1. Introduction Complete models of interactive uncertainty summarize the infinite regress of “beliefs about beliefs”. Since the work of Mertens and Zamir [9], it is well known that, under some conditions, the interactive uncertainty over a space S can be represented by a type space of countable hierarchies of beliefs about beliefs. To fix ideas, say that two agents face uncertainty described by a topological space S and have beliefs about events on S; in turn, the beliefs of the other agent are uncertain for a given agent, so she/he has beliefs over events describing the beliefs of the other; and so on. Let B(X) denote the set of beliefs on events in a space X. Let T represent the set of hierarchies of one agent: T ∈ (B(S) × B(S × B(S)) × ...). The literature has provided some conditions on the spaces S and B(·) ensuring that T ∼ B(S × T ), where “∼” means “homeomorphic to”. The space T is then called the type space associated with S. An alternative perspective is provided by the concept of an S-based belief structure: a pair (T, R), where T is a set and R is a mapping R : T → B(S × T ). We say that a belief structure (T, R) is complete whenever R is surjective. Hence, when the set of hierarchies T of one agent satisfies T ∼ B(S ×T ), the implied belief structure is complete. In the category of S-based belief structures, a couniversal S-based belief structure is a pair (T , R) such that for every (T, R) there exists a unique mapping ϕ such that the following diagram commutes: Date: April 14, 2010. This is a preliminary and incomplete version, please do not cite. 1

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T

ϕ

R

B(S × T )

T R

(IdS ;ϕ)B

B(S × T )

where IdS is the identity map on S and (IdS ; ϕ)B is the mapping between beliefs induced by IdS and ϕ. The literature has also provided some conditions on the spaces S and B(·) ensuring that the S-based belief structure constructed from countable hierarchies of beliefs about beliefs is couniversal. For instance, the prototypical result in the literature reads as follows: “Let S be a compact Hausdorff space. Then there exists a couniversal S-based belief structure whenever B(·) is the space of all regular Borel probability measures endowed with the weak* topology. Moreover, the couniveral object is constructed using countable hierarchies of beliefs about beliefs.”1 We investigate the possibility of dropping the use of conditions on S: in the language of category, a couniversal object does not depend on a particular space S used: the couniversal object outght to be defined for any topological space S. That is, we investigate whether it is possible to construct a couniversal S-based belief structure using countable hierarchies. Equivalently, our starting point is the following: If a particular situation requires that S be a non compact, non Hausdorff, non metric, non separable, etc., topological space, we would like to be able to determine whether it is or it is not possible to construct a type space representing interactive uncertainty using countable hierarchies of beliefs about beliefs. In the case of possibility models, where the beliefs of an agent are represented by a possibility set consisting of states regarded as possible, we present two results, one positive and one negative. The positive result refers to the case that beliefs are compact sets and the negative result refers to beliefs as closed sets. When beliefs are compact sets, we prove that for each topological space S the space T of coherent hierarchies of beliefs about beliefs satisfies T ∼ K(S × T ), where K(X) denotes the hyperspace of non empty compact subsets of a space X, provided that the topology on K(·) satisfies a compatibility condition. Moreover, the implied S-based compact possibility structure is couniversal. The compatibility condition is restrictive: some well known hyperspace topologies, like the Vietoris and the Fell topology, do not satisfy the compatibility condition. Other topologies, like the myopic topology, do. 1This is the result in [9]. The term used in the literature is “universal” instead of “couniversal”.

We follow the standard terminology in calling a “terminal” object couniversal and an “inital” object as ”universal”’. See Hungerford [10].

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When beliefs are closed sets, we show that it is impossible to obtain an analogous result. In order for a complete possibility structure to exist, one needs to impose restrictions on the space S. For instance, we show that whenever S is locally compact and second countable, the space T of coherent hierarchies of beliefs about beliefs satisfies T ∼ F(S × T ), where F(X) denotes the hyperspace of non empty closed subsets of a space X, provided that we use the Fell topology on F(·). It is well-known that some “regularity” requirement is needed for countable hierarchies to capture all interactive uncertainty (see Fagin, Geanakoplos, Halpern and Vardi [5]). Regularity here is a general term and has three components: (i) the topology in S, (ii) the choice of beliefs, and (iii) the topology on the space of beliefs. In possibility models with closed or compact beliefs, we already impose a significant restriction on component (ii). Our results show that enough regularity is provided by compact beliefs in (ii) and a compatible topology in (iii), with no need to impose any restriction in (i). For closed beliefs in (ii), restrictions in (i) are needed in addition to restrictions in (iii). The distinction between closed and compact beliefs is intriguing. The motivation for the use of compact beliefs in Mariotti, Meier and Piccione [7] is that agents ought not to distinguish sets with the same closure. In their framework, S is compact Hausdorff, so closed and compact sets are the same. In more general environments, the added regularity brought about by compactness is required. That is, what agents cannot distinguish are the “bounded” sets with the same closure. Unbounded sets with the same closure (or, in general, closed sets that are not compact) need not be regarded as the same. The heuristic idea here is that it requires too much in terms of reasoning to keep track of hierarchies of sets that are “too large”. The argument builds on the approach initiated by Brandenburger and Dekel [3], and applied in many subsequent papers, including Epstein and Wang[4], Ahn [1] and [7]. The argument goes as follows: using a suitable analog of Kolmogorov Extension Theorem, one first shows that the space of all coherent hierarchies of beliefs about beliefs (call it T1 ) is homeomorphic to the space of all beliefs over S × T0 , where T0 is the space of all hierarchies. After that, one shows that the space of all hierarchies where coherence is not only satisfied but it is also commonly known among the agents (call it T∞ ) is homeomorphic to the space of all beliefs over S × T∞ . In short, first one shows that T1 ∼ B(S × T0 ) and then uses it to show that T∞ ∼ B(S × T∞ ). Since we are ultimately interested in this last homeomorphism, we use an intermediate result showing that we have B(S × T0 ) ֒→ T1 , where “֒→” means “embedding”, whenever B(·) is endowed with a topology that satisfies the compatibility condition. That is, we do not use the full force of an analogue of the Kolmogorov Extension

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Theorem: by not restricting the topological spaces involved, we can only guarantee the existence of an embedding, which suffices for our needs: we show that the common knowledge of coherence requirement suffices to the existence of a space T∞ and a mapping R∞ : T∞ → B(S × T∞ ) such that R∞ is a homeomorphism. To verify the couniversality of (T∞ , R∞ ), we show that the assignment τB (X) = (B(X), τB(X) ) is functorial whenever X belongs to the category of topological spaces and τB(X) is a topology in B(X) that satisfies the compatibility condition. For the case of closed beliefs, the existence of the embedding is not guaranteed. When S is locally compact, Hausdorff and the Fell topology is used on F(·), then this latter topology is compatible and we have F(S × T0 ) ֒→ T1 . A similar, but more involved construction, then shows the existence of T∞ , such that T∞ ∼ F(S × T∞ ). 1.1. Related Literature. The literature on “beliefs about beliefs” is large. From [9] to [1], several authors have showed in different environments that countable hierarchies of beliefs about beliefs form a couniversal object in the relevant subcategory. Most results in the literature are special cases of Theorem 6.1 in [4]: the use of regular preferences in [4] is the most general choice of beliefs, and the vast majority of the other works in this literature assume that S is compact Hausdorff, as in [4]. Since we do not impose conditions on S, our results are new. In particular, [7] is a special case of our result under compact beliefs. In the case of probabilitic beliefs, there are a few papers that go beyond the compact Hausdorff case. In [3] S is Polish; in Heifetz [6] S is Hausdorff, and in Mertens, Sorin and Zamir [8], a list of allowed assumptions on S is provided, including the two cases just mentioned. The probabilistic beliefs, on the other hand, are assumed to be regular Borel measures: analogously to our choice of compact beliefs in component (ii), these papers assume enough regularity by the choice of beliefs as regular probability measures. S is assumed to be a Hausdorff space in these papers, so an answer to our starting point is not provided. Finally, our construction of a complete S-based closed possibility correspondences with S locally compact and using the Fell topology on the hyperspace of closed subsets is new in the literature. 2. The Model Let there be given two agents facing uncertainty described by a topological space S. A belief of a player is represented by subsets B ⊂ S, meaning that the player 2

2The construction carries through with a number I > 2, possibly infinite, of agents. We focus

on I = 2 for ease of notation.

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considers any point s ∈ B as possible. Equivalently, any point s ∈ / B is considered impossible. Interactive beliefs are captured by a possibility structure. Formally, for any given set X, let P(X) be the set of nonempty subsets of X: Definition 2.1. An S-based possibility structure is a pair (T, R) consisting of: (i) a nonempty set T ; (ii) a mapping R : T → P(S × T ). The idea is that an element t ∈ T describes an agent’s type, in that it fully characterizes what an agent considers possible in both S and T (the other agent’s types). [7] showed that R cannot be surjective if S and T are not singletons by checking the cardinalities of the sets involved. This means that if we want to represent T as space of subsets of S × T , we need to restrict the kinds of subsets that we consider as representatives of agents’ beliefs. An intuitive and natural restriction is that player cannot distinguish very “close” subsets. For instance, [7] assume that agents cannot tell apart two subsets in it with the same closure. In their set up, closed sets are compact, so the assumption therein is that agents only consider compact subsets as possible. In this paper, we shall assume agents consider either closed or compact subsets of a space X as possible. Let B(X) denote the set of either closed or compact subsets of X. When B(X) is taken to be the set of (nonempty) compact sets, we also denote it by K(X); when B(X) is the set of (nonempty) closed sets, we also denote it by F(X).3 There are infinitely many choices to topologize B(X), including the myopic topology adopted in [7]. We shall use (B(X), τB(X) ) to denote this space. When the topology τB(X) on B(X) is clear by context, we also use B(X) for short. When B(X) is taken to be the set of nonempty compact sets topologized with myopic topology as in [7], we also denote it by K(X), which was the notation therein. To give a general criterion on the choice of τB(X) instead of working with special cases, we introduce a compatible condition. Definition 2.2. An S-based closed/compact possibility structure (depending on the definition of B(·)) is a pair (T, R) consisting of: (i) a nonempty topological space T ; (ii) a mapping R : T → B(S × T ). 3Mathematically, most the results in this paper hold true in both the cases with or without the

empty set. But it’s more reasonable to exclude the empty set. When the results hold true for both the cases with or without the empty set, usually the notation B(·) is adopted. When the empty set makes a difference, we use B′ (·) to denote the case of including the empty set, and use B′′ (·) to denote the case of excluding the empty set.

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The main result in [7] is the construction of a “(co)universal possibility structure” in a subcategory of the category of S-based compact possibility structures with S compact Hausdorff. That is, [7] constructed a couniversal structure in “a family F of S-based compact possibility structures” restricted by T Hausdorff and S compact Hausdorff. Here we investigate the possibility of such construction without making any assumptions on the topological spaces S and T . For the standard terminology used in this paper, see Hungerford [10], Munkres [11], Molchanov [12], and other standard texts. 3. The Universal Possibility Structure The interactive beliefs can be captured by hierarchies of beliefs about beliefs. Let X0 = S, Xn+1 = Xn × B(Xn ) for all n ≥ 0, Q and view {βn }n≥1 ∈ n≥0 B(Xn ) as an agent’s type. It is an infinite sequence of increasing levels of beliefs about beliefs. Let T0 be the set of all types. A type is coherent if different levels of beliefs do not contradict each other. That is, a type {κn }n≥1 ∈ T0 is coherent if κn = ProjB Xn−1 (κn+1 ) for all n ≥ 1, where ProjB Xn−1 is defined as follows: When we take B(·) be all the (nonempty) n−1 ; when we take B(·) be all compact sets, ProjB X n−1 is the usual projection on X the (nonempty) closed sets, then ProjB X n−1 is the closure of the usual projection on X n−1 . The reason for such a treatment is that the projection of a compact set is still compact, but the projection of a closed set may not be closed. Let T1 be the set of all coherent types. Let T be the category of all the nonempty topological spaces. τB : T → T is a function such that τB (X) = (B(X), τB(X) ). When there is no ambiguity, τB(X) is also abbreviated as τ . If the choice of τB was unrestricted, it would be impossible to obtain anything significant from it. On the other hand, it is desirable to have the most general conditions as possible. In order to construct a couniversal S-based closed/compact possibility structure, τB has to be functorial, i.e., it can be extended into a (covariant) functor in a natural way. It is natural to define τB (f ) for any continuous map f : X → Y as follows: τB (f ) : B(X) → B(Y ) , K 7→ f (K)B

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where f (K)B means the closure of f (K), i.e. f (K), in the case of closed sets and f (K) in the case of compact sets. Since the object function, the morphism function and the functor are usually denoted by the same notation, whenever we say that τB is functorial we mean that τB is a functor, and coversely. Let {Zn }n≥0 be a collection of nonempty topological spaces. For each n, let Q Q = 0≤i≤n Zi and Z ∞ = n≥0 Zn . Define Y DB = {{βn }n≥1 |βn ∈ (B(Z n−1 ), τ ) and ProjB (β ) = β } ⊂ (B(Z n ), τ ), n−1 n+1 n Z

Zn

n≥0

where ProjB Z n−1 is defined similarly as above. Then we can define a function as follows: Q n fB : (B(Z ∞ ), τ ) → n≥0 (B(Z ), τ ) , B B 7→ {ProjZ n−1 (B)}n≥1 and we have the following lemma: Lemma 3.1. Let {Zn }n≥0 , Z n , Z ∞ , DB and fB be as above, then fB is injective and Im(fB ) ⊂ DB . So we readily see that and for any {βn }n≥1 ∈ Im(fB ), (3.1)

−1 ProjB Z m−1 (fB ({βn }n≥1 )) = βm , for all m ≥ 1.

Especially, when B(·) is taken to be the set of (nonempty) compact sets and Z ∞ is compact, we have Im(fB ) = DB . Proof. Consider the case B(·) = K(·) first. Notice that in this case, ProjB · is just Proj· which is continuous, and (3.2)

ProjZ n−1 (ProjZ n (K)) = ProjZ n−1 (K), for any K ∈ K(Z ∞ ),

so Im(fB ) ⊂ DB and (3.1) follows. Moreover, ProjZ n (K) = ProjZ n (K ′ ) for all n ≥ 1 if and only if K = K ′ , so fB is injective. The proof for Im(fB ) = DB when Z ∞ compact is not hard, interested readers are referred to [7]. For the case of closed sets, we need to show the following holds (3.3)

B ∞ B ProjB Z n−1 (ProjZ n (F )) = ProjZ n−1 (F ), for any F ∈ F(Z ),

and then Im(fB ) ⊂ DB and (3.1) follows. By an equivalent definition of continuous functions [11, §18], we have (3.4)

ProjZ n−1 (ProjZ n (F )) ⊂ ProjZ n−1 (ProjZ n (F )).

Since (3.2) is true, (3.4) becomes B ProjZ n−1 (ProjB Z n (F )) ⊂ ProjZ n−1 (F ) = ProjZ n−1 (F ).

So B B B ProjB Z n−1 (ProjZ n (F )) = ProjZ n−1 (ProjZ n (F )) ⊂ ProjZ n−1 (F ).

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On the other hand, B B ProjB Z n−1 (F ) = ProjZ n−1 (F ) = ProjZ n−1 (ProjZ n (F )) ⊂ ProjZ n−1 (ProjZ n (F )),

so (3.3) holds. Now take F 6= F ′ from (B(Z ∞ ), τ ), we need to show that fB (F ) 6= fB (F ′ ). Without loss of generality, we can take a point x = (xn )n≥0 ∈ F \F ′ . Since F ′ is closed in Z ∞ , there exists an open set U in Z ∞ containing x such that U ∩ F ′ = ∅. Q Furthermore, we can take U to be of a special type, i.e., U = Um × i>m Zi for some Um open in Z m . Since U ∩ F ′ = ∅, then Um ∩ ProjZ m (F ′ ) = ∅. Since Um ′ is open, then Um ∩ ProjZ m (F ′ ) = ∅, i.e., Um ∩ ProjB Z m (F ) = ∅. In particular, B ′ ′ (x0 , . . . , xm ) is in ProjB Z m (F ) but not ProjZ m (F ), which implies fB (F ) 6= fB (F ). So fB is injective. It is tempting to conjecture that in the case of closed sets, Im(fB ) = DB . That is, for any {βn }n≥1 ∈ DB , we need to find some F closed in Z ∞ such that fB (F ) = Q T {βn }n≥1 . It seems obvious that we should take F = n≥1 (βn × i≥n Zi ): Indeed, F is closed in Z ∞ , and the only thing left to check is that ProjZ n−1 (F ) = βn for all n ≥ Q 1. We see that ProjZ n−1 (F ) ⊂ ProjZ n−1 (βn × i≥n Zi ) = βn , so ProjZ n−1 (F ) ⊂ βn for all n. But unfortunately, that is all we can have. In the Appendix, we construct a “pathological” example to show why fB fails to be onto DB . As we have pointed out, we need to introduce restrictions on τB in order to obtain meaningful results. We will focus on two types of restrictions: first we require that τB be meaningfully compatible with the underlying product structure (Definition 3.2 below), and later we will require that τB be functorial. Definition 3.2. We say that τB is compatible with products for the family of nonempty topological spaces {Zn }n≥0 , if the following two conditions are satisfied: Qk−1 Q (i) For any Pk open in (B(Z k ), τ ), k ≥ 0, fB−1 (( n=0 (B(Z n ), τ ))×Pk ×( n>k (B(Z n ), τ ))) is open in (B(Z ∞ ), τ ); n f open in Q (ii) For any W open in (B(Z ∞ ), τ ), there exists a W n≥0 (B(Z ), τ ), f , i.e., fB (W ) is open in Im(fB ). such that fB (W ) = Im(fB ) ∩ W

When the above two conditions hold for any family of nonempty topological spaces {Zn }n≥0 , then we say τB is compatible with products in the category T . Our main result is the following:

Theorem 3.3. Let {Zn }n≥0 , Z n , Z ∞ , DB and fB be as above. Without causing ambiguity, we also denote the resulting function by fB when its codomain is restricted to DB . If τB is compatible with products for the family {Zn }n≥0 , then fB is a

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topological embedding. Especially, when B(·) is taken to be the set of (nonempty) compact sets and Z ∞ is compact, fB : (B(Z ∞ ), τ ) → DB is a homeomorphism. Proof. Immediate from Lemma 3.1 and Definition 3.2.

Notice that Theorem 3.3 is not analogous to Kolmogorov’s Extension Theorem, for we are not able to guarantee the surjectiveness of fB . It’s important to note that compatibility with products for the family of nonempty topological spaces {Zn }n≥0 is the minimal, actually the necessary and sufficient, condition for fB being an embedding or homeomorphism. If we do not require τB to be a functor, then in general, for given family of spaces {Zn }n≥0 , there are uncountably many ways to topologize B(Z n ) and B(Z ∞ ) which satisfy the compatibility with products for family {Zn }n≥0 . Though we do need τB to be a functor in order to construct the (co)universal beliefs space, sometimes it is too strong to require that the functor τB is compatible with products in the whole category T . There are a few ways to define τB compatible with products. A typical example is the myopic topology. But, as we will see shortly, the well-known Vietoris and Fell topologies are not compatible with products. Let us first introduce the following notations: For A ⊂ X, B(X)A = {B ∈ B(X)|B ∩ A 6= ∅} denotes the family of sets in B(X) hitting A; B(X)A = {B ∈ B(X)|B ∩ A = ∅} is the family of sets in B(X) missing A. We use G(X) to denote the open sets in X. Now let’s list some of the topologies here as examples: (i) Taking B(X) = F(X), the Vietoris topology (also called exponential topology) is generated by subbasis sets F(X)G for all G ∈ G(X) and F(X)F for all F ∈ F(X); (ii) Taking B(X) = F(X), the Fell topology (sometimes called vague topology) is generated by subbasis sets F(X)G for all G ∈ G(X) and F(X)K for all K ∈ K(X); (iii) Taking B(X) = K(X), the myopic (or narrow) topology is generated by subbasis sets K(X)G for all G ∈ G(X) and K(X)F for all F ∈ F(X); (This is the case studied in [7].) (iv) The topology on B(X) generated by subbasis sets B(X)G for all G ∈ G(X). And we have the following lemma:

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Lemma 3.4. All the topologies listed above satisfy 3.2.(i) in the category T ; topologies (iii) and (iv) satisfy 3.2.(ii) in the category T . Proof. We will typically check the case of myopic topology, and other cases can be checked similarly. In this case, B(·) = K(·). As mentioned before, fB : K(Z ∞ ) → DB is defined by fB (K) = {βn }n≥1 , where K ∈ K(Z ∞ ) and βn = ProjZ n−1 (K). Let V be an open subset of DB , that is, V = G ∩ DB where G is an open subset Q Q in n≥0 K(Z n ). Since the product topology of n≥0 K(Z n ) is generated by the subbasis S = {(

k−1 Y

K(Z n )) × Pk × (

n=0

then we have G =

S

Y

K(Z n )) : Pk open in K(Z k ), k ≥ 0},

n>k α Gα ,

where Gα is a finite intersection of sets in S, i.e., we have G=

[

kα [ \ Gα = ( Gα,j ),

α

α j=1

where Gα,j is in S for any α and j. S T α −1 fB (Gα,j ), fB−1 (V ) is open in K(Z ∞ ) if Since fB−1 (V ) = fB−1 (G) = α kj=1 fB−1 (P ) is open for every P ∈ S. For any given P ∈ S, let P =(

k−1 Y

K(Z n )) × Pk × (

n=0

Y

K(Z n )),

n>k

for some k ≥ 0. We see that Pk =

[ γ

Pγ =

kγ [\

Pγ,j ,

γ j=1

where Pγ,j is of type KF or KG in K(Z k ) for any γ and j. By the similar logic as before, we see that fB−1 (P ) is open in K(Z ∞ ) if fB−1 (Q) is open for every Q of the Q Qk−1 K(Z n )) × Qk × ( n>k K(Z n )), where Qk is of type KF or KG in K(Z k ). form ( n=0 Now suppose Qk is of type KF in K(Z k ), i.e., Qk = {K ∈ K(Z k )|K ⊂ U } for some U open in Z k . Then we see that Y fB−1 (Q) = {K ∈ K(Z ∞ )|ProjZ k (K) ⊂ U } = {K ∈ K(Z ∞ )|K ⊂ U × ( Zi )}. i>k

The last set in the above expression is of type KF in K(Z ∞ ), so it’s open in K(Z ∞ ). Similarly, we can check the case when Qk is of type KG in K(Z k ). Thus we finish showing that condition (i) is satisfied. Now we check condition (ii). It suffices to show that fB takes every subbasis set W in K(Z ∞ ) to an open set in Im(fB ), since fB is an injection. Suppose W is a type KF set in K(Z ∞ ), then W = {K ∈ K(Z ∞ )|K ⊂ U } for some U open in Z ∞ .

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Q For each x ∈ U , there exists an open set Ux of the form Ux = Ux,k × ( i>k Zi ) for some Ux,k open in Z k , k ≥ 0, such that x ∈ Ux ⊂ U . So {Ux }x∈U forms an open cover of K for each K ∈ W . For fixed K, since K is compact, then there exists a S finite subcover {Uxi }ni=1 of K. Let V = ni=1 Uxi , we see that V can be written as Q V = Vj × ( i>j Zi ), where Vj is open in Z j . So we see that [ W = {K ∈ K(Z ∞ )|ProjZ j (K) ⊂ Vj }, V ⊂U,V =Vj ×(

Q

i>j

Zi )

and f, fB (W ) = Im(fB ) ∩ W Q Q j−1 n n j f= Q where W V ⊂U,V =Vj ×( i>j Zi ) ( n=0 K(Z ))×{K ∈ K(Z ))|K ⊂ Vj }×( n>j K(Z )). So fB (W ) is open in Im(fB ). We can also check the case when W is a type KG set in K(Z ∞ ). Thus, we finish showing that condition (ii) is satisfied. S

Remark 3.5. From the proof of that myopic topology satisfies 3.2.(ii), we see that compactness is needed in order to show fB maps KF sets to open sets. That’s why the proof for the case of Vietoris topology on F(·) can’t get through. In the appendix, we construct a counterexample showing that the Vietoris topology on F(·) fail to satisfy 3.2.(ii). For the Fell topology case, 3.2.(ii) is satisfied whenever Zn is locally compact Hausdorff for each n > 0. Note that the locally compact Hausdorff condition is general and includes important cases such as open sets in Rd , which are not compact. Also, note that if S is locally compact Hausdorff then so is Zn for each n > 0. Lemma 3.6. The Fell topology satisfies 3.2.(ii) when Zn is locally compact and Hausdorff, for each n ≥ 0. The proof of Lemma 3.6 uses the following result from [12, Appendix B] (refer to the references therein for proofs and further information). Lemma 3.7. Let X be any space. Then F ′ (X) equipped with the Fell topology is compact. If X is a locally compact Hausdorff space, then F ′ (X) is a compact Hausdorff space and F ′′ (X) is a locally compact Hausdorff space when they are equipped with the Fell topology.4 Proof of Lemma 3.6. The case of F ′ is immediate: fB is a continuous bijection beQ tween the compact space F ′ (Z ∞ ) and the Hausdorff space Im(fB ) ⊂ n>0 F ′ (Z n ), so fB must be open. 4Recall that F ′ (X) includes the empty set and F ′′ (X) does not.

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In terms of possibility models, the result above reads as follows: Proposition 3.8. Suppose τB is compatible with products (for a family of spaces specified below or in the whole category T ), then there exists a topological embedding fB : (B(S × T0 ), τ ) ֒→ T1 such that for any {βn }n≥1 ∈ Im(fB ), ProjXm−1 (fB−1 ({βn }n≥1 ) = βm , for all m ≥ 1. When S is compact Hausdorff, fB is a homeomorphism. Proof. Follows from Theorem 3.3 above, by putting Z0 = X0 , Zn = (B(Xn−1 ), τ ) for each n ≥ 1, so that Z n = Xn and Z ∞ = S × T0 , and DB = T1 . That is, the set of closed/compact subsets in S × T0 is homeomorphic to a subset of T1 . The second step of the construction in [3] is to impose common knowledge of coherence. That is, the candidate for a (co)universal beliefs space is the space of hierarchies of beliefs about beliefs where all agents are sure that only coherent hierarchies are considered. Define inductively the sets Tk+1 = fB (B(S × Tk )), for k ≥ 1. T

Let T∞ = k≥1 Tk , as the set of types where coherence is common knowledge. That 2 have the property that each player believes that other players’ types is, points in T∞ are coherent, believes that the other players believe that other types are coherent, and so on ad infinitum. As mentioned before, players cannot distinguish sets that are very “close”, so we take the closure in the case of closed sets. Lemma 3.9. Let X be a topological space and A a nonempty subspace of it, then: (i) For topologies (iii) and (iv) in the previous list, K(A) is a subspace of K(X); (ii) If A is closed in X, then for topologies (i) and (ii) in the previous list, F(A) is a subspace of F(X). Proof. Under the above conditions, in both the cases of closed and compact sets, we can check that B(A) ⊂ B(X) as a set. We need to show that B(A) is actually a subspace of B(X). Here we will typical show the case of Vietoris topology on F(·). Suppose A is closed in X. It is easy to check that for any G ∈ G(X), F(X)G ∩ F(A) = F(A)G∩A and for any F ∈ F(X), F(X)F ∩ F(A) = F(A)F ∩A . Then we can readily see the result.

HIERARCHIES OF BELIEFS

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Lemma 3.10. When S is a nonempty locally compact Hausdorff space, B(·) = F(·) and τB(·) is taken to be the Fell topology, then T1 is closed in T0 . Proof. To be completed.

Then we have a corollary. Corollary 3.11. We have the following: (i) For topologies (iii) and (iv) in the previous list, K(S × Tk )(k ≥ 1) and K(S × T∞ ) are subspaces of K(S × T0 ); (ii) If S is nonempty locally compact Hausdorff, then for Fell topology, F(S × Tk )(k ≥ 1) and F(S × T∞ ) are closed subspaces of F(S × T0 ). Proof. The proof for (i) is not hard, since compactness is unchanged in the original and relative topologies and the open sets in a subspace are induced from the original space. The proof for (ii) will be completed later. Now let fB,∞ : B(S × T∞ ) → T1 be the restriction of fB to B(S × T∞ ). By Corollary 3.11, we know fB,∞ is well defined, and it is a continuous injection by Proposition 3.8. In addition, we have: Proposition 3.12. Under the conditions of Corollary 3.11, fB,∞ (B(S ×T∞ )) = T∞ . Proof. Since Tk+1 = gB (B(S × Tk )), k ≥ 1, so \ fB (B(S × Tk )) T∞ = T 1 ∩ k≥1

= fB (

\

B(S × Tk ))

k≥1

= fB (B(S ×

\

Tk ))

k≥1

= fB (B(S × T∞ )), where the second equality holds because the image of fB is in T1 and fB is an embedding, and the third holds because {Tk }k≥1 is a nested sequence of (closed) subspaces of T0 and compactness/closedness is unchanged in the original and relative topologies. Let’s restrict the codomain of fB,∞ to T∞ and denote the resulting function by gB , then it follows that gB : B(S × T∞ ) → T∞ is a homeomorphism under the conditions of Corollary 3.11. This contains the canonical homeomorphism in [7] as a special case, and it provides a stronger justification for the use of types to represent

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interactive uncertainty, since it is not tied up to a particular topological assumption on S. We can now invoke the results in Sec. 3.2 of [7] to argue that the space T∞ is universal. Actually, the standard term should be couniversal, since (T∞ , gB−1 ), or equivalently, (T1∗ , gB∗ ) in terms of *-hierarchy which can be defined similarly as in [7], is the terminal object in the category of S-based closed/compact possibility structures. Notice that the terminal object is unique in a category. Here the uniqueness reveals the logical equivalence of the hierarchies of beliefs defined herein and the completeness of the possibility structure. References [1] Ahn, David (2007): Hierarchies of Ambiguous Beliefs, Journal of Economic Theory, 136, 286-301. [2] Boge, W. and T. Eisele (1979): On Solutions of Bayesian Games, International Journal of Game Theory, 8, 193-215. [3] Brandenburger, A. and E. Dekel (1993): Hierarchies of Beliefs and Common Knowledge, Journal of Economic Theory, 59, 189-198. [4] Epstein, L. and T. Wang (1996): Beliefs about Beliefs without Probabilities, Econometrica, 64, 1343-1374. [5] Fagin, R., J. Geanakoplos, H. Halpern and M. Vardi (1999): The hierarchical Approach to Modeling Knowledge and Common Knowledge, International Journal of Game Theory, 28, 331-365. [6] Heifetz, A. (1993): The Bayesian Formulation of Incomplete Information: The Non-Compact Case, International Journal of Game Theory, 21: 329-338. [7] Mariotti, T., M. Meier and M. Piccione (2005): Hierarchies of Beliefs for Compact Possibility Models, Journal of Mathematical Economics, 41, 303-324. [8] Mertens, J.-F., S. Sorin and S. Zamir (1994): Repeated Games. Part A. Background Material , CORE Discussion Papers 9420. [9] Mertens, J.-F. and S. Zamir (1985): Formulation of Bayesian Analysis for Games with Incomplete Information, International Journal of Game Theory, 14, 1-29. [10] Hungerford, T. (1974): Algebra (3rd Printing), (GTM 73), Springer-Verlag New York, Inc. [11] Munkres, J. R. (2000): Topology (2nd Ed.), Prentice Hall, Inc. [12] Molchanov, I (2005): Theory of Random Sets, Springer-Verlag London Ltd. Department of Economics, University of Rochester, Rochester, NY 14627 E-mail address: [email protected] Department of Mathematics, University of Rochester, Rochester, NY 14627 E-mail address: [email protected]