AXIOMATIC FOUNDATIONS OF HARSANYI’S TYPE SPACE AND
SOLUTION CONCEPTS
JEFFREY C. ELY AND MARCIN PESKI
Abstract. The standard model of games with incomplete information is the Harsanyi’s
type space. We argue that the standard model has some interpretational di¢ culties: the
de…nition of a type is not clear, and it is not clear how to discuss the restrictions imposed
by the standard model. In order to address these di¢ culties, we propose an alternative way
to describe games in a language that refers only to objective outcomes like states of Nature
and actions. The objective description is more general than the type space in the sense
that all type spaces have an objective description, but not all objectve descriptions have a
type space representation. We provide conditions that are necessary and (almost) su¢ cient
for the objective description to be derived from a type space. Finally, we argue that that
the objective description contains all (and not more) information that is necessary for the
decision theoretic analysis. In particular, we argue that many standard solution concepts
can be understood as an application of Common Knowledge of Rationality on di¤erent kinds
of objective descriptions.
1. Introduction
We compare two ways of describing decision problems and games with incomplete information.1 In the single-player case, the standard model consists of space of states of Nature
; set of types T , and distribution 2 (
T ) : Nature chooses state and type from
distribution ; the decision maker learns about his type (but not state), and he chooses an
action a 2 A: In the multiple player case, the standard model was introduced by Harsanyi
(Harsanyi (1967-68)) and it includes for each player i = 1; :::; N; a set of types Ti and a belief
function i : Ti ! (
T i ) : The belief function assigns types with beliefs over the states
of Nature and the types of the other players.2 The contingent plan of actions given types is
called a strategy.
Due to its simplicity and versatility, the type space model became one of the most important tools of game theory. Nevertheless, there are few di¢ culties with the interpretation of
the standard model. First, the model is not clear about the meaning of a type. In many
1
EXTREMELY PRELIMINARY. Department of Economics, Northwestern University.
Email:
je¤[email protected].
Department of Economics, University of Texas at Austin.
Email:
[email protected].
2 Here, and elsewehere, X denotes the space of probability distributions over measure space X.
1
2
JEFFREY C. ELY AND MARCIN PESKI
applications, a type of the player is well-de…ned as a physical form of the signal that the
player received. In other applications (like auctions with independent values), a type corresponds to a preference parameter. However, the interpretation of a type as a physical
signal or utility function is too narrow for many other applications. In particular, in order
to discuss the strategic impact of higher-order uncertainty, it is essential to treat a type, at
least partially, as a belief about other players’types. Unfortunately, the de…nition of a type
as a belief about states of Nature and other players’types is too self-referential to be useful.
Alternatively, one could de…ne a type as a model of a player’s state of mind (which includes
the player’s beliefs about other players’ states of mind). Such de…nition presumes that it
is, at least in theory, possible to ask players about their types. However, since there is no
natural language shared by the modeler and the player that includes a reference to a type,
it is not clear how to do it.3
Second, the standard model contains information that is super‡uous for many decision
theoretic purposes. For example, to check the rationality of the single player, one needs to
know the joint distribution over actions and states, not the joint distribution over actions,
states, and types. Finally, given the absence of a more substantive description, it is not
clear what are the restrictions imposed by the standard model and whether there are any
alternative models.
The goal of this paper is to provide a foundation for the type space that avoids the
di¢ culties mentioned above and that allows one to discuss restrictions that the type space
imposes on the analysis of the games with incomplete information. We propose a description
of games in a language that contains references only to objective outcomes like states and
actions. We show that our description is strictly more general than the one derived from the
type space, and we provide necessary and (almost) su¢ cient conditions under which the two
descriptions are equivalent. Finally, we use the description to provide a uni…ed approach to
the solution concepts.
3
In many applications, a type of the player is well-de…ned as a physical form of the signal that the player
received. In other applications (like auctions with independent values), a type corresponds to a utility
function. In all these cases, one can imagine a common language that is shared by the modeler and the
player that enables both of them to communicate about the player’s type. However, the interpretation of a
type as a physical signal or utility function is too narrow for many other applications. In particular, in order
to discuss the strategic impact of higher-order uncertainty, it is essential to treat a type, at least partially,
as a belief about other players’types.
FOUNDATIONS OF TYPE SPACE
3
Single player
Multiple players
Type space
(T; ),
(T; ),
where 2 (
T ) ; where T = T1 ::: TN , and
(
T i) ;
i : Ti !
Strategies
:T !A
i : Ti ! Ai
Situation
s2S= (
A)
s i 2 S i ' Ai
(
S i)
Frame
set of situations
set of situations
Table 1. Two ways of describing decision problems and games.
The objective description has two elements, a situation and a frame (see Table 1). A
situation is a complete description of a single state of the decision problem or game. In
the single player case, we take an ex ante perspective and we de…ne a situation as joint
distribution over states and actions. In the multi-player case, we de…ne situation from the
interim point of view of a single player. In such a case, a situation consists of an action, a
belief about the states of Nature, and a belief about other players actions and beliefs (i.e.,
other player’s situations). We construct the universal space of situations Si so that the
isomorphism holds:
S i ' Ai
(
S i) :
The construction closely follows the construction of the universal type space (Mertens and
Zamir (1985)).
A set of situations is called frame. We interpret frame as a complete list of every situation
that may happen in the decision problem or game.
A frame can be derived from a model. In the single-player case, each strategy of the
decision maker induces a joint distribution over states and actions s 2 (
A). In the
multi-player case, there are multiple ways of deriving a frame:
For each type ti , any pro…le of strategies can be represented by a situation: an action
of type ti and an induced hierarchy of beliefs states of Nature, actions, and the beliefs
of the other players. A set of all situations derived in such a way is called a derived
frame of type ti :
If the strategy pro…les must satisfy some measurability constraints, we say that frame
is derived with restrictions.
The notion of strategy can be generalized to allow for subjective beliefs about actions.
For example, suppose that player i uses strategy i? and believes that player j uses
strategy ji ; and that player j believes that player k uses strategy kij ; and so on
4
JEFFREY C. ELY AND MARCIN PESKI
Derived frame with restrictions
Introspection and Invariance to relabelings
+Invariance to
situation-based relabelings
+ Countable action hierarchy
+ Countable action hierarchy
Ex ante subjective frame w. restrictions
+Invariance to
situation-based relabelings
+ Invariance to subjective relabelings
Subjective frame w. restrictions
Derived frame
Ex ante subjective frame
+ Invariance to subjective relabelings
+Invariance to
situation-based relabelings
Subjective frame
Figure 1. Characterization of derived frames in the multi-player case.
... . We call such strategies as ax ante subjective, and say that frames are ex ante
subjective derived.
Di¤erent types of players may have di¤erent beliefs about the opponents strategies.
For example, player i type ti chooses action i? (ti ) and believes that player j type tj
tt
chooses action jti (tj ) ; and that player k type tk chooses action ki j (tk ) ; and so on
... . We refer to such strategies and corresponding frames as (interim) subjective.
Finally, we can distinguish between strategies that always take pure values, and
strategies which may take randomized (mixed) values.
Not all frames can be derived. The main results of the paper characterize frames that
can be derived from some model. In the single-player case, we show that frame F is derived
(almost) if and only if it satis…es two conditions:
Invariance to relabelings says that for any situation s that belongs to the frame, if
situation s0 is obtained from s by relabeling of actions, then s0 also belongs to the
frame.
Introspection requires that any two situations in the frame can be connected by a
certain consistency condition.
In the multiple player case, there are many natural procedures that can be used to derive a
frame from a given model. We show that Introspection together with appropriate versions of
the Invariance axiom characterize condition frames derived according to di¤erent procedures.
We leave the details for the main body of the paper; Fig. 1 explains the relations between
assumptions imposed on frames and the derivation procedure.
FOUNDATIONS OF TYPE SPACE
5
We use the objective description for an uni…ed presentation of various solution concepts
in games with incomplete information. The idea is simple: For any game (payo¤ function),
one can identify situations in which each player’s action is the best response against his
beliefs about states and actions of the opponent, each player believes that his opponents
best respond, each player believes that his opponent believe, and so on ... . The set of is
such situations is called Common Knowledge of Rationality (CKRat).4 For each frame, the
intersection of the frame with CKRat yields the set of situations that, on one hand, can
plausibly happen in the game (as a part of the frame), and on the other hand, everybody is
commonly known to best respond. It turns out that various solution concepts as Bayesian
Nash equilibrium, ex ante or interim rationalizability, can be interpreted as intersections of
appropriately derived frames. In particular, information contained in the derived frames is
su¢ cient to solve the game.
This has a couple of implications. First, we suggest that frames are the natural domain on
which the solution concepts are de…ned. Second, we argue that one can shift the attention
the literature devotes to the di¤erences between solution concepts, towards the di¤erences
in modelling the interactive uncertainty. In our interpretation, there is only one solution
concept, CKRat; but there are many di¤erent ways of thinking about interactive uncertainty.
Literature. TBA
The next section de…nes some notation. Section 3 is devoted to the single-player case.
Section 4 describes situations, frames, and di¤erent types of relabelings in the multi-player
case. The next three sections (5, 6, and 7) representation theories for, respectively, derived,
ex ante subjective, and subjective frames. Section 8 discusses randomization. Section 9
relates classes of derived frames to solution concepts.
2. Mathematical preliminaries and notation
If (X; X ) is a measure space, then (X; X ) is a space of set functions : X ! [0; 1] that
are countably additive, hence probability measures. Unless we say otherwise, all topological
spaces are treated as measurable spaces equipped with Borel -algebra. Also, if X is a
topological space, then X is the space of probability measures on X equipped with the
weak topology and with Borel -algebra. If X is compact and metrisable, then X is
compact and metrisable as well.
Let X be measurable spaces and 2 X is a measure. For each measure 2 X; and
R
measurable function f : X ! R; let [f ] = f (x) d (x) denote the expectation of f w.r.t.
:
4 Whenever
we say "knowledge," we mean, more precisely, belief with probability 1.
6
JEFFREY C. ELY AND MARCIN PESKI
Some notation on the transport measures is very useful. For any two measurable
X; Y; and measurable function g : X ! Y; there exists an associated operator g :
Y on the spaces of probability measures so that for each measure 2 X; ( g)
(g 1 (:)) : More generally, for each measurable function g : X ! Y; we de…ne g :
Y so that or each measure 2 X; and for each measurable subset E Y ,
Z
(( g) ) (E) = [g (:) (E)] = g (x) (E) d (x) :
spaces
X!
(:) =
X!
In the paper, we …x measurable space : Then, for all measurable spaces X; Y and measurable mapping g : X ! Y , we de…ne
g :=
(id
g) .
We consider pro…les of objects (oi ) indexed with the names of players, i = 1; :::; N . For all
such pro…les, we use notation o = i oi to denote an appropriate product of all the elements
of the pro…le. For example, if oi are sets, then o is the Cartesian product of the sets; if oi are
functions, then o is the function from product of the domains to the product of the target
spaces de…ned coordinatewise by functions oi . Similarly, we de…ne o i = j6=i oj .
3. Single-player decision problem
A single player takes action a 2 A, where A is the set of all feasible actions. The choice
of action may be correlated with the state of Nature ! 2 . We assume that A and
are compact and separable spaces5. Additionally, we assume that the space of actions are
su¢ ciently rich:
Assumption 1 (Rich actions). Each Polish space B is homeomorphic to a measurable subset
of A:
The Assumption is satis…ed if A is su¢ ciently rich: For example, A contains a homeomorphic image of a Hilbert cube [0; 1]N .6 We interpret set A as the "union" of all (complete and
separable) action sets.
The standard model of decision-making under uncertainty consists of a measurable space
T of types and a prior distribution over types and states of the world, 2 (
T ) : The
decision maker observes the type (but not the state of the world) and chooses an action. A
complete and measurable plan of actions : T ! A contingent on observed types is called
a strategy.
5I
think can be safely made Polish (without compact), but it requires some caution in the proofs.
we work hard enough, we can get rid of compactness of A: Then, the Assumption will be satis…ed if A
is countably in…nite.
6 If
FOUNDATIONS OF TYPE SPACE
7
For now, we assume that the strategies take only pure values. There are two reasons: First,
the type space model is su¢ ciently general so that it allows for an explicit randomization
with extra types corresponding to the coin ‡ips). Second, it is instructive to consider a
model with only pure strategies …rst, and then compare it with the model in which mixed
actions are allowed. We pursue this goal in Section 8
A probability distribution over the states of Nature and the choices of actions is called
a situation, s 2 S = (
A). A situation is interpreted as an outcome of the decision
problem. A set of situations is called frame F
(
A) : A frame is interpreted as a
complete description of all possible outcomes of the decision problem. We say that frame is
closed, if set F is a closed of (
A).
In a typical application, frames are derived from a Bayesian model of decision-making
2 S be the situation induced by strategy : for each
under uncertainty. Let s =
measurable set E
A;
Z
(t) fa : f!; ag 2 Eg d (!; t) :
s (E) =
T
We say that frame F is derived from model (T; ) if F consists of all situations s st.
: T ! A: We say that the frame is derived if and only if it is derived from some model.
Not all frames are derived. Consider the following examples.
Example 1. Suppose that = fx; yg, fa; bg A; and the frame F
(
A) contains
0
two situations s; s 2 F : situation s assigns probability 1 to state ! = x; and situation s0
assigns probability 1 to state ! = y:
In Example 1, frame F consists two situations s and s0 that di¤er with respect to their
marginal distributions over the state of Nature: Because Nature’s choice is a …xed parameter
of the model, the two situations s and s0 cannot be derived from the same decision problem.
Example 2. Suppose that fa; bg
action a:
A; and let F = fsg ; where s assigns probability 1 to
The frame F in Example 2 contains a situation in which the player takes action a; but
does not contain any situation in which the player takes action b: This violates the property
that the derived frame contains situations obtained from all possible strategies.
Example 3. Suppose that
s ; ; s0 ; :
s
;
= fx; y; zg, and fa; bg
(x; ) = s
;
(y; ) = s
A: For each ;
;
s0 ; (x; ) = s0 ; (y; ) = s0 ;
1
(z; ) = ; and
3
1
(z; ) = :
3
2 A; de…ne situations
8
JEFFREY C. ELY AND MARCIN PESKI
Let F = s
;
0
;s
: ;
2A :
In Example 3, frame F consists of two kinds of situations: First, there are situations in
which the same action is played if the state of Nature is x and y; and a possibly di¤erent
action is played when the state of Nature is z: That means that the player can di¤erentiate
between state z and the remaining states. Second, there are situations in the same action
is played if the state of Nature is z and y; and a possibly di¤erent action is played when
the state of Nature is x: That means that the player can di¤erentiate between state x and
the remaining states. On the other hand, the player is not expected to di¤erentiate between
state y and the remaining states.
If the model allows the player to tell states fx; yg and fzg as well as fxg and fy; zg apart,
introspection should allow him to distinguish state y from the other states. Thus, frame
F cannot describe the decision problem faced by a rational player.
We describe two conditions Invariance to Relabelings and Introspection that (almost)
characterize derived frames. A measurable mapping l : A ! A is called a relabeling: A
relabeling l induces a mapping on situations that we denote as l : S ! S and that is
de…ned as
l
(s) =
l s:
Informally, mapping l takes each situations and renames actions according to relabeling
l: If the modeler believes that the player is free to choose between any actions, a situation
obtained from an element of the frame by the relabeling of actions should remain in the
frame.
Invariance to RelabelingSP : For each relabeling l; each situation s, if s 2 F;
then l (s) 2 F:
The interpretation is that in the decision problem, the decision maker is free to choose
any action. In particular, he is free to replace a for b for each pair of actions a and b.
Invariance to RelabelingSP is violated by Example 2, but it is satis…ed in Example 3.
IntrospectionSP : For any two situations s0 ; s00 2 F; there exists relabelings
l0 ; l00 and situation s 2 F such that l0 (s) = s0 and l00 (s) = s00 :
The interpretation is that the modeler thinks that two situations s0 and s00 can plausibly
occur in the decision problem. The modeler should be able to perform a thought experiment
in which he imagines two parallel decision problems: one described by situation s0 and
the other described by s00 : More formally, the modeler should be able to imagine a proxy
"situation" s^ 2 (
A A) over action space A A; where the marginal over
and
0
the …rst coordinate of the action space is equal to s and the marginal over the second
coordinate is equal to s00 : Because the proxy "situation" s^ is not a proper situation, it cannot
FOUNDATIONS OF TYPE SPACE
9
be a part of frame F: However, due to the Rich Actions and the existence of homeomorphism
h : A A ! h (A A) A, there exists a situation
s=
h s^
and relabelings
l0 = proj
A(1)
h
1
and l00 = proj
A(2)
h
1
that ful…l the requirements of Introspection. (Here, proj A(i) is the projection from
A A
on the space of states of Nature and ith coordinate of the product A A.) Relabelings l0
and l00 are chosen so that to make the diagram from Figure 2 commute.
l'
s
l'
h
s'
s''
Figure 2. Introspection
Example 3 violates IntrospectionSP exactly because it does not contain a situation that
distinguishes between all three states of Nature.
Theorem 1. Assume Rich Actions. If frame F is derived, then it satis…es Invariance
to RelabelingSP , and IntrospectionSP . If frame F is closed and it satis…es Invariance to
RelabelingSP and IntrospectionSP , then it is derived.
The Theorem provides an (almost) complete characterization of the frames that are
derived from some decision problem: A frame satis…es Invariance to RelabelingSP and
10
JEFFREY C. ELY AND MARCIN PESKI
SP
Introspection if and (almost) only if it is derived. The only di¤erence between the su¢ cient and the necessary part is that the former additionally requires the frame to be closed.
In general, not all derived frames must be closed.78
The argument that the above conditions are necessary is straightforward. To see why each
derived frame must satisfy Invariance to RelabelingSP , suppose that s is a situation derived
from strategy : Then, for each relabeling l; l
is a well-de…ned strategy, and the derived
situation is equal to situation obtained from relabeling of actions, sl = l (s) : Second,
00
for any two measurable strategies 0 and 00 , construct a strategy = h ( 0
) ; where
h : A A ! A is a measurable injection. Then, situations s 0 and s 00 can be obtained
from situation s by appropriate relabelings. It follows that any derived frame has to satisfy
Introspection.
To establish su¢ ciency, we take T = AF to be the set of all (not necessarily continuous or
measurable) functions from the frame F to set of actions A: We assume that T is equipped
with the product topology and the Borel -algebra. Finally, we construct a distribution
2 (
T ) so that its marginals over
AJ for subsets J
F are appropriately
consistent with situations in F: Precisely, we require that satis…es two conditions:
(1) for each situation s 2 F; the marginal of over
A(s) is equal to s (here, A(s)
denotes the coordinate of product F ). This should ensure that the frame derived from
the decision problem (T; ) contains F: Indeed, for each situation s 2 F; the derived
0
frame contains a situation s s = s obtained from strategy s a(s ) ; s0 2 F = a(s) :
(2) Because of Rich Actions, for each …nite set J
F; there exists a homeomorphism
J
J
J
(s)
h : A ! A onto A: Denote
to be the marginal of over
s2J A : Then,
sJ =
h J is a proper situation. Notice that sJ is equal to the situation obtained
from strategy ;J a(s) ; s 2 F = hJ a(s) ; s 2 J .
We choose so that sJ belongs to frame F: This ensures that the situation obtained
from strategy J belongs to F: Because hJ is a homeomorphism, this ensures that
each situation derived from strategy that continuously depends on the J-coordinates
7 To
see an example, consider a decision problem (T; ) such that the state of Nature is uniformly
distributed over interval = [0; 1] and the player knows the state of Nature: (t; ! 2 [x; y]) = jx yj for
each 0 x y 1: Fix any two di¤erent actions a 6= b. Consider a sequence of strategies
(
a, if t 2 2kn ; k+1
for some even k < 2n
2n
n (t) =
b,
otherwise.
Then, the closure of the derived frame contains a limit of situations s n that is equal to the uniform
distribution on
fa; bg : Such a situation cannot be derived from any measurable strategy.
8
The closedness of a frame has a behavioral interpretation. Recall that a frame is a set of situations that
are considered plausible by the modeler. Closedness means that the modeler does not …nd reasons to reject
the possibility of limits of situations that are already recognized as plausible.
FOUNDATIONS OF TYPE SPACE
11
F
of the product A belongs to F: Finally, the closedness of F implies that frame F
contains all derived situations.
The proof of the existence of measure is based on the Kolmogorov’s Extension Theorem.
The details can be found in Appendix A
4. Situations and frames in the multi-player case
4.1. Universal space of situations. There are I players. Let be a compact and Polish
space of states of Nature. For each player i; let Ai be action spaces. We assume that each
Ai satis…es Assumption 1. Recall that A i = j6=i Aj and A = (Ai ) denotes the entire pro…le
of the action spaces.
A player i’s situation is a description of the state of the game from the (interim) point
of view of the player. The situation should include the action chosen by player i. Also, it
should include i’s beliefs about the states of Nature as well as descriptions of the state of
the game from the point of view of other players, i.e., situations of other players.
Denote the space of all situations Si (A) of player i. (The notation emphasizes that Si (A)
is built over actions that belong to spaces (Ai ).) Space Si (A) should be large enough to
contain all possible situations, but not too large, so that it does not contains objects that
are not situations, and that it does not contain two elements that correspond to the same
situation. Because each situation is completely characterized by an action and a belief, the
above implies that the space of situations must be isomorphic to the product of actions and
beliefs over and situations of the other players:
Si (A) ' Ai
(
S
i
(A)) :
(4.1)
In Appendix B, we show that there exists the smallest space characterized by the …xedpoint identity (4.1). We refer to Si (A) as the universal space of situations. The notion, the
construction of such space and the proof of the Theorem follows parallels the notion, the
construction, and the proof of the existence of universal type space by Mertens and Zamir
(1985) (see also Brandenburger and Dekel (1993) and Mertens, Sorin, and Zamir (1994)). In
particular, we construct a sequence of spaces Si0 (A) = Ai and by induction on k > 0;
Sik (A) = Ai
S k i 1 (A) ;
Q
and we identify Si (A) as the subset of in…nite Cartesian product k Sik (A) that satis…es
appropriate consistency restrictions. In particular, each situation si is uniquely (and continuously) identi…ed by the sequence of situations ski 2 Sik : We refer to ski as the kth order
Q
situation. The universal space of situations inherits the product topology from k Sik (A).
The di¤erence between the universal type space the universal space of situations lies in the
…xed-point formula (4.1): In the former case, the universal type space is isomorphic to the
12
JEFFREY C. ELY AND MARCIN PESKI
beliefs over and universal types of the opponents; in the latter, the space of situations is
isomorphic to the product of actions and beliefs. In fact, if Ai = f i g are spaces that consist
of single element, then one can identify the universal type space (over space of uncertainty
) with the universal space of situations Si (A ). Thus, if ui : Si (A) ! Ai is the pro…le
of trivial mappings, then i u maps player i’s situations into Mertens-Zamir hierarchies of
beliefs over .
A frame is a subset of the universal space of situations. Frame F is closed, if F is a closed
subset of Si (A) :
The interpretation is that frame describes all possible situations that may happen in the
game. Not all frames adequately model games. In fact, most of the issues identi…ed in the
single-player case arises as well when there are multiple players.
4.2. Invariance to relabelings. As in the single-player case, we assume that players’
frames are invariant to relabelings of actions. However, here, we distinguish di¤erent classes
of relabelings:
(1) A relabeling of i’s action is a measurable mapping
li : Ai ! Ai :
Each pro…le of relabelings l = (li ) induces a pro…le of relabeling of situations
Si (A) ! Si (A) that can be characterized by a …xed-point formula
i l (ai ; ui )
= li (ai ) ;
il
ui :
il
:
(4.2)
In Appendix B.2, we show that the induced relabeling i l as well as the relabelings
de…ned below are uniquely characterized by the equations. To understand formula
(4.2), recall that each situation s = (ai ;i u) consists of action ai and belief ui about
states of Nature and situations of the other players. Then, (4.2) means that relabeling
i l changes action ai into action li (ai ) and changes the beliefs so that the situations
of the other players are transformed through relabelings
i l. The interpretation is
that two di¤erent actions of the same player correspond to two di¤erent states of
mind of the player. Because players, or, more precisely, their states of mind, are free
to choose any action they want, that should mean that
(2) The above de…nition ensures that player j believes that player i changes his action
according to the same relabeling li , regardless of the state of mind of player j: It is
convenient to consider a class of subjective relabelings, in which player i beliefs about
how the opponents relabels their actions depend on i’s state of mind. A subjective
FOUNDATIONS OF TYPE SPACE
13
relabeling of i’s actions is a mapping
li : Ai
[
j
A ! Ai :
The interpretation is that according to player j playing action aj , player i changes
his action ai to li (ai ; aj ). Each pro…le of subjective relabelings induces a pro…le of
S
mappings on situations i l : Si (A)
A ! Si (A) de…ned by
j
i l (ai ; ui ; a)
= li (ai ; a) ;
i l (:; ai )
ui :
(3) In each of the two cases above, two situations with the same actions but two di¤erent
beliefs are treated as one subject of choice. On the other hand, recall that we interpreted a situation as a complete description of the state of the game from the point
of view of a player. Thus, it is reasonable to a class of transformations, in which
a relabeling of an action may depend on the original situation. A situation-based
relabeling is a measurable mapping
li : Si (A) ! Ai :
The pro…le of situation-based relabelings induces a pro…le of mappings on situations
i l : Si (A) ! Si (A) de…ned by
i l (ai ; ui )
= li (ai ; ui ) ;
il
ui :
Finally, we can state the Invariance axiom.
Invariance to (simple, subjective, situation-based) relabeling: For
each pro…le of (simple, subjective, situation-based) relabelings l; each situation
s, if s 2 F; then l (s) 2 F:
4.3. Introspection. Additionally, we assume that the frames satisfy a multi-player version
of Introspection.
Introspection: For any two situations s0 ; s00 2 F; there exists relabelings
l0 ; l00 2 L and situation s 2 F such that l0 (s) = s0 and l00 (s) = s00 :
The interpretation is that for each pair of situations s0 and s0 in the frame, the player
should be able to imagine two parallel games that are jointly described by a proxy "situation"
s^ 2 Si (A A) for which s0 and s00 are marginals. The proxy "situation" is mapped to a
proper situation s 2 F via homeomorphism given by the Rich Actions.
14
JEFFREY C. ELY AND MARCIN PESKI
4.4. Common Knowledge. An important example of a frame is a set of situations that
satisfy knowledge, or common knowledge of another frame. Let Wi Si (A) for each player
i be a pro…le of measurable subsets of situations. Let W = Wi be the product of the pro…le
of sets. For each player j; de…ne
Kj0 W = Wj , and for each k
Kjk+1 W = Kjk W \ sj :
u
j
0;
(sj ) (
W j) = 1 :
Thus, Kjk W is the event that Wj , and that player j knows (more precisely, believes with
probability 1) that W j ; and that player j knows that W j and that everybody believes that
W i , and so on ... (k times). Set of frames
\
CKj W =
Kjk W
k
is called common knowledge of W .
5. Harsanyi’s type space
The standard model of games with incomplete information was introduced by Harsanyi
(Harsanyi (1967-68)). A type space over
is a tuple T = (Ti ; Ti ; i )i I ; where Ti is a
measurable space of i’s types with -…eld Ti ; T i = j6=i Tj and i : Ti ! (
T i ) is
a belief mapping. Harsanyi (1967-68) introduced type spaces as a model of games with
incomplete information.
Player i’s behavioral strategy is a measurable mapping i : Ti ! Ai : Let be the collection
of strategy pro…les. For each pro…le of behavioral strategies; there exists mappings si; :
Ti ! Si (A) so that for each player i and type ti , the recursive formula holds:
si; (ti ) = idAi
s
i;
i
(ti )
The existence and the uniqueness of the situation is ensured by the universal property of
space Si (A) and the discussion following Theorem 8.
A (derived) frame of type ti is a set of situations that are obtained from type ti by varying
strategy pro…les
F (ti ) = fsi; (ti ) :
2 g:
Say that frame F is derived, if it is equal to a derived frame from of some type in some type
space.
We enrich the Harsanyi’s model by allowing for restrictions on the measurability of strategies. A restriction is a collection of -…elds B = (Bi ) such that Bi
Ti . Let B be the
subset of strategy pro…les = ( i ) of strategies i that are B measurable. A B-restricted
FOUNDATIONS OF TYPE SPACE
15
(derived) frame of type ti is a set of situations that are obtained from type ti by varying
strategy pro…les that respect measurability restrictions B,
F B (ti ) = si; (ti ) :
2
B
:
Say that frame F is derived with restrictions, if it is equal to a derived frame from some
type in some type space with some restrictions. Thus, frame F is derived if it is equal to a
derived frame from some type in type space with restrictions Bi = Ti .
One can provide a behavioral interpretation of the restrictions. The idea is that a player
may have some information about the state of Nature or the type of the other player, but
for whatever reasons, she is unable, and she is known to be unable such to use such an
information. Consider the following example.
Example 4. There are two players. Player i’s type is a 0-1 pair t1i ; t2i 2 f0; 1g where t1i are
drawn independently and uniformly. Conditionally on ft1i g ; t2 i is independently chosen to
be equal to t1i with probability p > 12 and 1 t1i with probability 1 p: The interpretation
is that is player is either Nice (t1i = 1) or Mean (t1i = 0) and t2i codes the beliefs of player i
about the Niceness of player i. Player 1; when asked, will say that he is commonly known
to have non-trivial beliefs about the type of the opponent. On the other hand, it might be
commonly known that player 1’s behavior does not depend on his beliefs about player 2 type
(1’s behavior may depend on his known to be unwilling to vary his behavior with the beliefs )
The next result characterizes derived frames.
Theorem 2. Any frame derived with restrictions satis…es Invariance to Relabeling and Introspection. If frame F is closed and it satis…es Invariance to Relabeling and Introspection,
then it is derived with restrictions.
Any derived frame satis…es Invariance to Situation-Based Relabeling and Introspection. If
frame F is closed and it satis…es Invariance to Situation-Based Relabeling and Introspection,
then it is derived.
There is a small gap between the necessary and su¢ cient conditions: on one hand, the
su¢ cient conditions require that the frame is closed, on the other hand, there are derived
frames that are not closed.
As in the single-player case, it is straightforward to show that each derived frame must
satisfy Introspection and Invariance to relabeling. Frames that are derived without restrictions must also satisfy Invariance to subjective-based relabelings. Indeed, take any strategy
pro…le . Then, for each player j; mapping sj; : Tj ! Sj (A) is Tj -measurable. In particular,
for any measurable lj : Sj (A) ! Ai ; mapping lj sj; : Tj ! Aj is a measurable strategy.
Finally, notice that the relabeled situation ( i l) si; (ti ) is equal to the situation si;l s (ti )
16
JEFFREY C. ELY AND MARCIN PESKI
obtained from the strategy pro…le l s : Because the derived frame includes the latter, it
must also include the former.
We sketch the argument for su¢ cient conditions. In the case of the …rst part of Theorem
2, the argument follows the same lines as the proof of Theorem 1. In the …rst step, we
construct an "universal space of situations" Si over the states of Nature and action space
AF : The pro…le of spaces (Si ) satis…es the …xed point property (4.1). In particular, each
element si 2 Si is associated with beliefs over
S i and the pro…le of spaces (Si ) together
with the beliefs is a type space.
We carefully describe the topology and the measurability structure on Sj : Recall that for
each player j; Sj is the space of "situations " over product space AF : For each …nite J F;
there are natural projections jJ : Sj ! Sj AJ ; where Sj AJ is the space of situations
over compact metrisable space AJ : Let TjJ be the -algebra on Sj that consists of all sets
1
J
(B) ; where B is a measurable subset of Sj AJ : Then, the Borel -algebra on Sj is
j
induced by the union of TjJ for all …nite J F:
We describe the -algebra of restrictions. For each …nite J F , let BiJ be the -algebra
on player j’s space of situations Sj induced by sets B AF nJ
S i , where B is a
measurable subset of AJ . Let Bj be the -algebra induced by the union of algebras BjJ for all
…nite J F: Thus, if j (:) is a strategy that is Bj -measurable, then its value depends only
on the "action" part of the situation sj = (aj ; uj ), but it does not depend on the "belief"
part. (Notice that the beliefs are not measurable with respect to -…eld B).
We show that there exists an element si 2 Si such that two conditions are satis…ed:2
(1) for each s 2 F; the appropriately taken marginals of ti over the states of Nature
and A(s) are equal to s: (Here, A(s) is the (s)th coordinate of product AF .) That
ensures that frame F is contained in the frame derived from type si 2 Si ,
(2) for each …nite J and each B J -measurable strategy pro…le ; frame F contains the
derived strategy si; (ti ). Together with a version of the dominated convergence and
the fact that frame F is closed, this ensures that F contains the frame derived from
type ti .
For the second part, the idea is to take the same construction of type space and type
ti 2 Si . We argue that if frame F satis…es Invariance to Situation-Based Relabeling, then
it contains all situations that are derived from strategy pro…les that are T J -measurable for
each …nite J F . Because of the choice of the measurability structure on Si ; the closedness
of frame F together with the dominated convergence imply that all derived situation belong
to frame F:
To show the claim, …rst observe that by Rich Actions, AJ can be homeomorphically
mapped into the action space. Abusing notation, AJ
Aj , and each element of AJ can
FOUNDATIONS OF TYPE SPACE
17
0
j;J , in which player j
aJj : Such strategies are
be associated to an action of player j: Consider a strategy pro…le
with type aJj ; uJj 2 Sj AJ chooses an action that is associated
BjJ -measurable, which by Theorem 2 implies that the situation si; J0 (ti ) associated with
pro…le pro…le J0 belongs to frame F: Notice that si; J0 (ti ) is equal to aJj ; uJj (up to the
homeomorphisms mapping AJ into Aj ).
Second, recall that Sj AJ is compact metrisable, hence Polish, and by Rich Actions,
it can also be homeomorphically embedded in the action space of player j; Aj : Consider a
strategy pro…le J in which all players choose actions associated with their J-hierarchies.
We check the associated situation si; J (ti ) can be obtained from situation si; J0 (ti ) by a
situation-based relabeling. Thus, by Invariance, si; J (ti ) belongs to frame F:
Finally, we check that any strategy pro…le that is T J -measurable can be obtained from
situation si; J (ti ) by a simple (non-situation-based) relabeling of actions:
The details can be found in Appendix C.
6. Ex ante subjective type space
In the construction of derived frames (with or without restrictions), we always assumed
that situations are derived from commonly known pro…les of strategies. Next, we present
two procedures that relax this assumption.
In this section, we assume that player i chooses action according to strategy i? ; that i
believes that player j chooses action from strategy ji ; that i believes that j believes that
player k chooses action from strategy kij , and so on ... . In other words, each player has
a subjective (i.e., not common) beliefs about other players strategies and their (subjective)
beliefs.
Formally, let I = f(i1 ; :::; in )g be the collection of all …nite sequences of player’s names,
including the empty sequence ?. An ex ante subjective strategy of player i is a measurable
mapping i : I Ti ! Ai with the interpretation that player i1 believes that .... in believes
that player i chooses an action from strategy ii1 :::in . For each sequence (i1 ; :::; in ) 2 I, de…ne
mapping si; : I Ti ! Sin (A) so that
(i ;:::;in )
si; 1
=
(i1 ;:::;in )
i
(i1 :::in i)
j6=i sj;
i:
(One can easily show that such mappings exists and they are uniquely de…ned. We omit the
argument.)
Let B be a pro…le of restrictions on type space (T; ) : Say that ex ante subjective strategy
(i :::i )
pro…le
= ( i ) respects the restrictions B if i 1 n is Bi -measurable for each i; and
(i1 ; :::; in ) 2 I: Let A;B be the collection of such pro…les. De…ne
F A;B (ti ) = s?
i; (ti ) : for each
2
A;B
:
18
JEFFREY C. ELY AND MARCIN PESKI
A;B
We refer to F
(ti ) as an frame ex ante subjective derived from type ti with restrictions B:
We say that the frame is derived without restrictions, if B = T ; in such a case, we drop the
reference to the -…eld of restrictions.
Let a = aii1 :::in be a collection of actions aii1 :::in 2 Ai : We say that collection a consists
0
1
of distinct actions if for any two sequences i0 ; i1 2 I; aii 6= aii . De…ne
[
(ai ; ui ) 2 Si (A) : ai = aii1 :::in and ui 8j6=i aj = aij1 :::in i = 1 :
Bi (a ) =
i1 ::::in
Thus, Bi (a ) is the set of situations of player i in which i plays action aii1 :::in and believes
that each player j 6= i plays action aij1 :::in i . Then,
\ CKi (B (a ))
Bi (a ) = (ai ; ui ) 2 Si (A) : ai = a?
i
i
is the set of situations in which player i plays a?
i ; he believes that players j 6= i play aj and
that they believe that players j 0 6= j play aij
j 0 ; and so on ... .
Countable action hierarchy. There exists a collection of distinct actions
a = ai st. F \ Bi (a ) 6= 0:
It is easy to see that each ex ante subjective frame must have countable action hierarchy
a = aii1 :::in . Indeed, such frame contains a situation derived from ex ante subjective
(i :::i )
(i ;:::;i )
strategy i 1 n (ti ) = ai 1 n ; and any such situation belongs to set Bi (a ) : The next result
shows that, additionally to the conditions from Theorem 2, Countable Action Hierarchy is
su¢ cient for the frame to be ex ante subjective derived.
Theorem 3. Any frame that is ex ante subjective derived with restrictions satis…es Invariance to Relabeling, Introspection, and Countable Action Hierarchy. If frame F is closed and
it satis…es Invariance to Relabeling, Introspection, and Countable action hierarchy, then it
is ex ante subjective derived with restrictions.
Additionally, Any frame that is ex ante subjective derived (without restrictions) satis…es Invariance to Situation-Based Relabeling, Introspection, and Countable Action Hierarchy. If
frame F is closed and it satis…es Invariance to Situation-Based Relabeling, Introspection,
and Countable action hierarchy, then it is ex ante subjective derived (without restrictions).
We explain the argument for the su¢ cient conditions. Notice that if frame F satis…es
Invariance to Relabeling and Introspection, then Theorem 2, and the frame is derived with
restrictions from some type and type space ti 2 Ti . In the same time, frame F contains a
countable action hierarchy s 2 Bi (a ). Because frame F is derived, there exists a strategy
pro…le that gives rise to the countable action hierarchy s :
(i :::i )
Take any ex ante subjective strategy pro…le
= i 1 n : We construct a (standard)
strategy pro…le
according to which each player j type tj chooses action
i1 :::in
j
(tj ) if
j
(tj ) =
FOUNDATIONS OF TYPE SPACE
19
aij1 :::in :
We show that the situation derived from strategy pro…le
derived from ex ante strategy pro…le :
is equal to the situation
7. Interim subjective type space
In the construction from the previous sections, all types of a player have the same subjective beliefs about the strategies used by the other players. Here, we allow the beliefs to
(i )
depend on types. We assume that player i1 type ti1 chooses action i11 (ti1 ) ; believes that
player i2 type ti2 chooses action from pro…le (ti2 ; ti1 ) ; believes that type ti2 believes that
player i3 type ti3 chooses action (ti3 ; ti1 ti2 ), and so on ... . In other words, each type of each
player has a subjective (i.e., not shared by other types) beliefs about other players’ types
strategies and their (subjective) beliefs.
Formally, let T = f(t1 ; :::; tn )g be the collection of all …nite sequences of players’types. A
(interim) subjective strategy of player i is a measurable mapping i : T Ti ! Ai For each
(t ;:::;t )
sequence (t1 ; :::; tn ) 2 T , each player i; de…ne mapping si; 1 n : Ti ! Si (A) so that
(t :::tn )
si; 1
(ti ) =
(ti ; t1 :::tn )
(t1 :::tn ti )
j6=in sj;
:
(As in the previous section, one can easily show that such mappings exists and they are
uniquely de…ned. Again, the argument is omitted).
Let B be a pro…le of restrictions on type space (T; ) : Say that subjective strategy pro…le
respects the restrictions B if for each player i, i is measurable with respect to -…eld
S
::: Bin Bi : Let I;B be the collection of such pro…les. De…ne
i1 :::in Bi1
F I;B (ti ) = s?
i; (ti ) : for each
2
I;B
:
We refer to F I;B (ti ) as an frame (interim) subjective derived from type ti with restrictions
B: Again, we say that the frame is derived without restrictions, if B = T ; in such a case, we
drop the reference to the -…eld of restrictions.
Theorem 4. Any frame that is subjective derived with restrictions satis…es Invariance to
Subjective Relabelings and Introspection. If frame F is closed and it satis…es Invariance to
Subjective Relabelings and Introspection, then it is subjective derived with restrictions.
Additionally, Any frame that is subjective derived (without restrictions) satis…es Invariance
to Situation-Based Relabelings, Invariance to Subjective Relabelings and Introspection. If
frame F is closed and it satis…es Invariance to Situation-Based Relabeling, Invariance to
Subjective Relabelings and Introspection, then it is subjective derived (without restrictions).
It is easy to see why the Invariance to subjective relabelings is necessary. Suppose that F
is subjective derived frame from type and type space ti 2 Ti . Take any situation s 2 F: Then,
20
JEFFREY C. ELY AND MARCIN PESKI
there exists a subjective strategy
such that s = s?
i; (ti ) : Take any subjective relabeling
S
lj : Al Aj ! Aj and de…ne a new subjective strategy 0 so that for each player j;
l
0
j
(tj ; t1 :::tn ) for n
[
0
Al :
j (tj ; ?) = l (a; j (tj ; ?)) for some a 2
(tj ; t1 :::tn ) = l
in
(tn ; t1 :::tn 1 ) ;
j
0, and
l
One checks that s?
i; 0 (ti ) = ( i l) si; (ti ) = ( i l) s: Because the former belongs to frame F; it
must be that the frame contains ( i l) s as well.
The argument that the Invariance to subjective relabelings is also su¢ cient (together with
conditions of Theorem 2) is based on the induction with respect to the lag of subjective
strategy. We say that subjective strategy pro…le
has a lag k; if for all i (ti ; t1 :::tn ) =
0
0
0
n k: Then, a subjective strategy pro…le with lag
i (ti ; t1 :::tn ) whenever tm = tm for all m
equal to 0 is also a standard (and not only subjective) strategy pro…le. In particular, because
Theorem 2 applies, frame F is derived from some type and some type space, it contains all
subjective strategies of lag 0. Using induction and subjective relabelings, we show that frame
F must contain all subjective strategies with …nite lag. The argument is concluded from the
fact that F is closed and that any subjective strategy can be approximated by strategies
with …nite lag.
8. Randomization
TBA
9. Games and solution concepts
In this section, we argue that many popular solution concepts used in the games with
incomplete information can be described as an application of the Common Knowledge of
Rationality to an appropriate class of frames. This has couple of implications. First, we
suggest that frames are the natural domain on which the solution concepts are de…ned.
Second, we argue that one can shift the attention the literature devotes to the di¤erences
between solution concepts, towards the di¤erences in modelling the interactive uncertainty.
In our interpretation, there is only one solution concept that can be applied to di¤erent
classes of frames.
De…ne a game G = (gi ) as an I-tuple of measurable functions gi :
A ! R: Form now
on, we …x game G:
9.1. Common Knowledge of Rationality. We say that an action ai of player i is a best
response against (…rst-order) beliefs 2 (
A i ) ; if for each action a0i ;
[(gi (ai ; :)
gi (a0i ; :))]
0:
FOUNDATIONS OF TYPE SPACE
21
For each player i, de…ne the set of i’s situations in which player i’s behaves rationally,
Rat0i = (ai ; ui ) 2 Si (A) : ai is a best response against u1i :
For each k 0; de…ne the set of i’s situations in which player i’s behaves rationally and he
believes that the opponents behave rationally, and that .... they behave rationally (k + 1
times):
Ratk+1
= Ratki \ (ai ; ui ) 2 Si (A) : ui Rk i = 1 :
i
Let
CKi Rat =
\
k 0
Ratki :
We say that CKi R is the set of player i’s situations that satisfy the Common Knowledge of
Rationality.
9.2. Bayesian Nash Equilibrium. For any strategy pro…le = ( i ) ; say that it prescribes
best response for type ti of player i if i (ti ) is a best response against the belief induced by
strategy pro…le ;
i
(:) [
i
(ti )] :
Recall that a pro…le of strategies = ( i ) is a (pure strategy) Bayesian Nash Equilibrium
(BNE) if it prescribes best response for each type of each player. Say that a strategy pro…le
is a local Bayesian Nash Equilibrium at ti if there exists pro…le of subsets Sj Tj for each
player j such that ti 2 Si ; and for each j; each type tj , (a) j (S j jtj ) = 1 and (b) prescribes
best response for tj .
Theorem 5. Fix type and type space ti 2 Ti . A strategy pro…le
Equilibrium at ti if and only if si; (ti ) 2 CKi Rat.
is a local Bayesian Nash
Proof. Suppose that si; (ti ) 2 CKi R.. For each player j; let Sj0 be the set of types of player
j for which prescribes best response. By induction on k; de…ne
Sjk+1 = Sjk \ tj :
j
S k j jtj = 1 :
It is easy to check that for each player i; each type ti , si; (ti ) 2 Ratki if and only if ti 2 Sik :
The claim follows.
Thus, the class of pure strategy local BNEs at ti corresponds to the intersection of the
frame derived (without restrictions) from type ti with the common knowledge of rationality
Comment on mixed strategies. Without loss of generality - because we can always assume
that they form a part of the set of actions.
22
JEFFREY C. ELY AND MARCIN PESKI
9.3. Ex ante rationalizability. We describe an iterative solution concept that we call ex
ante rationalizability. Its distinguishing feature is that is players reason at the ex ante stage,
when their choose their strategy (i.e., a contingent plan of actions given types). 9
Fix a type space (T; ) with restrictions B: Consider a sequence of sets of strategies: Let
0;B
k;B
= Bi contains all Bi -measurable strategies. For each k 0; let k+1;B
consists
i
i
i
k;B
of all strategies i such that there exists a strategy pro…le i 2
i so that for each type
ti ; i (ti ) is the best response of type ti if the other players follow strategies i
i
(ti ) 2 arg max
a
i
(ti ) [gi (a;
i
(:) ; :)] :
In other words, i is the best response contingent play, if the other players follows
i.
A;k;B
Notice that each strategy i 2 i
has a k-element sequence of rationalizations through
B-measurable strategies: there are strategy pro…les l 2 k l;B such that i is the best
response against 1 ; 1 is the best response against 2 ; ..., k 1 is the best response against
k
: Let
\ A;k;B
A;B
=
:
i
i
k
be the set of strategy pro…les that have in…nite rationalization. Say that action ai is (pure
strategy) ex ante rationalizable with restrictions B for type ti if there exists a strategy i 2
A;B
such that i (ti ) = ai .
i
Theorem 6. Action ai is ex ante rationalizable with restrictions B for type ti if and only if
there exists situation (ai ; ui ) 2 F A;B (ti ) \ CKi Rat.
Proof. Suppose that for some ex ante subjective B-measurable strategy pro…le , (ai ; ui ) =
i1 :::in
Ti such that for each
s?
i; (ti ) 2 CKi Rat. Then, there exists sets of types of sets Tj
i1 :::in j
i1 :::in
, j (tj ) T j
player j, each sequence i1 :::in , and each type tj 2 Tj
= 1; (b) j (tj ) is
i1 :::in j
the best response of type ti against strategies j : We can modify strategy pro…le so
that for each type tj 2
= Tji1 :::in ; we assume that j (tj ) is the best response of type tj against
strategies i1j:::in j : Then, we can check by induction on k that ji1 :::;in 2 A;k;B
for all players
j
A;B
?
j and sequences i1 :::in . It follows that i 2 i . Thus, ai is ex ante rationalizable with
B-restrictions.
Suppose that ai is ex ante rationalizable with restrictions B: Then, there exists a rationalization, i.e., sequence of strategies ji1 :::in 2 A;B
such that ji1 :::in is the best response
j
i1 :::in
against i1j:::in j : Construct a subjective strategy
=
. It is easy to check that
j
?
si; (ti ) 2 CKi Rat.
9 The
earliest mention of ex ante rationalizability that we know comes from the Fudenberg-Tirole game
theory textbook and it applies only to spaces with prior. Our de…nition applies to general type spaces.
CHECK HISTORY AND LITERATURE TBA.
FOUNDATIONS OF TYPE SPACE
23
9.4. IIR. We describe an iterative solution concept that we call interim (independent) rationalizability. Its distinguishing feature is that is players reason at the interim stage, when
type separately is choosing his action.
Fix a type space (T; ) with restrictions B: Consider a sequence of sets of measurable
k;B
correspondences from k;B
: Ti
Ai . Let i0;B = Ti Ai . For each k 0; let k+1;B
i
i
i
consists of all pairs (ti ; ai ) such that ai is the best response of type ti against some Bmeasurable selections j : Tj ! Aj such that (tj ; j (tj )) 2 k;B
for each j 6= i. Let
j
\
A;B
A;k;B
=
:
i
i
k
A;B
i
Then, each action ai 2
(ti ) can be rationalized as the best response against some Bti
measurable strategy pro…le i and such that for each player j 6= i and type tj ; jti (tj ) can
be rationalized against some strategy pro…le ti tj ; and so on ... .Say that action ai is (pure
strategy) interim rationalizable with restrictions B for type ti if ai 2 A;B
(ti ).
i
Theorem 7. Action ai is ex ante rationalizable with restrictions B for type ti if and only if
there exists situation (ai ; ui ) 2 F A;B (ti ) \ CKi Rat.
Proof. The proof follows the same steps as the proof of Theorem 6. We omit the details.
References
Brandenburger, A., and E. Dekel (1993): “Hierarchies of Beliefs and Common Knowledge,”Journal of Economic Theory, 59, 189–198.
Harsanyi, J. C. (1967-68): “Games with Incomplete Information Played by Bayesian
Players, I, II, III,”Management Science, 14, 159–182, 320–334, 486–502.
Mertens, J.-F., S. Sorin, and S. Zamir (1994): “Repeated Games,” CORE Discussion
Papers 9420,9421,9422; Louvain-la-Neuve.
Mertens, J. F., and S. Zamir (1985): “Formulation of Bayesian Analysis for Games with
Incomplete Information,”International Journal of Game Theory, 14, 1–29.
Appendix A. Proof of Theorem 1
A.1. Necessity. Let F be a frame that is derived from the decision problem (T; ). To see
that F satis…es the Invariance to RelabelingSP , take any situation s 2 F and relabeling
l 2 L: Then, l
is a strategy and sl = l (s ) : To see that F satis…es IntrospectionSP ,
24
JEFFREY C. ELY AND MARCIN PESKI
take any two measurable strategies 0 and 00 : Let h : A
that is homeomorphic on its image. De…ne strategy
(t) = h ( 0 (t) ;
De…ne relabelings: for each a 2 h (A
l0 (a) = projA(1) h
00
A ! A be a continuous mapping
(t)) :
A) ; let
1
(a) and l00 (a) = projA(2) h
1
(a) :
(Here, A(i) denotes the ith coordinate of the product A A.) The value of the relabelings
for a 2
= h (A A) is not important. Then, it is easy to check that
l0
(s ) = s
0
and
l00
(s ) = s 00 .
A.2. Su¢ ciency. Take any frame F: Let J be a collection of …nite subsets of F: For each
J 2 J ; let AJ denote the set of functions from J to A (which is equivalent to a Cartesian
product of jJj copies of A). Similarly, let AF denote the set of functions from F to A:
We assume that those sets are equipped with the product topology. By the Tychono¤’s
Theorem, those spaces are compact.
For any J F; let T J be the -algebra on T that consists of sets of form B AF nJ ; where
B is a measurable subset of B AJ : Then, the Borel -algebra on T is induced by the union
of -algebras T J across all …nite J F:
We use two preliminary results. First notice that because of Assumption 1, there exists a
homeomorphism hJ : AJ ! hJ AJ
A: Let
hJ =
hJ :
AJ !
(
A)
be the induced mapping. Also, de…ne set of proxy "situations"
JF
=
s st.
AJ :
: A ! AJ is measurable and s 2 F
Lemma 1. If frame F is closed and it satis…es Invariance to RelabelingSP , then set
closed and non-empty.
JF
is
Proof. Let J
(
A) consists of all situations such that s
hJ AJ = 1: Then,
J is a closed set of situations. Moreover, there is a homeomorphism between
J and
J
A : By Invariance to Relabeling, ( hJ ) ( J F )
J \ F: On the other hand,
1
1
( hJ ) ( J \ F )
( J \ F ) : Because J \ F is closed, J F
J F: Thus, J F = ( hJ )
is closed.
Second, for each J; H 2 J such that J
H; de…ne natural projections pHJ : AH ! AJ
and qHJ =
pHJ : Also, de…ne pJ : AF ! AJ and qJ :
AF !
AJ : Notice
that by de…nition, pHJ ( H F )
H.
J F for each J; H 2 J such that J
FOUNDATIONS OF TYPE SPACE
25
SP
SP
Lemma 2. If frame F satis…es Invariance to Relabeling and Introspection
each J 2 J; there exists sJ 2 J F such that for each f 2 J; pJff g (sJ ) = f:
, then for
Proof. The proof goes by induction on the size of set J: Suppose that the Lemma holds for
some J 2 J . Find sJ 2 J F such that for each f 2 J, pJff g (sJ ) = f: Let s0 = ( hJ ) (sJ ) 2
(
A) : By Invariance to RelabelingSP , s0 2 F:
Take any f0 2 F nJ: By IntrospectionSP , there exists situation s 2 F and relabelings l0
and lf such that l00 (s) = s0 and lf (s) = f: We can assume that l0 (A) hJ AJ :
Construct mapping : A ! AJ[ff0 g so that for each a 2 A,
(a) =
Let sJ[ff0 g =
hJ 1 l0 (a) ; lf (a) .
s. Then,
pJ[ff0 g;J sJ[ff0 g = sJ , and pJ[ff0 g;ff0 g sJ[ff0 g = f:
Q
De…ne S 0; = J2J
AJ : One can think about an element s 2 S as a net of
probability measures directed by collection J : Say that net s = (sJ ) is consistent if for each
J; H 2 J such that J H, pHJ sH = sJ . Say that net s is faithful if for each f 2 F; sf = f:
Assume that frame F is closed and it satis…es Invariance to RelabelingSP and IntrospectionSP .
We show that there exists a consistent and faithful net s = (sJ ) such that sJ 2 AJ F for
each J 2 J . Indeed, space S 0; is compact in the product topology. Consider a collection of
subsets
SJ0; = fs : sJ 2
JF g
\ fs : sH = pJH sJ for each H
Jg
\ s : sff g = f for each f 2 J .
Notice that if s 2 SJ0; , then sH = pJH (sJ ) 2 H F for each H J.
By Lemma 1, sets J F are closed and non-empty. Thus, set SJ0; is closed, hence compact.
By Lemma 2, SJ0; is non-empty. Moreover, for any …nitely many sets J1 ; :::; Jm ; the intersection SJ0;1 \ ::: \ SJ0;m is non-empty as it contains a non-empty set SJ0;1 [:::[Jm . Thus, family
of compact sets SJ0; J2J has the …nite intersection property. It follows that there exists s
T
that belongs to the all members of the family, s 2 J SJ0; .
Let T = AF be equipped with the product topology. Then, T is a compact Hausdor¤
space. By the Kolmogorov’s Extension Theorem, there exists a measure 2 (
T ) such
that for each J 2 J , pJ = sJ 2 AJ F .
26
JEFFREY C. ELY AND MARCIN PESKI
Let F be a frame that is derived from the decision problem (T; ) : First, we show that
F F . Indeed, for each s 2 F; let s : T ! A be de…ned so that s (as0 )s0 2F = as . Then,
because net s is faithful, s s = s:
Next, we show that F
F: Let 0 be the collection of strategies
: T ! A that
depend only on …nitely many coordinates of T (i.e., there exists …nite J 2 J such that for
all (as ) ; (a0s ) 2 T; if as = a0s for all s 2 J; then (a: ) = (a0: )). De…ne F 0 = fs : 2 0 g :
We show that F 0 F . Indeed, take s 2 F 0 such that 2 0 and suppose that it depends
only on the coordinates from …nite set J 2 J . Let 0 : AJ ! A denoted the same function
with the domain restricted to the relevant coordinates. Find s 2 F and measurable mapping
: A ! AJ such that sJ =
(s). Consider relabeling l = 0
: A ! A. Then,
s = l (s) 2 F:
By the de…nition of the topology on T; all measurable functions : T ! A are pointwise
limits of the elements of F 0 : By the Lebesgue Theorem, F = cl F 0 . Because F is closed, it
follows that F
F:
Appendix B. Universal spaces of situations
B.1. Proof of Theorem 8. We show the existence and the universal properties of the
universal space of situations.
Theorem 8. For each pro…le of compact metrisable spaces A = (Ai ) ; there exists a pro…le
of compact metrisable spaces Si (A) and a pro…le of homeomorphisms i : Si (A) ! Ai
(
S i (A)) so that for any pro…le of measure spaces (Ti ) and measurable mappings
(
T i ), there exists a unique pro…le of mappings si : Ti ! Si (A) such
i : Ti ! Ai
that the recursive formula holds:
i
si = idAi
s
i
i.
(B.1)
In particular, each situation si is associated through the isomorphism with an action
ai 2 Ai and belief ui 2 (
S i (A)), (si ) = (ai ; ui ) 2 Si (A) : Moreover, the space of
situations Si (A) has the following universal property: any other space (Ti ) that satis…es the
…xed point property
Ti ' Ai
(
T i) :
can be uniquely immersed into the space Si (A) in a way that preserves the …xed point
identity.
B.1.1. Spaces Si (A). For each player i; de…ne a sequence of spaces
Si0 (A) = Ai and for each k > 0;
Sik (A) = Ai
S k i 1 (A) .
FOUNDATIONS OF TYPE SPACE
Sik
27
For each k > l; there are natural projections i;kl :
(A) !
is a projection on the …rst coordinate, and for each l < k;
i;kl
Sil
(A) so that for each k;
k0
i;k 1;l 1 .
= idAi
De…ne the space of sequences of elements of Sik (A)
Si0; (A) =
Y
k
Sik (A) :
Say that sequence s = ski 2 Si0; (A) is consistent, if for each k > l;
(
i;kl ) sk
= sl .
Let Si (A) Si0; (A) be the subset that consists of all consistent sequences.
We assume that for each k;
S k i 1 (A) has the weak topology and Sik (A) has the
product topology. Then, spaces Sik (A) are compact, complete and metrisable. Similarly
Si0; (A) is compact, complete, and metrisable in the product topology. Because projections
i;kl are continuous, Si (A) is compact, complete, and metrisable.
Let i;k : Si0; (A) ! Sik (A) be the natural projection. Notice that projections i;k restricted to the subset of consistent sequences are "onto", i;k (Si (A)) = Sik (A).
B.1.2. Measurable structure on Si (A). For each k; let Sik be the Borel -algebras on Sik (A)
and let Si ;k = ( i;k ) 1 Sik be the induced -algebra on Si (A). Because Si (A) consists of
consistent sequences, Si ;k is a re…nement of Si ;l for each k > l: For each k > 0; let S ;ki 1 be
a -algebra on
S i (A) that is equal to the product of -algebras on and Sj ;k 1 for
j 6= i.
[ ;k
Si : Then, Si is the
Let Si be the -algebra generated by the union of algebras
k
Borel -algebra on Si (A).
B.1.3. Homeomorphism. Recall that (X; X ) is the space of probability measures on measurable space (X; X ) : For each si = ski 2 Si (A) ; de…ne a measure i ;k (s ) 2
S i (A) ; S
so that for each measurable subset E
;k
i
(s ) (id
S k i 1 (A),
i;k 1 )
1
(E) = si
;k
(E) ;
(B.2)
;k
where si ;k is the "measure" part of element ski = sA;k
2 Ai
S k i 1 (A) .
i ; si
Because projection i;k is "onto", the identity (B.2) uniquely and well-de…nes a measure.)
;k 1
i
28
JEFFREY C. ELY AND MARCIN PESKI
Measures
S
i
k
i
;k 1
i
k
l 1
S i (A),
(s ) are consistent with respect to the …ltration S
(A). (Indeed, for each k > l; and each measurable E
;k
i
(s ) (id
= si
;k
=
;l
i
i;k 1 )
(id
1
(id
1
i;k 1;l 1 )
(s ) (id
1
(E)
(E) = si ;l (E)
1
i;l 1 )
i;k 1;l 1 )
on the space
(E) :)
By the Bochner Theorem (Theorem 14.23 from IDA), there exists an extension i (s ) 2
(
S i (A)) of measures i ;k (s ) such that for each k and each measurable subset E
S k i 1 (A),
i
(s ) (id
1
i;k 1 )
(E) = si
;k
(E) :
(notice that the existence of the appropriate compact class is ensured because spaces
S k i (A) are compact and mertrisable and measures i ;k (s ) are regular.) Let ai sa ; s =
sa ; and let i = ai ; i : It is easy to check that i is a bijection and that i and its inverse
are continuous. Thus, i determines the required homeomorphism.
Also, for each k; and s 2 Si (A), i;k (s ) = sk , and, because
i;k 1
(s ) = si ;k ,
i
it must be that
i;k
= idA
(B.3)
i:
i;k 1
B.1.4. Embeddings. Take any pro…le of measure spaces (Ti ) and measurable mappings i :
Ti ! Ai
(
T i ).
For each player i and k 0, construct a sequence of mappings si ;k : Ti ! Sik (A) so that
si ;0 (ti ) = margAi
i
(ti ) and for each k > 0;
si ;k (ti ) = idAi
s
The mappings induce a mapping si =
consistent; hence, si (Ti ) Si (A).
si ;k
;k 1
i
i
(ti ) :
: Ti ! Si0; (A) : One checks that si (ti ) is
B.1.5. Uniqueness. We show that there exists a unique pro…le of mapping (si ) that satis…es
formula (B.1). Indeed, suppose that si : Ti ! Si (A) is a pro…le of measurable mappings
such that for each player i; each ti 2 Ti ,
(si (ti )) =
For each k
1
i
idAi
ski (ti ) =
i;k
s
i
0; de…ne
si (ti ) :
i
(ti ) :
FOUNDATIONS OF TYPE SPACE
29
Then, by (B.3),
ski (ti ) =
i;k
si (ti )
= idA
i;k 1
i
= idA
i;k 1
i
= idA
i;k 1
idAi
sk i 1
= idAi
si (ti )
1
i
i
s
idAi
i
i
s
i
i
(ti )
(ti )
(ti ) :
We will show that ski (ti ) = si ;k (ti ) : Indeed, s0i (ti ) = si ;0 (ti ) : By induction on k
ski (ti ) = si ;k (ti ) ; then
sk+1
(ti ) = idAi
i
sk i
i
(ti ) = idAi
s
;k 1
i
i
0; if
(ti ) = si ;k (ti ) :
B.2. Uniqueness of mappings. We use the universal property of spaces Si (A) to de…ne
a class of mappings. Suppose that (Bi ) is a pro…le of compact separable spaces. Let (li )
be a pro…le of measurable mappings li : Si (A) ! Ai from the universal space of player i’s
situations over A to action spaces Ai for each player i: Then, pro…le l induces a pro…le of
mapping i l : Si (A) ! Si (A) uniquely de…ned by the recursive formula
i
il
= idAi
il
The existence and the uniqueness of mapping
8.
Recursive de…nitions.
il
li
id
i;
follows from the second claim of Theorem
B.3. Dominated Convergence. The following version of the Dominated Convergence
Theorem for derived situations turns out to be quite useful.
Theorem 9. Suppose that A = (Ai ) is a pro…le of Polish spaces; and T = (Ti ; i ) is a type
space over : Then, for each sequence of strategy pro…les k = ik that converges pointwise
to strategy pro…le ; (i.e., for each player i and each type ti ; in ! i (ti )), for each player i
and type ti ; the sequence of situations derived from pro…les n and type ti converges to the
situation derived from pro…le ,
lim si;
n
n!1
Let si;
n
(ti ) = ski;
n
(ti ) = si; (ti ) .
(ti ) and si; (ti ) = ski; (ti ) . We show that for each k
lim ski;
n!1
n
(ti ) = ski; (ti ) .
The Theorem follows from the de…nition of the topology on Si (A).
0;
30
JEFFREY C. ELY AND MARCIN PESKI
The above claim is trivially true for k = 0: Suppose that it is true for some k
claim for k + 1 follows from the fact that
lim
n!1
ski;
n
i
ski;
(ti ) =
n
i
0: The
(ti ) ;
S k i (A) :
where the limit takes place in the weak topology on
B.4. Extension Theorem. Theorem 8 establishes the existence of the universal space of
situations over compact, complete, and metrisable space of actions A: Next, we show that the
construction carries over to the case where A is a (possibly uncountably in…nite) product
of compact, complete and metrisable spaces. Suppose that Af = (Ai;f ) is a collection of
pro…les of compact, complete, and metrisable spaces indexed with some in…nite set T: Let
J denote the collection of all …nite subsets of F: In order to shorten the notation, for each
J 2 J , let AJ = f 2J Af . For each H J; J; H 2 F; let pJH : AJ ! AH and pJ : A ! AJ
be the natural projections. Also, let
qiJH = pi;JH
be the projection from Si AJ onto Si AH :
De…ne A = f 2F Af and
Y
Si0; =
J2J
q
i;JH
S i AJ
be the product of universal spaces of situations equipped with the product topology. Then,
Si0; is a compact Hausdor¤ space. Each element s 2 Si0; can be interpreted as a net
directed by algebra of …nite subsets of F: Say that net s is consistent, if for each H
J;
such that J; H 2 J ,
(qi;JH ) sJ = sH .
Let Si
Si0; contains all consistent nets. Of course, Si is a closed, hence compact.
Let qi;J : Si ! Si AJ to be the natural projection. For each J 2 J ; let i;J : Si AJ !
AJ
S i AJ be the homeomorphisms from the proof of Theorem 8. We use the
A
notation that i;J = A
i;J ; i;J ; where
i;J is the …rst coordinate and
i;J is the second
coordinate of homeomorphism i;J .
Theorem 10. There exists a homeomorphism
each J 2 J ;
pi;J
qJ i
i
i
: S i ! Ai
=
i;J
qiJ :
S
i
such that for
(B.4)
We say that Si is the universal space of situations over a pro…le of compact (but not
necessarily Polish) spaces A
FOUNDATIONS OF TYPE SPACE
31
Proof. Fix J 2 J : In order to shorten the notation, de…ne wJ = id q i;J : Let XJ be the
Borel -algebra on
S i AJ and let XJ = wJ 1 (XJ ) be a -algebra on
S i . Let CJ
J
J
S i A is Polish, for each
be a collection of compact subsets of
S i A . Because
J
measure J 2
S i A
; and each measurable set E 2 XJ ;
(E) = sup f
J
J
E and C 2 CJ g :
(C) : C
Mapping wJ induces a homeomorphism J between the spaces of probability measures
S i ; XJ :
S i AJ ; XJ and
S i AJ =
Let
CJ = wJ 1 (C) : C 2 CJ :
S
Then, C = J2J CJ is a compact class (see the de…nition from IDA). Moreover, for each
measure J 2
S i ; XJ and each measurable subset E XJ ;
(E) = sup f (C) : C E and C 2 CJ g :
S
Let X be the -algebra generated by sets J2J XJ . Then, X is the Borel -algebra on
S i : By the Bochner Theorem (Theorem 14.23 from IDA), for each s = (sJ ) 2 Si , there
exists a unique probability measure u 2
S i such that for each J 2 J and each
E 2 XJ ;
J
u
wJ 1 (E) =
J
(sJ ) (E) :
Similarly, there exists a such that for each J 2 J pi;J a
Let
i
=
i;J
(sJ ).
(si ) = a ; u :
Clearly, mapping i is bijective and formula (B.4) holds.
We check that mapping i is continuous. Fix J 2 J and a continuous function h :
S i AJ ! R: Then, for each s = (sJ ) 2 Si ,
;
i
(s ) [f
wJ ] =
and the latter expression is continuous in sJ :
The continuity of the inverse mapping ( i )
i;J
qiJ
( i)
1
=
1
1
i;J
i;J
(sJ ) uJ [f ] ;
follows from the fact that for each J
pi;J
qJ i :
and the mapping on the right-hand side is continuous.
Appendix C. Proof of Theorem 2
C.1. Type space with restrictions.
32
JEFFREY C. ELY AND MARCIN PESKI
C.1.1. Necessity. Let F = F B (ti ) be a frame that is derived from type and type space
ti 2 Ti and restrictions B: To see that F satis…es Invariance to Relabeling, take any Bmeasurable strategy pro…le and relabeling l 2 L: Then, l
is a B-measurable strategy
pro…le and si;l (ti ) = ( i l) (si; (ti )) 2 F: To see that F satis…es Introspection, take any
two B-measurable strategy pro…les 0 and 00 : For each player j; let hj : Aj Aj ! Aj be a
continuous mapping that is homeomorphic on its image. De…ne Bj -measurable strategy
j
=h (
De…ne relabelings: for each aj 2 hj (Aj
0
00
):
Aj ) ; let
lj0 (a) = projA(1) hj 1 (aj ) and lj00 (aj ) = projA(2) hj 1 (aj ) :
j
j
(i)
(Here, Aj denotes the ith coordinate of the product Aj Aj .) The value of the relabelings
for aj 2
= hj (Aj Aj ) is not important. Then, it is easy to check that
( i l0 ) (si; (ti )) = si; 0 (ti ) and ( i l00 ) (si; (ti )) = si;
00
(ti ) .
C.1.2. Su¢ ciency. Fix frame F: Let J be a collection of …nite subsets of F: For each J 2 J ;
let AJi denote the set of functions from J to Ai (which is equivalent to a Cartesian product
of jJj copies of Ai ). Similarly, let AFi denote the set of functions from F to Ai : We assume
that those sets are equipped with the product topology. By the Tychono¤’s Theorem, those
spaces are compact. Let Si AJ be the space of situations over AJ and let i;J : Si AJ !
AJ
S i AJ be the homeomorphisms from the proof of Theorem 10. For each
J
H
! Si AJ be the
J; H 2 J such that J
H; let pi;HJ : AH
i ! Ai and qi;HJ : Si A
natural projections so that
qi;HJ = pi;HJ
p
i;HJ :
We start with some preliminary remarks. Because of Assumption 1, for each J 2 J ,
there exist homeomorphisms hi;J : AJi ! hi;J AJi
Ai : Because each AJi is compact, its
homeomorphic image hi;J AJi is compact as well.
Let Vi;J denote all measurable mappings from Ai to AJi and let Vj = i Vi;J denote all
pro…les of such mappings. For any frame F; de…ne a set of situations over AJ
VJ F = f(
i
) (s) st.
Notice that by de…nition, qHJ ( VH F )
2 VJ and s 2 F g
S i AJ :
VJ F for each J; H 2 J such that J
H.
Lemma 3. If frame F is closed and it satis…es Invariance to Relabeling, then set Vj F is
closed and non-empty.
FOUNDATIONS OF TYPE SPACE
33
Proof. Let SJ
Si (A) consists of all situations such that it is common knowledge that
the actions of each player i belongs to hi;J AJi . Then, SJ is a closed set of situations.
Moreover, there is a homeomorphism between SJ and Si AJ : By Invariance to Relabeling, ( i hi;J ) ( VJ F ) Si AJ \ F: On the other hand, ( i hi;J ) 1 (SJ \ F )
VJ F: Thus,
1
VJ F = ( i hi;J ) (SJ \ F ) : Because SJ \ F is closed, VJ F is closed.
Lemma 4. If frame F satis…es Invariance to Relabeling and Introspection, then for each
J 2 J; there exists sJ 2 VJ F such that for each f 2 J; qJff g (sJ ) = f:
Proof. The proof goes by induction on the size of set J: Suppose that the Lemma holds for
some J 2 J . Find sJ 2 VJ F such that for each f 2 J, qJff g (sJ ) = f: Let s0 = ( hJ ) (sJ ) 2
Si (A) : By Invariance to Relabeling, s0 2 F:
Take any f0 2 F nJ: By Introspection, there exists situation s 2 F and relabelings l0 and
lf such that ( i l0 ) (s) = s0 and ( i lf ) (s) = f: We can assume that l0 (A) hJ AJ :
J[ff g
Construct a pro…le of mappings i : Ai ! Ai 0 so that for each a 2 A,
i
Let sJ[ff0 g = (
i
(a) =
hi;J1 li;0 (a) ; li;f (a) .
) (s). Then,
qJ[ff0 g;J sJ[ff0 g = sJ , and qJ[ff0 g;ff0 g sJ[ff0 g = f:
Let
Si0; =
Y
J2J
S i AJ
be the Cartesian product of universal spaces of situations over AJ : We assume that Si0;
equipped with the product topology. By the Tychono¤’s theorem, Si0; is a compact Hausdor¤ space. De…ne pi;J : Ai ! AJi and qi;J : Si0; ! Si (Aj ).to be the natural projections.
Each element s 2 Si0; can be interpreted as a net directed by algebra of …nite subsets of
T: Say that net s is consistent, if for each H J; such that J; H 2 J ,
(qi;JH ) sJ = sH .
Let Si
Si0; contains all consistent nets. Of course, Si is a closed, hence compact. Theorem
10 shows that there exists a homeomorphism i : Si ! Ai
S i such that
pi;J
q
i;J
i
=
i;J
qi;J :
(C.1)
Additionally, say that net s is faithful if for each f 2 F; sf = f:
We move to the proof of Theorem 2. Assume that frame F is closed and it satis…es
Consistency, Invariance to Relabeling, and Introspection. Let Ai = AFi be the Cartesian
product of F copies of spaces A i :
34
JEFFREY C. ELY AND MARCIN PESKI
Lemma 5. If frame F satis…es Invariance to Relabeling and Introspection, then there exists
a consistent and a faithful net s = (sJ ) 2 Si such that for each J 2 J , sJ 2 VJ F .
Proof. Indeed, consider a collection of sets
SJ0; = fs : sH 2 VH F for each H
Jg
\ fs : sH = qJH sJ for each H
Jg
\ s : sff g = f for each f 2 J
Notice that if s 2 SJ0; , then sH = qJH (sJ ) 2 VH F for each H J.
By Lemma 3, sets VJ are closed, which implies that set SJ0; is closed, hence compact. By
Lemma 4, SJ0; is non-empty. Moreover, for any …nitely many sets J1 ; :::; Jm ; the intersection
SJ0;1 \:::\SJ0;m is non-empty as it contains a non-empty set SJ0;1 [:::[Jm . Thus, family of compact
sets SJ0; J2J has the …nite intersection property. It follows that there exists s that belongs
T
to the all members of the family, s 2 J SJ0; .
We construct a type space Ti = Si with continuous belief mapping: for each ti = (ai ; ui ) 2
Ti ; let
i
(ai ; ui ) = ui 2
S
i
:
Let Bi be the -algebra on Ti generated by the Borel -algebra on Ai : Let ti = si be equal
to the consistent and faithful net from Lemma 5.
Let F be a frame that is derived from the pointed type space (Ti ; ti ) with restrictions Bi .
First, we show that F
F . Indeed, for each s 2 F; let s : T ! A be de…ned so that
s (as0 )s0 2F = as . Then, because net s is faithful, s s = s:
Next, we show that F
F: For each J 2 J ; let J be a collection of strategy pro…les
: T ! A that depend only J-coordinates of A (i.e., there exists …nite J 2 J such that for
all (as ) ; (a0s ) 2 A ; if as = a0s for all s 2 J; then (a: ) = (a0: )). De…ne F J = s : 2 J :
We show that F J
F . Indeed, take any s 2 F L ; where 2 J . Let 0 : AJ ! A
denoted the same function with the domain restricted to the relevant coordinates. Find
s 2 F and measurable mapping pro…le 2 VJ such that sJ = ( i ) (s). De…ne relabeling
l= 0
: A ! A. Then, s = ( i l) (s) 2 F:
By the de…nition of the topology on T; all measurable functions : T ! A are pointwise
S J
limits of the elements of
: By the Lebesgue Theorem, F = cl F 0 . Because F is closed,
it follows that F
F:
C.2. Type space without restrictions.
FOUNDATIONS OF TYPE SPACE
35
C.2.1. Necessity. Let F = F (ti ) be a frame that is derived from type and type space
ti 2 Ti : The proof that the frame satis…es Invariance to Relabeling and Introspection is the
same as the corresponding proof of Theorem 2. We show that frame F satis…es Invariance
to Situation-Based Relabeling. Indeed, let T the measurable -algebra on the type space
(T; ) : Then, for any strategy pro…le , the mappings that assigns situations to player j’s
types, sj; : Tj ! Sj (A) are T -measurable. Take any situation-based pro…le of relabelings
lj : Sj (A) ! Aj for each player j: De…ne measurable strategies
^j = lj
sj; .
Then, it is easy to see that
( i l) (si; (ti )) = si;^ (ti ) 2 F .
C.2.2. Su¢ ciency. Suppose that frame F satis…es Invariance to Situation-Based Relabeling
and Introspection. Because Invariance to Situation-Based Relabeling implies Invariance to
Relabeling, the proof of Theorem 2 holds. In what follows, we use the same notation as in
the proof of Theorem 2. Let S and si be the type space and the type de…ned in the proof
of Theorem 2 and let Bj for each player j be the identi…ed behavioral restrictions.
For each player j; let Tj be the Borel -algebra on Sj : For each …nite J 2 J ; let qj;J :
Sj ! Sj AJ be the projection from the de…ne …nite-order versions of the situation spaces
over action space AJ .
For each player j; …nite J 2 J ; let gj0;J : AJ ! Aj and gjJ : Sj AJ ! Aj be continuous
mappings that are homeomorphic on their images. Let
0
j;J
= gj0;J
j;J
= gjJ
J
j
J
j
be player j’s strategies, and let J0 and J be the corresponding strategy pro…les. In other
0
words, strategy j;J
chooses an action that, up to homeomorphism gj0;J , is equal to player j’s
J-coordinate actions, and strategy j;J chooses an action that, up to homeomorphism gjJ , is
equal to player j’s situation.
0
Notice that j;J
is measurable with respect to Bj for each player j: In particular, si; J0 (ti ) 2
F:
Lemma 6. For each J 2 J ; si;
J
(ti ) 2 F .
Proof. For each J 2 J and k, let hJj : gjJ AJj ! AJj be inverse of gjJ on its …rst coordinate:
hJj = projAJj
gjJ
1
36
JEFFREY C. ELY AND MARCIN PESKI
De…ne situation-based relabelings lj : Sj (A) ! Aj : for each (aj ; uj ) 2 Sj (A), let
ljJ (aj ; uj ) = gjJ
1
g 0;Jj
hJj (aj ) ;
u1j
:
Notice that relabeling ljJ depends only on the action aj and the …rst-order beliefs u1j of the
situation (aj ; uj ).
We show that
J
sj; J0 (tj ) = sj; J (tj ) :
jl
Indeed, consider the belief about the state of the world and situations of type tj = aj ; uj 2
Sj ,
s
(:)
0
J
j;
(tj ) 2
j
(
S
j
(A)) .
Its …rst-order part is equal to
1
s
0
J
j;
0
=
j;J
(:) [
j
(tj )]
(:) [
j
(tj )]
gj0;J
=
J
j
(:) [
j
(tj )] ;
and
ljJ
0
j;J
= gjJ
(tj ) ;
s
j;
J
j
projAJj
= gjJ (aj;J ; uj;J ) =
0
J
(:)
j
(tj )
j;J
1
g 0;Jj
(aj;J ) ;
gj0;J
J
j
(:) (
j
(tj ))
(tj ) :
(Notice that f
g = (f g) for functions f : X ! Y and g : Y ! Z.) Due to the
uniqueness part of Theorem 8, we can use a recursive representation
jl
J
sj;
0
J
(tj ) =
jl
= ljJ
J
0
j
0
j;J
(tj ) ;
jl
=
j;J
= sj;
J
(tj )
J
(tj )
s
s
j;
0
J
s
j;
0
J
jl
J
j;
0
J
(:)
(:)
j
(:)
s
j
(tj )
j
j;
0
J
(tj )
(tj )
j
(tj )
(tj ) :
Finally, by Invariance to Situation-Based Relabeling and the inductive assumption,
si;
J
(ti ) = ( i l) si;
0
J
(ti ) 2 F:
FOUNDATIONS OF TYPE SPACE
Lemma 7. For each J 2 J , any pro…le
1
TjJ , si; (ti ) 2 F .
to -algebra jJ
Proof. For each player j; let
de…ne relabeling:
j
of strategies
j
that are measurable with respect
: Sj AJ ! Aj be such that
lj =
j
gjJ
1
37
j
=
j
J
j.
For each player j;
:
Then, by Invariance to Relabeling,
( i l) (si;
J
(ti )) = si; (ti ) 2 F .
The Theorem follows from Theorem 9, the fact that frame F is closed, and the fact that
any measurable strategy j : Sj ! A is a pointwise limit of strategies that are measurable
1
with respect to -algebras jJ
TjJ .
Appendix D. Proof of Theorem 3
We prove the …rst part only. The proof is based on the construction from Theorem 2 (The
second part follows from similar arguments based on Theorem ??.) Let (T; ) be a type
space.
D.1. Uniqueness of si mapping. We begin the proof with showing that the subjective
situation mapping is unique.
Theorem 11. For each pro…le of measurable mappings i : T ! Ai , there exists a unique
pro…le of measurable mappings si : I Ti ! Si (A) such that for each t1 ; :::; tn ; each player
i; each type ti
sii;1 :::in (ti ) =
i1 :::in
(ti )
i1 :::in i
j6=i sj;
j
(tj ) :
(D.1)
Proof. TBA.
D.2. Necessity. Take any collection a = aii1 :::in of actions aii1 :::in 2 Ai : Consider a subjective strategy pro…le
such that ii1 :::in (ti ) = aii1 :::in for each sequence (i1 :::in ) 2 I and
each player i: Then, it is easy to check that the derived situation si; (ti ) belongs to set
Bi (a ) : In particular, if a contains distinct actions (and one can ensure that due to Rich
Actions), then F satis…es Countable action hierarchy.
38
JEFFREY C. ELY AND MARCIN PESKI
D.3. Su¢ ciency. Suppose that F satis…es Invariance to relabelings, Introspection, and it
has a countable action hierarchy associated with collection a = aii1 :::in of distinct actions
aii1 :::in 2 Ai : Then, Theorem 2 applies and there exists a type space (T; ) with restrictions
B and a type ti such that frame F is derived from type ti , F = F (ti ) :
Because frame F is derived, there exists a B-measurable strategy pro…le
such that
i1 :::;in
si; (ti ) 2 CKi (B (a )) : In particular, there exists sets of types Tj
Tj such that
i1 :::in
i1 :::in
?
(a) ti 2 Ti , (b) for each tj 2 Tj
, j (tj ) = aj
; and (c) for each tj 2 Tji1 :::in ,
i1 :::in j
= 1: For each action aj 2 aij1 :::in , let i (a) be a …nite sequence so
j
j 0 6=i Tj 0
S i1 :::in
i(a)
that aj = aj . Notice that i ( j (:)) is a Bj -measurable on
Tj
.
i1 :::in
Take any B-measurable ex ante subjective strategy
S i1 :::in
that for each tj 2
Tj
;
: Find a B-measurable strategy
so
i1 :::in
j
(tj ) =
i(
j
j (tj ))
(tj ) :
Then, is a standard strategy pro…le that is B-measurable. In particular, si; (ti ) 2 F:
We show that for each player j and each type tj 2 T i1 :::in j ;
(i :::in )
sj; (tj ) = sj; 1
(tj ) :
Indeed, de…ne mappings sj : Tj ! Sj (A) so that for each tj 2
(i :::in )
sj = sj; 1
S
i1 :::in
Tji1 :::in ;
(tj ) :
Then,
sj =
=
(i1 :::in )
j
j
(i1 :::in j)
j 0 6=j sj 0 ;
(s j )
j
j:
But the above implies that sj satis…es the …xed point de…nition of mapping sj; : By the
uniqueness of such mapping, sj = sj; :
Appendix E. Proof of Theorem 4
We prove the …rst part only. The proof is based on the construction from Theorem 2 (The
second part follows from similar arguments based on Theorem ??.) Let (T; ) be a type
space.
E.1. Uniqueness of st mapping. We begin the proof with showing that the subjective
situation mapping is unique.
FOUNDATIONS OF TYPE SPACE
39
Theorem 12. For each pro…le of measurable mappings i : T ! Ai , there exists a unique
pro…le of measurable mappings si : T Ti ! Si (A) such that for each t1 ; :::; tn ; each player
i; each type ti
(t :::tn )
si; 1
(ti ) =
(t1 :::tn ti )
j6=i sj;
(ti ; t1 :::tn )
:
(E.1)
Proof. TBA.
E.2. Necessity. TBA.
E.3. Su¢ ciency. Suppose that F satis…es Invariance to subjective relabelings and Introspection. Because all (standard) relabelings are also subjective, Theorem 2 applies. Let
(T; ) and ti the type and the type space with restrictions constructed in the proof of Theorem 2. Thus, Ti = Si AF : Then, F = F (ti ) :
We construct a new type ti 2 Si AF and we will show that
F = F (ti )
Notice that F (ti )
F I (ti )
F (ti )
F:
F I (ti ) is immediate from de…nition.
S
S
E.3.1. Construction of type ti . For each set J F; and each k 0; let AJ;k = n k i1 :::in AJi1
S
S
::: AJin be the space of k-sequences of elements of j AJj and let A = AF;k be the space
k
S
of all …nite sequences of elements of j AFj . We assume that A contains an empty sequence
?: For any two elements, a; a0 2 A, let a a0 2 A be the concatenation of sequences. By
convention, we take ? a = a:
By Rich Actions, spaces A can be mapped homeomorphically into a measurable subset of
F
Aj (notice that AFj is a F -product of Polish spaces, and A is a countable union of F -products
of Polish spaces). Let hj : A ! hj (A) Aj be the homeomorphism.
For each player j, construct measurable mappings Dj : A Sj AF ! Sj (h (A)) so that
Dj (aj ; uj ; a) = hj (a
aj ) ;
(D
j
(:; a
aj )) uj :
One shows that the mappings are unique and well-de…ned (TBA). Let ti = Di (ti ; ?) :
E.4. Subjective and regular strategies. We show that there is a correspondence between
subjective strategies evaluated from the point of view of type ti and (standard) strategies
evaluated from the point of view of type ti :
First, suppose that
is a B-measurable pro…le of subjective strategies. For each player
j; …nd Bj -measurable strategy j h so that
(h ( ))j (hj (a1
:::
an
aj ) ; uj ) =
j
((a1 ; u1 ) ; :::; (an ; un ) ; (aj ; uj )) :
40
JEFFREY C. ELY AND MARCIN PESKI
(This is well-de…ned on Sj AF because of B-measurability of ). Second, suppose that is
a B-measurable pro…le of (standard) strategies. For each player j; construct Bj -measurable
1
subjective strategy jh so that
1
h
( )
(a1 ;u1 );:::;(an ;un )
j
(aj ; uj ) =
(hj (a1
j
:::
an
aj ) ; uj ) :
Then,
h
1
(h ( )) =
:
Lemma 8. For each sequence ((a1 ; u1 ) ; :::; (an ; un )) 2 T , and type (aj ; uj ) 2 Sj (A)
sj;(h(
((a1 ;u1 );:::;(an ;un ))
(Dj ((aj ; uj ) ; a1
:::
an )) = sj;
sj; (Dj ((aj ; uj ) ; a1
:::
an )) = sj;(h1
))
, and
((a ;u1 );:::;(an ;un ))
:
1(
))
Proof. Notice that
sj;h(
)
=h
=
j
(Dj ((aj ; uj ) ; a1
j
(hj (a1
:::
:::
an
an ))
aj ) ; uj )
s
((a1 ; u1 ) ; :::; (an ; un ) ; (aj ; uj ))
s
j;h(
j;h(
D
)
)
D
j
j
(:; a1
:::
(:; a1
an
:::
an
aj ) u j
aj ) u j
((a1 ;u1 );:::;(an ;un ))
Thus, sj;h( ) (Dj (:; a1 ::: an )) satis…es the …xed point de…nition of sj;
the uniqueness of the latter, if follows that
sj;h(
E.4.1. F I (ti )
)
(Dj ((aj ; uj ) ; a1
((a1 ;u1 );:::;(an ;un ))
:::
an )) = sj;
(aj ; uj ) :
F (ti ). By Lemma 8, for each subjective strategy pro…le
s?
i; (ti ) = si;h(
)
(Di (ti ; ?)) = si;h(
)
: By
;
(ti ) 2 F (ti ) :
E.4.2. F (ti )
F . We will show that F (ti )
F: For each …nite J
F; and each k
0;
J;k
let j be the class of measurable mappings k : AJ;k AJj ! A: For each aj 2 AFj , let
aJj = pJj (aj ) 2 AJj :
Let J;k
be a class of strategies j of player j for which there exists a mapping j 2 kj so
j
that for each n; a1 ::: an aj 2 A and uj ,
j
(hj (a1
We say that strategy
not more than k:
j
:::
an
aj ) ; uj ) =
is factorized by
Lemma 9. For each …nite J
F; each k
j
j
aJmax(1;n+1
J
J
k) ; :::; an ; aj
and we say that strategy
0, and each
2
J;k
j
.
has a lag of length
; si; (ti ) 2 F:
FOUNDATIONS OF TYPE SPACE
41
J;0
Proof. The argument proceeds with induction on k: To start the induction, take any 2
and suppose that is factorized by mappings 2 0 . Construct a strategy pro…le
that for each player j; and type (aj ; uj ) 2 Sj (A) ;
(aj ; uj ) =
j
Then,
so
(aj ) :
= h ( ) and si; (ti ) 2 F: By Lemma 8,
si; (ti ) = si; (ti ) 2 F:
Suppose now that Lemma holds for some k 0: Due to Rich Actions, there exists homeomorphisms on measurable set gjJ;k : AJ;k AJj ! Aj : Let
k
j
= hJ;k
j ;
and let jk be a strategy that can be factorized by by kj : By the inductive step, si; jk (ti ) 2 F:
Take any strategy pro…le 2 k+1;J and …nd its factorization : Find a subjective relaS
beling lj : j 0 Aj 0 Aj ! Aj so that ;
lj hJ;k aJmax(1;n
J
k) ; :::; an
; hJ;k aJimax(1;n
k+1)
; :::; aJin ; aJj
=
j
aJmax(1;n
J
J
k) ; :::; an ; aj
:
Such a relabeling exists because of the invertibility of g J;k : By Invariance to subjective
relabelings,
jl
g J;k aJmax(1;n
J
k) ; :::; an
sj;
k
(ti ) 2 F:
Thus, in order to …nish the inductive claim, it is enough to show that
jl
g J;k aJmax(1;n
J
k) ; :::; an
sj;
k
(Dj (:; a1
:::
an )) = sj; (Dj (:; a1
To prove the last claim, we consider a (subjective) strategy pro…les
((ai1 ;ui1 );:::;(ain ;uin ))
j
((ai1 ;ui1 );:::;(ain ;uin ))
j
Then,
k
(aj ; uj ) =
j
aimax(1;n
k)
(aj ; uj ) =
k
j
aimax(1;n
k+1)
and
:::
an )) .
k
so that
; :::; ain ; aj ;
; :::; ain ; aj :
= h ( ) : By Lemma 8,
sj; (Dj ((aj ; uj ) ; a1
sj;
k
(Dj ((aj ; uj ) ; a1
:::
((ai ;ui );:::;(ain ;uin ))
an )) = sj; 1 1
((aj ; uj )) ;
:::
((ai ;ui );:::;(ain ;uin ))
an )) = sj; k1 1
((aj ; uj ))
(E.2)
42
JEFFREY C. ELY AND MARCIN PESKI
By de…nition of mappings
g J;k aJmax(1;n
jl
=
jl
jl
=
J
k) ; :::; an
((ai1 ;ui1 );:::;(ain ;uin ))
jl
:::
an ) ;
((ai ;ui );:::;(ain ;uin ))
sj; k1 1
((aj ; uj ))
J
k) ; :::; an
g J;k aimax(1;n
j
and Dj (:; a1
J
k) ; :::; an
g J;k aJmax(1;n
= l g J;k aJmax(1;n
jl
sj;
k
(Di ((aj ; uj ) ; a1
; g J;k aimax(1;n
k+1)
k+1)
:::
an ))
; :::; ain ; aj
; :::; ain ; aj
sj;
; :::; ain ; aj
((ai ;ui );:::;(ain ;uin );(aj ;uj ))
sj; k1 1
:
k
(Di ((aj ; uj ) ; a1
:::
an
aj ))
aJj
g J;k aimax(1;n
k+1)
By Theorem 12,
jl
g J;k aJmax(1;n
J
k) ; :::; an
((ai ;ui );:::;(ain ;uin ))
((aj ; uj ))
sj; k1 1
((ai ;ui );:::;(ain ;uin ))
((aj ; uj )) :
= sj; 1 1
The claim follows from (E.2).
The result follows from the Dominated Convergence and the fact that each strategy 2
can be approximated by sequences of strategies in J;k for su¢ ciently large J and k:
B