Topologies on Types

8/14/2019 Topologies on Types

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1 Introduction

The universal type space proposed by Harsanyi (1967/68) and constructed by Mertens

and Zamir (1985) shows how (under some technical assumptions) any incomplete informationabout a strategic situation can be embedded in a single "universal" type space. As a practicalmatter, applied researchers do not work with that type space but with smaller subsets of the universal type space. Mertens and Zamir showed that nite types are dense under theproduct topology, but under this topology the rationalizable actions of a given type maybe very dierent from the rationalizable actions of a sequence of types that approximateit. 1 This leads to the question of whether and how one can use smaller type spaces toapproximate the predictions that would be obtained from the universal type space.

To address this question, we dene and analyze "strategic topologies" on types, underwhich two types are close if their strategic behavior is similar in all strategic situations.There are three ingredients that need to be formalized in this approach: how we vary the"strategic situations," what is meant by "strategic behavior" (i.e., what solution concept),and what is meant by "similar."

To dene "strategic situations," we start with a given space of uncertainty, , and a typespace over that space, i.e. all possible beliefs and higher order beliefs about . We then studythe eect of changing the action sets and payo functions while holding the type space xed.

We are thus implicitly assuming that any "payo relevant state" can be associated with anypayos and actions. This is analogous to Savages assumption that all acts are possible,and thus implicitly that any "outcome" is consistent with any payo-relevant state. Thisseparation between the type space and the strategic situation is standard in the mechanism-design literature, and it seems necessary for any sort of comparative statics analysis, butit is at odds with the interpretation of the universal type space as describing all possibleuncertainty, including uncertainty about the payo functions and actions. According to thislatter view one cannot identify "higher order beliefs" independent of payos in the game. 2

In contrast, our denition of a strategic topology relies crucially on making this distinction.Our notion of "strategic behavior" is the set of interim-correlated-rationalizable actions

1This is closely related to the dierence between common knowledge and mutual knowledge of order n

that is emphasized by Geanakoplos and Polemarchakis (1982) and Rubinstein (1989).2See the discussion in Mertens, Sorin and Zamir (1994, Remark 4.20b).

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that we analyzed in Dekel, Fudenberg and Morris (2005). This set of actions is obtained bythe iterative deletion of all actions that are not best responses given a types beliefs overothers types and Nature and any (perhaps correlated) conjectures about which actions are

played at a given type prole and payo-relevant state. Under interim correlated rationaliz-ability, a players conjectures allow for arbitrary correlation between other players actions,and between other players actions and the payo state; in the complete information case,this denition reduces to the standard denition of correlated rationalizability (e.g., as inBrandenburger and Dekel (1987)). A key advantage of this solution concept for our purposesis that all type spaces that have the same hierarchies of beliefs have the same set of interim-correlated-rationalizable outcomes, so it is a solution concept which can be characterized byworking with the universal type space.

It remains to explain our notion of "similar" behavior. Our goal is to nd a topologyon types that is ne enough that the set of interim-correlated-rationalizable actions hasthe continuity properties that the best-response correspondences, rationalizable actions, andNash equilibria all have in complete-information games, while still being coarse enough tobe useful.3 A review of those properties helps clarify our work. Fix a family of complete-information games with payo functions that depend continuously on a parameter , where

lies in a metric space . Because best responses include the case of exact indierence, theset of best responses for player i to a xed opponents strategy prolea j , denoted BR i (a j ; ),

is upper hemicontinuous but not lower hemicontinuous in , i.e. it may be that n ! ; anda i 2 BR i (a j ; ); but there is no sequence ani 2 BR i (a j ;

n ) that converges to a i . However, theset of "-best responses, 4 BR i (a j ; " ; ); is well behaved: if n ! ; and a i 2 BR i(a j ; " ; ); thenfor any ani ! a i there is a sequence "n ! 0 such that ani 2 BR i (a j ; " + "n ;

n ). In particular,the smallest " for which a i 2 BR i (a j ; " ; ) is a continuous function of . Moreover the sameis true for the set of all "-Nash equilibria (Fudenberg and Levine (1986)) and for the set of "-rationalizable actions. That is, the " that measures the departure from best response orequilibrium is continuous. The strategic topology is the coarsest metric topology with this

3Topology P is ner than topology P 0 if every open set in P 0 is an open set in P . The use of a veryne topology such as the discrete topology makes continuity trivial, but it also makes it impossible toapproximate one type with another; hence our search for a relatively coarse topology. We will see that ourtopology is the coarsest mertrizable topology with the desired continuity property; this leaves open whetherother, non-metrizable, topologies have this property.

4An action is an " -best response if it gives a payo within " of the best response.

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continuity property.Thus for a xed game and action, we identify for each type of a player, the smallest " for

which the action is " interim correlated rationalizable. The distance between a pair of types

(for a xed game and action) is the dierence between those smallest ". In our strategictopology a sequence converges if and only if this distance tends to zero pointwise for anyaction and game. Thus a sequence of types converges in the strategic topology if, for anygame and action, the smallest " does not jump either up or down in the limit, so that themap from types to " is both upper semicontinuous and lower semicontinuous. We show thata sequence has the upper semi-continuity property if and only if it converges in the producttopology (theorem 2), that a sequence has the lower semi-continuity property if and only if it converges in the strategic topology, and that if a sequence has the lower semi-continuity

property then it converges in the product topology (theorem 1). A version of the electronicmail game shows that the converse is false: the lower semi-continuity property and thusconvergence in the strategic topology are strictly stronger requirements than convergence inthe product topology. An illustration of the extra strength of the strategic topology is thefact that nite common prior types are dense in the product topology (Lipman (2003)) butare not dense in the strategic topology (as we show in section 7.3).

Our main result is that nite types are dense in the strategic topology (theorem 3). Thusnite type spaces do approximate the universal type space, so that the strategic behavior

dened as the"-correlated-interim-rationalizable actionsof any type can be approximatedby a nite type. However, this does not imply that the set of nite types is large. In fact,while nite types are dense in the strategic topology (and the product topology), they aresmall in the sense of being category I in the product topology and the strategic topology.

Our paper follows Monderer and Samet (1996) and Kajii and Morris (1997) in seekingto characterize "strategic topologies" in incomplete-information games. These earlier pa-pers dened topologies on priors or partitions in common-prior information systems with acountable number of types, and used equilibrium as the solution concept. 5 We do not have

a characterization of our strategic topology in terms of beliefs, so we are unable to pin downthe relation to these earlier papers.

5For Monderer and Samet (1986), an information system was a collection of partitions on a xed statespace with a given prior. For Kajii and Morris (1997), an information system was a prior on a xed typespace.

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= 1 t21 t22 t23 t24 t21t11 0 0 0 0 0

t12 (1 ) (1 ) (1 ) (1 )2 0 0 0

t13 0 (1 ) (1 )3 (1 ) (1 )4 0 0

t14 0 0 (1 ) (1 )5 (1 ) (1 )6 0...

......

...... . . .

...

t11 0 0 0 0

where , 2 (0; 1). There is a sense in which the sequence (t1k )1k=1 converges to t11 . Observethat type t12 of player 1 knows that = 1 (but does not know if player 2 knows it). Typet13 of player 1 knows that = 1 , knows that player 2 knows it (and knows that 1 knows it),but does not know if 2 knows that 1 knows that 2 knows it. For k 3, each type t

1kknows

that = 1 , knows that player 2 knows that 1 knows... ( k 2 times) that = 1 . But fortype t11 , there is common knowledge that = 1 . Thus type t1k agrees with type t11 up to2k 3 levels of beliefs. We will later dene more generally the idea of product convergence of types, i.e., the requirement that kth level beliefs converge for every k. In this example,(t1k )1k=1 converges to t11 in the product topology.

We are interested in the "-interim-correlated-rationalizable actions in this game. We willprovide a formal denition shortly, but the idea is that we will iteratively delete an actionfor a type at round k if that action is not an "-best response for any conjecture over theaction-type pairs of the opponent that survived to round k 1.

Clearly, both N and I are 0 interim correlated rationalizable for types t11 and t21 of players 1 and 2, respectively. But action N is the unique "-interim-correlated-rationalizableaction for all types of each player i except t i1 , for every " < 1+2 (note that

1+2 >

12 ). To

see this x any" < 1+2 . Clearly, I is not "-rationalizable for type t11 , since the expectedpayo from action N is 0 independent of player 2s action, whereas the payo from action I is 2. Now suppose we can establish that I is not "-rationalizable for types t11 through t1k .Type t2k s expected payo from action I is at most

12

(1) +1

2( 2) =

1 +2

>>>>>>>>=>>>>>>>>>;

:

The correspondence of " best replies in game G for all types given a subset of actions for all types is denoted BR (G; ") : 2A i T i

i2 I ! 2A i T i

i2 I and dened by BR (G; ") (E ) =

(BR i (t i ; E i ;G;")) t i 2 T i i2 I ; where E = (E t i )t i 2 T i i2 I 2 2A i T i

i2 I .11

Remark 3 In cases where E j is not measurable, we interpret [f (t j ; ; a j ) : a j 2 E j g] = 1 as saying that there is a measurable subset E 0 E j such that v [ E 0] = 1. Because A iis nite, and utility depends only on actions and conjectures, the set of " best responses in G

11 We abuse notation and write BR both for the correspondence specifying best replies for a type and forthe correspondence specifying these actions for all types.

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given some E i , BR i (t i ; E i ;G;"), is non-empty provided there exists at least one measurable

i that satises (i). Such i exist whenever E i is non-empty and measurable, and more generally whenever E i admits a measurable selection.

The solution concept and closely related notions with which we work in this paper aregiven below.

Denition 6 For a given game G = ( Ai ; gi )i2 I and ", we use the following notation:

1. The interim-correlated-rationalizable set, R(G; ") = (R i (t i ;G;")) t i 2 T i i2 I 2 2A i T i

i2 I ,

is the largest (in the sense of set inclusion) xed point of BR.

2. The kth -order interim-correlated-rationalizable sets, k = 0 ; 1; 2;:::1 ; are dened as

follows:

R0 (G; ") (R i; 0 (G; ")) i2 I (R i; 0 (t i ;G;")) t i 2 T i i 2 I = (( Ai )t i 2 T i )i2 I

Rk (G; ") (R i;k (G; ")) i 2 I (R i;k (t i ;G;")) t i 2 T i i 2 I BR (G; ") (Rk 1) ;

R1 (G; ") (R i; 1 (G; ")) i 2 I (R i; 1 (t i ;G;")) t i 2 T i i 2 I \ 1k=1 Rk (G; ").

Dekel, Fudenberg and Morris (2005) establish that the sets are well-dened, and show thefollowing relationships among them for the case " = 0; the extensions to general non-negative" are immediate.

Result 1 1. R (G; ") equals R1 (G; ").

2. R i;k (G; ") and R i; 1 (G; ") are measurable functions from T i ! 2A i =; , and for each action a i and each k the sets f t i : a i 2 R i;k (t i ;G;")g and f t i : a i 2 R i; 1 (t i ;G;")g are closed.

To lighten the paper, we will frequently drop the "interim correlated" modier, and

simply speak of rationalizable sets and rationalizability whenever no confusion will result.In dening our strategic topology, we will exploit the following closure properties of R as

a function of ".

Lemma 1 For each k = 0 ; 1;:::, if "n # " and a i 2 R i;k (t i ;G;"n ) for all n, then a i 2R i;k (t i ;G;").

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Proof. We will prove this by induction. It is vacuously true for k = 0Suppose that it holds true up to k 1. Let

i;k (t i ; ) = 8>>>>>>>>>:2 ( A j ) :

(a j ; ) =

R T

j

[dt j ; ; a j ]

for some 2 T j A j such that [f (t j ; ; a j ) : a j 2 R j;k 1 (t j ;G; )g] = 1and marg T = i (t i )

9>>>>>=>>>>>;.

The sequence i;k (t i ; "n ) is decreasing in n (under set inclusion) and converges (by -additivity) to i;k (t i ; "); moreover, in Dekel, Fudenberg and Morris (2005) we show thateach i;k (t i ; "n ) is compact. Let

i;k (t i ; a i ; ) =

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5 The Strategic Topology

5.1 Basic Denitions

The most commonly used topology on the universal type space is the "product topology"on the hierarchy.

Denition 7 tni ! t i if, for each k, k (tni ) ! k (t i ) as n ! 1 .

Here, the convergence of beliefs at a xed level in the hierarchy, represented by ! , is withrespect to the topology of weak convergence of measures. 12

However, we would like to use a topology that is ne enough that the "-best-response cor-respondence and "-rationalizable sets have the continuity properties satised by the "-best-response correspondence , "-Nash equilibrium and "-rationalizability in complete informationgames with respect to the payo functions. The electronic mail game shows that the producttopology is too coarse for these continuity properties to obtain, which suggests the use of aner topology. One way of phrasing our question is whether there is any topology that isne enough for the desired continuity properties and yet coarse enough that nite types aredense; our main result is that indeed there is: this is true for the "strategic topology" thatwe are about to dene.

Ideally, it would be nice to know that our strategic topology is the coarsest one withthe desired continuity properties, but since non-metrizable topologies are hard to analyze,we have chosen to work with a metric topology. Hence we construct the coarsest metric topology with the desired properties.

For any xed game and feasible action, we dene the distance between a pair of types asthe dierence between the smallest " that would make that action "-rationalizable in thatgame. Thus for any G = ( Ai ; gi )i2 I and a i 2 Ai

h i (t i ja i ; G) = min f " : a i 2 R i (t i ;G;")g

di (t i ; t0i ja i ; G) = jhi (t i ja i ; G) h i (t0i ja i ; G)j

In extending this to a distance over types, we allow for a larger dierence in the hs ingames with more actions. Thus, the metric that we dene is not uniform over the number of

12 It is used not only in the constructions by Merens and Zamir (1985) and Brandenburger and Dekel(1993) but also in more recent work by Lipman (2003) and Weinstein and Yildiz (2003).

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actions in the game. When studying a game with a large or unbounded number of actions,we think there should be a metric on actions and accompanying constraints on the set of admissible payo functions (such as continuity or single-peakedness) that make "nearby"

actions "similar." Requiring uniformity here would require that strategic convergence beuniform over the number of actions in the game, and this seems too strong a requirementgiven that we allow arbitrary (bounded) payo functions.

For any integer m; there is no loss in generality in taking the action spaces to be Am1 =Am2 = f 1; 2; : ;mg. Having xed the action sets, a game is parameterized by the payo function g. So for a xedm, we will write g for the game G = ( f 1; 2; : ;mg ; f 1; 2; : ;mg ; g) ;and Gm for the set of all such games g.

Now consider the following notion of distance between types:

d (t i ; t0i ) = Xm m sup

i2f 1;2g;a i 2 A mi ;g2G mdi (t i ; t0i ja i ; g) , (1)

where 0 < < 1.13

Lemma 2 The distance d is a pseudo-metric :

Proof. First note d is symmetric by denition. To see that d satises the triangle inequality,note that for each action a i and game g;

d (t i ; t00i ja i ; g) = jhi (t i ja i ; g) hi (t00i ja i ; g)j

j hi (t i ja i ; g) hi (t0i ja i ; g)j + jh i (t0i ja i ; g) h i (t00i ja i ; g)j= d (t i ; t0i ja i ; g) + d (t0i ; t00i ja i ; g)

hence

d (t i ; t00i ) = Xm m supa i 2 A mi ; g2G m d (t i ; t00i ja i ; g)Xm m supa i 2 A mi ; g2G m (d (t i ; t0i ja i ; g) + d (t0i ; t00i ja i ; g))

Xm

m supa i 2 A mi ; g2G m

d (t i ; t0i ja i ; g) + supa i 2 A mi ; g2G m

d (t0i ; t00i ja i ; g)

!= Xm m supa i 2 A mi ; g2G m d (t i ; t0i ja i ; g) + Xm m supa i 2 A mi ; g2G m d (t0i ; t00i ja i ; g)= d (t i ; t0i ) + d (t

0i ; t

00i ) .

13 We do not need to take the supremum over i in this dention, as we can instead consider a game withthe indices switched on the action spaces and payo functions :

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Theorem 1 below implies that d (t i ; t0i ) = 0 ) t i = t0i , so that d is in fact a metric.

Denition 8 The strategic topology is the topology generated by d.

To analyze and explain the strategic topology, we characterize its convergent sequencesusing the following two conditions. 14

Denition 9 (Strategic Convergence)

1. ((tni )1n =1 ; t i ) satisfy the upper strategic convergence property (written t

ni ! U t i) if for

every m, a i 2 Ami ; and g 2 Gm , hi (tni ja i ; g) is upper semicontinuous in n.

2. ((tni )

1n =1 ; t i ) satisfy the lower strategic convergence property (written t

ni ! L t i ) if for

every m, a i 2 Ami , and g 2 Gm h i (tni ja i ; g) is lower semicontinuous in n.

Recall that by the denition of upper semicontinuity if tni ! U t i then for each m andg 2 Gm , there exists "n ! 0 such that h i (t i ja i ; g) < h i (tni ja i ; g) + "n for all n and a i 2 Ami .The statement for lower semicontinuity is analogous (switching tni and t i in the implication).Lemma 11 in the appendix states that for each m the sequence "n can be chosen inde-pendently of g, so that upper and lower strategic convergence have the following strongerimplications: for each m,

tni ! U t i ) 9 "n ! 0 s.t. h i (t i ja i ; g) < h i (tni ja i ; g) + "

n

tni ! L t i ) 9 "n ! 0 s.t. h i (tni ja i ; g) < h i (t i ja i ; g) + "

n

for every n, a i 2 Ami , and g 2 Gm .15

14 In metric spaces convergence and continuity can be assessed by looking at sequences (Munkres (1975),p.190). Since we show in section 6.1 that each convergence condition coincides with convergence accordingto a metric topology (the product and strategic topologies respectively) we conclude that the open setsdened directly from the convergence notions below do dene these topologies. We do not know if there arenon-metrizable topologies with the same convergent sequences.

15 If we dened convergence of t ni by upper (and lower) hemicontinuity of the correspondence R i (G; " ) :T i ! 2A i ? , then whether a sequence t ni converged to t i would depend on whether h i (t ni ja i ; g) " impliedh i (t i ja i ; g) " (and conversely). We did not adopt this approach because by analogy with nite-dimensionalcases we do not expect R to be everywhere lower hemi-continuous. Note though that t ni ! U t i does implythat if for " n ! 0, a i 2 R i (t ni ; g ; " n ) for each n , then a i 2 R i (t i ; g; 0). Also, t ni ! L t i implies that if a i 2 R i (t i ; g; 0), then there exists " n ! 0 such that a i 2 R i (t ni ; g ; " n ) for each n .

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We do not require convergence uniformly over all games, as an upper bound on thenumber of actions m is xed before the approximating sequence "n is chosen. Requiringuniformity over all games would considerably strengthen the topology, as briey discussed

in section 7.5.

Lemma 3 d (tni ; t i ) ! 0 if and only if tni ! U t i and tni ! L t i .

Proof. Suppose d (tni ; t i ) ! 0. Fix m and let

"n = m d (tni ; t i ) .

Now for any a i 2 Ami ; g 2 Gm ,

m jh i (tni ja i ; g) h i (t i ja i ; g)j Xm m sup

a 0i 2 Ami ; g

02G md (tni ; t i ja0i ; g0) = d (tni ; t i ) ;

sojh i (tni ja i ; g) h i (t i ja i ; g)j

m d (tni ; t i ) = "n ;

thus tni ! U t i and tni ! L t i .Conversely, suppose that tni ! U t i and tni ! L t i . Then 8m, 9 "n (m) ! 0 and "n (m) !

0 such that for all a i ; G 2 Gm ,

h i (t i ja i ; g) < h i (tn

i ja i ; g) + "n

(m)and hi (tni ja i ; g) < h i (t i ja i ; g) + "

n (m) .

Thus

d (tni ; t i ) = Xm m supa i ;G 2G m d (tni ; t i ja i ; G)Xm m max( "n (m) ; "n (m))

! 0 as n ! 1 .

The strategic topology is thus a metric topology where sequences converge if and only if they satisfy upper and lower strategic convergence ,and so the strategic topology is the coars-est metric topology with the desired continuity properties. We now illustrate the strategictopology with the example discussed earlier.

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5.2 The E-Mail Example Revisited

We can illustrate the denitions in this section with the e-mail example introducedinformally earlier. We show that we have convergence of types in the product topol-ogy, t1k ! t11 , corresponding to the upper semicontinuity noted in section 2, whiled (t1k ; t11 ) 6! 0, corresponding to the failure of lower semicontinuity.

The beliefs for the types in that example are as follows.

1 (t11) [(t2; )] = (1, if (t2; ) = ( t21; 0)0, otherwise1 (t1m ) [(t2; )] = 8>>>:

12 , if (t2; ) = ( t2;m 1; 1)12 , if (t2; ) = ( t2m ; 1)0, otherwise

, for all m = 2 ; 3;::::

1 (t11 ) [(t2; )] = (1, if (t2; ) = ( t21 ; 1)0, otherwise2 (t21) [(t1; )] = 8>>>:

12 , if (t1; ) = ( t11 ; 0)12 , if (t1; ) = ( t12; 1)

0, otherwise

2 (t2m ) [(t1; )] = 8>>>:1

2 , if (t1; ) = ( t1m ; 1)12 , if (t1; ) = ( t1;m +1 ; 1)0, otherwise

for all m = 2 ; 3;::::

2 (t21 ) [(t1; )] = (1, if (t1; ) = ( t11 ; 1)0, otherwiseOne can verify that t1k ! t11 . This is actually easiest to see by looking at the table

presenting the beliefs: t1m assigns probability 1 to = 1 , to 2 assigning probability 1 to= 1 , to 2 assigning probability 1 to 1 assigning probability 1 to = 1 ; and so on up to

iterations of length m 1. As m ! 1 ; the kth level beliefs converge to those of t11 whereit is common knowledge that = 1 .

Note that the only nite types here are t i1 .

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Denote by bG the e-mail game described earlier. For any " >>:

2, if k = 11+2 , if k = 2 ; 3;:::0, if k = 1

while h1 t1k N; bG = 0 for all k.Thus d (t1k ; t11 ) 2 1+2 for all k = 2 ; 3; :: and we do not have d (t1k ; t11 ) ! 0.6 Results

6.1 The relationships among the notions of convergence

We rst demonstrate that both lower strategic convergence and upper strategic conver-

gence imply product convergence.

Theorem 1 Upper strategic convergence implies product convergence. Lower strategic con-vergence implies product convergence.

These results follow from a pair of lemmas. The product topology is generated by themetric

ed(t i ; t0i) =

Xk

k

edk (t i ; t0i )

where 0 < < 1 and edk is a metric on the kth level beliefs that generates the topology of weak

convergence. One such metric is the Prokhorov metric, which is dened as follows. For anymetric space X; let F be the Borel sets, and for A 2 F set A = f x 2 X j inf y2 A jx yj g:Then the Prokhorov distance between measures and 0 is dP ( ; 0) = inf f j (A) 0(A )+ for all A 2 Fg , and ~dk (t i ; t0i ) = dP ( k (t i ); k (t0i )):

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Lemma 4 For all k and c > 0 , there exist " > 0 and m such that if edk (t i ; t0i ) > c , there

exist G 2 Gm and a i such that h i (t0i ja i ; G) + " < h i (t i ja i ; G).

Proof. To prove this we will construct a variant of a "report your beliefs" game, and showthat any two types whose kth order beliefs dier by will lose a non-negligible amount byplaying an action that is rationalizable for the other type.

To dene the nite games we will use for the proof, it is useful to rst think of a very largeinnite action game where the action space is the type space T . Thus the rst componentof player is action is a probability distribution over : a1i 2 ( ) . The second componentof the action is an element of ( ( )) , and so on. The idea of the proof is to start witha proper scoring rule for this innite game (so that each player has a unique rationalizable

action, which is to truthfully report his type), and use it to dene a nite game where therationalizable actions are close to truth telling.

To construct the nite game, we have agents report only the rst k levels of beliefs, andimpose a nite grid on the reports at each level. Specically, for any xed integerz1 let A1

be the set of probability distributions a1on such that for all 2 ; a1( ) = j=z 1 for someinteger j; 1 j z1. Thus A1 = f a 2 R j j : a = j=z for some integer j; 1 j z1;

Pa = 1 g; it is a discretization of the set ( ) with grid points that are evenly spaced inthe Euclidean metric.Let D

1

= A1

. Note that this is a nite set. Next pick an integer z2

and let A2

be theset of probability distributions on D 1 such that a2(d) = j=z 2 for some integer j; 1 j z2.Continuing in this way we can dene a sequence of nite action sets A j , where every elementof each A j is a probability distribution with nite support. The overall action chosen is avector in A1 A2 ::: Ak .

We call the am the " m th -order action." Let the payo function be 16

gi (a1; a2; ) = 2 a1i ( )

X0

a1i (0) 2+

k

Xm =2

264

2ami a1 j ;::;am 1 j ;

Xea 1j ;::;ea m

1

j ;eami

ea1 j ;::;

eam 1 j ;

e2375Note that the objective functions are strictly concave, and that the payo to the m th -order

action depends only on the state and on actions of the other player up to the (m 1)th

16 The payo function given in the text is independent of the payo bound M , and need not satisfy it if M is small - in that case we can simply multiply the payo function by a suciently small positive number.

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metric, if ed2(t i ; t0i ) > c there is a Borel set A in ( ) such that 2(t i )(A) > 2(t0i )(Ac)+ c.

Because the rst-order actions a1 converge uniformly to 1 as z1 goes to innity, ( ; a 1 ( 1)) 2Ac for every ( ; 1) 2 A, so for all such that c=2 > > 0 there is a z2 such that 2(t i )(Ac)

2(t i )(A) > 2(t i )(A) > 2(t0i )(Ac) + c 2(t0i ) (Ac) 2 + c > 2(t0i ) (Ac), where the rstinequality follows from set inclusion, the second and fourth from the uniform convergence of the a1, and the third from ed

2(t i ; t0i ) > c .As with the case of rst-order beliefs and actions , this implies that when ed

2(t i ; t0i ) > cthere is a z2 and "2 > 0 such that

hi (t i ja2(t i ); G) + "2 < h i (t0i ja2(t i ); G)

for all z2 > z2. We can continue in this way to prove the result for any k.

Lemma 5 Suppose that t i is not the limit of the sequence tni in the product topology, then (tni ; t i ) satises neither the lower convergence property nor the upper convergence property.

Proof. Failure of product convergence implies that there exists k such that edk (tni ; t i ) does

not converge to zero, so there exists > 0 such that for all n in some subsequence

edk (tni ; t i ) > .

By lemma 4, there exists " and m such that, for all n,

9a i 2 Ami ; g 2 Gm s.t. h i (t i ja i ; g) + " < h i (tni ja i ; g) and (2)

9a i 2 Ami ; g 2 Gm s.t. h i (tni ja i ; g) + " < h i (t i ja i ; g) . (3)

Now suppose that the lower convergence property holds. Therefore

9 n ! 0 s.t. h i (tni ja i ; g) < h i (t i ja i ; g) +n , a i 2 Ami ; g 2 G

m .

This combined with (2) gives a contradiction.Similarly, upper convergence implies the following.

9 n ! 0 s.t. h i (t i ja i ; g) < h i (tni ja i ; g) +n , a i 2 Ami ; g 2 G

m .

This gives a contradiction when combined with (3).Lemma 5 immediately implies theorem 1.

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Theorem 2 Product convergence implies upper strategic convergence.

Proof. Suppose that tni product-converges to t i . If upper strategic convergence fails there isan m; a

i2 Am

i; g 2 Gm such that for all "n ! 0 and N , there is n0 > N such that

h i (t i ja i ; g) > h i tn0

i ja i ; g + "n 0 .

We may relabel so that tni is the subsequence where this inequality holds. Pick so that

h i (t i ja i ; g) > h i (tni ja i ; g) +

for all n. Since, for each n and tni , a i 2 R i (tni ;G;h i (t i ja i ; g) ), there exists n 2T j A j such that

(i) n

[(t j ; ; a j ) : a j 2 R j (t j ; g ; hi (t i ja i ; g) )] = 1(ii) marg T

n = i (tni )(iii) R ( t j ; ;a j ) [gi (a i ; a j ; ) gi (a0i ; a j ; )] d n h i (t i ja i ; g) + for all a0i 2 Ami

Since under the product topology, T is a compact metric space, and since A j and arenite, so isT A j . Thus ( T A j ) is compact in the weak topology, so thesequence n has a limit point, .

Now since (i), (ii) and (iii) hold for every n and = lim n n , we have

(i*) [f (t j ; ; a j ) : a j 2 R j (t j ; g ; hi (t i ja i ; g) )g] = 1(ii*) marg T = i (t i )(iii*) R (t j ; ;a j ) [gi (a i ; a j ; ) gi (a0i ; a j ; )] d hi (t i ja i ; g) + for all a0i 2 Ami .

Here (i*) follows from the fact that f (t j ; ; a j ) : a j 2 R j (t j ; g ; hi (t i ja i ; g) )g is closed. Tosee why (ii*) holds, note that marg T n ! marg T , since n ! . It remains toshow that i (tni ) ! i (t i), i.e., t i = 1i (lim i (tni )), whenever tni ! t i . This can beinferred from the Mertens-Zamir homeomorphism and standard results about the continuityof marginal distributions in the joint, but a direct proof is about as short: Recall that k (t)isthe kth level belief of type t, that marg X k 1 (t) = k (t), and that (by denition) t i =

1i (lim i (tni )) i for all k, k (t i) = k

1i (lim i (tni )) . Now k

1i (lim i (tni )) =

marg X k 1 lim i (tni ) = lim marg X k 1 i (tni ) = lim k (tni ) = k (t i ) by product convergence.This proves (ii*); (iii*) follows from n ! .

This implies a i 2 R i (t i ; g ; hi (t i ja i ; g) ), a contradiction.

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The recent paper of Ely and Peski (2005) similarly identies types with sets of rationalizableactions, although for their dierent purpose (constructing a universal type space for theinterim-independent-rationalizability solution concept), no approximation is required.

Lemma 6 For any t i , there exists a sequence of nite types ~tni such that ~tni1n =1 ; t i satisfy

lower strategic convergence.

The two critical stages in the proof are as follows. We rst prove that there is a nite setof m-action games, G

m, that approximate the set Gm .

Lemma 7 For any integer m and " > 0, there exists a nite collection of m action games Gm such that, for every g 2 Gm , there exists g0 2 Gm such that for all i, t i and a i 2 Ami ,

jh i (t i ja i ; g) h i (t i ja i ; g0)j ".

Next we use this to prove that there is a nite approximating type space.

Lemma 8 Fix the number of actions m and > 0. There exists a nite type space ~T i ; ~i

i=1 ;2and functions (f i )i=1 ;2, each f i : T ! ~T i , such that hi (f i (t i ) ja i ; g) h i (t i ja i ; g)+

for all t i , g 2 Gm and a i .

The key step in this proof is constructing the type space, so we present that here. Theremaining details are provided in the appendix.

Fix a nite set of games Gm . Write hxi for the smallest number in the set f 0; ; 2 ; ggreater than x; and let m be the set of all maps from Ami G

minto n0; ;::::; h2M i o. We

build the type spaces ~T i ; ~ii=1 ;2

using subsets of m as the types. Specically, dene the

function f i : T i ! m by f i (t i ) = hh i (t i ja i ; g)ia i ;g

.

Let eT i be the range of f i ; note that eT i is a nite set. Thus for given each type of i in the universal type space is mapped into a function that species for each one of thenitely many games and actions of i the smallest multiple j of under which that action is

j -rationalizable. These functions constitute the types in a nite type space. The beliefs inthis nite type space are dened next.

Dene~i : eT i ! eT j as follows. For each ~t i 2 eT i , x anyt i 2 T such thatf i (t i ) = ~t i . Label this type i ~t i and let~i ~t i ~t j ; = i i ~t i (t j ; ) : f j (t j ) = ~t j .

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We discuss relaxing the uniform bound on payos that we have used throughout thepaper.

Finally, we emphasize that caution is needed in working with nite types despite ourresult.

7.2 Is the set of nite types "generic"?

Our denseness result does not imply that the set of nite types is "generic" in the universaltype space. While it is not obvious why this question is important from a strategic point of view, we nonetheless briey report some results showing that the set of nite types is notgeneric in either of two standard topological senses.

First, a set is sometimes said to be generic if it is open and dense. But the set of nitetypes is not open. To show this, it is enough to show that the set of innite types is dense.This implies that the set of innite types is not closed and so the set of nite types is notopen.

Let T n be the collection of all types that exist on nite belief closed subsets of theuniversal type space where each player has at most n types. The set of nite types is thecountable union T F = [ n T n . The set of innite types is the complement of T F in T .

Theorem 4 If # 2, innite types are dense under the product topology and the strategic topology.

Thus the "open and dense" genericity criterion does not discriminate between nite andinnite types. A more demanding topological genericity criterion is that of "rst category".A set is rst category if it is the countable union of closed sets with empty interiors. In-tuitively, a rst category set is small or "non-generic". For example, the set of rationals isdense in the interval [0; 1] but not open and not rst category.

Theorem 5 If # 2, the set of nite types is rst category in T under the product topology and under the strategic topology.

Proof. Theorem 4 already established that the closure of the set of innite types is the wholeuniversal type space. This implies that each T n has empty interior (in the product topologyand in the strategic topology). Since the set of nite types is the countable union of the T n ,

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it is then enough to establish that each T n is closed, in the product topology and thus in thestrategic topology.

By Mertens and Zamir (1985), ! corresponds to the weak topology on the compact set

T . We will repeatedly use the following implications of weak convergence. If a sequence of measures k on a metric space X converges weakly to ; and the support of every k has nor fewer elements, then (a) the support of has at most n elements; (b) every element x of the support of is the limit of a sequence of elements xk 2 support( k ); moreover, (c) thereexists an integer K and, for each k > K and x 2 support ( ), k (x) 2 support k , suchthat, (i) for all x 2 support ( ), k (x) converges to x and (ii) x; x0 2 support ( ) and x 6= x0

implies k (x) 6= k (x0) for all k > K .Now recall from remark 2 that, for k even, we write M (k; t1) for the set of types of player

1 reached in k or less steps from t1 of player 1; and, for k odd, we write M (k; t1) for the setof types of player 2 reached in k or less steps from t1. This implies that for even k,

M (k; t1) = f t1g [ [ t 2 M (k 1;t 1 )r (t1)

and for odd k,M (k; t1) = [ t 2 M (k 1;t 1 )r (t1) .

Now x ann, and suppose tk1 ! t1 and tk1 2 T n for all k. So tk1 2 T k1 , where T k1 T k2 isa belief-closed type space with # T ki n. We will establish inductively the following claim.

Claim: Fix any integer L. (1) M L; t 1 has at most n elements. (2) There exists K L suchthat for every k > K L and every t 2 M (L; t 1), there exists k (t) 2 T ki such that k (t) ! tand k (t) 6= k (t0) for all t; t 0 2 M (L; t 1) with t 6= t0 (where i = 1 if L is even and i = 2 if L is odd).

We rst establish the claim for L = 1 . Since tk is a sequence of measures convergingto (t), we have that (a) M 1; t1 has at most n elements; (b) there is a sequence k2 (t2) 2T k2 s.t. k2 (t2) ! t2; and (c) there exists K 1 such that for all k > K 1, k2 (t2) 6= k2 (t02) if t2 6= t02.

Now suppose that the claim holds for all L L 1, where L is even. We establish theclaim for L. For k > K L 1, we know that

ntk1o[ [ t 2 2 M (L 1;t 1 )r k2 (t2) T k1 n.Now ntk1o[ [ t 2 2 M (L 1;t 1 )r k2 (t2) n implies t1 [ [ t 2 2 M (L 1;t 1 )r (t2) n since k2 (t2) ! t2 and supports cannot grow. Thus M L; t 1 has at most n elements. Also observe

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that for each t1 2 M L; t 1 , there is a sequence k1 (t1) 2 T k1 s.t. k1 (t1) ! t1. This is true byassumption if t1 = t1; otherwise t1 is the limit of a sequence of types in r tk2 for some tk2 2 T k2 .Since M L; t 1 is nite, there exists K L such that for all k > K L and all t1 2 M L; t 1 ,

there is a sequence of k1 (t1) 2 T k1 converging to t1 with with k1 (t1) 6= k1 (t01) if t1 6= t01.Now suppose that the claim holds for all L L 1, where L is odd. We establish the

claim for L. For k > K L 1, we know that

[ t 1 2 M (L 1;t 1 )r k1 (t1) T

k2 n.

Now [ t 1 2 M (L 1;t 1 )r k1 (t1) n implies [ t 1 2 M (L 1;t 1 )r (t1) n since

k1 (t1) ! t1

and supports cannot grow. Thus M L; t 1 has at most n elements. Also observe that foreach t1 2 M L; t 1 , there is a sequence k

2(t2) 2 T k

2s.t. k

2(t2) ! t2. This is true as each t2

is the limit of a sequence of types in r tk1 for some tk1 2 T k1 . Since M L; t 1 is nite, thereexists K L such that for all k > K L and all t2 2 M L; t 1 , there is a sequence of k2 (t2) 2 T k2converging to t2 with with k2 (t2) 6= k2 (t02) if t2 6= t02.

We have now established the claim for all L by induction, and thus that type t1 is anelement of T n .

Thus the set of nite types is not generic under two standard topological notions of genericity.

Heifetz and Neeman (2004) use the non-topological notion of "prevalence" to discussgenericity on the universal type space. Their approach builds in a restriction to commonprior types, and it is not clear how to extend their approach to non common prior types.They also show that the generic set has the property that any convex combination of anelement in the set with an element of its complement will be in the set. This property issatised here as well: convex combinations of nite types with innite types are innite.

7.3 Types with a Common Prior

We say that t is a nite common-prior type if it belongs to a nite belief-closed subspace(T i ; i )

2i=1 , and there is a probability distribution on T (the common prior), that assigns

positive probability to every type of every player, such that i (t i ) = ( jt i ) : The set of nite common-prior types is not dense in the set of nite types, and thus is not dense in theset of all types. Intuitively, this is because the strategic implications of a common prior (such

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as certain no-trade theorems) do not extend to general types. Since Lipman (2003) showsthat the set of nite common prior types are dense in the product topology, this observationdemonstrates a general distinction between the product topology and the strategic topology.

To demonstrate this observation, it is enough to give a single game that separates commonprior types from non common-prior types. Suppose there are two states, L and R. Eachplayer has two actions, Y (trade) and N (no trade). If a player says N she gets zero, if theyboth say Y they get (1; 2) and ( 2; 1) in states L and R respectively (a negative-sum trade)and in case one says Y and the other says N the one who said Y gets 2 (independent of thestate). Clearly both players saying Y is 0-rationalizable for some non-common prior type:if a player believes there is common certainty that each player believes the state is the onefavorable to him, they may both say Y . But with common priors the only "-rationalizable

action, for " small enough, is N : Let ~T be the set of common-prior type pairs for whom Y is1=4-rationalizable; for each t1 2 ~T 1, the probability of state L and ~T has to be at least 7=12(since 712 2

512 =

14 ). However, a similar property, replacing R for L, must also hold for

all t2 2 ~T 2, so there cannot be a common prior on ~T f L; R g with such conditionals.

7.4 Alternative Solution Concepts

We used the solution concept of interim correlated rationalizability to dene the strategic

topology. We noted two reasons for doing this: the set of interim correlated rationalizable ac-tions depend only on hierarchies of beliefs and the solution concept captures the implicationsof common knowledge of rationality.

One might wonder what would happen with alternative solution concepts, such as Nashequilibrium or interim independent rationalizability. But the set of Nash equilibrium ac-tions or interim-independent-rationalizable actions depend in general not just on the belief hierarchies but also on "redundant" types: those that dier in their ability to correlatetheir behavior with others actions and states of the world. In dening a topology for these

solution concepts, one would have to decide what to do about this dependence.Suppose one wanted to dene a topology on hierarchies of beliefs (despite the fact thathierarchies of beliefs do not determine these other solution concepts). One approach wouldbe to examine all actions that could be played by any type with a given hierarchy of beliefsunder that solution concept (allowing for all possible type spaces and not only the universal

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type space as we do here). Dekel, Fudenberg and Morris (2005) show that, given any gameand xed hierarchy of beliefs, the unionover all type spaces that contain a type with thathierarchy of beliefsof the equilibrium actions equals the interim-correlated-rationalizable

actions for that hierarchy of beliefs. The same arguments can be used to prove the sameconclusion for interim-independent-rationalizability.

Another approach would be to x a solution concept and construct a larger representationof types, that included hierarchies of beliefs but also incorporated the redundant types thatare relevant for the solution concept. One would then construct a topology on this largerspace. The rst part of this approachconstructing the larger type space that incorporatedthe redundant types relevant for the solution conceptwas carried out by Ely and Peski(2005) for the solution concept of interim-independent-rationalizability for two-person games.

7.5 A Strategic Topology that is Uniform on Games

The upper and lower convergence conditions we took as our starting point are not uniformover games. As we said earlier, when the number of actions is unbounded, there is usuallya metric on actions and accompanying constraints on the set of admissible payo functions.Uniformity over all games without such restrictions seems too demanding and hence of lessinterest. Despite this, it seems useful to understand how our results would change if we did

ask for uniformity over games.Let Ai(G) denote the set of actions for player i in a given game G. A distance on typesthat is uniform in games is:

d (t i ; t0i ) = supa i 2 A i (G );G

d (t i ; t0i ja i ; G) .

This metric yields a topology that is ner than that induced by the metric d, so the topologyis ner than necessary for the upper and lower convergence properties that we took as ourgoal. Finite types are not dense with this topology. To show this we use the fact that

convergence in this topology is equivalent to convergence in the following uniform topologyon beliefs:

d (t i ; t0i ) = supk

supf 2 F k

jE (f j (t i )) E (f j (t0i ))j ,

where F k is the collection of bounded functions mapping T that are measurable withrespect to kth level beliefs.

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Proposition 2 The metrics d and d are equivalent.

An argument of Morris (2002) implies that nite types are not dense in the uniform topology

on beliefs.18

Together with the preceding proposition this implies that nite types are notdense in the uniform strategic topology generated by d .

7.6 Bounded versus Unbounded Payos

We have studied topologies on the class of games with uniformly bounded payos. If arbitrary payo functions are allowed, we can always nd a game in which any two typeswill play very dierently, so the only topology that makes strategic behavior continuous is thediscrete topology. From this perspective, it is interesting to note that full surplus extractionresults in mechanism design theory (Cremer and McLean (1985), McAfee and Reny (1992))rely on payos being unbounded. Thus it is not clear to us how the results in this paper canbe used to contribute to a debate on the genericity of full surplus extraction results. 19

7.7 Interpreting the denseness result

That any type can be approximated with a nite type provides only limited supportfor the use of simple nite type spaces in applications. The nite types that approximate

arbitrary types in the universal type space are quite complex. The approximation resultshows that nite types could conceivably capture the richness of the universal type space,and does not of course establish that the use of any particular simple type space is withoutloss of generality.

18 For a xed random variable on payo states, we can identify the higher-order expectations of a type,i.e., the expectation of the random variable, the expectation of the other players expectation of the randomvariable, and so on. Convergence in the metric d implies that there will be uniform convergence thesehigher order expectations. Morris (2002) showed that nite types are not dense in a topology dened interms of uniform convergence of higher order expectations. Thus nite types will not be dense under d .

19 Bergemann and Morris (2001) showed that both the set of full surplus extraction types and the set of non full surplus extraction types are dense in the product topology among nite common prior types, andthe same argument would establish that they are dense in the strategic topology identied in this paper.But of course it is trivial that neither set is dense in the discrete topology, which is the "right" topology forthe mechanism design problem.

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In particular, applying notions of genericity to the belief-closed subspace of nite typesmust be done with care. Standard notions of genericity for such nite spaces will not ingeneral correspond to strategic convergence. Therefore, results regarding strategic interac-

tions that hold on such "generic" subsets of the nite spaces need not be close to the resultsthat would obtain with arbitrary type spaces. For example, our results complement thoseof Neeman (2004) and Heifetz and Neeman (2004) on the drawbacks of analyzing genericitywith respect to collections of (in their case, priors over) types where beliefs about deter-mine the entire hierarchy of beliefs, as is done, for instance, in Cremer and McLean (1985),McAfee and Reny (1992), and Jehiel and Moldovanu (2001).

8 AppendixFor some proofs we use an alternative characterization of the interim-correlated-rationalizable

actions.

Denition 10 Fix a game G = ( Ai ; gi ) and a belief-closed subspace (T i ; i )i2 I . Given S =(S 1; S 2), where each S i : T i ! 2A i ? , we say that S is an " best-response set if S i (t i )BR i (t i ; S j ;G;").

In Dekel, Fudenberg and Morris (2005) we prove that if S is an 0 best-response set thenS i (t i ) R (t i ; G; 0) ; the extension to positive " is immediate.

LetD (g; g0) = sup

i;a;jgi (a; ) g0i (a; )j .

Lemma 9 : For any integer m and " > 0, there exists a nite collection of m action games G

msuch that, for every g 2 Gm , there exists g0 2 G

msuch that D (g; g0) ".

Proof. Assume without loss of generality that M is an integer. For any integer N , we write

GN = (g : f 1;:::;mg2 ! M; M + 1N ; M + 2N ;:::;M 1N ; M 2).For any game g, choose g0 2 GN to minimize D (g; g0). Clearly D (g; g0) 12N .

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Lemma 10 For all i, t i ,m, a i 2 Ami , and g, g0 2 Gm

hi (t i ja i ; g) h i (t i ja i ; g0) + 2 D (g; g0)

Proof. By the denition of R, we know that R (g0; ) is a -best response set for g0. ThusR (g0; ) is a ( + 2 D (g; g0))-best response set for g. So R i (t i ; g0; ) R i (t i ; g ; + 2 D (g; g0)).Now if a i 2 R i (t i ; g0; ), then a i 2 R i (t i ;g ; + 2 D (g; g0)). So h i (t i ja i ; g0) implies + 2 D (g; g0) h i (t i ja i ; g). So hi (t i ja i ; g0) + 2 D (g; g0) h i (t i ja i ; g).

Lemma 11

1. If for each m and each g 2 Gm , there exists "n ! 0 such that hi (t i ja i ; g) < h i (tni ja i ; g)+"n for every n and a i 2 Ami ; then for each m there exists "n ! 0 such that h i (t i ja i ; g) 0, there exists a nite collection of m action games Gm such that, for every g 2 Gm , there exists g0 2 Gm such that for all i, t i , and a i 2 Ami

jh i (t i ja i ; g) h i (t i ja i ; g0)j ".

Proof. By lemma 9, we can choose nite collection of gamesGm

such that, for every g 2 Gm ,there exists g0 2 G

msuch that D (g; g0) "

2. Lemma 10 now implies that we also have

h i (t i ja i ; g) hi (t i ja i ; g0) + "

andh i (t i ja i ; g0) h i (t i ja i ; g) + ".

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Lemma 12 Fix any nite collection of m action games Gm

and > 0. There exists a nite type space ~T i ; ~i

i=1 ;2and functions (f i )i=1 ;2, each f i : T ! ~T i , such that R i (t i ; g ; ")

R i (f i (t i ) ; g ; ") for all t i 2 T and " 2 f 0; ; 2 ; :::: g.

Proof. Let ~T i ; ~i and (f i )i=1 ;2 be as constructed in subsection 6.2. Fix " 2 n0; ;::::; h2M i o.Let S i ~t i = R i i ~t i ; g ; " .We argue that S is an "-best response set on the type space ~T i ; ~i

i=1 ;2. To see why,

suppose thata i 2 R i (t i ; g ; ")

and let

et i = f i (t i ). Because " 2 n0; ;::::; h2M i o, we have R i (t i ; g ; ") = R i i ~t i ; g ; "and thus

a i 2 R i i ~t i ; g ; " .

This implies that there exists 2 ( T A j ) such that

[f (t j ; ; a j ) : a j 2 R j (t j ; g ; ")g] = 1

marg T = i i ~t i

Z (T j A j )

[gi (a i ; a j ; ) gi (a0i ; a j ; )] d " for all a0i 2 Ai

Now deneev 2 ~T j A j by

ev~t j ; ; a j = (t j ; ; a j ) : f j (t j ) = ~t j

By construction,

ev~t j ; ; a j : a j 2 S j ~t j = 1

marg ~T j

ev =

ei (

et i )

X~T j A j

[gi (a i ; a j ; ) gi (a0i ; a j ; )]

ev ~t j ; ; a j " for all a0i 2 Ai

So a i 2 BR i (S ) et i .Since S is an "-best response set on the type space ~T i ; ~i i=1 ;2, S i ~t i R i ~t i ; g ; " .Thus a i 2 R i (t i ;g ; ") ) a i 2 R i (f i (t i ) ; g ; ").

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Lemma 13 Fix any nite collection of m action games Gm and > 0. There exists a nite type space ~T i ; ~i

i=1 ;2and functions (f i)i=1 ;2, each f i : T ! T i , such that hi (f i (t i ) ja i ; g)

h i (t i ja i ; g) + for all t i , g 2 Gm

and a i 2 Ami .

Proof. We will use the type space from lemma 12, which had the property that

R i (t i ; g ; ") R i (f i (t i ) ; g ; ") (4)

for all " 2 f 0; ; 2 ; :::: g. By denition,

a i 2 R i (t i ; g ; hi (t i ja i ; g)) .

By monotonicity,a i 2 R i t i ; g; hhi (t i ja i ; g)i .

By (4),R i t i ; g; hh i (t i ja i ; g)i R i f i (t i ) ; g; hh i (t i ja i ; g)i

Thush i (f i (t i) ja i ; g) h hi (t i ja i ; g)i h i (t i ja i ; g) + .

Lemma 8: Fix the number of actions m and > 0. There exists a nite type space (T i ; bi )i=1 ;2 and functions (f i )i=1 ;2, each f i : T !

~T i , such that hi (f i (t i ) ja i ; g) h i (t i ja i ; g)+ for all t i , g 2 Gm and a i 2 Ami .

Proof of lemma 8 . Fix m and > 0. By lemma 7, there exists a nite collection of m-actiongames G

msuch that for every nite-action game g, there exists g0 2 G

msuch that

hi (t i ja i ; g) h i (t i ja i ; g0) + 3

(5)

and h i (t i ja i ; g0) h i (t i ja i ; g) +

3

(6)

for all i, t i and a i . By lemma 13, there exists a nite type space ~T i ; ~ii=1 ;2

and functions

(f i )i=1 ;2, each f i : T ! ~T i , such that

hi (f i (t i ) ja i ; g) h i (t i ja i ; g) + 3

(7)

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for all t i , g 2 Gm

and a i 2 Ami .Now x anyi, t i , a i and g. By (5), there exists g0 such that

h i (t i ja i ; g0

) hi (t i ja i ; g) 3

andh i (f i (t i ) ja i ; g) h i (f i (t i) ja i ; g0)

3

.

By (7),

hi (f i (t i ) ja i ; g0) h i (t i ja i ; g0) 3

.

So

hi (f i (t i ) ja i ; g) h i (t i ja i ; g) 8>>>:(hi (f i (t i ) ja i ; g) h i (f i (t i ) ja i ; g0))+ ( h i (f i (t i ) ja i ; g0) hi (t i ja i ; g0))

+ ( h i (t i ja i ; g0) h i (t i ja i ; g)) 9>>=>>; .Theorem 4: If # 2, innite types are dense under the product topology and the strategic topology.Proof. It is enough to argue that for an arbitrary n and t 2 T n , we can construct a sequence

tk

which converges to t in the strategic topology (and thus the product topology) such thateach tk =2 T F . Let T 1 = T 2 = f 1;:::;ng and i : f 1;:::;ng ! ( f 1;:::;ng ) (as beforethese are to be viewed as a belief-closed subspace of the universal type space.) Without lossof generality, we can identify t 2 T n with type 1 of player 1.

The strategy of proof is simply to allow player i to have an additional signal about T j(which will require an innite number of types for each player) but let the informativenessof those signals go to zero. Thus we will have a sequence of types not in T F but convergingto t in the strategic topology (and thus the product topology).

We will now dene a sequence of type spaces T ki ;

ki i2 I for k = 1 ; 2;:::;1 . Let

us suppose each player i observes an additional signal zi 2 f 1; 2;:::g, and dene T ki =T i f 1; 2;:::g, with typical element (n i ; zi ). ) Fix 2 (0; 1) and for each k = 1 ; 2;:::, choose

ki : T ki ! T k j to satisfy the following two properties:

ki ((n j ; z j ) ; j (n i ; zi )) (1 )

z j 1i (n j ; jn i )

1k

(8)

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for all n j ; z j ; ;n i ; zi ; andki ( j (n i ; zi )) 6=

ki (

0j (n0i ; z0i )) (9)

for all ; 0; and all (n i ; zi ) 6= ( n0i ; z0i ).For k = 1 , set

1i ((n j ; z j ) ; j (n i ; zi )) = (1 )

z j 1i (n j ; jn i ) ,

where (9) is not satised (but holds instead with equality).We distinguish the dierent copies of T ki with the superscript k because we identify a

type (n i ; zi ) in T ki as potentially distinct from (n i ; zi ) 2 T k0

i when viewed as types in theuniversal type space . Note that in the type space (T 1i ; 1i ), for each n i 2 T i every type

(n i ; zi ) 2 T i f 1; 2;:::g corresponds to the same type in the universal type space, namelythe type in the universal type space that corresponds to type n i in the type space (T i ; i ).On the other hand, from (9) we see that for all other type spaces with k 6= 1 , each distinctpair of types (n i ; zi ) 6= ( n0i ; z0i ) in T ki ; ki correspond to distinct types in the universal typespace. Let tk 2 T be the type in the universal type corresponding to type (1; 1) in the typespace T ki ; ki i=1 ;2.

Now (9) also implies that each tk =2 T F .We will argue that the sequence tk converges to t in the strategic topology. To see why, for

any (n i ; zi ) 2 T ki = T i f 1; 2;:::g, let S i (n i ; zi ) = R i (n i ;G; ) (i.e., the set of -rationalizable

actions of type n i of player i in game G on the original type space). First observe that S is an -best-response set in game G on the type space (T 1i ; 1i )i=1 ;2. This is true because,as noted, the type space (T 1i ; 1i )i=1 ;2 and the original type space (T i ; i)i=1 ;2 correspond tothe same belief-closed subspace of the universal type space. (The only dierence is that in(T 1i ; 1i )i=1 ;2 it is common knowledge that each player i observes a conditionally independentdraw with probabilities (1 ) z i 1 on f 1; 2;:::g.) But now by (8), S is an + 2M k bestresponse set for G. Thus if a i 2 R i (t ;G; ), then a i 2 R i tk ;G; + 2M k . Thus the sequence

tk

; t satises the lower strategic convergence property. By corollary 1, this implies strategicconvergence. By theorem 1, we also have product convergence.Proposition 2: d is equivalent to d .Proof. First, observe that if d (t i ; t0i ) ", then for any measurable f : T j ! [ M; M ],

jE (f j (t i )) E (f j (t0i)) j 2". (10)

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To see this, for any k, we will denef k : T j ! [ M; M ] that is measurable with respectto is kth level beliefs. Let f k (t j ; ) = R n( t j ; ):(^ jl (t j ))k 1l =0 = (^ jl (t j ))k 1l =0 of (t j ; ) d ( (t i)) , thatis, f k is the expected value according to (t i ) of the function f evaluated over all t j withthe same rst k levels. Now f k ! f pointwise so by the bounded convergence theoremE (f k j (t0i )) ! E (f j (t0i )). By the denition (and iterated expectations) E (f k j (t i )) =E (f j (t i )). Since d (t i ; t0i ) " we know that for all k, jE (f k j (t i )) E (f k j (t0i ))j ".

Now suppose that a i 2 R i (t i ;G; ). Then there exists 2 T j A j such that

(i) [f (t j ; ; a j ) : a j 2 R j (t j ;G; )g] = 1(ii) marg T = i (t i )(iii)

R ( t j ; ;a j )

[gi (a i ; a j ; ) gi (a0i ; a j ; )] d for all a0i 2 Ai

Let 0 be a measure whose marginal on T is i (t0i ) and whose probability on A j ,conditional on (t j ; ) is the same as . Since T j A j is a separable standard measure spacethere exist conditional probabilities (see Parthsarathy (1967, Theorem 8.1) jT j 2( A i ), measurable as a function of T j . Dene 0 2 T j A i , by setting,

for measurable F T j ; 0(F f ; a j g) = R F ( (a j jt j ; ) i (t i)) [dt j ; ].20 Let f a i ;a 0i be afunction taking the value R A j (gi (a i ; a j ; ) gi (a0i ; a j ; ))d a j jT j at each (t j ; ). Soby (10),Z (t j ; ;a j ) [gi (a i ; a j ; ) gi (a0i ; a j ; )] d 0

Z ( t j ; ;a j )

[gi (a i ; a j ; ) gi (a0i ; a j ; )] d 2" (2) 4"

Thus a i 2 R i (t0i ;G; + 4 "). Since this argument holds for every game G independent of the cardinality of the action sets, we have d (t i ; t0i) 4".

On the other hand, if d (t i ; t0i) " then there exists k such that

jE (f j (t i)) E (f j (t0i ))j "2

for some bounded f that is measurable with respect to kth level beliefs. Now lemma 4 statesthat that we can construct a game G and action a i such that h i (t0i ja i ; G) + "2 < h i (t i ja i ; G).Thus d (t i ; t0i ) "2 .

20 A similar construction appears in Dekel, Fudenberg and Morris (section 7.1, 2005).

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We conclude that12

d (t i ; t0i ) d (t i ; t0i ) 4d (t i ; t

0i )

or equivalently14

d (t i ; t0i ) d (t i ; t0i ) 2d (t i ; t0i ) .

Thus d (tni ; t i ) ! 0 if and only if d (tni ; t i ) ! 0.

References

[1] Battigalli, P. and M. Siniscalchi (2003). Rationalization and Incomplete Information,Advances in Theoretical Economics 3, Article 3.

[2] Bergemann, D. and S. Morris (2001). Robust Mechanism Design,http://www.princeton.edu/~smorris/pdfs/robustmechanism2001.pdf.

[3] Bergemann, D. and S. Morris (2005). Robust Mechanism Design, Econometrica 73 ,1771-1813.

[4] Brandenburger, A. and E. Dekel (1987). Rationalizability and Correlated Equilibria,Econometrica 55 , 1391-1402.

[5] Brandenburger, A. and E. Dekel (1993). Hierarchies of Beliefs and Common Knowl-edge, Journal of Economic Theory 59, 189-198.

[6] Cremer, J. and R. McLean (1985). Optimal Selling Strategies Under Uncertainty fora Discriminating Monopolist when Demands are Interdependent, Econometrica 53 ,345-361.

[7] Dekel, E., D Fudenberg, and D. K. Levine (2004) . "Learning to Play Bayesian Games,"Games and Economic Behavior , 46 , 282-303.

[8] Dekel, E., D. Fudenberg and S. Morris (2005). Interim Rationalizability," mimeo.

[9] Ely, J. and M. Peski (2005). "Hierarchies of Belief and Interim Rationalizability,"http://ssrn.com/abstract=626641, forthcoming in Theoretical Economics .

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[10] Fudenberg, D. and D.K. Levine (1986). "Limit Games and Limit Equilibria," Journal of Economic Theory 38, 261-279.

[11] Geanakoplos, J. and H. Polemarchakis (1982). We Cant Disagree Forever, Journal of Economic Theory 28, 192-200.

[12] Harsanyi, J. (1967/68). Games with Incomplete Information Played by Bayesian Play-ers, I-III, Management Science 14, 159-182, 320-334, 486-502.

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[15] Heifetz, A. and Z. Neeman (2004). On the Generic Infeasibility of Full Surplus Extrac-tion in Mechanism Design, forthcoming in Econometrica .

[16] Jehiel, P. and B. Moldovanu (2001). Ecient Design with Interdependent Valuations,Econometrica 69 , 1237-59.

[17] Kajii, A. and S. Morris (1997). Payo Continuity in Incomplete Information Games,

Journal of Economic Theory 82, 267-276.

[18] Lipman, B. (2003). Finite Order Implications of Common Priors, Econometrica 71 ,1255-1267.

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[20] Mertens, J.-F., S. Sorin and S. Zamir (1994). "Repeated Games: Part A BackgroundMaterial," CORE Discussion Paper #9420.

[21] Mertens, J.-F. and Zamir, S. (1985). Formulation of Bayesian Analysis for Games withIncomplete Information, International Journal of Game Theory 14, 1-29.

[22] Monderer, D. and D. Samet (1996). "Proximity of Information in Games with CommonBeliefs," Mathematics of Operations Research 21, 707-725.

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[23] Morris, S. (2002). Typical Types, http://www.princeton.edu/~smorris/pdfs/typicaltypes.pdf.

[24] Munkres, J. (1975). Topology . Prentice Hall.

[25] Neeman, Z. (2004). The Relevance of Private Information in Mechanism Design, Jour-nal of Economic Theory 117 , 55-77.

[26] Rubinstein, A. (1989). The Electronic Mail Game: Strategic Behavior under AlmostCommon Knowledge, American Economic Review 79, 385-391.

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[28] Wilson, R. (1987). Game-Theoretic Analyses of Trading Processes, in Advances in Economic Theory: Fifth World Congress , ed. Truman Bewley. Cambridge: CambridgeUniversity Press chapter 2, pp. 33-70.

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