Every hierarchy of beliefs is type

Every hierarchy of beliefs is type∗

arXiv:0805.4007v1 [cs.GT] 26 May 2008

Mikl´os Pint´er Corvinus University of Budapest† September 25, 2008

Abstract Any model of any incomplete information situation has to handle the players’ hierarchies of beliefs, which can make the modeling very cumbersome. Hars´ anyi [14] suggested that the hierarchies of beliefs can be replaced by types, i.e., a type space can substitute for the hierarchies of beliefs (henceforth Hars´ anyi program). In the purely measurable framework Heifetz and Samet [18] formalized the concept of type space, and proved that there is universal type space, i.e., the most general type space exists. Later Meier [20] showed that the universal type space is complete, in other words, the universal type space is a consistent object. After these results, only one step is missing to prove that the Hars´ anyi program works, that every hierarchy of beliefs is in the complete universal type space, put it differently, every hierarchy of beliefs can be replaced by type. In this paper we also work in the purely measurable framework, and show that the types can substitute for all hierarchies of beliefs, i.e., the Hars´ anyi program works. The key point of our result is a generalization of the Kolmogorov Extension Theorem.

1

Introduction

It is recommended that the models of incomplete information situations to be able to handle the players’ hierarchies of beliefs, e.g. player 1’s beliefs about the parameters of the game, player 1’s beliefs about player 2’s beliefs about the parameters of the game, player 1’s beliefs about player 2’s beliefs about player 1’s beliefs about the parameters of the game, and so on. However the explicit use of hierarchies of beliefs1 makes the models very cumbersome, hence it is desirable to evade that they appear explicitly in the model. In order to make the models of incomplete information situations more handy, Hars´ anyi [14] suggested that the hierarchies of beliefs could be replaced by types. He wrote2 “It seems to me that the basic reason why the theory of games with incomplete information has made so little progress so far lies in the fact that these games give rise, or at least appear to rise, to infinite regress in ∗ Thanks.

This work was supported by grant OTKA 72856. of Mathematics, Corvinus University of Budapest, 1093 Hungary, Budapest, F˝ ov´ am t´ er 13-15., [email protected] 1 In this paper we use the terminology hierarchy of beliefs instead of the longer coherent hierarchy of beliefs. 2 [14] pp.163–167. † Department

1

reciprocal expectations on the part of the players. . . . The purpose of this paper is to suggest an alternative approach to the analysis of games with incomplete information. . . . As we have seen, if we use the Bayesian approach, then the sequential-expectations model for any given I-game G will have to be analyzed in terms of infinite sequences of higher and higher-order subjective probability distributions, i.e., subjective probability distributions over subjective probability distributions. In contrast, under own model, it will be possible to analyze any given I-game G in terms of one unique probability distribution R∗ (as well as certain conditional probability distributions derived from R∗ ). . . . Instead of assuming that certain important attributes of the players are determined by some hypothetical random events at the beginning of the game, we may rather assume that the players themselves are drawn at random from a certain hypothetical population containing the mixture of different “types”, characterized by different attribute vectors (i.e., by different combinations of relevant attributes). . . . Our analysis of I-games will be based on the assumption that, in dealing with incomplete information, every player i will use Bayesian approach. That is, he will assign a subjective joint probability Pi to all variables unknown to him − or at least to all unknown independent variables, i.e., to all variables no depending on the players’ own strategy choices.” In other words, Hars´ anyi’s main concept was that the types can substitute for the hierarchies of beliefs, and all types can be collected into an object on which the probability measures are for the players’ (subjective) beliefs. Henceforth, we call this method of modeling Hars´anyi program. However, at least two questions come up in connection with the Hars´anyi program: (1) is the concept of type itself appropriate for the proposes under consideration? (2) can every hierarchy of beliefs be a type? Question (1) consists of two subquestions. First, can all types be collected into one object? The concept of universal type space introduced by Heifetz and Samet [18] formalizes this requirement: the universal type space in a certain category of type spaces is a type space (a) which is in the given category, and (b) into which, every type space of the given category can be mapped in a unique way. In other words, the universal type space is the most general type space, it contains all type spaces (all types). In the purely measurable framework Heifetz and Samet [18] formalized the idea of type space, and proved that the universal type space exists and unique. Second, can every probability measure on the object of the collected types (universal type space) be a (subjective) belief? Brandenburger [6] introduced the notion of complete type space: a type space is complete, if the type functions in it are surjective (onto). Put it differently, a type space is complete, if all probability measures on the object consisting of the types of the model are corresponded to types. Quite recently Meier [20] showed that the purely measurable universal type space is complete. Summing up the above discussion, the answer for question (1) is affirmative, i.e., in the purely measurable framework the complete universal type space exists. Question (2) is on that whether the universal type space contains every hierarchy of beliefs or not. Mathematically the problem is the following: every hierarchy of beliefs defines an inverse system of measure spaces, and the question is that: do these inverse systems of measure spaces have inverse limits? Kolmogorov Extension Theorem is on this problem, however it calls for topological concepts, e.g. for inner compact regular probability measures. Therefore up to 2

now, all papers on this problem (e.g. Mertens and Zamir [23], Brandenburger and Dekel [8], Heifetz [15], Mertens et al. [24], Pint´er [27]) used topological type spaces instead of purely measurable ones. Although these papers give positive answer for question (2) (i.e. their type spaces contain all “considered” hierarchies of beliefs), very recently Pint´er [28] showed that there is no universal topological type space (there is no such a topological type space that contains every topological type space), therefore the answer for question (1) is negative i.e., in the topological framework the Hars´anyi program breaks down. In the above mentioned papers the authors answer question (2) (affirmatively) by constructing an object consisting all considered hierarchies of beliefs, called beliefs space, and show that the constructed beliefs space defines (is equivalent to) a complete topological type space. In this paper we work with the category of type spaces introduced by Heifetz and Samet, i.e., in the purely measurable framework. It is our main result that (in the purely measurable framework) every hierarchy of beliefs is type, put it differently, the Hars´ anyi program works. The strategy of the proof is the same as in the papers Mertens and Zamir, Brandenburger and Dekel, Heifetz, Mertens et al., Pint´er [27], i.e., we construct such an object that contains every hierarchy of beliefs (see definition 14.) and generates a type space. More exactly, it is showed that the (purely measurable) beliefs space is equivalent to the complete universal type space. As we have already mentioned the above strategy depends on the Kolmogorov Extension Theorem. Since we work in the purely measurable framework, therefore we avoid the direct use of topological concepts and generalize the Kolmogorov Extension Theorem in this (non-topological) direction. Mathematically, we use a new result to show that the inverse systems of measure spaces under consideration have inverse limits (see lemma 20.). Actually, the key point in our main result is the concept of pseudo completeness (see definition 17.), which is a well interpretable peculiarity of hierarchies of beliefs from the viewpoints of both economics and game theory. In nutshell (for the details see the discussion after definition 17.), pseudo completeness characterizes the fact that no new information enters into the model via the hierarchies of beliefs. Since the hierarchies of beliefs are the simple representatives of the players’ states of mind, therefore pseudo completeness seems to be a very natural feature. One important remark, our result does not contradict with Heifetz and Samet’s [19] counterexample, since their hierarchy of beliefs is not in the (purely measurable) beliefs space (for the the details see section 6.). The paper is organized as follows: in the first section we introduce an example illustrating our main result. Section 5. presents the technical setup and some basic results of the field. Our main result (theorem 15.) comes up in section 4. Section 5. is on the proof of theorem 15. Section 6. is for a detailed discussion of the connection between our result and two other papers Heifetz and Samet [19], and Pint´er [28]. The last section briefly concludes.

2

An example

Consider a 2 × 2 game in strategic form, two players: Player 1 and Player 2, both have two actions U , D and L, R respectively, there are two states of the

3

nature in the model: s1 and s2 (S = {s1 , s2 }) with the payoffs in tables 1 and 2

Player 1

Player 2 L R (2,3) (4,2) (3,4) (5,5)

U D

Table 1: The payoffs at the state of nature s1

Player 1

Player 2 L R (4,5) (3,4) (5,3) (2,2)

U D

Table 2: The payoffs at the state of nature s2 In this example that an action is rationalizable for a certain player means that there is such a state of the world that the common belief of rationality implies that the player under consideration plays the given action. Although the usual (see e.g. Osborne and Rubinstein’s textbook [26]) and the above rationalizability concepts differ, both catch the same intuition of rationalizability, and the only reason for introducing a new terminology is that the ”new” concept of rationalizability reflects the main massage of the example more and makes us possible to keep the discussion quite simple. It is easy to verify that in this example for both players both actions are rationalizable. If player 1 believes with probability 1 that the state of the nature is s1 then her rationality implies that she plays action D. Furthermore, if player 1 believes with probability 1 that the state of the nature is s2 and that player 2 believes with probability 1 that the state of the nature is s1 and that she (player 1) also believes with probability 1 that the state of the nature is s1 , then that she believes that player 2 is rational, and that player 2 believes that she is rational imply that she believes with probability 1 that player 2 believes with probability 1 that she plays action D, hence she believes with probability 1 that player 2 plays action R, therefore she plays action U . If player 2 believes with probability 1 that the state of the nature is s2 then her rationality implies that she plays action L. If it is mutually believed with probability 1 that the state of the nature is s1 then that player 2 believes that player 1 is rational implies that she believes with probability 1 that player 1 plays action D, hence she plays action R. Summing up the above discussion, an adequate type space must reflect the fact that for both players both actions are rationalizable. If this does not happen then the given type space is inappropriate for modeling the incomplete information situation under consideration. In the following we look into the question of what kind of type spaces can be appropriate for the modeling proposes under discussion. Case 1: The type space is neither complete nor universal.

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Consider the type space (see definition 5.) (S, (Ω, Mi )i=0,1,2 , g, {fi }i=1,2 ) ,

(1)

where Ω ⊜ S × {t1 } × {t2 }, f1 ⊜ f2 ⊜ δs2 ×t1 ×t2 (the Dirac measure concentrated at point s2 × t1 × t2 ), g : Ω → S is the coordinate projection, Mi ⊜ P(Ω) (the class of all subsets of Ω) i = 0, 1, 2. It is easy to verify that in model (1) at every state of the world both players believe that they play the game at state of the nature s2 , hence e.g. for player 2 action R is not rationalizable. Case 2: The type space is complete but not universal. Consider the type space ({s2 }, (Ω, Mi )i=0,1,2 , g, {fi }i=1,2 ) ,

(2)

where Ω ⊜ {s2 } × {t1 } × {t2 }, g : Ω → S is g ⊜ s2 (the natural embedding of Ω into S), f1 ⊜ f2 ⊜ δs2 ×t1 ×t2 , and Mi ⊜ {∅, Ω}, i = 0, 1, 2. It is easy to verify that this type space is complete (see definition 12.), and at every state of the world (there is only one in this model) it is commonly believed (known) that the state of the nature is s2 , hence e.g. for player 1 action U is not rationalizable. Case 3: Complete universal type space. From Heifetz and Samet [18], and Meier [20]: the complete universal type space (see definitions 8. and 12.) exists. Therefore in this example it contains the type space (S, (Ω, Mi )i=0,1,2 , g, {fi }i=1,2 ) ,

(3)

where T1 ⊜ T2 ⊜ [0, 1], Ω ⊜ S × T1 × T2 , g : Ω → S, i = 1, 2: pri : Ω → Ti are coordinate projections, ∀x ∈ [0, 1]: µ(x) ∈ ∆(S) is such that µ({s1 }) = x, M0 ⊜ σ(P(S)⊗{T1 }⊗{T2 }) (σ-field generated by the sets P(S)⊗{T1 }⊗{T2 }), M1 ⊜ σ({S} ⊗ B(T1 ) ⊗ {T2 }) (B(T1 ) is for the Borel σ-field of T1 ), M2 ⊜ σ({S} ⊗ {T1 } ⊗ B(T2 )), and last f1 (ω) ⊜ µ(pr1 (ω)) × δpr1 (ω)×1 (the product measure of the measures µ(pr1 (ω)) and δpr1 (ω)×1 ), f2 (ω) ⊜ µ(pr2 (ω))×δ1×pr2 (ω) . In this model at every state of the world every player believes that the other player believes that the state of the nature is s1 and that the given player believes that the state of the nature is s1 , hence, as we have already discussed, for both players both actions are rationalizable. Therefore, in this example the complete universal type space reflects the main intuitions of the modeled situation. Case 4: Complete universal type space that does not contain every hierarchy of beliefs. Only one question has remained, whether the complete universal type space contains every hierarchy of beliefs or not. Although in this example the universality implies that the model reflects the main intuitions of the situation we considered, in general, if the complete universal type space misses some hierarchies of beliefs then it is possible to construct a game in which the missing hierarchy(ies) of beliefs is(are) important, i.e., there is a game for which the complete universal type space is not appropriate (as in e.g. Case 1). 5

Therefore if the above mentioned failure happens then the Hars´anyi program breaks down, since the complete universal type space cannot reflect all important details of incomplete information situations. The main result of this paper (theorem 15.) argues that in the purely measurable framework Case 4 cannot happen, i.e., the complete universal type space contains every hierarchy of beliefs, in other words, the Hars´anyi program works. Our result heavily depends on that we work in the purely measurable framework, i.e. with the measurable structure introduced in definition 1. However, doing so is not restrictive at all, in contrary the richer structures bring only irrelevant details into the model, hence they are useless and more, as Pint´er’s result [28] shows, they can be harmful.

3

Type space

First some notations. Let N be the set of the players, w.l.o.g. we can assume that 0 ∈ / N , and let N0 ⊜ N ∪ {0}, where 0 is for the nature as an extra player. In this paper we use probability measures, hence if we do not indicate differently every measure is a probability measure. Let A be arbitrary set, then #A is for the cardinality of set A. For any A ⊆ P(X): σ(A) is the coarsest σ-field which contains A. Let (X, M) and (Y, N ) be arbitrary measurable spaces. Then (X × Y, M ⊗ N ) or briefly X ⊗ Y is the measurable space on the set X × Y equipped by the σ-field σ({A × B | A ∈ M, B ∈ N }). Let (X, M, µ) be an arbitrarily fixed measure space and µ∗ be the following set function on P(X): µ∗ (A) ⊜ inf µ(B) (therefore µ∗ is an outer measure). A⊆B∈M

Let (X, M, µ) be arbitrarily fixed measure space, where (X, τ ) is a topological space as well. µ is inner τ -closed / compact regular if ∀A ∈ M and ∀ε > 0: ∃C ∈ M such τ -closed / compact set that C ⊆ A and µ(A \ C) < ε. Let (X, τ ) be arbitrarily fixed topological space. The Borel σ-field of X is the smallest σ-field that contains the τ -open sets of X. The measurable spaces (X, M) and (Y, N ) are measurable isomorphic if there is such a bijection f between them that both f and f −1 are measurable. Let the measurable space (X, M) and x ∈ X be arbitrarily fixed. Then δx is for the Dirac measure on (X, M) concentrated at point x. Furthermore, we use the terminology of inverse system as Bourbaki [4] uses, hence the mappings in the inverse system are not necessarily surjective (onto). In the following, practically, we work with Heifetz and Samet’s [18] terminologies. Definition 1. Let (X, M) be arbitrarily fixed measurable space, and denote ∆(X, M) the set of the probability measures on it. Then the σ-field A∗ on ∆(X, M) is defined as follows: A∗ ⊜ σ({{µ ∈ ∆(X, M) | µ(A) ≥ p}, A ∈ M, p ∈ [0, 1]}) . In other words, A∗ is the smallest σ-field among the σ-fields that contain the sets {µ ∈ ∆(X, M) | µ(A) ≥ p}, where A ∈ M and p ∈ [0, 1] are arbitrarily chosen. 6

In incomplete information situations it is necessary to handle the events like player i believes with probability at least p that an event occurs (beliefs operator see e.g. Aumann [2]). For this reason {µ ∈ ∆(X, M) | µ(A) ≥ p} must be an event (measurable set). To keep the class of events as small (coarse) as possible, we use the A∗ σ-field3 . Notice that A∗ is not a fixed σ-field, it depends on the measurable space on which the probability measures are defined. Therefore A∗ is similar to the weak ∗ topology, which depends on the topology of the base (primal) space. Corollary 2. Let (X, M)Sbe arbitrarily fixed measurable space. Then ∆(X, M) −1 prA (B([0, 1]))), where prA : ∆(X, M) → [0, 1] is ⊆ [0, 1]M , and A∗ = σ( A∈M

coordinate projection, and B([0, 1]) is the Borel σ-field of [0, 1]. Furthermore, ∆(X, M) is a subset of the compact topological space [0, 1]M , hence as a subspace of [0, 1]M it is a completely regular topological space, and ∀µ ∈ ∆(∆(X, M), A∗ ), ∀A ∈ A∗ , and ∀ε > 0: ∃C ∈ A∗ closed set in the completely regular (subspace) topology such that C ⊆ A and µ(A \ C) < ε. −1 Proof. Let µ ∈ ∆(∆(X, M), A∗ ) be arbitrarily fixed, B ⊜ {prA (B) | B ∈ B([0, 1]), A ∈ M} (it is clear that σ(B) = A∗ ) and

Br ⊜ {B ∈ σ(B) | ∀ε > 0 : ∃C ∈ σ(B), C closed set in the subspace topology, C ⊆ B such that µ(B \ C) < ε} . First B ⊆ Br , hence σ(B) ⊆ σ(Br ) .

(4)

Br ⊆ σ(B) .

(5)

On the other hand

Let {An }n ∈ Br be such that ∀n: An ⊆ An+1 , andSε > 0 be arbitrary fixed. µ is a σ-additive set function, hence lim µ(An ) = µ( An ), i.e. ∃n∗ such that n→∞ n S ε µ( An ) − µ(An∗ ) < . Moreover ∀n: ∃Cn ∈ σ(B) closed set, Cn ⊆ An such 2 n S ε that µ(An \ Cn ) < . Summing up the above discussion, Cn∗ ⊆ An and 2 n [ µ( An \ Cn∗ ) < ε , i.e.

S

n

An ∈ Br .

n

3 For a more detailed argument see Meier [22] p. 56. “Why should the knowledge operators of the players just operate on measurable sets and not on all subsets of the space? The justification for this is that we think of events as those sets of states that the players can describe, and only those can be the objects of their reasoning. In view of this interpretation a statement saying “player i knows that the actual state of the world is in E,” where E is an entity of states he cannot represent in his mind, is meaningless. Of course, it might well be that in some knowledge–belief spaces all subsets of the space of states of the world can be described by the players (for example in the finite knowledge–belief spaces), but we do not want to assume this in general.”

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In the similar T way, it is also easy to see that if {An }n ∈ Br is such that ∀n: An+1 ⊆ An then An ∈ Br . To sum up, Br is a monotone class, hence n

σ(Br ) = Br .

(6)

From equations (4), (5), and (6) A∗ = σ(B) = Br . µ was arbitrarily fixed, hence the proof is complete.

Q.E.D.

From the above corollary A∗ has topological “ancestors,” i.e. it is not independent from topological concepts. However, it depends only on the Euclidean topology of [0, 1], the topology of the base space, if that has any at all, has no direct impact on it (indirectly, through the measurable space, that can have). Assumption 3. Let (S, A) be a fixed parameter space. Henceforth we assume that (S, A) is the fixed parameter space that contains all states of the nature. For instance in the example of section 2. S has two elements: S = {s1 , s2 }, and A = P(S). Definition 4. Let Ω be the space of states of world, and ∀i ∈ N0 : Mi be a σ-field on Ω. The σ-field Mi represents player i’s information, M0 is for the information available for the nature, hence itSis the representative of A, the σMi ), the smallest σ-field which field of the parameter space S. Let M ⊜ σ( i∈N0

contains all σ-fields Mi . Every point in Ω provides a complete description of the actual state of the world. It includes both the state of the nature and the players’ states of mind. The different σ-fields are for modeling the informedness of the players, they have the same role as the partitions in e.g. Aumann’s [1] paper have. Therefore, if ω, ω ′ ∈ Ω are not distinguishable 4 in the σ-field Mi then player i is not able to discern difference between them, i.e., she knows, believes the same things, and behaves in the same way at the two states ω and ω ′ . M represents all information available in the model, it is the σ-field got by pooling the information of the players and the nature. For the sake of brevity, henceforth - if it does not make confusion - we do not indicate the σ-fields. E.g. instead of (S, A) we write S, or ∆(S) instead of (∆(S, A), A∗ ). However, in some cases we refer to the non-written σ-field: e.g. A ∈ ∆(X, M) is a measurable set in A∗ , i.e., in the measurable space (∆(X, M), A∗ ), but A ⊆ ∆(X, M) keeps its original meaning: A is a subset of ∆(X, M). Definition 5. Let (Ω, M) be the space of states of world (see definition 4.). The type space based on the parameter space S is a tuple (S, {(Ω, Mi )}i∈N0 , g, {fi }i∈N ), where 1. g : Ω → S is M0 -measurable, 2. ∀i ∈ N : fi : Ω → ∆(Ω, M) is Mi -measurable. 4 Let

(X, T ) be arbitrarily fixed measurable space, and x, y ∈ X be also arbitrarily fixed. x and y are measurably indistinguishable if ∀A ∈ T (x ∈ A) ⇔ (y ∈ A).

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Some papers in the literature consider the so called Hars´anyi type space (see discussion in Heifetz and Mongin’s [16] paper), where the above properties are supplemented by the following point: 3. ∀ω ∈ Ω and ∀i ∈ N : fi (ω)|(Ω,Mi ) = δω . Point 3. is the formalization of Hars´anyi’s intuition it “means” that every player knows her own type. The difference between the two concepts of type spaces is not relevant, the same results are valid for both formalizations of type space. However, by using the version in the above definition, we can keep the model as simple as possible, therefore, in this paper the type spaces do not need to meet point 3. Put definition 5. differently, S is the parameter space, it contains the ”types” of the nature. Mi represents the information available for player i, hence it corresponds to the concept of type (Hars´ anyi [14]). fi is the type function of player i, it maps the player i’s types to her (subjective) beliefs. The above definition of type space differs from Heifetz and Samet’s concept, but it is similar to Meier’s [20], [22] type space. We do not use Cartesian product space, but refer only to the σ-fields. By following strictly Heifetz and Samet’s paper, if one takes the Cartesian product of the parameter space and the type sets, and defines the σ-fields as the σ-fields induced by the coordinate projections (e.g. M0 is induced by the coordinate projection pr0 : S × ×i∈N Ti → S, for the notations see their paper) then she gets at our concept. However, if the Cartesian product is not used directly then it is necessary to connect the parameter space into the type space in some way. For this we use g (Mertens and Zamir [23] and Meier [22] use a similar formalism), hence g and pr0 have the same role in the two formalizations, in this and in Heifetz and Samet’s paper respectively. A further difference between the two formalizations lies in the role of the parameter space. While in Heifetz and Samet the entire parameter space must appear in the type space, in our approach this is not required (see Case 2 in the example of section 2.). We emphasize that this difference is not relevant. Definition 6. The type morphism between the type spaces (S, {(Ω, Mi )}i∈N0 , g, {fi }i∈N )

and

(S, {(Ω′ , M′i )}i∈N0 , g ′ , {fi′ }i∈N )

ϕ : Ω → Ω′ is such an M-measurable function that 1. diagram (7) is commutative (i.e. ∀A ∈ S: g −1 (A) = (g ′ ◦ ϕ)−1 (A)) Ω

g

ϕ

-

? Ω′

(7)

g′ S

2. ∀i ∈ N : diagram (8) is commutative (i.e. ∀ω ∈ Ω, ∀A ∈ M′ : fi′ ◦ ϕ(ω)(A) = fi (ω)(ϕ−1 (A)))

9

Ω

fi

- ∆(Ω, M) ϕ

ϕ ? Ω′

(8)

? - ∆(Ω′ , M′ )

fi′

ϕ type morphism is a type isomorphism, if ϕ is a bijection and ϕ−1 is also a type morphism. The above definition is practically the same as Heifetz and Samet’s, hence all intuitions, they discussed, remain valid, i.e., the morphism maps type profiles from a type space to type profiles in an other type space in a way that the corresponded types induce equivalent beliefs. In other words, the type morphism reserves the players’ beliefs. For the consequences of the (irrelevant) differences in the formalizations of type space see the discussion after definition 5. Corollary 7. The type spaces that are based on the parameter space S as objects and the type morphisms form a category. Let C S denote this category of type spaces. Proof. It is a direct corollary of definitions 5. and 6.

Q.E.D.

Heifetz and Samet introduced the concept of universal type space. Definition 8. The type space (S, {(Ω, Mi )}i∈N0 , g, {fi }i∈N ) is universal, if for any type space (S, {(Ω′ , M′i )}i∈N0 , g ′ , {fi′ }i∈N ) there is a unique type morphism ϕ from

(S, {(Ω′ , M′i )}i∈N0 , g ′ , {fi′ }i∈N )

to

(S, {(Ω, Mi )}i∈N0 , g, {fi }i∈N ) .

In other words, the universal type space is the most general, the broadest type space among the type spaces. It contains all types that appear in the type spaces of the given category. Corollary 9. The universal type space is terminal (final) object in C S . Proof. It comes directly from definition 8.

Q.E.D.

From the viewpoint of category theory the uniqueness of universal type space is really straightforward. Corollary 10. The universal type space is unique up to type isomorphism. Proof. Every terminal object is unique up to isomorphism.

Q.E.D.

The only question is the existence of universal type space. Proposition 11. There is universal type space, in other words, there is terminal object in C S . Proof. See Heifetz and Samet Theorem 3.4.

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Q.E.D.

As we have already mentioned, Heifetz and Samet’s formalization of type space is a little bit different from ours. However the difference between the two approaches is quite slight, and we prove a stronger result in theorem 15., hence we have omitted the formal proof of the above proposition. Next, we turn our attention to an other property of type spaces, the completeness. Definition 12. The type space (S, {(Ω, Mi )}i∈N0 , g, {fi }i∈N ) is complete if ∀i ∈ N : fi is surjective (onto). The above concept was introduced by Brandenburger [6]. Completeness recommends that for any player every probability measure on (Ω, M) be in the range of the given player’s type function. In other words, all measures on (Ω, M) must belong to types. Proposition 13. The universal type space is complete. Proof. See Meier [20] Theorem 4.

Q.E.D.

We can say again that Meier’s type space is a little bit different from ours, however the difference is really slight, and we prove a stronger result in theorem 15., hence we have omitted the formal proof of the above proposition.

4

Beliefs space

In the following we formalize the intuition of hierarchy of beliefs, i.e., the “infinite regress in reciprocal expectations.” First we give a rough description (Mertens and Zamir’s [23]): T0 T1 T2 T3

⊜ ⊜ ⊜ ⊜ = .. .

S T0 ⊗ ∆(T0 )N T1 ⊗ ∆(T1 )N = T0 ⊗ ∆(T0 )N ⊗ ∆(T0 ⊗ ∆(T0 )N )N T2 ⊗ ∆(T2 )N T0 ⊗ ∆(T0 )N ⊗ ∆(T1 )N ⊗ ∆(T0 ⊗ ∆(T0 )N ⊗ ∆(T1 )N )N

Tn

⊜ Tn−1 ⊗ ∆(Tn−1 )N = T0 ⊗ = T0 ⊗ .. .

n−2 N

n−1 N

∆(Tm )N

m=0 n−3 N

∆(Tm )N ⊗ ∆(T0 ⊗

m=0

∆(Tm )N ⊗ ∆(Tn−2 )N )N

m=0

The above formalisms can be interpreted in the following way. T0 describes the basic uncertainty of the modeled situation, it consists of the states of nature. T1 is for T0 and the first order beliefs of the players ∆(T0 )N (N is the players’ set), i.e., what the players believe about the states of the nature. In general, Tn describes Tn−1 and the nth order beliefs of the payers ∆(Tn−1 )N , i.e., what the players believe about Tn−1 .

11

However, there is some redundancy5 in the above description. E.g. ∆(T0 ⊗ ∆(T0 )N )N determines ∆(T0 )N and so does ∆(Tn−1 )N ∀(0 ≤ m ≤ n − 2): ∆(Tm )N , therefore we can rewrite the above formalisms into the following form: T0 T1 T2 T3

Tn

⊜ ⊜ ⊜ ⊜ .. .

S T0 ⊗ ∆(T0 )N T0 ⊗ ∆(T0 ⊗ ∆(T0 )N )N T0 ⊗ ∆(T0 ⊗ ∆(T0 )N ⊗ ∆(T1 )N )N

⊜

T0 ⊗ ∆(T0 ⊗

(9) n−3 N

∆(Tm )N ⊗ ∆(Tn−2 )N )N

m=0

.. .

Take player i, and examine the situation from her viewpoint. ∀n ∈ N: let i #Θi−1 = 1, Θin ⊜ ∆(Tn ), and q−10 : Θi0 → Θi−1 . Moreover, ∀n ∈ N, and ∀µ ∈ n n−1 N N ∆(Tm )N ): µ can be naturally defined on T0 ⊗ ∆(Tm )N Θin+1 = ∆(T0 ⊗ m=0

m=0

i as a restriction of µ, i.e. let qnn+1 : Θin+1 → Θin be as follows ∀µ ∈ Θin+1 : i (µ) ⊜ µ| qnn+1 T0 ⊗

N

.

n−1

∆(Tm )N

m=0

i qnn+1

is measurable and continuous w.r.t. the completely regular subspace Then topology. Furthermore, it is worth noticing that for any n the measures in ∆(T0 ⊗ n−3 N ∆(Tm )N ⊗ ∆(Tn−2 )N ) are defined on a product space. This feature has a

m=0

very important role in the proof of our main result (see lemma 21.). The next definition, which is a variant of Mertens et al.’s [24], summarizes the above discussions and formalizes the “infinite regress in reciprocal expectations.” It differers from Mertens et al.’s original one in the point that we do not use Hars´ anyi’s type space (see definition 5.). Definition 14. In the diagram (10) Θi

∆(S ⊗ ΘN )

pin+1

idS

? ? ∆(S ⊗ ΘN n)

? Θin+1

⊜

i qnn+1

? Θin

pN n

idS

(10)

N qn−1n

? ? ∆(S ⊗ ΘN n−1 )

⊜

• i ∈ N is an arbitrarily fixed player, 5 This

redundancy is called coherency and consistency in the literature of game theory and mathematics respectively.

12

• n ∈ N, • S is the fixed parameter space, moreover ∀i ∈ N : • #Θi−1 = 1, i • q−10 : Θi0 → Θi−1 ,

• ∀m, n ∈ N, m ≤ n, ∀µ ∈ Θin : i qmn (µ) ⊜ µ|S⊗ΘN . m−1 i Therefore qmn is a measurable and continuous (w.r.t. the completely regular subspace topology see corollary 2.) mapping. i • Θi ⊜ lim(Θin , N ∪ {−1}, qmn |m≤n ), ←−

• ∀n ∈ N ∪ {−1}: pin : Θi → Θin is canonical projection, N j • ∀m, n ∈ N∪{−1}, m ≤ n: qmn is the product of the mappings qmn , j ∈ N, j N and so is pn of pn , j ∈ N , therefore both mappings are measurable and continuous, N j N j • ∀n ∈ N ∪ {−1}: ΘN Θn , and ΘN ⊜ Θ . n ⊜ j∈N

j∈N

Then T ⊜ S ⊗ ΘN is called beliefs space. The interpretation of the beliefs space is the following. For any θi ∈ Θi : θ = (µi1 , µi2 , . . .), where µin ∈ Θin−1 is the nth order belief of player i. Therefore every point Θi defines an inverse system of measure spaces i

i i N ((S ⊗ ΘN n , pn+1 (θ )), N ∪ {−1}, (idS , qmn )|m≤n ) .

(11)

We call inverse system of measure spaces like (11) player i’s hierarchy of beliefs 6 . To sum up, T consists of all states of the world: all states of nature, the points in S, and all states of mind, the points in the set ΘN , therefore T contains all players’ all hierarchies of beliefs. Our main result: Theorem 15. The complete universal type space contains all players’ all hierarchies of beliefs. All next section is devoted to the proof of the above theorem. 6 In the literature this system is usually called coherent hierarchy of beliefs. Since it does not make confusion, in this paper we omit the adjective coherent.

13

5

The proof of theorem 15.

The strategy of the proof is to show that the beliefs space (see definition 14.) generates (is equivalent to) the complete universal type space (in category C S ). The key point of the proof is to demonstrate that in the diagram (10) ∀i ∈ N : Θi = ∆(S ⊗ ΘN ) , i.e. they are measurable isomorphic. Therefore in the following we focus on this point. Lemma 16. The beliefs space T generates a type space in the category C S . The following mathematical concept has a key role in our proof. Definition 17. The inverse system of measure spaces ((Xn , Mn , µn ), N, fmn |m≤n ) is pseudo complete, if ∀(m ≤ n) and ∀A ⊆ Xm : −1 (µ∗n (fmn (A)) = 0) ⇒ (µ∗m (A) = 0) .

Put the above definition differently, an inverse system of measure spaces is pseudo complete, if the σ-fields in it are getting richer and richer “continuously,“ we mean, there is no ”qualitative change” in them. It is an usual assumption in the theory of stochastic processes that the filtration is complete. If a filtration is complete then the inverse system of measure spaces on it is pseudo complete as well, but the reverse is not true (this is the reason for the naming). An interpretation of pseudo completeness for stochastic processes can be the following: let N be for the time. Pseudo completeness says that if it is possible to exclude some states at time n + 1, i.e., outer measure of the set of the given −1 states (fnn+1 (A)) is zero, and those states are recognizable at time n (A), then it must be also possible to exclude those (A) at time n as well. In other words, no outer information appears in the model between time n and n + 1. In the case of hierarchies of beliefs the interpretation is as follows: if a −1 player believes at order n + 1 that some states of world (fnn+1 (A)) occur at outer measure zero, and those states of world can be identified at order n as well (A), then she must believe at order n that those states (A) occur at outer measure zero. Put it differently, the player does not change her belief, get new / outer information. It is easy to verify that the counterexample given by Heifetz and Samet [19], actually, its mathematical core (see Halmos [13] exercise (3) pp. 214–215) is not a pseudo complete inverse system of measure spaces. The pseudo completeness is neither necessary nor sufficient condition for the existence of inverse limit. For the non-necessity it is enough to see the following system of measure spaces: (([0,

1 1 1 ), {∅, [0, )}, δ0 ), N, id[0, n+1 ) |m≤n ) . n+1 n+1

(12)

It is easy to verify that (12) is an inverse system of measure spaces, and its inverse limit is ({0}, {∅, {0}}, δ0). However, (12) is not pseudo complete. It is a little bit more cumbersome to see that the pseudo completeness is not sufficient condition for the existence of inverse limit.

14

Example 18. This counterexample is a variant of Rao’s [29] example (example 6. (a) p. 40). S { 2jn | j ∈ N, 0 ≤ j ≤ 2n }, and ∀(m ≤ n): fmn = idX . Moreover Let X ⊜ n∈N

let Σ0 Σ1 Σ2

= = = .. .

{[0, 1] ∩ X} {[0, 21 ] ∩ X, ( 12 , 1] ∩ X} {[0, 41 ] ∩ X, ( 14 , 12 ] ∩ X, ( 21 , 34 ] ∩ X, ( 34 , 1] ∩ X}

Σn

= .. .

{[0, 21n ] ∩ X, . . . , (1 −

1 2n , 1]

∩ X}

Let Mn ⊜ σ(Σn ), and µn be such a probability measure that ∀A ∈ Σn : µn (A) = 21n . From the definition of fmn ((X, Mn , µn ), N, fmn |m≤n ) is a surjective (onto) inverse system of measure spaces, but it does not admit inverse limit. Indirectly assume that (X, M, µ) inverse limit exists, hence µ is σ-additive set function. Then ∀x ∈ X: {x} ∈ M and µ({x}) = 0. However, this is a contradiction, because the measure of the whole space is 1, and X consists of countably many points. On the other hand, since ∀(m ≤ n): fmn = idX , and ∀n: (µ∗n (A) = 0) ⇒ (A = ∅), therefore ((X, Mn , µn ), N, fmn |m≤n ) is pseudo complete. Definition 19. Let ((Xn , Mn , µn ), N, fmn |m≤n ), ((Yn , Nn , νn ), N, gmn |m≤n ) be such inverse systems of measure spaces that ∀(m ≤ n): 1. Xn ⊆ Yn , 2. νn∗ (Xn ) = 1, 3. Mn = {A ∩ Xn | A ∈ Nn }, 4. µn = νn |Mn , 5. fmn = gmn |Xn . Then ((Xn , Mn , µn ), N, fmn |m≤n ) is called a restriction of ((Yn , Nn , νn ), N, gmn |m≤n ). The next lemma is the mathematical core of this paper and the proof of our main result theorem 15. Lemma 20. Let ((Xn , Mn , µn ), N, fmn |m≤n ) and ((Yn , Nn , νn ), N, gmn |m≤n ) be such inverse systems of measure spaces that 1. (Y, N , ν) ⊜ lim((Yn , Nn , νn ), N, gmn |m≤n ) exists, ←− 2. ((Xn , Mn , µn ), N, fmn |m≤n ) is pseudo complete, 3. ((Xn , Mn , µn ), N, fmn |m≤n ) is a restriction of ((Yn , Nn , νn ), N, gmn |m≤n ). Then (X, M, µ) ⊜ lim((Xn , Mn , µn ), N, fmn |m≤n ) exists. ←− 15

The above lemma says that if the given inverse system of measure spaces is “dense” in an other system having inverse limit, and it is not leaking (pseudo completeness) then it has inverse limit. −1 Proof. ∀n ∈ N: let Ln ⊜ {x ∈ Xn | fnn+1 ({x}) ∩ Xn+1 = ∅}, therefore that ((Xn , Mn , µn ), N, fmn |m≤n ) is pseudo complete implies that µ∗n (Ln ) = 0. S −1 ∗ ∗ Moreover ∀n: νn (Ln ) = 0, hence ν ( pn (Ln )) = 0 and ν ∗ (X) = 1, where n

pn : Y → Yn is canonical projection in (Y, N , ν). Finally, ν ∗ (X) = 1 implies that ((Xn , Mn , µn ), N, fmn |m≤n ) admits inverse limit, actually µ = ν|X . Q.E.D. Next we show that the inverse system of measure spaces (11) is pseudo complete. Lemma 21. ∀i ∈ N , ∀θi ∈ Θi : the inverse systems of measure spaces (11) is pseudo complete. Proof. Let i ∈ N , θi ∈ Θi , and n ∈ N be arbitrarily fixed. Then from (9) S ⊗ ΘN n−1 = S ⊗ ∆(T0 ⊗

n−2 O

∆(Tm )N )N .

m=0

Moreover ∀µ ∈ ∆(T0 ⊗

n−2 N

∆(Tm )N ) and ∀ν ∈ ∆(∆(Tn−1 )N ):

m=0

µ × ν ∈ ∆(T0 ⊗

n−2 O

∆(Tm )N ⊗ ∆(Tn−1 )N ) ,

m=0

where µ × ν is the product measure. n−2 N Then ∀µ ∈ ∆(T0 ⊗ ∆(Tm )N ): m=0

i {ν|∆(Tn−1 )N | ν ∈ (qn−1n )−1 ({µ})} = ∆(∆(Tn−1 )N ) . −1 N N (A) ⊆ B then Therefore ∀A ⊆ S ⊗ ΘN n−1 : if B ∈ S ⊗ Θn , (idS , qn−1n ) N −1 N N ∃C ∈ (idS , qn−1n ) (S ⊗ Θn−1 ), i.e. C is in the σ-field induced by (idS , qn−1n ) N from S ⊗ Θn−1 , such that N (idS , qn−1n )−1 (A) ⊆ C ⊆ B . N Put it differently, if (pin+1 (θi ))∗ ((idS , qn−1n )−1 (A)) = 0 then (pin (θi ))∗ (A) = 0, i.e. the inverse system of measure spaces (11) is pseudo complete. Q.E.D.

In order to prove that the inverse system of measure spaces (11) is a restriction of an inverse system of measure spaces having inverse limit, we need the following result. Lemma 22. Let (M, AM , µM ), (N, AN , µN ) be probability measures spaces, and µ be additive set function on A ⊆ P(M ×N ) the field generated by the coordinate −1 projections pM and pN . If µ ◦ p−1 M = µM and µ ◦ pN = µN then µ is σ-additive.

16

Proof. It is easy to verify that the sets of A have the form:

m S

(Mj × Nj ),

j=1

where m ∈ N, Mj ∈ AM , Nj ∈ AN . It is well known that µ is σ-additive on A if T and only if for arbitrary sequence of sets (An )n such that ∀n: An ⊇ An+1 and An = ∅: lim µ(An ) = 0. n→∞

n

∀n ∈ N: let kn ∈ N such that An =

k Sn

j=1

(Mjn × Njn ), and F ⊜ {f ∈ NN | ∀n :

f (n) ≤ kn }. Then \

[ \

An =

n

(Mfn(n) × Nfn(n) ) .

f ∈F n

Moreover \ \ ( An = ∅) =⇒ (∀f ∈ F : (Mfn(n) × Nfn(n) ) = ∅) . n

(13)

n

T Split the sets (Mfn(n) × Nfn(n) ) into two groups. Let the first one F1 contain T n f such that Mfn(n) = ∅, and the others be in F2 . Furthermore, let Mn ⊜ n S T

f ∈F1 j=1

n

Mfj (j) ,

where n is arbitrarily fixed.

Then ∀n: Mn ∈ AM , and ∀n: Mn ⊇ Mn+1 , hence (Mn )n∈N is a monotone sequence of sets. T S T n T Furthermore, Mn = Mf (n) , and from (13) ∀f ∈ F1 : Mfn(n) = ∅, n n f ∈F1 n T hence Mn = ∅. n

Moreover \ [ \ (Mn × N ) ⊇ (Mfn(n) × Nfn(n) ) . n

f ∈F1 n

µM is σ-additive, hence µM (Mn ) → 0 and lim µM (Mn ) = lim µ(Mn × N ) ≥ lim µ(

n→∞

n→∞

n→∞

[ \

(Mfn(n) × Nfn(n) )) ,

f ∈F1 n

therefore µ(

[ \ (Mfn(n) × Nfn(n) )) → 0 .

(14)

f ∈F1 n

The case of F2 is the same, therefore [ \ µ( (Mfn(n) × Nfn(n) )) → 0 .

(15)

f ∈F2 n

µ is additive, hence S T n S S T n µ(( (Mf (n) × Nfn(n) )) ( (Mf (n) × Nfn(n) ))) n f ∈FS f ∈F2 S 1 nT T n ≤ µ( (Mfn(n) × Nfn(n) )) + µ( (Mf (n) × Nfn(n) )) . f ∈F1 n

f ∈F2 n

17

(16)

Then from (14), (15), (16) µ(An ) → 0 . Q.E.D. Next we show that (11) is a restriction of an inverse system of measure spaces having inverse limit. Lemma 23. ∀i ∈ N and ∀θi ∈ Θi : there is such an inverse system of measure spaces 1. that has inverse limit, 2. (11) is the restriction of that. Proof. We use the same trick as that Pint´er [27] used, i.e., we cut the parameter space off the inverse system, and examine the retained part. Later, we put them together again. Let i ∈ N and θi ∈ Θi be arbitrarily fixed. The truncated (11) is the following: N (ΘN n , ptn+1 , N ∪ {−1}, qmn |m≤n ) ,

(17)

where ptn+1 ⊜ pin+1 (θi )|ΘN . n Corollary 2. implies that ∀n ∈ N∪{−1}: ΘN n has a σ-field which is the subset of the Borel σ-field of a completely regular topological space, and the ptn+1 are inner closed regular measures. It is well known that, completely regular topological space is characterized by the fact that it can be embedded into a ˆ compact space as an everywhere dense set (Cech-Stone compactification7 ). N Therefore ∀n ∈ N ∪ {−1}: Θn can be mapped into a compact space Cn as an everywhere dense set. Let emn be this embedding, and Mn ⊜ {A ⊆ N Cn | em−1 n (A) ∈ Θn }, and µn+1 be such a measure on (Cn , Mn ) that µn+1 ⊜ −1 ptn+1 ◦ emn . Then µn+1 is an inner compact regular probability measure on the measurable space (Cn , Mn ). Furthermore, ∀n ∈ N ∪ {−1}: let fnn+1 : Cn+1 → Cn be such a continN = fnn+1 |ΘN and (Cn , N ∪ uous and Mn+1 -measurable function that qnn+1 n+1 {−1}, fmn|m≤n ) is an inverse system of topological spaces (it is easy to verify that there are such functions). Then ((Cn , Mn , µn+1 ), N∪{−1}, fmn|m≤n ) meets the conditions of 3.2. Theoreme in Metivier [25] pp. 269–270., hence it has a unique inverse limit. Moreover from lemma 22. lim((S × Cn , A ⊗ Mn , νn+1 ), N ∪ {−1}, (idS , fmn )|m≤n ) , ←− also exists, where νn+1 ⊜ pin+1 (θi ) ◦ (idS , emn )−1 . Furthermore (17) is the restriction of the inverse system of measure spaces ((Cn , Mn ), µn+1 ), N ∪ {−1}, fmn|m≤n ), hence (11) is a restriction of ((S × Cn , A ⊗ Mn , νn+1 ), N ∪ {−1}, (idS , fmn )|m≤n ) . Q.E.D. ˆ of using the Cech-Stone compactification it is possible to employ the compact spaces from corollary 2. 7 Instead

18

The proof of lemma 16. Let ∀i ∈ N : pri : T → Θi , pr0 : T → S be coordinate projections, and ∀i ∈ N ∪ {0}: the σ-fields M∗i be induced by pri . Lemmata 20., 21., 23. imply that (11) has measure inverse limit, hence in the diagram (10) ∀i ∈ N : Θi = ∆(S ⊗ ΘN ) ,

(18)

i.e., they are measurable isomorphic. Furthermore, let g ∗ ⊜ pr0 , and ∀t ∈ T : fi∗ (t) ⊜ pri (t). Then (S, {(T, M∗i )}i∈N , g ∗ , {fi∗ }i∈N ) is a type space in category C S .

Q.E.D.

Corollary 24. In the type space (S, {(T, M∗i )}i∈N , g ∗ , {fi∗ }i∈N ) ∀i ∈ N : if pri (t) 6= pri (t′ ) then fi∗ (t) 6= fi∗ (t′ ). Proof. It is the direct corollary of that different inverse systems of measure spaces have different inverse limits, and Θi consists of different inverse systems of measure spaces (hierarchies of beliefs). Q.E.D. Proposition 25. The type space (S, {(T, M∗i )}i∈N , g ∗ , {fi∗ }i∈N ) is complete. Proof. It is the direct corollary of (18).

Q.E.D.

Proposition 26. The type space (S, {(T, M∗i )}i∈N , g ∗ , {fi∗ }i∈N ) is a universal type space. Proof. Let (S, {(Ω, Mi )}i∈N , g, {fi }i∈N ) be an arbitrarily fixed type space (an object in C S ), and i ∈ N and ω ∈ Ω be also arbitrarily fixed. The first order belief of player i at state of the world ω vi1 (ω) is the measure defined as follows ∀A ∈ S: v1i (ω)(A) ⊜ fi (ω)(g −1 (A)) . fi is Mi -measurable, hence vi1 is also Mi -measurable. The second order belief of player i at state of the world ω v2i (ω) is the measure defined as follows ∀A ∈ S ⊗ ΘN 0 : v2i (ω)(A) ⊜ fi (ω)((g −1 , (v1N )−1 ))(A) , where v1N is the product of the mappings v1j , j ∈ N . Since fi is Mi -measurable, hence vi2 is also Mi -measurable. Let n > 1 be arbitrarily fixed, then the nth order belief of player i at state of the world ω vni (ω) is the measure defined as follows ∀A ∈ S ⊗ ΘN n−2 : N vni (ω)(A) ⊜ fi (ω)((g −1 , (vn−1 )−1 ))(A) .

Since fi is Mi -measurable, hence vni is also Mi -measurable. To sum up, there is a well defined mapping φ : Ω → S ⊗ T as follows ∀ω ∈ Ω: ψ(ω) ⊜ (g(ω), (v1i (ω), v2i (ω), . . .)i∈N ) . Then it is easy to verify that (1) φ is M-measurable. 19

(19)

(2) The above construction implies that ∀ω ∈ Ω, ∀i ∈ N , ∀A ∈ T : fi∗ ◦ φ(ω)(A) = fi (ω)(φ−1 (A)) , i.e., φ is a type morphism. (3) From corollary 24. φ is the unique type morphism from the type space (S, {(Ω, Mi )}i∈N , g, {fi }i∈N ) to (S, {(T, M∗i )}i∈N , g ∗ , {fi∗ }i∈N ). Q.E.D. It is worth noticing that φ in the above proof is not injective (one to one). If there are duplicate types in a type space, i.e. such types that generate the same hierarchy of beliefs (see e.g. Ely and Peski’s [11] example), then the φ image of this duplication is one point in the universal type space. Therefore, it is not surprising at all that there are no duplicates in the universal type space, i.e., it can be complete. The proof of theorem 15. From proposition 26. (S, {(T, Mi )}i∈N , g ∗ , {fi∗ }i∈N )

(20)

is a universal type space. Then corollary 10. implies that Heifetz and Samet’s [18] universal type space and (20) coincide (they are type isomorphic). From proposition 25. (20) is complete (Meier [20] also proved this). Finally, from definition 14. (20) contains all hierarchies of beliefs. Q.E.D.

6

Related papers

In this section theorem 15. is compared to the results of Heifetz and Samet [19], and Pint´er [28]. These papers seem to contradict our main result, however in the following we show that it is not the case at all. Heifetz and Samet in their paper “Coherent beliefs are not always types,” as the title indicates, give an example, a hierarchy of beliefs, that can not be type in any type space. Mathematically, their counterexample is based on an exercise of Halmos’s book [13], an example for an inverse system of measure spaces without inverse limit. As we have already mentioned, this inverse system of measure spaces is not pseudo complete, hence the beliefs space introduced in definition 14. does not contain it. In other words, mathematically, our result does not contradict Heifetz and Samet’s [19] counterexample. From the viewpoint of games with incomplete information, the question is that whether the hierarchy of beliefs of Heifetz and Samet’s example is relevant or not, i.e., it describes a really possible state of mind or not. First, the measurable structure A∗ (see definition 1.) does reflect the sentences used for modeling incomplete information situations (see the logic e.g. in Meier [20]). Therefore every richer structure (see e.g. Mertens and Zamir [23], Brandenburger and Dekel [8], Heifetz [15], Mertens et al. [24], Pint´er [27]) deals with some hierarchies of beliefs that cannot be expressed by the language under consideration. Furthermore, as we have already mentioned the pseudo-completeness of an inverse system of measure spaces can be interpreted as no new information appears in the model after the process starts, i.e., during the player looks over her hierarchy of beliefs she cannot get new / outer information, she only looks / 20

thinks through the information available for her, structures her beliefs and gets a pseudo complete inverse system of measure spaces as her hierarchy of beliefs. From this viewpoint, Heifetz and Samet’s example models a situation, where the player is more informed when she forms her second order belief than when she does her first order belief (and so on). Put it differently, the model is not closed, something relevant and important “thing” not in the parameter space has a huge impact on the modeled incomplete information situation. However, since by definition the parameter space consists of all parameters having impact on the modeled situation, hence this “thing” must be contained rather in the parameter space than in the space of the state of mind. To sum up, from a modeler’s point of view Heifetz and Samet’s example does not cancel our result, because that mismodels an incomplete information situation. At last but far not least, Heifetz and Samet’s hierarchy of beliefs is a good example for that how important the premises of the modeling are. Quite recently, Pint´er provided a negative result, he argued that there is no universal topological type space in the category of topological type spaces. Actually, this non-existence is got by topological argument, hence his negative result does not contradict this paper’s positive one. On the other hand, Pint´er’s result clearly shows that irrelevant details, brought in the model by topological concepts, can make real difficulties, which culminate in that the goal proving that the Hars´anyi program works is unreachable in the topological approach.

7

Conclusion

The main result of this paper is theorem 15. concludes that in the purely measurable framework the Hars´ anyi program works, i.e., the incomplete information situations can be modeled by type spaces. In this sense, this paper ends the sequence of papers focusing on the Hars´anyi program: e.g. Heifetz and Samet [18], and Meier [20]. Theorem 15. with Pint´er’s [28] result raise the problem that although in the literature mostly the topological models are popular, the purely measurable and not the topological framework is appropriate for modeling incomplete information situations. Can every result in the topological framework be translated into the purely measurable one? For this question future research can answer.

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