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Massachusetts Institute of Technology
Department of Economics
Working Paper Series
WISHFUL THINKING
IN
STRATEGIC ENVIRONMENTS
Muhamet
Yildiz
Working Paper 04-31
September 1, 2004
Room
E52-251
50 Memorial Drive
Cambridge,
MA 021 42
This paper can be downloaded without charge from the
Social Science Research Network Paper Collection at
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,
I
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
LIBRARIES
|
WISHFUL THINKING IN STRATEGIC ENVIRONMENTS
MUHAMET YILDIZ
Abstract. Towards developing a theory
gies, I
of systematic biases about strate-
analyze strategic implications of a particular bias: wishful thinking
about the
strategies. Considering canonical state spaces for strategic uncer-
tainty, I identify a player as a wishful thinker at
a state
if
she hopes to enjoy
the highest payoff that
is
consistent with her information about the others'
strategies at that state.
I
develop a straightforward elimination process that
characterizes the strategy profiles that are consistent with wishful thinking,
mutual knowledge of wishful thinking, and so on. Every pure-strategy Nash
equilibrium
is
common knowledge
consistent with
generic two-person games,
I
further
of wishful thinking.
For
show that the pure Nash equilibrium
strategies are the only strategies that are consistent with
common
knowl-
edge of wishful thinking, providing an unusual epistemic characterization for
equilibrium strategies.
1.
Self-serving biases, such as
common among economic
Introduction
optimism and wishful thinking, are reportedly
agents.
1
There
is
a burgeoning theoretical literature
that investigates the role of such optimistic and heterogeneous beliefs in eco-
nomic applications, such as
financial
markets (Harrison and Kreps (1978), Mor-
(1996)), bargaining (Posner (1972), Landes (1971), Yildiz (2003,2004), Ali
ris
(2003)), collective action (Wilson (1968), Banerjee
and Somanathan
(2001)),
Date: July, 2004.
I
thank Daron Acemoglu, Haluk Ergin, Casey Rothschild, Marzena Rostek and Jonathan
Weinstein for detailed comments.
!
The
(1980),
literature
is
large.
For some examples, see Larwood and Wittaker (1977), Weinstein
and Babcock and Loewenstein
ments that control
(1997). Self-serving biases disappear in
for strategic reporting (Bar-Hillel
(2004)).
1
and Budescu
(1995),
some
experi-
Kaplan and Ruffle
MUHAMET YILDIZ
2
lending (Manove and Padilla (1999)), and theory of the firm (van den Steen
(2002)).
Another
line of research
has already set out to test these theories em-
Farber and Bazerman (1989)).
pirically (e.g.,
players hold heterogeneous priors about
These models
assume that
all
some underlying parameters, and then
apply equilibrium analysis. But their equilibrium analysis assumes away any
strategic uncertainty
—a suspect assumption when equilibrium
is
not implied
by dominance conditions. Indeed, Dekel, Fudenberg, and Levine (2004) demonstrate that
it
hard to interpret such equilibria as outcomes of a learning
is
process.
Most observers agree that
gic interactions
often harder to predict the
than predicting some physical
in physical sciences
sciences,
it is
reality.
After
outcome of
all,
strate-
present theories
have much sharper predictions than the theories in
social
and well-grounded game-theoretical solution concepts have weak
dictive power.
This suggests that
it
model players having heterogenous
pre-
would be methodologically preferable to
beliefs
about strategic uncertainty when-
ever they have heterogenous priors about the underlying parameters of the real
world.
In particular,
self-serving biases
it
seems more
likely that players
about the strategies of others instead of about physical
Metaphorically, one would expect that an individual
believing in her luck
ticket, believing
have more substantial
would also drive over the speed
who buys a
limit
reality.
lottery ticket
and get a speeding
that a police officer would forgive her. This paper takes a step
towards incorporating self-serving biases about the other players' strategies, by
considering "wishful thinking"
,
which the dictionary defines as identification of
Taking
one's wishes or desires with reality.
this
term to
refer precisely to the
this case provides
which allows
extreme case of optimism. The analysis of
an alternative benchmark to equilibrium.
Towards describing the problem,
sis,
this definition literally, I will use
for
let
me emphasize that non-equilibrium analy-
heterogeneous priors about the others' strategies,
the most developed areas in
game
theory.
We
is
one of
have solution concepts, such
as rationalizability (Bernheim (1984), Pearce (1984))
and subjective correlated
WISHFUL THINKING
(Aumann
equilibrium
(1974),
Brandenburger and Dekel (1987)), with
We
understood epistemic foundations.
strategic uncertainty
(Aumann
well-
have various frameworks to capture
under various assumptions about knowledge, and we know
how to check whether the players
gies
3
share a
common prior about
(1976,1987), Feinberg (2000)).
Nevertheless,
the others' strate-
we do not have a
theory that incorporates systematic deviations from the common-prior assumption about strategic uncertainty, such as optimism about the others' strategies.
Towards such a theory of optimism,
form of optimism. Let
certainty
no
me
I will
analyze wishful thinking, the extreme
do contain types with optimism, pessimism,
special attention has
models of strategic un-
reiterate that the standard
been paid to these types.
or wishful thinking, but
I will
simply identify these
types and investigate their behavior.
I will
use
Aumann's
(1976) canonical partition model for strategic uncertainty
to identify whether a player
there
is
a unique strategy
is
a wishful thinker. In this model, at any
profile to
state,
be played. Players do not necessarily know
the state. At each state, each player has an information
cell
states that she cannot rule out at that state.
represents the set of
This
correct assumptions that she takes as given, which
her outside information.
on
this cell,
which
is
is
cell
consisting of the
sometimes referred to as
She also has a (conditional) probability distribution
taken to represent her "subjective" beliefs (about other
players' strategies, etc.)
Consider an information
profiles
cell
of a player. This player takes the set of strategy
played by the other players on this
cell
she can choose any strategy from the set of her
these two sets
cell.
I
is
the set of
all
as given. She also
own
strategies.
own
The product
of
possible outcomes according to the information
identify a player as a wishful thinker at a state
(according to her
knows that
if
her expected payoff
probability distribution) at that state coincides with the
highest possible expected payoff one can ever expect within the set of these
possible outcomes.
This formalization has two noteworthy properties.
is
a property of the information
cell.
Firstly, wishful thinking
Therefore, whenever a player
is
to be
MUHAMET YILDIZ
4
knows that an outside observer
identified as a wishful thinker, she
her as such (although she would probably not accept that she
is
identifies
afflicted
with
the psychological anomaly of wishful thinking).
Secondly, a wishful thinker's
strategy-belief pair maximizes her expected payoff
among
all
her strategy maximizes her expected payoff given her
ticular,
must be
While wishful thinking
rational.
of irrationality involving self deception
is
such pairs. In parbeliefs.
Hence, she
popularly considered to be a form
and misperception of the world, wishful
thinking formally implies the standard game-theoretical notion of rationality.
~In application,
is
and so on.
are mostly interested in the strategic implications of wish-
We also want to know how these implications change when wishful
ful thinking.
thinking
we
mutually known, when this mutual knowledge
It is especially
mon knowledge
sistent
mutually known,
important to understand the implications of com-
of wishful thinking. This
that are consistent with
is
is firstly
common knowledge
because the strategy
of wishful thinking
2
profiles
are also con-
with mutual knowledge of wishful thinking at arbitrary order, and hence
these are the strategy profiles that remain possible as
knowledge of wishful thinking. More importantly,
we
it is
allow
more and more
highly desirable from a
methodological point of view to disentangle the implications of heterogeneous
priors or self-serving biases
from those of asymmetric information. Researchers,
such as those in the literature discussed above,
assuming that there
is
commonly accomplish
no asymmetric information.
3
this
by
(This also allows the re-
searcher to focus on belief differences without having to deal with asymmetric
information.) In keeping with this methodology,
implications of wishful thinking
it
—
i.e.,
when
wishful thinking
Unfortunately,
it
is
when
is
often too
there
common
is
it is
desirable to examine the
no asymmetric information about
knowledge.
cumbersome to determine such implications
through an epistemic model, which consists of a state space, partitions and
2
A
there
strategy profile
is
is
consistent with
(common knowledge
a model and a state at which the strategy
that) the players are wishful thinkers.
3
See Squintani (2001) for an exception.
profile is
of) wishful thinking
played and
(it is
means that
common knowledge
WISHFUL THINKING
conditional probability distributions.
More
5
problematically, the set of impli-
cations often depends on the model, as models often contain
knowledge" assumptions. To overcome
this,
I
many "common
develop a straightforward iterated
elimination process on the strategy profiles that characterizes the strategies that
are consistent with wishful thinking, mutual knowledge of wishful thinking, and
The
so on.
thinking
application of this procedure allows a researcher to use wishful
—the extreme form of optimism—
as
an alternative benchmark to equi-
librium.
~It turns out that there
wishful thinking
equilibrium
because
it
is
is
a strong relationship between
and equilibrium
are rational).
More
each strategy that
must be played
in
Firstly, every pure-strategy
Nash
of wishful thinking, simply
room for any strategic
uncertainty (and the players
interestingly, for generic,
is
of
common knowledge
consistent with
does not leave any
strategies.
common knowledge
consistent with
two-person games,
common knowledge
some pure-strategy Nash
terization of equilibrium strategies
4
I
show that
of wishful thinking
equilibrium. This yields a charac-
by common knowledge of wishful thinking.
This characterization illustrates that, unlike most existing iterative elimination
processes, such as iterated dominance, the elimination process above leads to
strong predictions. Notice, however, that the predictions are not so strong as
to imply equilibrium outcomes, as the equilibrium strategies played by
differ-
ent players need not match. For example, in the battle of the sexes, there are
three strategy profiles that are consistent with
thinking: the two pure strategy equilibria
common knowledge
and the
profile in
of wishful
which each player
plays according to his or her favorite equilibrium, believing that her favorite
equilibrium
possible to
4
is
to be played. It turns out that, in generic two-person games,
compute the
Since wishful thinking
and much milder than
are
(ii)
(i)
is
full
set of strategy profiles that are consistent with
an ordinal notion (Remark
2),
it is
common
the precise assumptions are ordinal
genericity assumption in normal form.
The
precise assumptions
each player has a unique best reply for each pure strategy of the other player and
player 1
is
never indifferent between two pure strategy profiles at which she plays a best
response to the strategy of player
2.
MUHAMET YILDIZ
6
knowledge of wishful thinking by applying the elimination process only once to
the product set of equilibrium strategies.
The above
characterization suggests that wishful thinking
mon knowledge
only in strategic situations in which there
wishful thinking,
when
strategic uncertainty
is
wishful thinking has
At the
thinking
is
states in
common
which equilibrium
not
much room
impact on players'
beliefs.
for
The
reduced to the uncertainty about the equilibrium that
to be played. This does not, however,
vanish.
little
is
may become com-
mean
which the outcome
is
that strategic uncertainty needs to
is
not an equilibrium (but wishful
knowledge) the players have substantial uncertainty about
is
,
played and exhibit a clear form of wishful thinking.
This characterization also provides partial support
for
the theoretical litera-
ture that uses equilibrium analysis to study the behavior of optimistic players.
It
shows that
ogy
if
a researcher allows wishful thinking but sticks to the methodol-
in this literature (for the
equilibrium strategies.
generic two-person
(ii)
motivation above), then she can simply focus on
The support
is
partial because
games while the above models are
(i)
this is true only for
typically non-generic
and
outcomes need not be equilibria due to mismatch, and economic implications
of these profiles
A
number
may
substantially differ.
of authors have developed general
game
theoretical
models that
incorporate deviations from expected utility maximization and psychological
motivations (such as Geanakoplos, Pearce, Stacchetti (1989)). These models led
to incorporation of non-traditional motivations, such as fairness (Rabin (1993)),
into
game
theory. Eyster
and Rabin (2000) propose an "equilibrium" notion in
which players underestimate the correlation between the other players' action
and private information while they correctly estimate the distribution of actions.
Rostek (2004) proposes some set-valued solution concepts and discusses various
decision rules
on these
sets.
though with a somewhat
I
One
of these decision rules reflects wishful thinking,
different formulation.
formulate the problem in the next section and investigate the strategic
implications of wishful thinking in Section
3.
Section 4 concludes.
I
present
,
WISHFUL THINKING
some
an appendix. In a technical appendix,
related concepts in
game
x
x Sn
•
is
(TV, S,
N=
u) where
{1, 2,
.
,
.
.
n}
the finite set of strategy profiles,
function of player
utility
i
text.
Formulation
2.
Consider a
more
present
I
about the elimination process and prove the results in the
results
Si
7
for each
£ N. In
i
5
the set of players,
is
S — R is the
players well may have
and
this set up,
S=
Ui
>
:
asymmetric information or heterogeneous priors about the physical parameters. 6
These aspects are suppressed
uncertainty. Strategic uncertainty
a model. Here,
called
fl is
on
in the notation, in order to focus
is
modeled through a
triple
strategic
I,p), which
(fi,
a state space that represents the players' uncertainty
about the strategies, where a generic member
£
uj
contains
ft
information
all
about players' strategies, including what each player knows. For each
Ii is
the information partition of player
that contains u. Here,
ui
according to player
ways. First,
it
An
may represent
interpretation, at
uj,
player
i
I
i;
write
Ii (u>)
information partition
all
interpreted in two
the "objective" information player
knows that one of the states
in
may
i
has. In this
/; (to)
occurs but
take an information
partition as a representation of possible sets of assumptions a player
as given. In this interpretation,
Ii (to)
Ki (F)
=
(F)
knowledge
I
is
=
C\ i&N
denoted by
-1
CK (F).
Write
use the notational convention of x
The
i
The
(F)).
e X-i =Uj ^ X
i
:j,
C F}
knows that
F
The mutual
occurs.
<r,
=
(xi
and
x=
priors.
=F
is
common
for the strategy played
by player
set of states at
(uj)
F
(F)
,
xn )
€
which
Xj x
•••
x
Xn
,
x_;
=
{x it x-i).
strategic-form representation here allows chance moves, about which players'
have non-common
take
Q,
m where K°
is represented by operator K
m>
Ki (iT™
(xi,...,Xi-i,x i+ i,...,x n )
6
FC
{u\Ii (y)
denotes the set of states at which player
and K m
may
corresponds to states at which a certain
assumptions holds. Given any event
knowledge at any order
£ N,
for the information cell
may be
cannot rule out any of these states. Alternatively, one
set of
i
the set of states that are indistinguishable from
7; (ui) is
i.
is
may
,
MUHAMET YILDIZ
8
i
at state
uj.
Each player knows her strategy so that
At u, player
i
to
pitU}
E
denoted by
is
itU1
payoff that
is
cell
representing
7, (uj),
respect
In this paper, the probability distribution pi>ai
.
probability distributions to identify whether a player
will define
on
The expectation operator with
taken to be a representation of player's "subjective"
I
on each
c^ is constant
also has a probability distribution pi tU}
her beliefs about the state of the world.
I will
beliefs.
is
use these
a wishful thinker.
is
a wishful thinker as a player who expects to enjoy the highest
possible given her information about the other players' strategies
and given that she can choose any of her own
and only
uj if
(2.1)
Ei^[ui(a)]= max E„
taken over
is
.
<r_j
(7
on
[u; (s)]
{s;},
=
v
which the player knows her own strategy
1
said to
x/i
all beliefs
distribution that puts probability
i is
if
u=S 3
where maximization
A player
strategies.
be a wishful thinker at
in
,
Sj.
5 Si
x
/j,
Here, 8 Si
and a-i
own
5,
7
x
(uj))
the probability
is
(7, (uj))
action profiles s_i of other players that are consistent with
the standard assumption that player knows her
on
is
the set of
{uj).
all
Notice that
strategy has no bite in this
definition. In fact,
max. Ui
(2.2)
(s)
= max^
s
where
s
is
<S a
maximized over
Si
.
x/i
x
tions are over the set of beliefs
s l.
x „ [u; (s)]
<t_, (ij
on
Si
= maxEv
kn
(s)l
v
(w)),
x
and the second and third maximiza-
cr_i (Ii (lo))
with and without the above
assumption, respectively. Clearly, one can rewrite the condition
player
i is
a wishful thinker at
E
(2.3)
i>U]
where maximization
Si
E
Si.
That
is,
is
uj if
[ui (a)]
and only
= max £„
taken over
[u { {s h
all beliefs fi
is
s^))
on
,
o"_j (Ii (uj))
and
strategies
the strategy and the beliefs of a wishful thinker are as
make
equivalent definitions are used interchangeably.
i
a wishful thinker
is
that a
if
chooses her strategies and beliefs in order to
a player
(2.1), so
The
denoted by Wi;
herself feel happy.
set of all states at
W
=
n, e 7vWi
is
if
she
These
which
the event
WISHFUL THINKING
that everybody
is
a wishful thinker.
and other related concepts.)
Notation
Write
belief
if /i
1.
BRi
(/i)
for
another possible
BR^
Write
strategies.
i
against his
BRi
(s_;) instead of
(/z)
assigns probability 1 on s_;. Write
Bi
for the
=
{(,§,,
1
A player
.
thinker in this paper:
;
)
\ii
e BRi
if
u£
Wi, then
notion.
The
2.
A
first
,
s_i
e S-i}
profiles.
Ii (lo)
Ki {Wi)
Remark
(s_j)
always knows whether she
(2.4)
is
5_
graph of BRi on pure strategy
Remark
she
A
for the set of best responses of player
about the other players'
/j,
Appendix
(See
formalization
9
=
C
is
to
be
identified as
a wishful
Wi. Therefore,
Wi.
equality in (2.2) shows that wishful thinking
is
an ordinal
player's attitudes towards risk are irrelevant in determining
whether
a wishful thinker at a given state. The strategic implications of wishful
thinking are invariant to monotonic transformations of payoff functions.
The
only non-generic situations one must ever rule out regarding wishful thinking are
indifferences
to the
between certain pure strategy
game
profiles.
as pure strategies, again, has at
Addition of mixed strategies
most only
trivial effect
on the
strategic implications of wishful thinking.
Remark 3.
Firstly,
it
it is
Use of Aumann's (1976) partition model is justified on two grounds.
considered to be the canonical model of interactive epistemology, as
reflects the
does not
most stringent assumptions about knowledge, namely: a person
know a
false
statement as truth; a person knows whether she knows;
and she can use conjunction to make further
inferences. Exploring the strate-
gic implications of wishful thinking within this
model allows
the effects of wishful thinking from the possible effects that
me
to disentangle
come form drop-
ping some of these standard assumptions of game theory. Nevertheless,
all
of
these assumptions have been challenged in modeling bounded rationality and
strategic uncertainty.
It will
therefore be important to extend the analysis to
MUHAMET YILDIZ
10
such more permissive models.
Secondly,
requires a set of assumptions or
my
formulation of wishful thinking
an objective assessment of the world as a
ref-
erence in addition to individuals' subjective assessments. Aumann's canonical
model
perfectly meets this need as
and conditional probability
it
consists of partitions of the state space
distributions. This allows a parsimonious
model of
wishful thinking. In purely subjective models of beliefs, such as the universal
type space of Mertens and Zamir (1985) on the strategy space, one needs to add
such sets of correct assumptions or reference
beliefs to
the model. Since these
reference beliefs are irrelevant to the players' decision problems, such a model
be
will
less
parsimonious and
3.
I
will characterize
is
7
the strategies that are consistent with wish-
mutual knowledge of wishful thinking, and so on. The character-
ization will be given
thinking
hoc.
Strategic Implications of Wishful Thinking
In this section,
ful thinking,
may appear ad
by an
iterative elimination process.
stronger than rationality,
Firstly, since wishful
common knowledge of wishful
lead to a refinement of rationalizability,
i.e., all
non-rationalizable strategies will
Nash equilibrium
eventually be eliminated. Secondly, any
survive the iterated elimination process.
More
thinking will
in
pure strategies
surprisingly, I will
show that
will
for
generic two person games, these are the only strategies that survive. This will
yield
an unlikely epistemic characterization
for
pure-Nash-equilibrium strate-
gies.
I
two lemmas that describe certain properties of the
will first present
played by a wishful thinker.
elimination process.
some pure strategy
A
I
will use these properties to define the iterated
wishful thinker always gives a best response against
profile (see
thinker plays a strategy
s;
Lemma
4 in the Appendix.) Hence,
knowing that some event
F is true,
Defining optimism does not require an objective reference, and one
optimism
for
strategies
if
a wishful
then she must be
may be
able to define
in a purely subjective model, using the other players' beliefs as the reference (see
example Yildiz
(2004)).
WISHFUL THINKING
11
targeting a best reply against a pure strategy s_; that
F. This idea
Lemma
(si,s-i)
is
FC
f} 7
G N, and any
G a {Ki (F)
s
Wi), there exists
fl
G BiDa(F).
Consider a wishful thinker
By Lemma
Si.
i
consistent with event
Lemma.
formalized in the following
For any
1.
is
1,
who knows an
event
F and plays
a strategy profile
she must be targeting a utility level by playing
strategy profile S-i of others that
profile s_i of others that
is
consistent with F.
Sj
against a
Now if there is
a strategy
possible in this information set, she
is
must not be able
to get a higher payoff by playing her best response to s_;. For otherwise, she
would obtain a higher payoff by believing that the other players play
yields the following
Lemma
2.
FC
Q,
max it;
(si,
i
e N, and any
s_;)
<
^
s
G a
max
{K (F)
fl
becomes a
special case of
side of the inequality
hand
side
is
not
Ui(si,s_i).
is
— oo,
Lemma
any
2; for
s,
i.e.,
in
is
— oo, Lemma
the expression on left-hand
there exists (§i,s_i) G B{
wishful thinkers. For any generic
G B{
set
a number, requiring that the expression on the right-
out most strategy profiles as strategic outcomes
s, s'
Wi),
(s 1 ,s_ l )6B i na(F)
Under the usual convention that maximum of an empty
1
This
lemma.
For any
(3.1)
s_j.
which player
i
fl
o{F).
when some
game and any two
(s)
<
Ui (s
1
)
2 rules
of the players are
distinct strategy profiles
plays a best response, either
eliminated depending on whether Ui
Lemma
or Ui (s)
(s^, s'_^j
>
or
Ui (s').
(s^,
s_;)
is
This leads
to a powerful elimination procedure, which characterizes the strategy profiles
that are consistent with wishful thinking, mutual knowledge of wishful thinking,
and so on.
Elimination Procedure.
(1) Initialization: Set
(2) Elimination:
(3.1) for
X" =
1
S.
m > 0, eliminate
~
and for a (F) = X m
For any
some
all
l
i
,
the strategy profiles s that
fail
using the usual convention that
MUHAMET
12
maximum
profile
YILDIZ
— oo.
over an empty set yields
Xm
Call the remaining strategy
.
(3) Iterate step (2).
Note that
no
s_i
this
is
such that
(3.1) fails for
if It is
(§i,
s_i
6 Bi
s_;)
~
f]X m
1
Note
profiles.
some strategy
for
each (s^s^^, and the entire strategy
is still
then
iteration,
s2
§i is
a best reply only to s_; and the strategy profile
some previous
at
an elimination of strategy
,
also that,
s_;)
—even
available. In this way, the elimination
That
eliminated.
(§i,
there
is
then the inequality
be eliminated
s 2 is to
if
is,
was eliminated
if
some part
of
procedure above contains the
iterated elimination of strategies that are not a best reply to a pure strategy.
More
formally,
<t>{X)
(3.2)
I
define a
Xn
—
mapping
slVi
<
:
X where
set yields
I
2
:
—>
s
2
s by setting
<
(s t s_i)
,
max
use the usual convention that the
— oo. The
sequence
(X
(si,S-i)
(si,s-i)€BinX
Si
[
at each
max itj
4>
_1
,X°,
.
.)
.
maximum
over the
recursively defined
is
by
empty
X =S
_1
and
Xm =
The
limit of the
sequence
<j)
(X™-
(m >
1
)
0)
.
is
Xm
X°°= f)
.
m=0
many strategy profiles, the elimination process stops
and we have X°° = X m for some m. I will now illustrate
Since there are only finitely
at
some
how
iteration
771,
the elimination procedure
Example
1.
is
applied.
Consider the following two-person game, where player
between the rows, and player 2 chooses between the columns:
a
j3
a
3*,0
-1,0
0,0
0,2*
b
0,0
2*,1*
0,0
0,0
c
0,0
0,0
r,2*
r,o
d
2,3*
1,0
0,0
0,0
7
S
1
chooses
WISHFUL THINKING
13
In this table, the asterisk after a player's payoff indicates that the player
Notice
playing a best reply to the other player's strategy at that profile.
that no strategy
is
weakly dominated, and hence
ble (and rationalizable)
m=
For Player
0.
1,
to any pure strategy.
max(()]S2
)
6 s 1 ui (b, S2)
No
eliminated.
ple, (a,
0)
=
the strategy d
For player
<
2
other profile
is
is
771
=
1,
player
2.
This
777
=
eliminated.
strategy
eliminated.
For
again, the profile
(3.
2,
is
a
X°
it is
(6,
a)
2,
among the strategy
P
a
3*,0
-1,0
0,0
0,2*
b
0,0
2*,r
0,0
0,0
c
0,0
0,0
r,2*
r,o
d
2,3*
1,0
0,0
0,0
(a, 5)
is
exam-
>
profiles that
u-i (b, j3);
a best reply only against
=
d,
but
X
eliminated for
is
1
(d,
—
a) has been
There are no more eliminations
for
strategy a
not a best reply for player
any remaining
is
no
5
7
(or simply the strategy profile (a, a))
a
are
(c, j3)
consists of the strategy profiles in bold:
a
because
a) and
at this stage. For
eliminated as U2
(a, /3) is
eliminated, as
targets the payoff of 3, while she
1
For player
1
not a best reply
is
(si, a). Similarly, (c,
eliminated for player
have not been eliminated already,
For
1
eliminated, as
is
= max Sl u\
3
can get only 2 by responding to
other profile
strategies are admissi-
Let us apply the above elimination procedure. Take
.
not eliminated because player
is
all
is
eliminated as a
This
profile in the first row.
is
is
777
1:
X°\{(a, a)}.
1 in
the only eUmination. Hence,
consists of the boldface strategy profiles in the second
= 3, strategy § is eliminated. The elimination
X = X" = = X°° = {(b, p) (b, 7) (c, 7)}.
777,
X
2
and third rows above. For
process stops here. Therefore,
3
•
•
,
Remark
4.
The
elimination process
(y). Hence, the limit set
strategies at
The next
,
some step
is
monotonic,
X°° does not change
if
i.e.,
one
X
fails
C Y
=> 4>{X)
C
to eliminate certain
or applies different orders.
result states that
Xm
is
precisely the strategies that are consistent
with mth-order mutual knowledge of wishful thinking.
Therefore, using the
MUHAMET YILDIZ
14
elimination procedure above, a researcher can investigate the strategic implications of wishful thinking directly from the strategy profiles
—without dealing
with abstract, complicated models of strategic uncertainty.
Proposition
1.
For any model
(f2,
I,p),
and any
a (K m (W)) C
Xm
m>
0,
;
in particular,
a(CK(W))CX
c
Moreover, there exist models (fl,I,p) in which the above inclusions are equalities.
The
first
statement
is
given by inductive applications of
The proof of the second part
X m where
profile in
thinking
,
is
involves constructing a
the strategy profile
is
Lemmas
submodel
for
1
and
each strategy
played at a state in which wishful
mth-order mutual knowledge. One constructs such a model using
information sets with only one or two states, and at such information
Lemmas
1
for the
Example
common knowledge
case in
Example
Q = X°° and a (lu) = u
also
=
I will
next present a
1.
1 (continued). In the following model, wishful thinking
knowledge, and each strategy profile in X°°
hence/1
cells,
and 2 characterize the wishful thinking behavior. Integrating these
models into one model, one obtains the desired model.
model
2.
{{(6,/5)
Pi,(6,/3) ((&, /3))
)
=
at each u.
=
=
1-
,
and
P2,{c,-y) (( c i7))
common
played at some state.
Each player knows her own
(6,7)},{(C 7 )}}and72
1
is
is
{{(b, 5)}
y
)
{(6,7),(c,
This model
is
Take
strategy;
7 )}}. Take
described in the
WISHFUL THINKING
following diagram,
Clearly, each player
K
2
(W)
=
where the information
is
Notice in Example
1
Player
cells of
a wishful thinker at each
CK (W) = fl = X°°.
=
15
state,
a
Therefore,
that every strategy that
is
are rectangular:
1
and hence
W=K
(CK (W)) =
1
is,
in fact, not
=
X°°.
part of a profile in X°°
a part of a pure-strategy Nash equilibrium. This
(W)
is
also
a coincidence.
Let
NE =
be the
in
set of all pure-strategy
which player
be the
Nash
i
Nash
B
p|
%
equilibria; recall that
Bi
is
the set of profiles
plays a best reply to others' strategies. Let also
NEi =
{si|3s_i
set of all strategies of
a player
:
i
(s i: s_i)
E
NE}
that are played in
some pure-strategy
equilibrium. Similarly, let
X°°
=
{si|3s_ i :( Si S _ J
,
be the set of all strategies of a player i that
of wishful thinking.
The next two
is
)eX 00 }
consistent with
common knowledge
propositions establish the close relationship
between the pure strategy Nash equilibria and
common knowledge
of wishful
thinking.
Proposition
mon
2.
Every pure strategy Nash equilibrium
knowledge of wishful thinking:
NE C X°°.
is
consistent with com-
MUHAMET
16
In a pure-strategy
YILDIZ
Nash equilibrium there
no strategic uncertainty, and
is
hence players do not have any freedom of entertaining different
all
players, independent of their level of optimism, hold the
Nash equilibrium
Formally, any pure-strategy
is
beliefs.
same
Hence,
correct beliefs.
consistent with a
model with
a single state at which each player plays according to the equilibrium. In such
a model,
common knowledge
it is
that each player
a wishful thinker. The
is
next proposition states a more surprising and substantive fact about two-player
games. The assumptions of this proposition generically hold.
Proposition
3.
For any two-person game, assume
a unique best reply
any two
Sj
£ BRi
(s-i),
distinct strategy profiles
strategies are consistent with
s, s'
common
X°°
(3.3)
and
(ii)
E B\
Player
(i)
for each S-i, there exists
1 is
not indifferent between
Then, only pure Nash equilibrium
.
knowledge of wishful thinking:
= NEi
(Vie
TV).
Moreover,
(3.4)
=
X°°
=
<f>
(NE
l
x
NE
2)
{(si,s 2 )\3(si,s 2 ),(s 1 ,s 2 )
e
NE
Ui(s 1 ,s 2 )
:
>
«i(si,s 2 ),
U2(S1,S 2 )
The
first
part of the proposition characterizes
thinking in terms of strategies.
It states
only strategies that are consistent with
are the pure equilibrium strategies.
common knowledge
of wishful
that in a generic two-person game, the
common knowledge
The second
of wishful thinking. It states that this set can be
the elimination procedure only once to the set
as follows.
of wishful thinking
part gives a practical charac-
terization for the strategy profiles that are consistent with
The proof can be summarized
> U 2 (s U S 2 )}.
common knowledge
computed by simply applying
NEi
First,
x
NE2
.
under assumption
(i),
one
can show that the restrictions of the best response functions to X°° (which are
defined from X°° to Xf°) are well-defined, one-to-one, and onto. Together with
assumption
is
(ii),
this allows
one to rearrange the strategies so that B\ D X°°
equal to the diagonal of X^° x X£° and the payoff of player
1 is
decreasing
—
WISHFUL THINKING
along this diagonal
—as
Example
in
As
1.
17
Example
in
1,
this implies that all
the strategy profiles that are under the diagonal and in X^° x X%° must have
been eliminated. Using arguments similar to the one used to eliminate strategy
profile (a, a),
one then shows that
Player 2 does not give a best reply in a
if
strategy profile on the diagonal, then the strategy of player 2 must have been
eliminated. Therefore, the diagonal of
Since the elimination process
(3.3).
would not change
if
is
X%° x X%°
is
monotonic,
first step,
NE.
This proves
(3.3) implies that the result
one started elimination from the
elimination stops at the
equal to
set
NE\ x NE2 The latter
-
yielding (3.4).
This provides an unusual epistemic characterization for pure-Nash-equilibrium
strategies in terms of
tantly,
it
common knowledge
suggests that there
the wishful thinking
strategic uncertainty
Nevertheless,
is
is
is little
common
left for
knowledge. In
More impor-
optimism or pessimism when
that the
fact, (3.4) establishes
reduced to uncertainty about the equilibrium played.
common knowledge
librium strategies
room
of wishful thinking.
of wishful thinking
not by equilibria.
As
characterized by equi-
is
stated in (3.4), two players
may
play
equilibrium strategies that correspond to two different equilibria at some state
in
which wishful thinking
is
common
knowledge. In such a state there
substantial strategic uncertainty remaining,
wishful thinking. For example, at state
(6,
and players exhibit a
7) in
Example
1,
rectly believes that they will play her favorite equilibrium.
fact.
By
(3.4), if
the outcome s
=
(si,.§2)
6 X°°
is
then there are two equilibria (si,s 2 ) and (si,S2) €
sider as possible
and
142(51,52)
>
and rank
in diagonally
U2(si,S2).
If
own
form of
clear
each player incorThis
is
a general
not an equilibrium already,
NE
that the players con-
opposing orders: ui(Si,
Each player plays according
equilibrium, believing that her
is still
favorite equilibrium
is
S2)
to her
>
u i( s i, £2)
'
own
favorite
to be played.
the equilibria are strictly Pareto-ranked, then the players cannot have such
opposing rankings. In that case, the outcome
Corollary
1.
Under
is
necessarily an equilibrium.
the (generic) assumptions of Proposition 3, if the equilibria
are Pareto-ranked with strict inequalities, then
X°°
= NE.
MUHAMET
18
YILDIZ
Proposition 3 has established already that for generic two-person games,
wishful thinking
common
is
knowledge, strategic uncertainty
certainty about which equilibrium strategies are played.
can indeed construct a model in which wishful thinking
all
strategy profiles that are consistent with
As
is
is
in
reduced to un-
Example
common
That
but they
may
is,
disagree about which equilibrium
Corollary
{CK (W)) =
which a
1, i.e.,
Under
2.
each player
is
is
one
knowledge,
1
to an equi-
the players are in agreement that an equilibrium
following corollary, which
1,
common knowledge of wishful think-
ing are played, and at each state each player assigns probability
librium.
when
is
played. This
is
is
played,
stated by the
suggested by Haluk Ergin.
the assumptions of Proposition
X°° and for each u G
certain that
CK (W)
an equilibrium
3,
there exists a
andi, pitU1 (a'
1
model in
(NE) n h
(u))
played.
is
Remark 5. Common knowledge of wishful thinking differs from usual epistemic
foundations for equilibrium, such as the sufficient conditions of
Brandenburger (1995). For example,
common knowledge,
sense of
at state (6,7) in
Example
Brandenburger and of
this paper),
other player's conjecture, but she
The next two examples show
when
first
is
(in
is
the
and these conjectures
do not form a Nash equilibrium. Notice that each player
The
rationality
but the players do not know each other's conjectures
Aumann and
superfluous.
1,
Aumann and
is
certain about the
wrong.
that the assumptions in Proposition 3 are not
one shows that the characterizations need not be true
there are three or more players.
Example
Consider the following three- player game where player 3 chooses
2.
the matrices (A and p denote the matrices on the
Below
Ei
:
S—
*
(0, 1), i
left
and
right, respectively).
G N, are arbitrary functions; arguments are suppressed.
I
r
t
2,— £2,— £3
-£i,0, -£ 3
b
— £1,— £2j — £3
1,1,1
I
— £i,£2,£3 — £1,— £2 ,£3
£l,£2,3
^1,
— ^2,— £3
WISHFUL THINKING
Check that
NE = {(&,
A)
r,
,
(b,
X°°
l,p)}, but
=
t is
Example
matching-penny game, X°°
that assumption
(i) is
and
r.
but X°°
=
Hence, assumption
The next example
whenever
Example
1,
A)}, so that the non-
=
NE =
S, while
In the
3.
thinking.
0, showing
game
r
I
{(£,/)},
{(6,
not superfluous in Proposition
'
NE =
S\
common knowledge of wishful
consistent with
equilibrium strategy
3. In the
19
2*
t
2*,
b
2*,0
i*,o
o,r
5\{(t, r)} containing non-equilibrium strategies b
5
(ii) is
not superfluous, either.
a generic two-person game, X°°
illustrates that, in
=
NE = 0.
4. Consider the following
game with no
Nash
pure-strategy
equilib-
rium.
/
r
0,2*
3*,1
2*,0
1,3*
r
t
Now, X°
=
and
(b,
respectively. Since
B2
{(tj),(t,r)} as
for players 2
and
and hence
r) is eliminated.
{(t, I)}.
(t,
But, 5i
1,
=
{(*, r)
,
(b,l)
No
(6, /)},
4.
strategy
r) are deleted at the first iteration
=
is
and hence
{(&, r)
,
(£,
I)},
X°
B2 =
(1
eliminated for player
X
l
f\B 1
=
1,
{(t, I)},
and
0. Therefore,
X =
1
X = 0.
2
Conclusion
Self-serving biases are reportedly
common. Moreover, given the
of strategic uncertainty, one would expect to have
biases about other players' strategies. It
is
more
What
is
substantial self-serving
surprising then such systematic bi-
ases about the other players' strategies are typically
even in the literature on optimism.
elusive nature
assumed away by modelers,
worse, there
is
no game theoretical
framework that incorporates such systematic deviations from the common-prior
MUHAMET YILDIZ
20
assumption
—although the use
ing mainstream in
game theory.
of such deviations, focusing
thinking.
gies.
I
player
I
of heterogenous priors about strategies
In this paper,
first
step towards a theory
on the extreme form of optimism, namely wishful
develop a framework for analyzing wishful thinking about strate-
is
a wishful thinker and develop a straightforward elimination process
on strategy
profiles to characterize the set of strategy profiles that are
consistent with wishful thinking,
I
take a
becom-
use the canonical model for strategic uncertainty to identify whether a
directly
on.
I
is
further
show that
mutual knowledge of wishful thinking, and so
in generic
two-person games, pure Nash-equilibrium
strategies are the only strategies that are consistent with
common knowledge
of wishful thinking. In such games, wishful thinking can be
common knowledge
only in cases in which strategic uncertainty
the equilibrium that
is
played,
and
information about whether a player
reduced to uncertainty about
a researcher assumes away asymmetric
if
is
is
a wishful thinker in order to disentangle
the effects of wishful thinking from that of informational differences, then he
may
as well focus
the hope that one
on equilibrium
may be
strategies.
My
analysis
is
elementary, raising
able to develop a tractable framework to incorporate
self-serving (or other) biases into
game
theory.
Appendix A. Other Related Concepts
In this section,
and
I will
relate wishful thinking to other concepts, such as
Notation
2.
Write
/.i
players' strategies at
A
develop another natural formalization of wishful thinking
player
is
i
lu>
=
pi iU o a~_\ for the belief of player
said to be rational at
which player
is
rational
the event that every player
is
rational.
states at
u
is
if
when her
o\ (u)
G BR{
denoted by Ri\
about the other
{u-iJ)-
R=
The
set of
Ci^^Ri denotes
Since a wishful thinker maximizes her
expected payoff over both her actions and
her expected payoff
i
rationality.
ui.
i
all
optimism and
beliefs,
her strategy
beliefs are fixed at the
must maximize
"optimum",
i.e.,
she must
WISHFUL THINKING
be
That
rational.
is,
for
each
i,
Wi C
(A.l)
A player
i is
21
IU.
u if she expects to enjoy
said to be fatalistically wishful thinker at
the highest payoff available at
/; (to)
the set of
,
the states that are consid-
all
ered possible according to the objective information player
i
has, or considered
consistent with the given assumptions. Write
=
Wi
u\Ei^
I
= max
[ui (a)]
u (a
for the set of all the states at
which player
i is
a
Notice that in the standard epistemic model that
her
own
The
action.
player takes her
make
own
herself happy.
even happier
Example
if
5.
(u/))
t
Lj'€li(w)
[
definition
>
J
fatalistically wishful thinker.
is
used here, a player knows
above therefore considers a situation
in
which a
action as predetermined and chooses a belief in order to
Hence the
She could expect to be
qualification fatalistic.
she did not take her
own
actions given
—as
in the next example.
Consider the following game in which the column player can be
taken to be Nature:
I
r
0,0
3,0
2,0
1,0
'
t
=
Consider Ix (u)
and a 2
=
(w')
r.
{u,
u>'}
with p 1>u>
Clearly, Player 1
to her belief that action
I
is
is
(uj)
=
1,
o\ (w)
rational at
u
=
as b
played with probability
o\ (a/)
is
1.
=
b,
a 2 (u)
(<j))
=
2
= m (a (w)) >
Nevertheless, there
is
ni (a (u/))
She
is
a fatalistically
all
particular,
it is
t
1 is
not truly a wishful thinker.
both actions of Player 2 are possible.
her actions are possible, as she
possible that she plays
(u>):
1.
a sense in which Player
Clearly, according to her information,
Moreover,
=
I,
the unique best reply
wishful thinker because she expects to enjoy the highest possible payoff at 7i
Ei,u (ui
=
is
free to choose
and Player
2 plays
between them. In
r. If
she had believed
MUHAMET
22
YILDIZ
that this outcome will be realized, she would have been happier (obtaining
expected payoff of
3).
Since a wishful thinker maximizes her expected payoff over both her actions
and
is
beliefs,
her chosen belief must maximize her expected payoff when her action
and her
fixed at her chosen action
beliefs are varied,
she must also be a
i.e.,
fatalistically wishful thinker:
W
t
Example
in
some
5 shows that the converse
C WiHRi.
is
not true,
the inclusion above
i.e.,
(generic) cases.
The next lemma
implies that a rational
and
fatalistically wishful thinker also
when
gives a best response against a pure strategy profile
in
normal form. Hence, the strategic implications of
and
is strict
the
game
generic
is
fatalistic wishful
thinking
rationality are stronger than iterated elimination of strategies that are not
a best reply to any pure strategy.
Lemma
3.
For any generic normal-form game and any
(A.2)
/x
i>u,
Proof. Consider a generic
Ui
(o~i (u>)
Since
,
S-i)
u 6 Wi,
Ui (dj (u)
,
s_i)
implies that
7^
Ui
({s_ 2 |a, (w) 6
game and any
(ctj (u>)
,
s'_j)
lo
>
Ui
(o"i (lo)
G BRi
,
The next example shows
iuJ )
= BR
=
=
1 for
t
some
€ a_i
6
s_j
(Ii (oj)).
o~-i
generic,
<x_i (/; (w)).
(U
Since
is
lu
(u>))
G
R
where
l:
this
(s_i).
that genericity requirement
hence the strategic implications of
1.
G Wi D Ri. Since the game
s_j) for each s_;
(/J.
(s_;)})
any two distinct s^jS^t G
this implies that pi >UJ (i-i)
er; (lo)
generic games.
for
BR,
weH^fl Ri,
fatalistic wishful
is
not superfluous, and
thinking
may
differ in
non-
.
WISHFUL THINKING
Example
Example
6. In
5,
replace the
23
game with the
following
game
against
Nature,
and
let
p lcJ
(ui)
=
Check that player
p\ <u (u')
1 is
=
I
r
t
0,0
3,0
m
3,0
0,0
b
2,0
2,0
'
example unchanged.
1/2, leaving the rest of the
and a
rational
fatalistically wishful thinker.
Wishful thinking can be defined equivalently in terms of extreme optimism.
To
this end, rewrite (2.3) as
E
i>ul
= max max E„
[ui (a)}
[ui (si, s_j)]
Hence, a wishful thinker can be considered as a person
maximize her payoff knowing that she
Hence, wishful thinking
beliefs.
is
A
player
is
similarly as the
i<UJ
[ui (a)}
= minmaxi^
[ui (s i;
chooses a belief to
with respect to her
rationality.
maximal pessimism under
s_ z )]
u> if
.
Si
\1
That
will act rationally
said to be extremely pessimistic at
E
(A.3)
who
maximal optimism under
Extreme pessimism can be defined
rationality.
.
she holds the most pessimistic belief under the constraint that she
is,
acts rationally. Notice that extreme pessimism
is
similar to (but distinct from)
ambiguity aversion, defined by
Ei U
[v,i
(a)]
= maxminE„
s,
Remark
6. It is
tempting to conjecture that a version of
extremely pessimistic rational players. The
this
is
not true, as Player
fact does not
a different mixed
when
=2+e
and p liW
b
and break the
belief.
(u)
=
tie,
+ e).
is
true for
Moreover, this
When we
slightly
she remains extremely pes-
For example, she
1/ (3
3
example shows, however, that
depend on the non-generic nature of the game.
simistic with
(6, 1)
last
Lemma
indeed extremely pessimistic.
1 is
change her payoffs for strategy
ui
[ut (sj, s_i)l
M
is
extremely pessimistic
.
MUHAMET
24
Appendix
YILDIZ
Technical Appendix and Proofs
B.
Basic Implications of Wishful Thinking. The following Lemma, which
B.l.
implies that wishful thinkers always play a best reply to a pure strategy, will
be very useful
\i
{
=
w
pi^j o
Lemmas
in proving
aZ\
1
and
the belief of player
is
(Recall from
2.
A
Appendix
that
about the other players' strategies at
i
u.)
Lemma
4.
For any
Jii~ {{s'Silai (w)
(B.l)
6 supp(/i iw ),
Proof. For each s_;
=
S-i
(=Wi,
ui
>
Uj (s_i)
If
u
%
>
(s-i)
would be
Ui {pi [to)
strict.
But
this
= max
define Ui (s_i)
Ui
(cr;
E^
(u)
,
S-i) for
> Ehul
[ui]
1-
[^
Sj
and
Ui (si, s_j),
let
each S-i G supp(yU iaJ ),
(a)}
.
with positive probability, then the above inequality
s_j)
,
impossible because, by (2.3),
is
> E5 ._.
Ei, u [ui (a)]
where
>
argmajcuj. Since Hi (s_,)
=
G BR, (*_;)})
[ui (s h s-i)]
=
Ui (s_;)
puts unit mass at s_j, and
5§_i is the probability distribution that
Sj
6
BRi(s-i).
Proo/ o/
Let
s
=
by
exists
ui
By
Lemma 1.
Lemma 4, there
definition, (s;,cr_j (w))
hence
(lj, <r_i
Proo/ o/
(w))
Lemma
=
2.
maxui
cr
cr
Pi
Lemma
D cr (P), and
C
7j
(w) such that
Sj
G PP;
G P;. Moreover, since u E Ki (P), p
a
Take any
(a))
(cr_i (w)).
C
P, and
(P).
wG
K
{
(P)
= maxE5s _.
Si
But, by
some w 6 Ki(F) C\Wi. Since w G Wi,
G supp^p^)
(w) G
(s^ s-i)
(u) for
n Wi
with 3
(s it s-i)]
[it*
= tr(u). By
< Ei>u
[m
(2.3),
(a)]
Si
4,
Eipl [m
(cr)]
= m (s,, s_j)
for
some
(s,,
s_j)
hence
•Ei,w
K
(o")l
Combining the two displayed
<
max
inequalities,
Ui(sj,s_j).
one obtains
(3.1).
G B, D o
(7i iU )
C
.
WISHFUL THINKING
B.2.
25
Characterization of Wishful Thinking through the Elimination
Process.
now prove Proposition
will
I
Proof of Proposition
m = 0,
For
1.
1.
the statement, a
(W) C X°,
is
immediately
~
~
implied by Lemmas 1 and 2. For any m, assume that a {K m
(W)) C X m
Take any s = a (ui) for some u E K m (W). Firstly, since K m (W) C iT"" (W),
l
l
.
1
1" 1
X"
6
s
where the
uEKi
By Lemma
.
1
max Ui
(X m
max
)
Fix any
.
Therefore,
.
E Xm
s
will construct
I
.
player
i, if
E Bi,
.
.
Bj-
and
m_1
l'
-
E Wi
such that Uj
1
(s
>
for
=
each such
set
Pj,oi €
w i,m-i, and
j
)
y™-
js
1
(s;,
i.
2
=
z,
(p,
=
s
,I ,p) in which s E
{wi,
m -i) =
a (wji7n_ 2 )
1.
=
'
new
(w 2
^
,
m_
and
If (^o)
m-2
,
m_1
(s}'
,
s^"
=
I? (u ii7n -i)
Ym Y
,
1.
By
and
con-
(ui,
,
.
.
,
Sj
2
'"1 - 1
g
~2
m
E Bj D X
2
)
If s
consider a
{u; iim _ 1 ,cj i
Conduct
m~ 1
m _i) =
operation again:
j,
be the set of states that are defined so
sets
such that
u>j >m _i
u^ m . x E Wj.
Once again w I]Tn -i £ Wj-
can define such a sequence of increasing
=
x )
cr,
last
For each such
).
state
1
set of states that are defined
yielding
m-2 =
s>>
sJ
2
B^X™-
E
)
conduct the
)
Uj (sj,s^™~
1
consider a
Let Y m_1 be the
.
(
s^"
{uo,Wi,m -i}, and p
Now for each j ^ i,
w im-1 = {^, m _!},
> maxSj
set
(cjj i7Tl _2)
let
s
i.
then there exists
0,
J,m ~ 2
member wiim _ 2 and
and
a (K m (W))
,
lf(uQ )
,
e Bjj then
m
p) in which
)
have been defined already.
s
(f2, /,
.
so far. Recall that, for each w» im _i, If
if
E
J
,
i,m-i
Thus,
s
Q,
7
struction, uiq
«;($;, s_v),
.
a model
m there exists s^ "" ee
If S g Bi, since s E X
l,m ~ 1
«i (s
) > max Sj Ui (s;, s_i). For each such
s
max
(Ji.s^jeBinX"1 - 1
Y m such that Y k C K k (W) for each k.) For any
then set I? (u = {u }. Clearly, u E Wi for such a player
define a sequence Y°,
a^u^m^) =
i,
s
(I will
set
,
= U S€X ™Q S and I = Uj e x-/ satisfies the
as the first member of Q and set a (uq) = s. Set Y m = {uq}.
Take some uq
s
Xm
now construct a model
will
I
a (Km (W)). Then, the model with
bill.
<
Ui(si,S-i)
a (K m (W)) C
any
for
1
2,
again by the induction hypothesis.
last inequality is
= Xm
1
For any given m,
Xm
Lemma
(ii.s-OeBiDCTCA'"— !(W))
where the
(f)
Wi. Hence, by
<
(si,3-i)
si
B^X™"
(W)) C
l
by the induction hypothesis. Moreover,
last inclusion is
(K m (W)) n
B na (K
there exists (sh s_,) E
1,
771 ' 1
m _ 2 },
this for
far.
Y°,
,
Clearly,
Y _1
new
each
one
following
MUHAMET
26
the above procedure.
which a
Ij
(idi-i)
=
(wi-i)
that if
1
G
s
l
For the states Wj_i 6
.
W
(W) 2
ym C
/C
Y
m
1
.
D
K
Similarly,
(iy),
k
l
\Y°, for
^
remains to be defined for j
set
i;
a wishful thinker at each state
is
Hence, for each
Y°.
Y~
need not be a wishful thinker or rational
Clearly, such j
{uii-i}.
Therefore,
in Y°.
6
s
l,_1
Q = Y~
X -1 = 5, 7|
But by construction each player
at oJi-\.
u
=
Set
YILDIZ
(W) 2
showing that
=
£
Y
K
i,
k
(W) D
{
;
D
7
1
,
so
< m. In particular,
e cr(iC m (H/)). Finally, for
for each
cr (u>
A' (Y°)
)
fc
the case of X°°, such a sequence of increasing sets could be defined indefinitely
without ever going out of X°°, and each player
common
state, so that wishful thinking is
the fixed profile £ 6 X°°
B.3.
is
will
be a wishful thinker at each
knowledge, and at the
played.
Wishful Thinking and Nash Equilibrium.
lationship between
initial state
Here,
I
Nash equilibrium and common knowledge
and prove Propositions 2 and
The next lemma
3.
states
but very useful facts about the elimination process and
will
explore the re-
of wishful thinking
some straightforward
its
relationship to equi-
librium.
Lemma
5.
The following are
true.
X
C Y
(1)
is
(2)
NE
is
a fixed point of
(j)
(i.e.,
C 4>(Y));
cp(NE) = NE);
(3)
X°°
is
a fixed point of
<p
(i.e.,
</>(X°°)
Lemma
monotonic
(i.e.,
=$ 4>(X)
=X°°).
5 immediately implies Proposition 2.
Proof of Proposition
2.
By Lemmas
NE =
00
(f)
5.2, 5.1,
(NE) C
4>°°
and the
(5)
=
definition of X°°,
X°°.
Recall that
X? = {
Si
\3s-i
:
(
Sl ,
5_i)
G
X
00
}
WISHFUL THINKING
is
the set of strategies for player
of wishful thinking.
Since
i
common knowledge
that are consistent with
=
(X°°)
4> a
27
X°°, each such strategy must be a best
reply to a surviving strategy:
Lemma
BiHX 00
For each
6.
i
and
G X°°
st
there exists s_j such that (sj,s_ 2 )
,
G
.
The next lemma
unique best
some
establishes
replies.
X°°
states that
It
when
applied to
NE\ x
and that
closed under best reply
is
the restriction of the best-response function to Xf°
part 3 states that,
games with
useful facts for two-player
NE2
,
a
is
Most notably,
bijection.
the elimination process stops
at the first iteration.
Lemma
For any two-player game assume
7.
=
\Xf°\
(2)
For each
i,
By Lemma
G BRi
<
1
(/of
(s^)
(sj)
NE
2)
6, for
= {p (sj)} for each
= 4>{NE x NE2 ).
{
each
£,
D
1
is
i
NE
2 ).
/ (JV£
2
x
Proposition
{s\,
Lemma
.
.
.
l
2
,
sf }
2,
)
3.
NE =
p{ yielding
is
,
=
4>
Sj
To prove
But
NE
2
(3.3),
{s 2
,
.
(ii),
:
i
)
X
for
(1) in
—
Xf°
To show
G NEi, there
>
Xf°
(3),
exists a
x 7VE2 ))
each
>
=
turn implies that
X*
is
well-defined.
check that,
unique
s_j)
NE2
^(iV.^ x
),
G
and
fc.
if
X°°
=
= NE. Assume
that
X°°
first
(.§;,
when
note that
0, then by
^
0. Us-
one can rename the strategies as
Xf =
0, and thus X°°
X™ =
(2).
^{^{NE
{NEx x
7 and assumption
and
—
X?°
:
1
exists a
Thus, p
also onto.
is
In that case,
NE
Proof of Proposition
ing
is
singleton- valued, for each
x
{
Sj;
arbitrary, this yields (1).
a bijection, so
is
(iV£?i
hence
singleton-
pj (s,) G Xj° such that
J
singleton- valued, p~ is one-to-one. Hence,
G Xf°, there
Sj
BRi
Since
.
\Xj°\. Since
Since p^
is
is
l
the one-to-one function p~
BR^
BRi
and onto mapping p
there exists a one-to-one
BRi
(NEi x
(3) <p°°
\Xf°\
i,
\X?\;
(1)
such that
Si
for each
Then, the following are true.
valued.
Proof.
that,
..,s 2 } for
some k >
1
so that Bi
r °°
DA
is
the
MUHAMET
28
Xf
diagonal of
X
x
°°, i.e.,
2
B,nX 0O ={ (s[, 4)
(B.2)
YILDIZ
|1
< <
Xf x X
C
k}
I
°°,
2
and
(B.3)
Ui (4) s 2)
Now,
any
for
maxuj
> m,
I
(sj,
is
strictly decreasing in
and (B.3) imply that
(B.2)
= ui (s™, s™) >
s™)
Now,
mathematical induction (on
will use
I
Together with (B.2), this implies
Si
Since
G
from
(
s i> s 2)
,
X°°. Therefore,
BR2
.
.
,4
.
Lemma
(by (B.4)),
it
=
/
Hence,
}.
7.2 that
is
p2
that
G
s2
n X°° C
ATE. Therefore, £1
G
s2
onto.
must be that
show that
4)
•
,
•
BR
•
,
But
BR
TVS.
{
(s[),
(7V£
the only strategy
s\ is
(4
_1
>
4
some
(4 \ 4)
•
BR
for
some
(si) for
2
Z
>
any
G {4>---,4}i reca U
sj
>
g"
s2
(i)),
NE
e
)
+
since
2
for
2 (si)
there exists
6,
Together with (B.2), this shows
-^2-
_1
(s\,
NE = Bid X°°.
Lemma
by
1,
singleton- valued (by assumption
is
-1
{5},
to
I)
X°°. But (B.4) states that
4) e NE. Assume
(s\,
For
(3.3).
Hence, (4,4) £
that can satisfy this.
1.
D
such that (si,4) S Bi
that
ui
X°°C {(s[,s^)\l<l<m<k}.
(B.4)
si
max
(s|,s2)ex M nBi
=
(4, s™) £ (f>(X°°)
in turn implies that
=
u \ (4> 4)
51
which
I.
•
showing that
C 5 a n X°° by
(4 4) } n X°° =
(4,4) £ B X C\B2 =
>
>
the
first
part of this
proposition.)
To prove
X°°
=
where the
and the
write
(3.4),
<f>
(X°°)
first
first
(NE
is
X
by
x
NE = 0°° (NE!
2)
Lemma
part of the proposition
by
Lemma 7.3,
is
by
definition.
the next inclusion
Proof of Corollary
—
<f>
equality
is
Jj (uj)
C
2.
{s G X°°\si
Let
=
o~{
Q =
5.3,
(i.e.,
is
(uj)} for
each
i
NE
2)
C
</>°°
the next inclusion
X°° C NEi x
again by
X°°, and
x
let
and
Lemma 5.1,
a be the
ui.
NE
By
2 ),
is
(5)
by
=
X°°,
Lemma
the next equality
and the
last equality
identity mapping.
(3.3), for
5.1
each
i
Set
and u, there
WISHFUL THINKING
exists
a (unique) Nash equilibrium
(3.4), for
any
G
s
Since tii(sj,s_j)
shows that
Uj (s)
with
s
(w), there exists (s i; s_j)
/;
= max
= max
s
«,
'.
(s'^,
s_j)
SiiS _ iG(7 _ i (/i (
s
e
29
(u).
/,
Set pi iU
G iVE such that
=
1
u; (s)
5$.
>
Now, by
u^ (s i; s_j).
(by definition of TVi?) for each s_;, this
w )) Uj
(s),
showing that
cj
D
G W{.
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.
27
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mam