Game Theory
Lecture 4:
Rationalizability
Christoph Schottmüller
University of Copenhagen
September 25, 2014
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Outline
1
Rationalizability
2
Iterative elimination of strictly dominated actions
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Rationalizability I
basic problem: which actions should we expect from a
rational player?
NE? how does a player guess the other player’s actions in a
one-shot game
other solution concepts that require less!
“rationality”: each player must hold some (arbitrary?)
belief about other players’ actions and play a best response
⇒ only actions that are best response to some belief will be
played
⇒ only best responses to action profiles that are themselves
best responses have to be considered; or best responses to
actions that are best responses to best responses etc.
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Rationalizability II
Example
a1
a2
L
1,2
0,1
R
3,1
2,3
Could a rational P2 play R?
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Rationalizability III
Example
a1
a2
a3
a4
L
5,14
3,0
6,1
4,0
M
4,-2
5,5
2,0
4,2
R
6,3
5,1
2,16
3,5
Could a rational P1 choose a1?
If P1 beliefs that P2 plays R, a1 is b.r.
R is b.r. if P2 thinks that a3 will be played
a3 is a best response to L
L is a best response to a1
a1 is the action we started with!
⇒ we cannot rule out that a1 is chosen by a rational player
though NE is (a2 , M )!
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Rationalizability IV
starting point: we want a theory of rational decision
making
requirement: self-enforcing (a player knowing the theory
should not do better by deviating from it)
self enforcing implies that the theory has to prescribe sets
of actions (matching pennies)
notation: theory recommends set of actions Zi to player i
what are rational beliefs?
probability zero on actions that are not in Z = ×i∈N Zi
any belief on Z should be allowed: otherwise player is
smarter than theory
self enforcing requires Z ⊆ B (Z ) (only best response
actions to recommended actions are rational)
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Rationalizability V
Definition
An action ai ∈ Ai is rationalizable if for each j ∈ N there is a
set Zj ⊆ Aj such that
(i) ai ∈ Zi
(ii) every action aj ∈ Zj is a best response to a belief µj (aj ) of
player j where the support of µj is a subset of Z−j .
if there are two families of sets (Zi1 ) and (Zi2 ) satisfying this
definition, then (Zi1 ∪ Zi2 ) satisfies the definition (check!)
set of rationalizable actions: Z
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Rationalizability VI
Example (continued)
a1
a2
a3
a4
L
5,14
3,0
6,1
4,0
M
4,-2
5,5
2,0
4,2
R
6,3
5,1
2,16
3,5
why is a1 rationalizable?
what could the sets Z1 and Z2 be?
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Rationalizability VII
Exercise
Consider a Bertrand model with homogenous goods:
two firms with 0 marginal costs set prices
one consumer buys from the firm with the lowest price
What is the set of rationalizable prices?
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Iterative elimination of strictly dominated actions I
Definition
An action ai ∈ Ai is strictly dominated if there is a mixed
strategy αi of player i such that U (a−i , αi ) > ui (a−i , ai ) for all
a−i ∈ A−i .
a rational player does not want to use a strictly dominated
action
⇒ a rational player should not expect other players to use
a strictly dominated action
Hence, a rational player should not use an action that is
strictly dominated given that all other players play only
actions that are not strictly dominated . . .
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Iterative elimination of strictly dominated actions II
Exercise
a1
a2
a3
L
6,5
3,0
4,1
M
4,2
3,5
3,0
R
3,3
6,1
4,16
Table: elimination of strictly dominated strategies
Show that a3 is strictly dominated. Which actions survive
iterated elimination of strictly dominated actions?
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Iterative elimination of strictly dominated actions III
Definition
The set X ⊆ A survives iterated elimination of strictly
dominated actions if X = ×j ∈N Xj and there is a collection
((Xjt )j ∈N )T
t=0 of sets such that for each j ∈ N
Xj0 = Aj and XjT = Xj
Xjt+1 ⊆ Xjt for t = 0, . . . , T − 1
every action in Xjt \Xjt+1 is striclty dominated in the game
hN , (Xit ), (ui )i i.e. the original game with actions restricted
to X t .
No action in XjT is strictly dominated in the game
hN , (XiT ), (ui )i.
Note: The set X does not depend on the order of elimination!
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Iterative elimination of strictly dominated actions IV
Lemma (“never-best responses” in finite games)
An action ai ∈ Ai that is not a best response for any belief
player i might have is strictly dominated.
Proof. skipped
Proposition
If X = ×j ∈N Xj survives iterated elimination of strictly
dominated action in a finite strategic game hN , (Ai ), (ui )i, then
Xj is the set of player j’s rationalizable actions.
Proof. (for 2 players) . . .
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Iterative elimination of strictly dominated actions V
Example (Cournot and rationalizable strategies)
2- firm Cournot oligopoly with linear demand
p = max (1 − a1 − a2 , 0)
zero marginal costs: profits are u1 (a1 , a2 ) = (1 − a1 − a2 )ai
Cournot equilibrium is a1 = a2 = 1/3
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Iterative elimination of strictly dominated actions VI
Example (Cournot and rationalizable strategies continued)
1−aj
2
1
best response to aj is ai (aj ) =
2
rational player chooses ai between 0 and 1
3
a1 > 1/2 is never optimal if P2 chooses an action in [0, 1]
4
hence a2 is at least the best response to 1/2, i.e. a2 ≥ 1/4
5
a1 is not higher than best response to 1/4, i.e. a1 ≤ 3/8
6
a2 is then at least 5/16
7
...
8
both bounds converge to 1/3!
The Cournot equilibrium is the only outcome rational players
can play!
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Iterative elimination of strictly dominated actions VII
Example (Cournot and rationalizable strategies continued)
Formally, 1/3 is the only rationalizable action:
the game is symmetric ⇒ both players have the same set of
rationalizable actions Zi (check!)
call the highest rationalizable action a h and the lowest a l
if j plays a mixed strategy, i ’s best response is
1−Ea
ai (aj ) = 2 j
as a h is rationalizable, it must be a best response to some
l
belief over Zi ⇒ a h ≤ 1−a
2
as a l is rationalizable, it must be a best response to some
h
belief over Zi ⇒ a l ≥ 1−a
2
using the two inequalities, show that a h ≤ 1/3 and a l ≥ 1/3
conclusion: as a l ≤ a h , we have a l = a h = 1/3
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Review Questions
Why might rational players not end up in a Nash
equilibrium?
What is a rationalizable action? Which conditions does it
have to satisfy?
What is a “strictly dominated” action?
Explain the method of iteratively eliminating strictly
dominated actions.
What is the relation between the set of rationalizable
actions and the set of actions surviving iterative
elimination of strictly dominated actions?
reading: OR ch. 4 (or MSZ 4.5, 4.7 and 4.11)
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Exercises I
1
2
Think about the following: Why might rational players not
play Nash equilibrium? Find an example game in which
rational players play Nash equilibrium for sure. Find an
example game where rational players might possibly choose
an action profile that is not a Nash equilibrium.
Two firms compete by setting prices. The demand of firm i
is Di (pi , pj ) = 5 − pi − 2/pj . Assume that both firms have
marginal costs of 1. Write down each firm’s profit function
and calculate their best response functions.
What is the lowest price a firm could set (with any belief
over the other firms’s prices)? What is the highest?
Given that the other firm never sets prices lower/higher
than the one in the last subquestion, what are the lowest
and highest price a firm might set?
continue to reason further. . .
Which actions are rationalizable?
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Exercises II
3
Is R a rationalizable action for P2 in the following game?
(use only the definition of rationalizable action)
T
B
L
2,1
1,3
R
1,1
0,1
Which actions of P1 are rationalizable in the following
game? Which of P2? (you can use all results from the
lecture)
T
M
B
4
L
2,1
4,2
1,3
R
1,1
0,2
3,1
*Show that only rationalizable actions are used with
positive probability in a mixed strategy Nash equilibrium.
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