GAMES AND ECONOMIC BEHAVIOR
ARTICLE NO.
17, 230]251 Ž1996.
0104
Hypothetical Knowledge and Games
with Perfect Information
Dov Samet
Faculty of Management, Tel A¨ i¨ Uni¨ ersity, Tel A¨ i¨ , Israel
Received August 30, 1994
Standard structures of information, in particular partition structures, are inadequate for the modeling of strategic thinking. They fail to capture the inner
structure of hypotheses players make about situations they know will not occur. An
extension of the partition structure is proposed in which such hypotheses can be
modeled in detail. Hypothetical knowledge operators are defined for extended
structures and are axiomatically characterized. The use of extended structures to
model games with complete information is demonstrated. A sufficient condition is
derived for players to play the backward induction in such games. Journal of
Economic Literature Classification Numbers: C72, D81, D82. Q 1996 Academic Press,
Inc.
1. INTRODUCTION
Finite games with perfect information are the simplest of games. Yet
they seem to be the source of continuing debate and conflicting intuition.
In particular, opinions diverge widely on the relation between rationality
and the backward induction equilibrium of these games Že.g., Aumann,
1995; Basu, 1990; Ben Porath, 1992; Bicchieri, 1989, 1994; Binmore, 1987;
Binmore and Brandenburger, 1990; Reny, 1992.. These opinions can be
divided, very roughly, into three groups. Some claim that common knowledge of rationality is possible in such games and that it implies the
backward induction equilibrium. Others maintain that it does not. Still
others insist that backward induction is inconsistent with common knowledge of rationality or even that common knowledge of rationality itself is a
contradictory term in such games. An adequate model in which we express
formally what players do, think, believe, and conjecture could enable a
rigorous formulation and comparison of these diverging opinions. The
basic elements of the existing simple models that would seem capable of
doing the job are the following:
Ž1. an information structure consisting of a state space and partitions
of the state space, one for each player, which define knowledge operators;
230
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Copyright Q 1996 by Academic Press, Inc.
All rights of reproduction in any form reserved.
HYPOTHETICAL KNOWLEDGE AND GAMES
231
Ž2. an assignment of one strategy combination to each state; and
optionally,
Ž3. players’ probability distributions over the state space.
In such models, which we shall call standard models, we can describe
players’ strategies, their knowledge and beliefs about others’ strategies,
their knowledge and beliefs about what others know and believe, and so
on.
This paper, in a nutshell, disputes the adequacy of the standard model
for describing games with perfect information and proposes a model which
is adequate for this purpose. We show that standard models fail to capture
an important structural aspect of strategic thinking and therefore leave
unformalized many intuitive arguments that depend on it. The model we
propose captures this aspect by giving a fuller and more faithful account of
strategic thinking. Using this model we re-examine the relation between
rationality and backward induction and give formal expression to statements about the reasoning of players in games with perfect information}statements that cannot be formalized in the standard model.
In the next section we explain in which way the said structural aspect of
strategic thinking is not represented in the standard models and why it
should be represented. For the time being, we shall explain what this
structural aspect is. A strategy of a player specifies an action at each of his
nodes in the game tree. When we assign a player’s strategy to a state, this
strategy specifies the actions of the player even in modes that he knows he
will not reach in the state. The simple intuitive meaning of this is that such
an action is what the player hypothesizes he would have done had he
reached his node. This last hypothesis is a compound, structured sentence.
It is formulated in terms of two clauses, one about reaching the node, the
other about taking an action. This structure, concerning the relation
between the clauses and the hypothesis, is absent from the standard
model.
In Section 3 we present an extended information structure in which the
relation between sentences expressing hypothetical thinking and their
components can be represented. In addition to a knowledge operator, it
provides for each player i a hypothetical knowledge operator that determines for each pair of events, H Žthe hypothesis. and E, another
event}that player i hypothesizes Žor simply thinks. that if H were true,
then he would know E. We define such operators axiomatically and
characterize them in terms of hypothesis transformations on the partitions.1
1
Extended information structures bear some resemblance to semantic models of logics of
counterfactuals. These logics were introduced in Stalnaker Ž1968. and further studied in
Stalnaker and Thomason Ž1970. and Lewis Ž1973.. Hypothetical knowledge operators differ
from counterfactuals in that they are epistemic operators. For a comparison between the
logic of counterfactuals and the logic of hypothetical knowledge, see Samet Ž1994..
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DOV SAMET
In Section 4 we define models for games with perfect information. For
an extended information structure to model a game, we assign to each of
its states a play in the game, that is, the sequence of actions taken in the
game. This is a major departure from the standard model, in which
strategies, rather than plays, are assigned to states. Thus, the primitive
notion here is purely behavioral. Strategies in this model are what they
should be, namely, cognitive constructions. The strategy of a player in a
state is determined by the hypotheses he holds about actions he would
take in nodes were they reached. We show that a strategy combination in a
state is uniquely defined and generates the actual play in that state.
Section 5 deals with rationality and backward induction. Here, rationality is a property of behavior, and when we say that a player is rational we
simply mean that his behavior is rational. For a given state and node we
say that the player at that node is rational at the node if: Ži. the node is
reached in the given state, Žii. there is no number x such that the player
knows that his action there yields him a payoff less than x, and he
hypothesizes that another action would yield him a payoff of at least x. A
player is rational in a state if he is rational at each of the nodes he reaches
in this state.
In our model, common knowledge of rationality does not imply that the
path is the one generated by backward induction. To see this, suppose that
in a certain state, which is common knowledge, the path is an equilibrium
path other than the backward induction path. Rationality of the players in
this state is determined by their behavior along the path. If players have
the ‘‘right’’ hypotheses there Žnamely, had they deviated from the path the
game would have developed according to the equilibrium strategies . then
they are rational and there is a common knowledge of rationality.
To explain backward induction in terms of rationality, in our model, we
need hypotheses about hypotheses about . . . about rationality rather than
common knowledge of rationality. For this purpose we define common
hypothesis, which is constructed iteratively, like common knowledge, except
that hypothetical knowledge operators are iterated rather than knowledge
operators. A common hypothesis, however, has more structure than common knowledge. The common hypothesis that if node ¨ is reached then e¨ ent
E holds true is the event obtained by iterating knowledge operators where
the sequence of hypotheses is the sequence of events corresponding to the
nodes leading to ¨ . The iteration ends with the event E. We say that node
rationality is a common hypothesis if for each node ¨ it is a common
hypothesis that if ¨ is reached, then the behavior of the player at ¨ is
rational. In Section 5 we prove Theorem 5.3 which gives epistemic conditions that guarantee the backward induction play.
THEOREM 5.3. If there is a common hypothesis of node rationality, then
players play the backward induction path.
HYPOTHETICAL KNOWLEDGE AND GAMES
233
To simplify the presentation we did not include in the model elements
that could give a fuller account of players’ reasoning and behavior. Thus,
time is absent from the model}we analyze the game at a point in time
before it is played. We do not introduce into the model beliefs Žwhich
would be given by probability distributions over the state space.. We do
not foresee any difficulties in replicating the results for games with beliefs.
Finally, we studied only games with perfect information, but we think that
extended information structures are the appropriate tool for the investigation of games in extensive form in general.
2. WHAT STANDARD MODELS LACK
The shortcomings of standard models are exhibited even in very simple
games. To demonstrate these shortcomings we use the following game,
which is used in Aumann Ž1995., as an illustration.
Bob
Alice
r
R
D
d
2, 2
1, 1
3, 3
Alice and Bob each have two strategies in this game: SR and SD for
Alice; S r and S d for Bob. The intuitive meaning of strategies equates them
with certain sentences. in Bob’s case:
S d s‘‘If Bob’s node is reached he plays d’’
Sr s‘‘If Bob’s node is reached he plays r ’’
To say that Bob’s strategy in a state v is S d simply means that it is true in
v that if Bob’s node is reached he plays d. Equivalently, in terms of
events, Bob’s strategy in a state v is S d if v is in the event w S d x.2
The sentences S d and Sr are compound sentences. For example, S d has
two component sentences, ‘‘Bob’s node is reached’’ and ‘‘Bob plays d.’’
Can S d be constructed from these component sentences? Rephrasing the
question in terms of events: Can we determine the event w S d x from the
events wBob’s node is reachedx and wBob plays d x? We show that in
standard models this is impossible. That is, the event w S d x is not con2
For a sentence S we denote by w S x the event that S is true, i.e., the set of all states in
which S is true.
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structed from the component events. Consider the following standard
model.
Model 1. There is only one state in this model, which is therefore
common knowledge. The strategies in this state are SD and S d .
In this model, the event w S d x consists of the single state v of the model,
while the event w S r x is the empty set. Bob’s node is not reached in v and
he plays neither d nor r. Thus, the component events of both w S d x and w Sr x
are the same}the empty set. But w S d x and w S r x are different events. Hence
these events cannot be constructed from their components.3 Thus, the
states in which S d is true are determined arbitrarily without consulting the
component sentences. The strategy S d is considered a primitive notion in
the standard model.
The intention of the previous discussion is by no means to claim that
sentences such as ‘‘If Bob’s node were to be reached, then he would play
d’’ have no meaning. The opposite is true. Such sentences describe
hypothetical thinking, which is essential to decision and game theory. We
only claim that the standard model does not lend such sentences any
meaning, since it takes them as primitive. There is nothing wrong with this,
since every theory must have some primitive notions Žsuch as actions and
payoffs in the standard model.. We show, however, that by taking strategy
as a primitive notion we fail to formalize the most basic ideas that relate
behavior to rationality. Consider the following model.
Model 2. There is only one state in this model, which is therefore
common knowledge. The strategies in this state are SR and Sr .
Models 1 and 2 both describe a Nash equilibrium. We can easily explain
the different behavior in these equilibria in terms of rationality. In Model
1, Alice knows that if Bob’s node were to be reached he would irrationally
play d, and being rational herself she plays D. In Model 2, Alice knows
that if Bob’s node were to be reached he would rationally play r, and being
rational herself she plays R.
This explanation Žor any other involving Bob’s rationality. cannot be
expressed in the standard model. Bob’s rationality in Model 1, for example,
is referred to in the component sentence ‘‘Bob irrationally plays d.’’ But
the specification of Bob’s strategy S d in this model is not related, as we
argued before, to this component sentence or the corresponding event. All
that can be said in Model 1 is that Alice knows that Bob’s strategy is S d .
3
To conform with English grammar, the sentence describing strategy S d , in Model 1,
should be ‘‘Were Bob’s node to be reached, he would play d,’’ since it is known in v that
Bob’s node will not be reached.
HYPOTHETICAL KNOWLEDGE AND GAMES
235
To conclude from this that she knows Bob would behave irrationally were
he to reach his node cannot be represented in the model.
Now, the strategies in Model 2 are the backward induction strategies
while those in Model 1 constitute another equilibrium. Thus we cannot
account for the difference between these two equilibria in terms of events
that depend on Bob’s rationality, let alone common knowledge of rationality.
There is a way to overcome this obstacle and express the difference
between the equilibria, in the standard model, in terms of rationality. To
this end we have to adopt a notion of rationality which is radically
different from the one we have been using until now. Instead of describing
actions as being rational or not we ascribe these properties to strategies.
Thus, instead of saying that if Bob were to reach his node he would
irrationally choose d, we say that choosing strategy S d , in Model 1, is
irrational on Bob’s part. With this definition of rationality, the event wBob
is irrational x is just v 4 and of course Alice knows that Bob is irrational.
Aumann Ž1995. uses a generalization of this type of rationality to prove
that common knowledge of rationality implies backward induction.
The difference between the two notions of rationality is immense.
According to the first notion, it is always an action actually taken by the
player, that is, behavior, which is judged to be rational or irrational. Thus,
in Model 1, where Bob does not take any action, he cannot be considered
irrational according to this notion of rationality, which is the reason we
cannot relate the equilibrium to Bob’s irrationality in the standard model.
The second notion applies rationality to strategies, which are primitive
entities that cannot be expressed in terms of actual actions Žin standard
models. and can be nonbehavioral and payoff irrelevant Žas in Model 1..
In summary, the standard models are inadequate to express the outcome
of the game in terms of player’s rationality, when rationality is the
elementary behavioral notion of rationality.
This is not the only drawback of standard models. As with strategy
sentences, no sentence expressing hypothetical thinking can be represented in terms of its components, in these models. Consider for example
the following sentence S.
S s‘‘Player i Ž whose chooses action a . thinks that had she chosen action
b, player j would have known she, player i , was irrational.’’
Such sentences are usually debated in the context of backward induction,
usually as an argument against its justification. Unfortunately, using the
standard model, we cannot test the validity of arguments concerning S, as
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S cannot be formalized in this model. That is, there is no way we can
define the event w S x in terms of the event wPlayer i chooses b x and wPlayer
j knows that player i is irrational x, for reasons similar to the ones given
above. We could have determined the states in which S is true arbitrarily,
in much the same way as we did for strategies, but this does not seem to be
fruitful, as there must be some nonarbitrary relations between the event
w S x and events describing strategies.
The conclusion from the above discussion is that an adequate model for
games in extensive form must provide a way to interpret as events
sentences describing hypothetical thinking in terms of their components.
In the next section we extend information structures to facilitate such
interpretation.
3. EXTENDED INFORMATION STRUCTURES
Information structures. An information structure, for a set of players I, is
a list
Ž V , Ž P i . igI . ,
where V is the set of states, and for each player i, P i is a partition of V.
Subsets of V are called e¨ ents. The set of all events is denoted by S. The
set theoretic operations of complementation, intersection, and union are
considered as the interpretation of ‘‘not,’’ ‘‘and’’ Žconjunction., and ‘‘or’’
Ždisjunction., respectively. The complement of event E is denoted by ! E.
We define a binary operator ‘‘ª ’’ by E ª F s Ž! E . j F, for any two
events E and F. The operator ‘‘ª ’’ interprets material implication
Ž‘‘if . . . , then . . . ’’..
For each state v , we denote by P i Ž v . the unique element of P i
containing v . Using the partitions, we define knowledge operators as
follows. For each i, the knowledge operator K i : S ª S is defined by
K i Ž E . s v < P i Ž v . : E4 ,
for each event E. We usually omit the parentheses and write K i E. The
event K i E is the event that i knows E. For each player i define an
operator L i by L i E s ! K i ! E, for each event E. The event L i E is the
event that i considers E possible. Clearly L i E s v < P i Ž v . l E / B4 . The
‘‘all know’’ operator, K I , is defined by K I E s F i g I K i E. The event that E
is common knowledge is F n G 1Ž K I . n E.
HYPOTHETICAL KNOWLEDGE AND GAMES
237
Hypothetical knowledge. We want to enrich information structures so
that hypothetical knowledge can be represented by events. For this purpose we define for each player i a binary operator K i Ž?, ? . such that for
any two events H / B and E, K i Ž H, E . is the event that i hypothesizes
that he would have known E had the hypothesis H been true. We write
K iH Ž E . or K iH E instead of K i Ž H, E ..
The following definition provides a list of axioms that we want the
hypothetical knowledge operator to satisfy. For the rest of this section we
omit the subscript i in K i , K iH , L i , and P i .
DEFINITION 3.1. An operator
K : Ž S _ B 4 . = S ª S,
is called a hypothetical knowledge operator if it satisfies for all events
H / B and E, in S, the following axioms:
ŽK1.
ŽK2.
ŽK3.
ŽK4.
ŽK5.
ŽK6.
K H E s KK H E,
K H E s K H KE,
! K H E s K H ! KE,
for any family of events Ea 4 , F a K H Ea s K H F a Ea ,
K H LH s V,
LH l K H E s LH l KE.
The intuition behind these axioms is as follows. We think of K H E as
representing a hypothesis of the player about his knowledge in a hypothetical situation H. Thus, two states of the player’s mind are involved. One is
the actual state of mind of which this hypothesis is a part. The other is a
hypothetical state of the player’s mind to which he refers in this hypothesis. In his actual state of mind, to produce the statement ‘‘Had H been the
case, I would have known E,’’ the player has to probe a hypothetical state
of mind of his, one that would have prevailed had H been the case. We
make the following two assumptions. Ži. The hypothetical state of mind is
not a special type of state of mind; it enjoys standard properties of states
of mind. The adjective ‘‘hypothetical’’ only defines its relation to the actual
state of mind. Žii. The hypothetical state of mind can be fully examined by
the player in his actual state of mind. That is, he can tell precisely what is
known and what is not known to him in this state of mind.
In light of this interpretation of hypothetical knowledge, the axioms are
rather natural. The first axiom ŽK1. expresses the idea that K H E is a
hypothesis made by the player in his actual state of mind and as such is
known to him. In other words, we consider the statement ‘‘Had H been
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the case, I would have known E’’ as having the same meaning as ‘‘I know
that had H been the case, I would have known E.’’
In hypothetical states of mind, as in all states of mind in a partition
model, the player knows E if and only if he knows that he knows E. If this
hypothetical state of mind is fully open to investigation by the player, then
stating ‘‘Had H been the case, I would have known E’’ must be true
precisely when ‘‘Had H been the case, I would have known that I know
E’’ is true. This is axiom ŽK2..
Axiom ŽK3. also reflects the full inspectability of the hypothetical state
of mind by the player. For each E, he either finds that in his hypothetical
state of mind he knows E or he finds there that he does not know it Žand
in the latter case he knows that he does not now it.. Thus, the player would
deny that if H had been the case he would have known E, precisely when
he finds that in this hypothetical situation he would not have known E
Žand therefore knows that he would not have known it.. This is what ŽK3.
says.
Axiom ŽK4. states that hypothetical states of mind have one of the
standard properties of states of mind. That is, knowing hypothetically that
certain events are simultaneously true is the same as simultaneously
knowing hypothetically that each one of them holds.
The following two axioms relate the hypothetical state of mind to the
hypothesis H. It is always the case, no matter what the hypothesis H, says
ŽK5., that one can state ‘‘Had H been the case I would have known it is
possible Žthat is, I would not have known it is false..’’
Axiom ŽK6. deals with the case in which H is not really a hypothesis
because it is indeed considered possible by the player 4 Žin his actual state
of mind.. In this case, the player does not have to go too far to find his
hypothetical state of mind. There is no reason Žand indeed it would be
strange. to assume that, if indeed H were the case, his state of mind would
be any different from the actual state of his mind, in which he considers H
possible. But then, hypothetical knowledge is precisely actual knowledge,
which is what axiom ŽK6. states.
Hypothesis transformations. In the previous subsection we formalized
hypothetical knowledge by an operator on events. In formulating the
axioms this operator should satisfy, we make no use of the structure of the
state space.
4
The hypothetical knowledge operator can be used when H is known to be false as well as
when it is not known to be so. In English, these two cases have different grammatical
expressions. In the first one, ‘‘Had H been true . . . ’’ is used. In the second case, when the
hypothesis H is considered possible, ‘‘If H is true . . . ’’ is used. There is no grammatical form
in English that covers both cases.
HYPOTHETICAL KNOWLEDGE AND GAMES
239
We start with an informal discussion which motivates the definition that
follows. In the previous subsection we did not formalize the notions of
actual states of mind and hypothetical ones. Rather, we used these notions
as intuitive guidelines in formalizing the properties of hypothetical knowledge operators. Now, we formalize actual and hypothetical states of mind,
and the relation between them, in terms of the agents’ partitions and then
show how this formalization is related to hypothetical knowledge operators.
The player’s state of mind in a state v can be identified with his
partition element P s P Ž v .. This event describes completely everything
that the player knows. Suppose now that P is the actual state of the
player’s mind. To find what he knows when hypothesis H holds true, the
player must have in mind a hypothetical state of mind, which is another
element in his partition. Let us denote this element by T Ž P, H ..
Obviously, as T Ž P, H . is the player’s state of mind when H is true, it
should be the case that the two events intersect non-trivially, because
otherwise the player knows in T Ž P, H . that H is false. In addition, if P
intersects H nontrivially, then the player considers H possible and therefore it is reasonable to assume that T Ž P, H . is simply H.
This leads us to the following definition.
DEFINITION 3.2. A hypothesis transformation on P is a map
T : P = Ž S _ B4. ª P that satisfies the following two conditions,
ŽT1. T Ž P, H . l H / B,
ŽT2. if P l H / B, then T Ž P, H . s P.
The following theorem shows that hypothetical knowledge operators are
those operators that can be represented by hypothesis transformations.
THEOREM 3.3. Let Ž V, P . be an information structure with set of e¨ ents
S, and K a binary operator K: Ž S _ B4. = S ª S. Then K is a hypothetical
knowledge operator iff there exists a hypothesis transformation T, on P, such
that
K HE s
D P <T Ž P , H . : E4 .
Ž 3.4.
PgP
Moreo¨ er, for a gi¨ en hypothetical knowledge operator, there exists a unique
hypothesis transformation T that satisfies Ž3.4..
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DEFINITION 3.5. An extended information structure, for a set of players
I, is a list,
Ž V , Ž P i , Ti . igI . ,
where V is the set of states and for each player i, P i is a partition of V,
and Ti is a hypothesis transformation on P i .
In the axioms defining the hypothetical knowledge operator we use the
knowledge operator K. But by axiom ŽK6., for each E, K V E s KE. Thus
the hypothetical knowledge operator is an extension of the knowledge
operator. Let ŽK1*. ] ŽK6*. be the axioms obtained from ŽK1. ] ŽK6. by
replacing each K with K V . Add also the axiom
ŽK7*. K V E : E.
Axioms ŽK1*. ] ŽK7*. can now be used to axiomatize both the knowledge
operator and the hypothetical knowledge operator, as follows.
THEOREM 3.6. Let V be a state space with set of e¨ ents S and K a binary
operator K: Ž S _ B. = S ª S. Then K satisfies the axioms ŽK1*. ] ŽK7*. iff
there exists an extended information structure Ž V, Ž P, T .., such that the
operator K V Ž?. is the knowledge operator corresponding to the partition P,
and K is the hypothetical knowledge operator defined by the hypothesis
transformation T.
4. MODELS FOR GAMES WITH PERFECT
INFORMATION
A game G with perfect information consists of a finite set of players I
and a finite tree with a set of nonterminal nodes Žor vertices. V, a set of
terminal nodes Z, and a root r. For each z g Z there is a unique path
called play leading to it from the root r. With some abuse of language we
call z a play when no confusion arises. For two nodes ¨ and u we write
¨ U u or u # ¨ when u is a node that follows ¨ , i.e., when u is a node in
the subtree the root of which is ¨ . The set of nonterminal nodes, V, is
partitioned into subsets Ž Vi . i g N , where Vi is the set of i’s nodes. For
¨ g V we denote by AŽ ¨ . the set a <Ž ¨ , a. is an arc of the tree4 . If ¨ g Vi ,
then the nodes in AŽ ¨ . are called i’s actions at ¨ . Player i’s payoff
function is a real-valued function h i : Z ª R.
A strategy for player i is a function si : Vi ª V j Z, such that for each
¨ g Vi , si Ž ¨ . g AŽ ¨ .. A strategy is a list of strategies s s Ž si . i g I . The set
of all strategies of i is denoted by S i and the set of all strategies is denoted
HYPOTHETICAL KNOWLEDGE AND GAMES
241
by S. We denote by sŽ ¨ . the terminal node that is reached by strategy s
from node ¨ . Thus, if ¨ is a terminal node, then sŽ ¨ . s ¨ . If sŽ a. is
defined for all a g AŽ ¨ . and ¨ g Vi , then sŽ ¨ . s sŽ si Ž ¨ ... The play that
results from playing strategy s is obviously sŽ r ..
In the next definition we use extended information structures as models
for games by introducing a function z : V ª Z. We use the following
notation for events that are defined in terms of z . The event that node ¨ is
Žor to be. reached, v < z Ž v . # ¨ 4 , is denoted for short by w ¨ x. Note that
w ¨ x s D ag AŽ ¨ .w ax, and w ax l w a9x s B for any a / a9 in AŽ ¨ ..
DEFINITION 4.1. A model for a game with perfect information G with
set of players I is a pair, Ž S , z ., where S is an extended information
structure Ž V, Ž P i , Ti . i g I ., and z is a map z : V ª Z, onto Z, such that for
each player i, node ¨ g Vi , and action a g AŽ ¨ .,
w ax : K i Ž w ¨ x ª w ax . .
Ž 4.2.
The map z specifies for each state the behavioral aspects of the state,
namely what players do in this state Ži.e., which terminal node they reach..
Using temporal terminology, z shows how the game evolves in time. We
intend the extended information structure to describe the epistemic status
of the players before the game is started. It tells us what the players know
or hypothesize in the beginning of the game about how the game will
evolve in time. Note that, as z maps V on Z, for each node ¨ , the event
w ¨ x is not empty and therefore can serve as a hypothesis for the hypothesis
transformations Ti .
To understand condition Ž4.2. note that w ¨ x ª w ax is the event that if ¨
is reached, then action a is taken there. Thus, K i Žw ¨ x ª w ax. is the event
that the player Žwho can take action a. knows that if ¨ is reached he takes
there action a, or simply, he knows that he takes a at ¨ . Now, Ž4.2. says
that if action a is indeed to be taken by the player at node ¨ , then the
player knows it. Observe, however, that it is not true that a player who
plays a in some state must know it. That is, we do not require that
w ax : K i Žw ax.. This is so, because at the beginning of the game, i may not
know for sure that his node ¨ will be reached.
A strategy map, which we define shortly, assigns a strategy to each state.
It is constructed in our model using the hypotheses players have about
their actions at all possible and all hypothetical nodes. In order to define
strategy maps we need the following proposition.
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PROPOSITION 4.3. For each player i and node ¨ g Vi , there exists a unique
action a, of player i, which i hypothesizes that he would know to be his action
at ¨ , if ¨ were to be reached. That is,
D
K iw ¨ x Ž w ¨ x ª w a x . s V ,
agAŽ ¨ .
and the e¨ ents in this union are disjoint in pairs.
The uniqueness of the hypothesized actions, guaranteed by Proposition
4.3, enables us to define a strategy in each state.
DEFINITION 4.4. Player i’s strategy map is the function si : V ª Si , such
that si Ž v . is the strategy that assigns for each ¨ g Vi the unique action a
for which v g K iw ¨ xŽw ¨ x ª w ax.. The strategy map s : V ª S is defined by
s Ž v . s Ž si Ž v .. i g I .
We denote the event that i’s strategy is si Žthat is, v < si Ž v . s si 4. by
w si x, and the event that the strategy is s Ži.e., v < s Ž v . s s4. is denoted by
w s x. Clearly
w si x s F K iw ¨ x Ž w ¨ x ª si Ž ¨ . .
¨ gVi
w s x s F w si x .
Ž 4.5.
igI
The following proposition shows that the strategy map is related to
behavior and knowledge in an appropriate way.
PROPOSITION 4.6. For each strategy s s Ž si . i g I , player i, and state v , if
the strategy in v is s, then the play in v is the play generated by s and each
player knows his strategy. That is,
w sx : sŽ r . ,
and
w si x s K i w si x .
5. RATIONALITY AND BACKWARD INDUCTION
We assume in this section that for each player i the payoff function h i is
nondegenerate, that is, it attains each value only once. The payoff maps hi
on V are defined for each player i by hi Ž v . s h i Ž z Ž v ... The event
HYPOTHETICAL KNOWLEDGE AND GAMES
243
v <hi Ž v . - x 4 is denoted by whi - x x. The event whi G x x is similarly defined.
Since the payoff functions are nondegenerate, it is easy to show by
induction that there is a unique strategy b s Ž bi . i g I which satisfies for
each i and ¨ g Vi the condition bi Ž ¨ . s argmax ag AŽ ¨ . h i Ž b Ž a... We call
b and b Ž r . the backward induction strategy and path, respectively.
DEFINITION 5.1. The event that player i is rational at node ¨ g Vi is
defined as
RŽ ¨ . s w ¨ x l
F F
x
! Ž K i Ž w ¨ x ª w hi - x x .
agAŽ ¨ .
lK iw ax Ž w a x ª w hi G x x . . .
That is, i is rational at ¨ if this node is reached and i does not know of
any bound x on his payoff, given that ¨ is reached, that could be
hypothetically surpassed by using some action a. Note that rationality at
node ¨ is a behavioral concept, as it applies only to states in which ¨ is
indeed reached.
We want to find sufficient conditions in terms of rationality that guarantee the backward induction play. Common knowledge of rationality does
not suffice. For example, if z is any equilibrium play, then it is possible to
construct a state v such that z Ž v . s z, v 4 is common knowledge and all
players are rational in v .
The following condition is stronger than the assumption of common
knowledge of rationality but it is still too weak. The event that i hypothesizes he is rational at all his nodes is F ¨ g V i K iw ¨ xŽw ¨ x ª RŽ ¨ ... Common
knowledge that each player hypothesizes he is rational in all his nodes is
not enough to guarantee the backward induction play. The reason is as
follows.
Suppose that i is the player who plays at the root r. His rationality at r
depends on his hypotheses of what would happen in each action he might
choose. Thus, the important events to look at are of the form K iw ax E, where
a is an action at r. Now the common knowledge assumption tells us that i
knows that the player who plays at a, say j, hypothesizes he is rational at a.
That is, K i Ž K jw axŽw ax ª RŽ a... is true. But there is nothing that bounds i to
hypothesize that if a is reached, j would behave rationally. Common
knowledge of the players’ hypotheses imposes no interpersonal consistency
on these hypotheses. Therefore what i knows about the hypotheses entertained by other players tells us nothing about what he hypothesizes their
244
DOV SAMET
behavior or hypotheses would be. This consideration leads us to the
following required condition. What we need is hypotheses about hypotheses about hypotheses . . . as we define next.
DEFINITION 5.2. For each pair of nonterminal nodes u and ¨ such that
¨ # u, and event E, the event that there is a common hypothesis of E from
u to ¨ , H Ž u, ¨ , E ., is defined inductively as
H Ž ¨ , ¨ , E . s E.
If u g Vi , ¨ # u, and H Ž a, ¨ , E . is defined for the Žunique. node a in
AŽ u. on the path from u to ¨ Ži.e., ¨ # a., then
H Ž u, ¨ , E . s K iw ax Ž w a x ª H Ž a, ¨ , E . . .
The event that there is a common hypothesis of node rationality is
F H Ž r , ¨ , RŽ ¨ . . .
¨ gV
THEOREM 5.3. If there is a common hypothesis of node rationality, then
players play the backward induction path, that is,
F H Ž r , ¨ , RŽ ¨ . . :
bŽ r. .
¨ gV
By the definition of common hypothesis it follows that a common
hypothesis of node rationality depends only on the hypotheses of the root
player. The reason that the hypotheses of one player determine the play of
the game is as follows. Clearly, only the hypotheses of the root player, and
nothing else, determine his action at the root. What he hypothesizes about
the consequence of his actual action at the root is knowledge Žsince the
antecedent is true. and, moreover, as the root player knows his action,
the consequences are indeed true. But among these consequences are the
hypotheses of the next player about what is true if his node is reached.
Since the next player’s node is indeed reached, his hypotheses are knowledge and so on.
Theorem 5.3 is of interest only if in some models of the game G, the
event that there is a common hypothesis of node rationality is not empty.
The following theorem guarantees the existence of such a model.
HYPOTHETICAL KNOWLEDGE AND GAMES
THEOREM 5.4.
model in which
245
For each game with perfect information there exists a
F H Ž r , ¨ , R Ž ¨ . . s B.
¨ gV
6. PROOFS
We first record several properties of hypothesis operators that we use in
the sequel.
PROPOSITION 6.1.
For all e¨ ents H, E and F:
Ža. If E : F, then K H E : K H F Ž monotonicity..
Žb. K H V s V.
Žc. K HB s B.
Žd. For any family of e¨ ents Fa 4 , D a K H Ž E ª KFa . s K H Ž E ª
D a KFa ..
Že. H l K H E s H l KE.
Proof. Ža. Using ŽK4., K H E s K H Ž E l F . s K H Ž E . l K H Ž F . :
K F ..
Žb. Follows from Ža. and ŽK5..
Žc. Using ŽK3. and Žb., K HB s K H ! K V s ! K H V s B.
Žd. It is easy to see that the knowledge operator K satisfies
HŽ
D a K Ž E ª KFa . s K Ž E ª D a KFa . .
Taking the complement of D a K H Ž E ª KFa . and using ŽK3. and ŽK4. we
conclude that
!D a K H Ž E ª KFa . s F a K H ! K Ž E ª KFa .
s K H F a ! K Ž E ª KFa .
s K H !D a K Ž E ª KFa .
s K H ! K Ž E ª D a KFa .
s ! K H Ž E ª D a KFa . .
Že. Intersect both sides of ŽK6. with H and note that H : LH. B
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DOV SAMET
Proof of Theorem 3.3. We denote by Fi the range, K S, of the knowledge operator K. It is closed under complementation and intersection Žof
any number of events. and for any event E, KE s E, if and only if, E g F.
Let K H Ž E . be a hypothetical knowledge operator. Consider event
H / B in S and P g P. Let Ea 4 be the family of all events for which
P : K H Ea . This family is not empty by ŽK5.. Thus P : F a K H Ea . Let
T Ž P, H . s F a Ca . Then by ŽK4.,
P : K HT Ž P, H . .
Ž 6.2.
By Ž6.2. and monotonicity we conclude that for any R g S,
P : K H R iff T Ž P , H . : R.
Ž 6.3.
By ŽK1., K H R g F and therefore P ­ K H R iff P : ! K H R. Hence Ž6.3.
is equivalent to
P : ! K H R iff T Ž P , H . :
u R.
Ž 6.4.
By Ž6.2., K H T Ž P, H . / B and thus by Proposition 6.1Žc., T Ž P, H . / B.
We show now that T Ž P, A. g P. First we prove that T Ž P, H . g F. This is
true if KT Ž P, H . s T Ž P, H . or, by the definition of T Ž P, H . and ŽK4., if
F a KEa s F a Ea . Clearly F a KEa : F a Ea . The inverse inclusion holds
if whenever P : K H Ea , then P : K H KEa , which is true by ŽK2.. Suppose
to the contrary that T Ž P, A. f P. Then, T Ž P, H . is a union of more than
one elements of P, and hence there is S g P such that T Ž P, H . ­ S and
T Ž P, H . ­ ! S. It follows by Ž6.4. that P : ! K H S and P : ! K H ! S.
Since KS s S it follows by ŽK3. that P : K H S and P : K H ! S. By ŽK4.
and Proposition 6.1Žc., P : K HB s B, which is a contradiction.
To prove ŽT1. note that by ŽK5., P : K H LH. By Ž6.3., this implies that
Ž
T P, H . : LH, i.e., T Ž P, H . l H / B.
To prove ŽT2., suppose that P l H / B, i.e., P : LH. Combining this
inclusion with Ž6.2. we have P : K H T Ž P, H . l LH. It follows by ŽK6. that
P : KT Ž P, H . l LH. But, as T Ž P, H . g P, KT Ž P, H . s T Ž P, H . and
thus P s T Ž P, H ..
Finally, Ž3.4. holds by Ž6.3., as K H E g F.
To see that T is unique, suppose T 9 is a hypothesis transformation
satisfying Ž3.4.. By applying Ž3.4. to T 9 we conclude that for all P g P and
H / B, P : K H T 9Ž P, H .. Hence, by Ž6.3., T Ž P, H . : T 9Ž P, H . and since,
by the definition of hypothesis transformations, both events are in P,
T Ž P, A. s T 9Ž P, A..
It is straightforward to show that if K is defined by Ž3.4., then it satisfies
ŽK1. ] ŽK6.; i.e., it is a hypothetical knowledge operator. B
HYPOTHETICAL KNOWLEDGE AND GAMES
247
Proof of Theorem 3.6. Suppose K satisfies axioms ŽK1*. ] ŽK7*.. By
substituting V for H in ŽK2*., ŽK3*., ŽK4*., and ŽK7*. we have the
following properties of K V Ž?.. For each E g S, K V E s K V K V E, ! K V E
s K V ! K V E, and for any family of events Ea 4 , F a Ž K V Ea . s
K V ŽF a Ea .. These properties, with ŽK7*., show that K V is a knowledge
operator and therefore there exists a partition P on V which represents
K V . By Theorem 3.3 there is a hypothesis transformation on P representing the binary operator K. The opposite direction of the theorem can be
easily checked. B
Proof of Proposition 4.3. We write K for K i in this proof. Unions and
intersections with respect to a are taken over all actions a g AŽ ¨ .. We
prove first that a is taken by i precisely when ¨ is reached and i knows
that he is playing a at ¨ . That is,
w ax s w ¨ x l K Ž w ¨ x ª w ax . .
Ž 6.5.
To see this, observe that, since w ax : w ¨ x, Ž4.2. implies
w ax : w ¨ x l K Ž w ¨ x ª w ax . .
Ž 6.6.
w ¨ x s D a w ax s D a Ž w ¨ x l K Ž w ¨ x ª w ax . . .
Ž 6.7.
Therefore,
By Ž6.6. and Ž6.7., it is enough to show that the events in the last union in
Ž6.7. are disjoint in pairs. Indeed, for w ax / w a9x in AŽ ¨ .,
w ¨ x l K Ž w ¨ x ª w a x . l K Ž w ¨ x ª w a9 x . s w ¨ x l K Ž w ¨ x ª Ž w a x l w a9 x .
s w ¨ x l K Ž w ¨ x ª B.
s w ¨ x l K Ž !w ¨ x .
s B.
Using Ž6.5. we have
D a K w¨ x Ž w ¨ x ª w ax . s D a K w¨ x Ž w ¨ x ª w ¨ x l K Ž w ¨ x ª w ax . .
s D a K w¨ x Ž w ¨ x ª K Ž w ¨ x ª w ax . . .
By Proposition 6.1Žd., this last event is K w ¨ xŽw ¨ x ª D a K Žw ¨ x ª w ax... But
by Ž6.7., w ¨ x : D a K Žw ¨ x ª w ax. and thus w ¨ x ª D a K Žw ¨ x ª w ax. s V.
Therefore, by Proposition 6.1Žb., K w ¨ xŽw ¨ x ª D a K Žw ¨ x ª w ax.. s V, and
we have proved the equality of the proposition.
248
DOV SAMET
To show that the events are disjoint in pairs, observe that by ŽK4., for
w ax / w a9x in AŽ ¨ .,
K w ¨ x Ž w ¨ x ª w a x . l K w ¨ x Ž w ¨ x ª w a9 x . s K w ¨ x Ž w ¨ x ª w a x l w a9 x .
s K w¨ x Ž w ¨ x ª B
. s K w¨ x Ž !w ¨ x . .
By ŽK2., ŽK5., ŽK4., and Proposition 6.1Žc.,
K w¨ x Ž !w ¨ x . s K w¨ x Ž K !w ¨ x . s K w¨ x Ž K !w ¨ x . l K w¨ x Ž ! K !w ¨ x .
s K w ¨ x Ž B . s B. B
Proof of Proposition 4.6. We prove by induction that for each node ¨ on
the play leading to sŽ r ., Ži.e., ¨ U sŽ r .., w s x : w ¨ x. For ¨ s r this is true,
since w r x s V. Suppose we have proved for ¨ g Vi on this play, and let
a s si Ž ¨ .. We must show that w s x : w ax. By Ž4.5., w s x : K iw ¨ xŽw ¨ x ª w ax. and
therefore, by the induction assumption, w s x : w ¨ x l K iw ¨ xŽw ¨ x ª w ax.. By
Proposition 6.1Že., w ¨ x l K iw ¨ xŽw ¨ x ª w ax. s w ¨ x l K i Žw ¨ x ª w ax.. But the
last event is w ax by Ž6.5..
The equality w si x s K i w si x follows from Ž4.5. and ŽK1.. B
To facilitate a proof of Theorem 5.3 by induction we provide in the next
lemma an inductive expression for a common hypothesis.
LEMMA 6.8.
For each u g Vi ,
F H Ž u, ¨ , R Ž ¨ . . s R Ž u . l F
¨ #u
K iw ax Ž w a x ª F ¨ # a H Ž a, ¨ , R Ž ¨ . . . .
agAŽ u .
Proof. By the definition of H ,
F H Ž u, ¨ , R Ž ¨ . . s R Ž u . l F H Ž u, ¨ , R Ž ¨ . .
¨ #u
¨ %u
s RŽ u. l
F F K iw ax Ž w a x ª H Ž a, ¨ , R Ž ¨ . . . .
agAŽ u . ¨ #a
The lemma follows now by applying ŽK4. to the last event. B
To prove the theorem we use the following lemma that states that if i’s
node u is reached and he hypothesizes that each of his actions leads to the
backward induction node that follows the action, then he knows in which
of these nodes he ends up. We denote
H Ž u, b . s
F
agAŽ u .
K iw ax Ž w a x ª b Ž a .
..
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HYPOTHETICAL KNOWLEDGE AND GAMES
For each player i and node u g Vi ,
LEMMA 6.9.
w u x l H Ž u, b . : D K i Ž w u x ª b Ž a. . .
agAŽ u .
Proof. Using Proposition 6.1Že., we have
w a x l H Ž u, b . : w a x l K iw ax Ž w a x ª b Ž a. .
s K i Ž w a x ª b Ž a.
..
Ž 6.10.
By assumption Ž4.2.,
w ax : K i Ž w u x ª w ax . .
Combining this with Ž6.10. yields
w a x l H Ž u, b . : K i Ž w u x ª w a x . l K i Ž w a x ª b Ž a. .
: K i Ž Ž w u x ª w a x . l Ž w a x ª b Ž a.
: K i Ž w u x ª b Ž a.
..
.
by ŽK4. and monotonicity ŽProposition 6.1Ža... Since w u x s D ag AŽ u.w ax, we
conclude
w u x l H Ž u, b . s D Ž w a x l H Ž u, b . .
agAŽ u .
:
D
K i Ž w u x ª b Ž a.
..
B
agAŽ u .
The next lemma claims that if the player at u is rational and he
hypothesizes that each of his actions leads to the backward induction node
that follows the action, then the play ends in the backward induction that
follows u.
LEMMA 6.11.
For each u g V,
R Ž u . l H Ž u, b . : b Ž u . .
Proof. Let x s h i Ž b Ž u... Then w b Ž a.x : whi - x x for each a g AŽ u.
such that a / bi Ž u., and w b Ž bi Ž u..x : whi G x x. From this we can conclude
that, for a / bi Ž u., K i Žw u x ª w b Ž a.x. l H Ž u, b . l RŽ u. s B, and K i Žw u x
ª w b Ž bi Ž u..x. l H Ž u, b . l RŽ u. : w b Ž u.x. Using Lemma 6.9 we have
R Ž u . l H Ž u, b . s R Ž u . l w u x l H Ž u, b .
: R Ž u . l Ž D a K i Ž w u x ª b Ž a.
: b Ž u. . B
...
250
DOV SAMET
Proof of Theorem 5.3. We prove by induction that for each node
u g V,
F H Ž u, ¨ , R Ž ¨ . . :
b Ž u. .
¨ #u
If AŽ u. : Z, then F ¨ # u H Ž u, ¨ , RŽ ¨ .. s RŽ u. : w b Ž u.x. Suppose we have
proved this for all a g AŽ u.. Combining Lemma 6.8, Lemma 6.11, and the
induction assumption we have
F H Ž u, ¨ , R Ž ¨ . . s R Ž u . l F
¨ #u
K iw ax w a x ª
agAŽ u .
F
: R Ž u. l
ž
F H Ž a, ¨ , R Ž ¨ . .
¨ #a
K iw ax Ž w a x ª b Ž a .
/
.
agAŽ u .
: b Ž u. . B
Proof of Theorem 5.4. We define an extended information structure as
follows. The state space V is the set of terminal nodes Z Žand thus for
each z g Z, z 4 s w z x.. Each player i’s partition is the finest, i.e., P i Ž z . s
z 4 for each z g Z. Define Ti for each z and H / B as follows. If z f H,
then Ti Ž t 4 , H . s b Ž ¨ .4 for some ¨ which is minimal Žwith respect to ).
in H. If z g H, then Ti Ž z 4 , H . s z 4 . It is easy to see that Ti is a
hypothesis transformation. Note that for each u g Vi and a g AŽ u.,
Ti Ž b Ž u.4 , w ax. s b Ž a.4 and therefore,
b Ž u . 4 : F K iw ax Ž w a x ª b Ž a. 4 . .
agAŽ u .
It is easy to see that from this inclusion it follows that for each u g V,
b Ž u. 4 : R Ž u. .
To show that F ¨ H Ž r, ¨ , RŽ ¨ .. is not empty we prove by induction that
for each u g V, b Ž u.4 : F ¨ # u H Ž u, ¨ , RŽ ¨ ... If AŽ u. : Z this inclusion
is reduced to b Ž u.4 : RŽ u., which clearly holds. Suppose we have proved
this for all a g AŽ u., then using Lemma 6.8, monotonicity, and the
induction assumption we have
F H Ž u, ¨ , R Ž ¨ . . s R Ž u . l F
¨ #u
K iw ax Ž w a x ª F ¨ # a H Ž a, ¨ , R Ž u . . .
agAŽ u .
: R Ž u. l
F
agAŽ u .
> b Ž u. . B
K iw ax Ž w a x ª b Ž a .
.
HYPOTHETICAL KNOWLEDGE AND GAMES
251
ACKNOWLEDGMENTS
The author thanks Robert Aumann, Dieter Balkenborg, Itzhak Gilboa, Joe Halpern,
Daniel Lehmann, David Schmeidler, and Robert Stalnaker for helpful comments. Financial
support of the Israel Institute of Business Research is gratefully acknowledged.
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