Static ggames: Rationalizabilityy
Lectures in Game Theory
Spring 2011, Part 2
10.12.2010
G.B. Asheim, ECON4240-2
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Questions to answer:

How do players behave in static strategic situations?

How should players behave in such situations?
Analysis:


Apply the normal form (as a representation of
games where all actions are taken simultaneously
and independently).
Model ‘rational behavior’.
10.12.2010
G.B. Asheim, ECON4240-2
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Strict dominance

Some examples:
Definition : A p
pure strategy
gy si of p
player
y i is strictlyy dominated
if there is a stragegy (pure or mixed)  i  Si such that
ui ( i , si )  ui ( si , si ) for all strategy profile si  S i of
his opponents. Write UDi  Set of undominate d strategies .

gy is strictlyy dominated if there
Comments: — A strategy
is another strategy that is better for all opponent choices.
— The dominating strategy may be mixed.
— Weak dominance is not sufficient.
10.12.2010
G.B. Asheim, ECON4240-2
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Simple theory of individually rational behavior:
Players do not use strictly dominated strategies.
Player i chooses some si UDi

First tension: Individually rational behavior may
not lead to collectively efficient outcomes.
The concept of efficiency
Definition : A strategy profile s  ( s1 , , si , , sn ) is
(Pareto) efficient if there is no other strategy profile
s  ( s1, , si, , sn ) such that ui ( s)  ui ( s ) for
every player i and u j ( s)  u j ( s ) for some player j.
10.12.2010
G.B. Asheim, ECON4240-2
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Equivalent
Alternative theory of individually rational behavior:
Players use strategies that can be best responses.
Definition : Suppose player i has a belief  i  S i
about the strategies played by his opponents.
Player i' s strategy si  Si is a best response to  i if
ui ( si ,  i )  ui ( si,  i ) for every si  Si .
Write BRi (  i )  Set of best respo nses to  i , and
Bi  {si | there
h is a belief
b l f  i  S i such
h that
h si  BRi (  i )}.
Player i chooses some si  Bi
10.12.2010
G.B. Asheim, ECON4240-2
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Strict dominance and best response compared
Observations for two-player games:
– If a strategy is not strictly dominated, then it is a
best response to some belief.
– If a strategy is strictly dominated, then it is not a
best response to some belief.

Result : In a finite two - player gam e, B1  UD1 and B2  UD2 .

In ggames with more than two players,
p y
the
equivalence between strict domianance and best
response is obtained if beliefs are correlated.
Result : In a finite game, Bi  Bic  UDi for each i  1,2, , n.
10.12.2010
G.B. Asheim, ECON4240-2
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Algorithm for finding Bi  UDi
in two-player games

Step 1: Strategies that are best
b responses to
simple (point mass) beliefs are in Bi.

Step 2: Strategies that are dominated by other
pure strategies are not in Bi.

Step 3: Other strategies can be tested for mixed
strategy dominance to see whether they are in Bi.
10.12.2010
G.B. Asheim, ECON4240-2
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Sophisticated theory of individually rational behavior:
Iterated strict dominance
(Iterated elimination of strictly dominated strategies)


Some examples:
In two-player games, iterated strict dominance is
equivalent to the procedure in which strategies that
are never best responses are removed in each round.
Rationalizability

SStrategies
i that
h survive
i iiterated
d strict
i ddominance
i
are
called rationalizable strategies.

Weak dominance is not sufficient.
10.12.2010
G.B. Asheim, ECON4240-2
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Two problems with iterated strict dominance
1. It is based on an assumption that it is commonly
players
y choose rationally.
y
believed that p
2. In many games there are no strictly dominated
strategies.

Second tension: Common belief of individually
rational behavior
b
may not lead
d to coordination.
d

Question: How to analyze games where iterated
strict dominance has no bite?
10.12.2010
G.B. Asheim, ECON4240-2
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