The Expressive Power of the Kierarchical Approach to Modeling Knowledge and Common Knowledge

One approach to representing knowledge or belief of agents, which has been explored independently by economists (BSge and Eisele; Mertens and Zamir; Brandenburger and Dekel; Tan and Werlang) and by computer scientists (Fagin, Halpern, and Vardi) involves an infinite hierarchy of beliefs. Such a hierarchy consists of an agent's beliefs about the state of the world, his beliefs about other agents' beliefs about the worlds, his beliefs about other agents' beliefs about other agents' beliefs about the worlds, etc. Economists and computer scientists differ, however, in the way they model beliefs. Economists prefer a probability-based framework, where belief is modeled as a probability distribution on the uncertainty space. In contrast, computer scientists prefer an information-based framework, where belief is modeled as a subset of the underlying space. The idea is that whatever is in the subset is believed to be possible, and whatever is not in the subset is believed to be impossible. We consider the question of when such an infinite hierarchy completely describes the uncertainty of the agents. We provide various necessary and sufficient conditions for this property. It turns out that the probability-based approach can be viewed as satisfying one of these conditions, which explains why the infinite hierarchy always completely describes the uncertainty of the agents in the probability-based approach. An interesting consequence of our conditions is that adequacy of an infinite hierarchy may depend on the "richness" of the states in the underlying state space. We also consider the question of whether an infinite hierarchy completely describes the uncertainty of the agents with respect to "interesting" sets of events and show that the answers depends on the definition of "interesting". 1 I n t r o d u c t i o n Reasoning about knowledge of agents and their knowledge of each other 's knowledge has now been recognized as a fundamental concern in game theory, computer science, artificial intelligence, and philosophy (see [Hal91] for a recent overview). The importance of finding good formal models that can represent the knowledge of the agents has also been long recognized. 230 Fagin, Geanakoplos, Halpern, and Vardi The original approach to representing knowledge and common knowledge in the game-theory l i terature is due to Aumann [Aum76]. Consider a situation with n agents. To model this, Aumann considers structures of the form A = (S, El , . . . ,En), where S is a set of states of the world, and each Ei is a part i t ion of S. We henceforth call these Aumann structures. 1 An agent "knows" about events, which are identified with subsets of S. Agent i's knowledge is modeled by Ei, his in]ormation partition. Given a state s E S, we use E~(s) to denote the set of states in the same element of the part i t ion as s; these are the states that agent i considers to be possible in state s. Agent i is said to know an event E at the state s if Ei(s) is a subset of E. The intuition behind this is that in state s, agent i cannot distinguish between any of the worlds in Ei(s) . Thus, agent i knows E in state s if E holds at all the states that i cannot distinguish from s. Using this intuition, we can define an operator Ki from events to events. Given an event E, the event Ki(E) (intuitively, the event "agent i knows E" ) can be identified with the set of states where agent i knows E according to our definition. We can also define the event O(E) ("everyone knows E" ) as the intersection of the events Ki(E), over all agents i = 1 , . . . , n. Finally, we can define the event C(E) (".E is common knowledge") as the intersection of the events O(E), O(O(E)), and so on. There is, unfortunately, a philosophical difficulty with this approach (cf. [Gi188, TW88, Aum89]). The problem is that it is not a priori clear what the relation is between a state in an Aumann s t ruc tu re -which is, after all, just an element of a s e t and the rather complicated reality tha t this state is t rying to model. If we think of a state as a complete description of the world, then it must capture all of the agents' knowledge. Since the agents' knowledge is defined in terms of the partitions, the state must include a description of the partitions. This seems to lead to circularity, since the parti t ions are defined over the states, but the states contains a description of the partitions. • Par t ly in response to these concerns, an alternative approach to modeling knowledge was investigated in a number of papers ]BE79, MZ85, TW88, BD92]. This approach, which involves an infinite hierarchy of beliefs, takes its cue from the work of Harsanyi [Har68]. We start with a set S of states of nature, which we take to be descriptions of certain facts about the world (e.g., in game theory, these may typically include values of parameters of the game, such as payoffs). Each agent has beliefs about the state of nature, modeled by a probabili ty distr ibution over S. These beliefs are clearly highly relevant to the agent's choice of strategy. But agents also have beliefs about other agents' beliefs, and beliefs about other agents' beliefs about their beliefs, and so on. Pursuing this line, one is natural ly led to associating with each agent a hierarchy of beliefs. We can build up this hierarchy level by level; for each natural number m, the (m + 1)St-order bel iefs of agent i are modeled by a probability distribution on the possible states of nature and the other agents' ruth-order beliefs (together with some consistency conditions described in [MZ85, BD92]). An agent's type is his infinite hierarchy of beliefs. We define a belie] structure to consist of a state of na ture and a description of each agent's type. Given a set S of states of the world, we take B(S) to be the set of belief s tructures where S is the set of states of nature. In belief structures, knowledge is identified with "believes with probability 1". Tha t is, roughly speaking~ agent i is said to know E C S in a given belief s t ructure b if, according to agent i's type in b, event E is assigned probability 1 at level 1 of agent i's hierarchy. Similarly, agent i knows tha t agent j knows E if the event "agent j knows E" is assigned probabili ty 1 at level 2 of agent 1The reader with a background in modal logic will recognize that an Aumann structure is nothing more than a Kripke frame ]or $5 [Kri59, HC78, HM92]. In Aumann's original paper, he also assumed that there was a probability distribution on S. Since the probability function plays no role in our discussion of knowledge in Aumann structures, we have decided to drop it here. This is consistent with Aumann's own discussion of knowledge in later papers (see, for example, [Aum89]) andwith the presentation of Aumann's framework in, for example, ]Wet89]. Expressive Power of Hierarchical Approach to Modeling Knowledge 231 i's type hierarchy. Finally, we say that E is common knowledge if all agents know E, all agents know they know E, and so on. We would like to think of a belief structure as describing a state of the world. It is not clear, however, that a belief structure is an adequate description of the state of the world. Even if we accept the doctrine that a state of the world can be adequately described by describing the actual state of nature and each agent's uncertainty about the state of nature and other agents' uncertainty (at all levels), it is not clear that the infinite hierarchy just described completely exhausts an agent's uncertainty. After all, an agent may have uncertainty as to the type of other agents. HarSanyi essentially assumed that there is an exogenously given probability distribution that describes each agent's probability distribution on the state of nature and the other agents' types. The key result proved in [BE79, MZ85] is that the hierarchy described above does exhaust an agent's beliefs: an agent's type determines a unique probability distribution on the states of nature and the other agents' types. This result also suggests that we can view the belief structures in 13(S) as the states in an Aumann structure, since each one completely describes a state of the world. If we take that view, then we might hope that the definitions of knowledge and common knowledge in Aumann structures and belief structures coincide. Unfortunately, this is not quite the case. Nevertheless, Brandenburger and Dekel [BD92] show that these notions do coincide if we interpret knowledge in Aumann structures probabilistically. Thus, we view 13(S) as an Aumann structure, with the information partit ion s being determined by the type (so that two belief structures b and b I are in the same equivalence class of ~i iff agent i has the same type in b and b~). In addition, we endow B(S) with probability measures #i (one probability for each agent i) based on information in the individual belief structures (for more details on the construction, see [BD92]). We then take the event "agent i knows E" to hold in state s if #i(E]/Ci(s)) = 1; similar modifications are necessary for common knowledge. (We identify the event E C_ S with the subset of B(S) consisting of all belief structures for which the state of nature is E.) Brandenburger and Dekel then show that an event E C S is common knowledge in a state b in the (probabilistically endowed) Aumann structure B(S) iff E is common knowledge in the belief structure b. A complementary result is proved in [TW88], where it is shown that given an Aumann structure A = (S , /Q , . . . , /Cn) and so E S, there is a belief structure b E B(S) such that an event E C_ S is common knowledge at so iff E is common knowledge in b. This may seem to pretty much complete the picture: the hierarchical approach provides the answer to the problem of circularity in Aumann structures, since the above results seem to indicate that belief structures are adequate in modeling the states in Aumann structures. Unfortunately, the situation is somewhat more complicated than these results suggest. The fundamental problem with these results is that they are trying to relate two incomparable concepts of knowledge: the information-theoretic concept in Aumann structures and the probability-theoretic concept in belief structures (which is why Brandenburger and Dekel had to recast Aumann's framework in a probabilistic setting). We would argue that the probabilistic framework masks some of the subtleties in the issue of the adequacy of the hierarchical approach. Since the circularity issue in Aumann structures arises in a non-probabilistic setting, we believe that the adequacy of the hierarchical approach is best examined in a non-probabilistic setting. A non-probabilistic setting for the hierarchical approach is described in [FHV91]. We again start with states of nature, and build a hierarchy, level by level. In this case, the (m + 1) storder knowledge of agent i is modeled by a set of possibilities, each of which is a description of a state of nature and each agent's ruth-order knowledge (satisfying certain consistency conditions). 232 Fagin, Geanakoplos, Halpern, and Vardi Intuitively, whatever is in the subset is believed to be possible, and whatever is not in the subset is believed to be impossible. Note that there is no probability distribution, just a set of possibilities. A knowledge structure consists of a state of nature and each agent's hierarchy of possibilities. We take .~(S) to be the set of knowledge structures where S is the set of states of nature. Knowledge and common knowledge are defined in knowledge structures in an informationtheoretic fashion, as in Aumann structures. That is, agent i is said to know E C S in a given knowledge structure if the set of states that i considers possible at level 1 is a subset of E; agent j knows that agent i knows E if the set of sequences of length 2 that j considers possible at level 2 is a subset of the set of sequences of length 2 where i knows E. Common knowledge is again defined in the standard way in terms of knowledge. In [FHV91], results connecting knowledge structures and Aumann structures analogous to those of [BD92] and [TWS8] are proved. Namely, it is shown that we can view ~ ( S ) as an Aumann structure, where the partitions are determined by the agents' types, and an event E C S is common knowledge in a knowledge structure f E .~'(S) according to Aumann's definition iff E is common knowledge at f according to the knowledge structures definition. Moreover, it is shown that given an Aumann structure A = (S, E 1 , . . . , E n ) and a state so E S, there is a knowledge structure f E .T'(S) such that an event E E S is common knowledge at so iff E is common knowledge in f. This seems to confirm the results of [TW88, BD92] and suggest that the hierarchical approach does address the circularity problem. Unfortunately, it is also shown in [FHV91] that knowledge structures are in general not an adequate description of the world, since they do not comple te ly describe an agent's uncertainty. In particular, an agent's type does not determine what other types the agent considers possible. The problem is that the hierarchy in knowledge structures (as well as in belief structures) contain only w levels, when in general we need to consider transfinite hierarchies. There seems to be an inconsistency here. How can it be the case that knowledge structures are not an adequate description of an agent's knowledge while belief structures are, and how we do reconcile the inadequacy of knowledge structures with the results relating knowledge structures to Aumann structures? Our goal in this paper is to address these questions by examining the adequacy of hierarchical structures and trying to make precise how expressive they are. We do this in the non-probabilistic framework of knowledge structures, but we go beyond [FHV91] in our focus on the adequacy issue, and in our logic-free setting. We start by considering more carefully the question of when a knowledge structure does completely characterize the agents' knowledge. We provide a necessary and sufficient condition fol this property. A surprising consequence of our condition is that a knowledge structure completely characterizes the agents' knowledge if and only if it characterizes the first w +w levels of knowledge. Another consequence of our condition is that the adequacy of knowledge structures may depend on the "richness" of the states in the underlying state space S. If the states of nature are modeled in enough detail, then knowledge structures do characterize the agents' knowledge. Our analysis also shows the importance of a certain limit-closure property, which says that what happens at finite levels determines what happens at the limit. This property holds for belief structures, because of the assumption that probability is countably additive. The fact that this property holds for belief structures is precisely why the result of [BE79, MZ85] holds. If we allowed finitely additive probabilities, then belief structures would not, in general, completely characterize the agents' beliefs. Since knowledge structures do not, in general, characterize agents' knowledge, it is inappropriExpressive Power of Hierarchical Approach to Modeling Knowledge 233 ate here to study their relationship to Aumann structures by simply defining partitions on .T'(S). We examine this relationship in a more general setting, by asking whether knowledge structures characterize agent's knowledge with respect to "interesting" sets of events. The answer, of course, depends on what is considered to be an "interesting" set of events. It turns out, for example, that if we consider only events that can be defined from "natural events" by knowledge and common knowledge operators, then knowledge structures are adequate. If, on the other hand, we are interested in common knowledge among coalitions of agents (rather than just common knowledge among all the agents), then knowledge structures are not adequate. In this case, a transfinite hierarchy is necessary, but w 2 levels suffice. This discussion gives the impression that the" only issue underlying the adequacy of the hierarchical approach is that of the "length" of the hierarchy. But it is easy to see that knowledge structures are also deficient in a way that no transfinite hierarchy can remedy. Aumann structures contain information about all conceivable states, even states that are commonly known not to hold. Thus, Aumann structures enable counter factual reasoning, such as "If Ron Fagin were the President, then he would not have stopped the war against Iraq so soon." A counterfactual statement can be viewed as a statement about a world commonly known not to be possible. (It is presumably common knowledge that Ron Fagin is not the President.) Knowledge structures, on the other hand, do not enable such reasoning, since situations commonly known to be impossible never appear as prefixes in knowledge structures. It turns out that this deficiency is not inherent in the hierarchical approach, but rather it is the result of the manner in which this approach was used in knowledge structures. Knowledge structures were designed to model knowledge; no more, no less. As we show, the hierarchical approach can also be used to define structures that do capture information about conceivable states. These results suggest that hierarchical structures can always serve as adequate models of the world. In general, however, we may need to capture more than just knowledge and we may need to continue the hierarchy into the transfinite ordinals, in order to completely capture the agents' uncertainty. What we choose to capture and how far into the ordinals we need to go depends on the events that we are interested in capturing. Thus, the question of whether knowledge or belief structures as defined are adequate models depends on both what features of the world we are trying to model, and on the events we are interested in describing. 2 K n o w l e d g e s t r u c t u r e s and b e l i e f s tructures : a r e v i e w In this section we review the definitions of knowledge structures and belief structures. We begin with knowledge structures. The following material is largely taken from [FHV91], slightly modified to be consistent with the rest of our presentation here. We start with a set S of states of nature, a fixed finite set {1 , . . . , n} of agents. We now define k-ary worlds, by induction on k. A Oth-order knowledge assignment, fo, is a member of S, that is, a state of nature (which, intuitively, corresponds to the "real world"). We call (f0) a 1-ary world (since its length is 1). Assume inductively that k-ary worlds (or k-worlds, for short) have been defined. Let Wk be the set of all k-worlds. A kth-order knowledge assignment (for k > 1) is a function that associates with each agent i a set fk(i) C_ Wk of "possible k-worlds"; we think of the worlds in .fk(i) as "possible" for agent i and the worlds in Wk fk(i) as "impossible" for agent i. A (k + 1)-sequence off knowledge assignments is a sequence ( ]0 , . . . , fkl, where fi is an ith-order knowledge assignment. A (k + 1)-world is a (k + 1)-sequence of knowledge assignments 234 Fagin, Geanakoplos, Halpern, and Vardi that satisfies certain conditions described below. These conditions enforce some intuitive properties of knowledge. An infinite sequence (f0, f l , f2 , . . . ) is called a knowledge structure if for each k, each prefix ( fo, . . . , f fk-1) is a k-world. Thus, a k-world describes knowledge of depth k 1, and a knowledge structure describes knowledge of arbitrary depth. We use .T'(S) to denote the set of knowledge structures over S. A (k+l)-world ( f0 , . . . , ffk) must satisfy the following restrictions for each agent i: (K1) C o r r e c t n e s s : (.f0,..-, fk-1) E fk(i), if k > 1. Intuitively, this condition says that knowledge is always correct (unlike belief, which can be incorrect). (K2) I n t r o s p e c t i o n : If (go,. . . , gk-1) E fk(i), and k > 1, then gk-l(i) = fkl ( i ) . This condition implies that our agents are introspective about their own knowledge; at each level k, they know exactly what they know and what they don't know at level k 1. (K3) E x t e n s i o n : (g0, . . . ,gk-2) E ffk-l(i) iff there is a (k 1)St-order knowledge assignment gk-1 such that (go, . . . ,gk-2,gk-1) E fk(i), i lk > 1. Intuitively, this condition says that the different levels of knowledge describing a knowledge world are consistent with each other. Let f be the knowledge structure ( f0 , ] l , . . . ) . Define i's view of f, or i's type in f, denoted ~i(f), to be the sequence (.fl(i), f2( i ) , . . . ) . We write f N i f ' if r i ( f ) = ~i(f'), that is, if i has the same type in f and f'. We can now ask whether knowledge structures as defined above are adequate to fully capture all of an agent's knowledge. As we shall discuss shortly, it is shown in [FHV91] that they are not adequate. That is, an agent's type does not necessarily determine the set of knowledge structures that he considers possible. At this point, it is not even clear what we mean by "the knowledge Structures that agent i considers possible". In a given knowledge structure f -(f0, .fl, . ) , it should be clear, for each finite k, what we mean by "the k-worlds that agent i considers possible": these are simply the worlds in fk(i). In the case of knowledge structures, rather than k-worlds, we would similarly like to say that agent i considers the knowledge structure g possible in f precisely if g E fw(i). In fact, this is exactly what we shall do shortly, when we extend the hierarchy into the transfinite ordinals. There is, however, no .f~ in a knowledge structure. So how do we make sense of "the knowledge structures that agent i considers possible in f"? One way is to say that agent i considers the knowledge structure g possible in f precisely if f ~i g. This says that agent i considers the knowledge structure g possible precisely if agent i has the same type in g as in f. Another way to say that agent i considers the knowledge structure g possible in f is if agent i considers every finite prefix of g possible. This second notion says that if g = (go,gx...), then agent i considers the knowledge structure g possible in f precisely if (go,. . . , gk-1) E fk(i) for every k_> 1. The next theorem says that these two notions are equivalent. T h e o r e m 2.1: [FHV91] Let f = ( fo, . f l , . . . ) and g = (go,g1,...) be knowledge structures. f '~i g iff (go,. . . , gk-1) E fk(i) .for every k > 1. Then We can think of the first notion in Theorem 2.1 (that is, f ~ i g) as an external notion of possibility. There can be uncountably many knowledge structures g such that f ~ i g. The other notion is internal: we consider every k-world that i considers possible, by "looking inside" the knowledge structure (at level k). Thus, this second notion is finitistic. Theorem 2.1 tells us that the external and internal notions coincide. Expressive Power of Hierarchical Approach to Modeling Knowledge 235 As we hinted above, to understand whether knowledge structures fully capture the agents' knowledge, we need to continue the construction of the hierarchy into the transfinite ordinals. We can view knowledge structures as we have defined them as w-worlds. To construct (w + 1)-worlds, we need a function that tells us, for each agent i, which w-worlds i considers possible. We can then continue on inductively to define A-worlds for every ordinal A. Suppose we have defined A-worlds for some ordinal A. Let W~ be the set of all A-worlds. A Ath-order knowledge assignment is a function that associates with each agent i a set f~(i) C_ W~. A A-sequence of knowledge assignments is a sequence (fo, ffl , . . . ) of length A, where fi is an ith-order knowledge assignment. A A-world f is a A-sequence of knowledge assignments satisfying conditions that are straightforward extensions of conditions K1-K3 above. For example, an (w + 1)-world is of the form (ff0, f f l , . . . , f~}, where fo~(i) is a set of w-worlds satisfying the appropriate conditions. In the sequel, when we speak of knowledge structures, we will mean w-worlds. Consider now an (w + 1)-world f = (f0, f l , . . . , foJI. There are now two ways we can define "the knowledge structures that agent i considers possible in f". One way is to say that agent i considers the knowledge structure g possible in f precisely if g E f~,(i). Note, however, that the first w levels constitute an w-world f ' = (f0, f l , . . . } . Thus, another way is to say that agent i considers the knowledge structure g possible in f precisely if agent i considers the knowledge structure g possible in f', i.e., precisely if f ' '~i g. It is shown in [FHV91] that these two ways are not equivalent; while we have that f~(i) C {g If ' ~i g}, equality need not hold. Thus, knowledge structures do not fully describe the agents' knowledge. Belief structures are defined along similar lines. We briefly sketch the definition here, and refer the reader to [MZ85, TW88, BD92] for more details. We start with S, which we assume is endowed with a topology that makes it a compact metric space. 2 Given a compact metric space X, let A(X) denote the set of Borel probability measures on X. If we endow A(X) with the topology of weak convergence of measures, then A(X) is also a compact metric space. Define a sequence of spaces Xk, k = 0, 1, 2 , . . . inductively by taking X0 = S and = x (xk) (= Xo x (Xo) x a(x ) × . . . × A belief structure b is a sequence {b0, bl , . . .} such that b0 E S, and bk E ~,(Xk_l) n for each k > 0. This means that for k > 0 we can view bk as a function such that for each agent i, we have bk(i) E A(Xk_i) . We have consistency conditions B1 and B2 on belief structures that correspond to K2 and K3: (B1) For all k > 1, the probability measure bk(i) assigns probability 1 to the subspace of Xk-1 consisting of sequences (co , . . . , ck-1) with Ck-l(i) = bk-l(i). This says that agent i knows his own probability assignment. (B2) For all k > 1, the probability measure bk-l(i) is the marginal of bk(i) on Xk-1. While we could in principle extend belief structures beyond the first w levels, just as we did in the case of knowledge structures, the result of [BE79, MZ85] assures us that there is no need; the first w levels determine the rest of the hierarchy. 2The assumption that S is a compact metric space is made in [TW88]. In [BD92] it is assumed that S is a complete separable metric space, white in [BE79] and [MZ85] it is simply assumed that S is compact. All these assumptions are trivially true if S is finite, which is often a reasonable assumption in practice. 236 Fagin, Geanakoplos, Halpern, and Vardi 3 Are knowledge structures adequate models of knowledge? As shown in [FHV91], knowledge structures do not in general completely describe the agents' knowledge, and to fully capture this knowledge we need to extend the hierarchy into the transfinite ordinals. This should be contrasted with the situation for belief structures, which completely describe the agents' beliefs [BE79, MZ85]. To understand this difference better, the first question we want to examine here is when knowledge structures completely describe the agents' knowledge. To answer this question, we first need to formalize it. If A > w is an ordinal, then we s a y that a knowledge structure f characterizes the agents' A-knowledge if there is a unique extension of f -('fo, 'fl, ' f2, . . .) to a (A + 1)-world ('f0, ' f l , ' f2 , . . , 'f~). In particular, f characterizes the agents' w-knowledge if the "next" level ,f~ is uniquely determined. We say that a knowledge structure f (completely) characterizes the agents' knowledge if it characterizes the agents' A-knowledge for every A > w, that is, if all extensions of f are determined. This definition captures the intuition that the first w levels determine the agents' knowledge. As we have already observed, the result of [BE79, MZ85] implies that all belief structures characterize the agents' beliefs in this sense. We now provide a necessary and sufficient condition for a knowledge structure to characterize the agents' knowledge. First, we give a necessary and sufficient condition for a knowledge structure to characterize the agents' w-knowledge. We say that a world (g0,gl , . . . ,g~-l) E .fk(i) is i-uniquely extendible in f if there is a unique knowledge structure g = (go, g l , . . , gk-1 , . . . ) such that (go, g l , . . . , gl) E ,fz+l(i) for all l. T h e o r e m 3.1: A knowledge structure f characterizes the agents' w-knowledge iff ,for each agent i ~ and each knowledge structure g = (go,gl,g2,. . .) different .from f such that f ,~ g, there exists some r such that (g0,gl ,-. ,gr) is i-uniquely extendible in f. We might hope that if a knowledge structure characterizes the agents' w-knowledge, then it completely characterizes the agents' knowledge. Unfortunately, this is not the case. For example, agent 1 might consider it possible that agent 2's w-knowledge is not characterized. As the next result shows, a knowledge structure characterizes the agents' knowledge iff it is common knowledge that the first w levels characterizes the agents' w-knowledge. To make this precise, we need some more definitions. Let f and g be knowledge structures. We say that g is reachable from f if there are knowledge structures h 0 , . . . , hr such that f = h0, g = hr, and for all j < r, we have hj "~i h j+l for some agent i. By Theorem 2.1, f ~i g iff agent i considers every finite prefix of g possible (according to f). Similarly, g is reachable from f iff, intuitively, according to f, some agent considers it possible that some agent considers it possible . . . that some agent considers every finite prefix of g possible. There is a close connection between reachability and common knowledge. For example, it can be shown that an event E C S is common knowledge in f iff E holds at each knowledge structure reachable from f. (See laura76, HM92] for analogous results in the context of Aumann structures, and [TW88] for an analogous result in the context of belief structures.) T h e o r e m 3.2: A knowledge structure f characterizes the agents' knowledge iff every knowledge structure reachable .from f characterizes the agents' w-knowledge. It is not hard to demonstrate knowledge structures that do and knowledge structures that do not characterize the agents' knowledge. For example, in [FHV91], there is a construction that , given Expressive Power of Hierarchical Approach to Modeling Knowledge 237 a k-world w, builds w*, the no-in,formation extension of w, a knowledge structure that, informally, is the knowledge structure where all each agent knows is what is already described by w. It is not hard to see from the construction there that w* does not characterize the agents' w-knowledge. We shall see another example later (Example 3.6) where the knowledge structure does not characterize the agents' w-knowledge. An example of a knowledge s t ruc tu re that characterizes the agents' knowledge is one where the state of nature is common knowledge. This is a knowledge structure f = ('f0, ' f l , . . . ) where every .fk(i) is a singleton set. We leave to the reader the straightforward verification that such a knowledge structure characterizes the agents' knowledge. As we noted, there exist knowledge structures that characterize the agents' w-knowledge, but do not completely characterize the agents' knowledge. Rather surprisingly, it turns out that if a knowledge structure f characterizes the agents' knowledge through the first w. 2 = w +w levels (that is, if f characterizes the agents' (w + k)-knowledge for every natural number k), then f completely characterizes the agents' knowledge. T h e o r e m 3.3: A knowledge structure characterizes the agents' knowledge iff it characterizes the agents' knowledge through the first w. 2 levels. To gain a better understanding of the issue of characterization of knowledge, we now consider a simpler sufficient condition on knowledge structures that guarantees characterization of the agents' knowledge. Let f be a knowledge structure. A world hereditarily appears in f if it is a prefix of a knowledge structure that is reachable from f. Intuitively, a world w hereditarily appears in f if some agent considers it possible that some agent considers it possible . . . that some agent considers w possible. Let k be a fixed natural number. We say that it is common knowledge in f how the state of nature determines the agents' knowledge if whenever w = (go,---, gr) and w ' = (g~),... ,get) hereditarily appear in f, and go = g~, then w = w ~. Using Theorem 2.1, it can be easily shown that it is common knowledge in f how the state of nature determines the agents' knowledge precisely if whenever g and g~ are reachable from f, and the state of nature is the same in g and g~, then g = gl. The next proposition gives us a simple sufficient condition on a knowledge structure that guarantees that it characterizes the agents' knowledge. P r o p o s i t i o n 3.4: Let f be a knowledge structure where it is common knowledge how the state of nature determines the agents' knowledge. Then f characterizes the agents' knowledge. The interest in Proposition 3.4 comes from the fact that the way an agent determines what states are possible (or, in the case of belief structures, the way an agent determines how to assign probabilities) clearly ultimately depends on circumstances external to the agent, including perhaps what the agent has observed, the agent's upbringing, and a myriad of other influences. In many applications, the most natural way to model the state of nature will capture these external circumstances, and therefore it is common knowledge how the state of nature determines the agents' knowledge. The following simple example may clarify this. E x a m p l e 3.5: There are three agents, 1, 2, and 3. Consider a fact p such as "the price of IBM stock is over $100". Suppose agents 1 and 3 discover whether or not p holds, and agent 2 does not. Agent 1 and 2 then start to communicate about p over an unreliable channel. First agent 1 tells agent 2 about p. If agent 2 receives the message, he sends an acknowledgement. If agent 1 238 Fagin, Geanakoplos, Halpern, and Vardi receives the acknowledgement, he acknowledges the acknowledgement, and so on. If at any point a message is not received, there is no further communication. There is never any communication between agent 3 and the other two agents. We also assume that agent 3 has no idea how much time has transpired since agent 1 found out about p, so that, in particular, he has no upper bound on the number of rounds of messages that may have passed between agents 1 and 2. Thus, we can take S to consist of pairs of the form (q, k), where q is either p or i~, and describes whether or not the fact p holds, and k describes how many messages have been exchanged between 1 and 2. In this situation, it is common knowledge how the state of nature (q, k) determines the agents' knowledge. Intuitively, this is because once we know how many messages have been exchanged, we can determine each agent's knowledge. For example, suppose that the state of nature is (p, 2), so that p holds and two messages have been exchanged (thus far) between 1 and 2 (i.e., 2 received l ' s initial message, and 1 received 2's acknowledgement). Then at the first level, agent 1 considers the states (p, 2) and (p, 3) possible (since agent 1 does not know whether his acknowledgement to agent 2's last message was received by agent 2) and 2 considers the states (p, 1) and (p, 2) possible (since agent 2 does not know whether agent 1 received the last acknowledgement he sent). Agent 3 considers all wor ldsof the form (2, k), k _> 0 possible, since he knows p holds, but has no idea how many rounds of communication there have been. In the full paper, we show how we can continue this construction in a unique way, level by level. The key point here is that it is common knowledge how the state of nature (q, k) determines the agents' knowledge. Therefore, by Proposition 3.4, each knowledge structure that arises in this scenario characterizes the agents' knowledge. Before leaving this example, let us consider exactly what knowledge the agents have in each of the knowledge structures that arise in this scenario. Let E be the event "p holds" (so that E consists of all knowledge structures where the state of nature is of the form (p, k)). It is easy to check that in (the knowledge structure that corresponds to) the state (p, 0), the' event K1 (E) holds but the event K2(Ki(E)) does not; in (p, 1), the event K2(Ki(E)) holds but the event Ki(K2(Ki(E))) does not; in (p, 2), the event Ki(K2(Ki(E))) holds but the event K2(Ki(K2(Ki(E)))) does not; and so on. Thus, at no point does common knowledge of E ever hold between agents 1 and 2, where agents 1 and 2 are said to have common knowledge of E if both 1 and 2 know that both 1 and 2 know . . . that E holds (cf. the discussion of the coordinated attack problem in [HM90]). Now consider agent 3. Then, informally, in every state agent 3 certainly knows that agents 1 and 2 do not have common knowledge of E (since they never attain common knowledge of E when communication is not guaranteed). He considers it possible, however, that agents 1 and 2 have arbitrarily deep knowledge of E (since agent 3 considers all the states (p, 0), (p, 1), (p, 2) , . . . possible). | While in simple examples it does seem reasonable to include enough information in the state of nature so that it is common knowledge how the state of nature determines the agents' knowledge, in more complicated examples this becomes a serious modeling problem. For example, even if we accept that the sum total of an agent's upbringing, together with hereditary factors, completely determines the agent's knowledge, it is not clear that we want to include all this information in the state of nature when modeling, say, a simple game. Once we leave it out, however, the knowledge structure may no longer adequately model the agents' knowledge, as the following example shows. E x a m p l e 3.6: Suppose we consider the same situation as in Example 3.5, but change the description of the state of nature. Instead of the state of nature describing not only whether p is true, but also how many messages arrive, suppose we simply take the state of nature to describe whether or not p is true. Thus, there are only two states of nature, p and ~. Intuitively, in this case, in the Expressive Power of Hierarchical Approach to Modeling Knowledge 239 worlds we construct, the state of nature no longer determines the agents' knowledge. In fact, for no knowledge structure f that arises in such a scenario does f characterize the agents' w-knowledge, as we now show. Assume that f = (f0, ffl, f 2 , . . , / . To capture the fact that agent 3 knows that agents 1 and 2 do not have common knowledge of E, it is necessary for f~(3) to contain no w-world where agents 1 and 2 have common knowledge of E. This gives one possible extension of f (the one that properly captures the knowledge of the agents in our example). Since, however, agent 3 considers it possible that agents 1 and 2 have arbitrarily deep knowledge of E, it is not hard to see that there is another legal extension of f, where f~(3) contains some w-world where agents 1 and 2 have common knowledge of E. | R e m a r k 3.7: Proposition 3.4 can be strengthened in a number of straightforward ways. One is as follows: We say that it is common knowledge how level k determines the agents' knowledge if whenever w -/ g o , . . . , g k , . . . , g r / and w' = (g~, . . . ,g~ , . . . ,g l r / hereditarily appear in f, and their prefixes (go, . . . ,gk/ and (g~,.. . ,g~) are identical, then w = w'. Then Proposition 3.4 still holds when we replace "it is common knowledge how the state of nature determines the agents' knowledge" by "for some k, it is common knowledge how level k determines the agents' knowledge." We can further strengthen Proposition 3.4 by further weakening the hypotheses: Let f be a knowledge structure, and let k be a fixed natural number. We say that agent i knows that level k determines the agents' knowledge in f if whenever g = (go,g1,. • ./ and g~ = (go,gl,.')~ are knowledge structures such that (a) f "~i g, (b) f ~-'i g~, and (c) the prefixes (go, . . . ,gk) and /g~, . . . , g~/are identical, then g = g~. Intuitively, this says that level k completely determines the knowledge structure, among those knowledge structures that agent i considers possible. Assume for now that f is a knowledge structure where for some k, each agent knows that level k determines the agents' knowledge. It turns out that this condition is not sufficient to guarantee that f completely characterizes the agents' knowledge, even if k = 0, that is, even if each agent knows that the state of nature determines the agents' knowledge. Nevertheless, we can show that this assumption (that for some k, each agent knows that level k determines the agents' knowledge) is sufficient to guarantee that the knowledge structure characterizes the agents' w-knowledge. We then see from Theorem 3.2 that if f is a knowledge structure where for some k, it is common knowledge that level k determines the agents' knowledge, then f characterizes the agents' knowledge. Notice that the definition of common knowledge that level k determines the agents' knowledge is different from our earlier definition of common knowledge how level k determines the agents' knowledge. It is common knowledge in f that level k determines the agents' knowledge if in every knowledge structure g reachable from f, every agent knows that level k determines the agents' knowledge. It is possible, however, that there are two different knowledge structures g and g~, both reachable from f, that have the same prefix through level k. This cannot happen if it is common knowledge how level k determines the agents' knowledge. It is not hard to show (by making use of Theorem 2.1) that "common knowledge how" implies "common knowledge that". | . How does the characterization of knowledge in knowledge structures relate to the characterization of beliefs in belief structures? To answer this question, we now consider an alternative way of capturing the intuition of when a knowledge structure captures the agents' w-knowledge. This time, we consider when it is the case that there is enough information in a knowledge structure to determine what other knowledge structures each agent considers possible. Consider a knowledge structure f. If f characterizes the agents' knowledge, then the knowledge structures that agent i considers possible in f are precisely the knowledge structures in f~~ = {g : 240 Fagin, Geanakoplos, Halpem, and Vardi f "~i g}On the other hand, if agent i has more information than is described in f, then he might consider only some proper subset of f~~ possible. We would expect, however, that a knowledge structure f would characterize agent i's knowledge at each finite level. More precisely, we would expect that if w E fk( i ) for some k, then there is a knowledge structure f~ that i considers possible such that w is a prefix of f~. We say that a set T" of knowledge structures is a coherent set of possibilities for agent i at f = (f0, f l , f2 , . . . ) if f C T" and