set of pure strategies for player 2 is S2 = {wy, wz, xy, wz}. A mixed strategy can be
P
(ii) In a game where player i has N information sets indexed
n = 1, ..., N and Mn possible actions
written as σ2 = (σ2 (wy), σ2(wz), σ2 (xy), σ2 (xz)) with s2 ∈S2 σ2 (s2 ) = 1 while a behavior
at information set n, how many strategies does player i have?
strategy can be written as b2 = ((b2 (w), b2 (x)), (b2 (y), b2(z))) with b2 (w) + b2 (x) = 1 and
b2 (y) + b2 (z) = 1.
Problem 3: Extensive form versus normal form
game Γ
a normal
form representation.
E has the
For each of Each
the extensive
followingform
games,
draw
normal-form
representation and find the set of Nash
equilibria.Example 4.3. (Normal Form Representation) Consider the following extensive form
games:
1
1
U
D
2
L
3,3
U
D
2
R
l
2,1
1,2
Game 3.1
2
r
L
4,4
3,3
18
R
L
2,1
1,2
Game 3.2
1
R
4,4
Problematic Information Sets
1
Problem 4: Information sets
2
Which one of the
1 following games looks like a dynamic imperfect-information game as the ones we
saw in class? Which one looks “problematic”? Are all these games valid?
Formal Review
Problematic Information Sets
A strategy of a player in an extensive form game is a
complete
plan
of actions, i.e. it specifies an action in
3
2
every
single information set
where2 that player moves.
2
1
Formally, it is a function that assigns an action to each
Game 4.1
Game
4.2
information
set.
3
2
1
1
3
1
3
2
3
Game 4.3
Problem 5: Predation game
Firm E
Out
" 0%
$ '
# 2&
In
Firm E
Accommodate
Fight
Firm I
!
Fight
#"3&
% (
$ "1'
Accommodate
Fight
&1#
$$ !!
% ' 2"
& ' 2#
$$ !!
% '1"
Accommodate
& 3#
$$ 1 !!
% "
!
(i) Draw the normal-form representation.
(ii) What outcomes does IWD predict?
(iii) What is the set of pure-strategy Nash equilibria?
(iv) Which prediction would you say is more reasonable? Why?
2
2
1
3
Answers
Problem 1
(i) If player i demands y > 100, then any strategy of player j with a demand of x > 0 is payoff
equivalent. Therefore, there are no strictly dominated strategies.
(ii) Any strategy demanding more than 100 dollars is weakly dominated. To show this, note that
if player 2 demands y > 100, then any strategy of player 1 with x > 0 is payoff equivalent. If
player 2 demands 0 6 y < 100, then player 1 could demand x = 100 − y and would obtain a
payoff of 100 − y. Demanding x > 100 will give player 1 a payoff of 0. Therefore, any strategy
demanding more than 100 dollars is weakly dominated.
(iii) Any pair (x, 100 − x) with 100 > x > 0 is a pure-strategy Nash equilibrium of this game.
There also exist equilibria of the form (x, y) where x, y > 100.
Problem 2
(i) Each player must have N information sets for the game to be one of perfect information. A
game is one of perfect information if each information set contains a single decision node.
Otherwise, it is a game of imperfect information.
(ii) Player i has M1 × M2 × M3 × ... × Mn strategies in this game.
Problem 3
The normal-form
givengames
below.
Normal formrepresentations
representations are
of these
are respectively given by
2
1
2
Ll
Lr
Rl
Rr
U
3,3
3,3
2,1
2,1
D
1,2
4,4
1,2
4,4
1
3.1 form game:
Consider the followingGame
extensive
L
R
U
3,3
2,1
D
1,2
4,4
Game 3.2
2
L
R
Now consider the set of Nash equilibria for Game 3.1. We first see if there are non-rationalizable
strategies that we can eliminate. Here, Rl is not 1rational for player 2 (note that it is strictly
dominated by Lr), so we can ignore it when U
looking forDthe Nash3,1equilibria. Next, we find the
pure-strategy Nash equilibria by underlying the payoff to player j’s best response to each of player
2
i’s feasible strategies, as we did last time. This
gives that there are three Nash equilibria in pure
3,0mixed-strategy Nash equilibria, note
strategies: (U, Ll), (D, Lr), and (D, Rr).l Finally,rto find the
that if player 1 plays U and D both with positive probability, then player 2 plays Lr for sure, but
then player 1’s best response is to play D for sure. Hence, there is no equilibrium where player 1
4,1
2,2
plays a totally mixed strategy. But there are equilibria where player 1 plays a pure strategy and
player Normal
2 plays form
a mixed
strategy. of
If this
player
1 plays
U by
with probability one, player 2 plays Ll with
representation
game
is given
probability p and Lr with probability (1 − p), where p must be such that player 1’s best response
is to play U :
2
1
EU1 (U ) > EU1 (D) ⇔ 3 > p + 4(1 −Ll
p) ⇔Lr
p > Rl Rr
U 4,1 2,2 33,1 3,1
1
D 3,0 3,0
3 3,1 3,1
By merging redundant pure strategies Rl and Rr into a single strategy R, we can obtain a
And if player 1 plays D with probability one, player 2 plays Lr with probability q and Rr with
probability (1 − q) where q ∈ [0, 1].
In sum, the set of Nash equilibria in this game is (U ; pLl, (1−p)Lr) where p > 31 and (D; qLr, (1−
q)Rr) where q ∈ [0, 1].
Next, consider the set of Nash equilibria for Game 3.2. There are two pure-strategy Nash
equilibria: (U, L) and (D, R). And, defining s as the probability with which player 1 plays U and t
as the probability with which player 2 plays L, there is a unique mixed-strategy Nash equilibrium
given by (s, t) = ( 21 , 12 ).
Problem 4
Game 4.3 has the structure of a dynamic imperfect-information game as the ones we will study
in this course.1 Game 4.1 is valid but “problematic,” since player 2 forgets his own previous
move. This is what is called a game of imperfect recall. We will in general impose that players
possess perfect recall. Loosely speaking, perfect recall means that a player does not forget what
she once knew, including her own actions. Finally, Game 4.2 is not valid since, by the definition of
information set, there cannot be a different number of possible actions in two nodes contained in
the same information set.
Problem 5
(i) The normal-form representation is
Out/Accommodate
Out/Fight
In/Accommodate
In/Fight
Accommodate
0,2
0,2
3,1
1,-2
Fight
0,2
0,2
-2,-1
-3,-1
(ii) We do iterative elimination of weakly dominated strategies as follows. We first eliminate
In/Fight for Firm E because it is weakly (and strictly) dominated by In/Accomodate. We next
eliminate Fight for Firm I as it is weakly dominated by Accommodate. Finally, we eliminate
Out/Accommodate and Out/Fight for Firm E as they are dominated by In/Accommodate.
Hence, IWD predicts (In/Accommodate, Accommodate).
(iii) There are three pure-strategy Nash equilibria: (Out/Accommodate, Fight), (Out/Fight,
Fight), and (In/Accommodate, Accommodate).
(iv) The only outcome that looks reasonable is the outcome predicted by IWD: (In/Accommodate,
Accommodate). The reason is that (Accommodate, Accommodate) is the sole Nash equilibrium in the one-shot simultaneous-move game that follows Firm E’s entry. Thus, the firms
should expect to play (Accommodate, Accommodate) if firm E enters the market. But if this
is the case, then firm E should choose to enter.
Another way to say this is that the Nash equilibria (Out/Accommodate, Fight) and (Out/Fight,
Fight) are supported by a “noncredible threat.” Firm E decides not to enter because it believes that Firm I is going to play Fight if it enters. But since Fight is not an equilibrium
strategy for Firm I once Firm E enters, any claim by the incumbent that it will fight is not
credible, and thus Firm E should anticipate that Firm I will accommodate.
1
To be precise, we should check that nodes contained in the same information set have the same possible actions.
We cannot verify this here as the actions are not specified on the tree.
4