Static games: Rationalizability
Lectures in Game Theory
Fall 2011, Part 2
24.07.2011
G.B. Asheim, ECON3/4200-2
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Questions to answer:

How do players behave in static strategic situations?

How should players behave in such situations?
Analysis:
y


Apply the normal form (as a representation of
games where all actions are taken simultaneously
and independently).
Model ‘rational behavior’.
24.07.2011
G.B. Asheim, ECON3/4200-2
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Strict dominance

Some examples:
Definition : A pure strategy si of player i is strictly dominated
if there is a stragegy (pure or mixed)  i  Si such that
ui ( i , si )  ui ( si , si ) for all strategy
gy p
profile si  S i of
his opponents. Write UDi  Set of undominate d strategies .

Comments: — A strategy is strictly dominated if there
is another strategy that is better for all opponent choices.
— The dominating strategy may be mixed.
— Weak dominance is not sufficient.
24.07.2011
G.B. Asheim, ECON3/4200-2
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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.
24.07.2011
G.B. Asheim, ECON3/4200-2
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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 is a belief  i  S i such that si  BRi (  i )}.
Player i chooses some si  Bi
24.07.2011
G.B. Asheim, ECON3/4200-2
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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 games with more than two players, 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.
24.07.2011
G.B. Asheim, ECON3/4200-2
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2
Algorithm for finding Bi  UDi
in two-player games

Step 1: Strategies that are best 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.
24.07.2011
G.B. Asheim, ECON3/4200-2
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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
r neverr b
bestt rresponses
p
are
r rremoved
d iin each
h rround.
d
Rationalizability

Strategies that survive iterated strict dominance are
called rationalizable strategies.

Weak dominance is not sufficient.
24.07.2011
G.B. Asheim, ECON3/4200-2
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Two problems with iterated strict dominance
1. It is based on an assumption that it is commonly
believed that players choose rationally.
2. In many games there are no strictly dominated
strategies
strategies.

Second tension: Common belief of individually
rational behavior may not lead to coordination.

Question: How to analyze games where iterated
strict dominance has no bite?
24.07.2011
G.B. Asheim, ECON3/4200-2
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3