Existence and definability of states of the world

Mathematical Social Sciences 49 (2005) 81 – 100 www.elsevier.com/locate/econbase

Existence and definability of states of the world Fernando Tohme´ * Departamento de Economı´a, Universidad Nacional del Sur, CONICET 12 de Octubre y San Juan, (8000) Bahı´a Blanca, Argentina Received 1 April 2003; received in revised form 1 January 2004; accepted 1 March 2004 Available online 28 July 2004

Abstract We present here a notion of state of the world general enough to embrace most circular phenomena in economics and game theory. We prove that it obtains by unfolding beliefs only if the process of belief generation has a fixed point. Otherwise, we are led to an unending transfinite hierarchy. This result indicates that Zermelo – Frenkel’s set theory cannot provide the modeling tools for the representation of states of the world. We apply, instead, a theory of non-well-founded sets. In that framework, states of the world are legitimate objects which can be seen as fixed points of the belief-generation operator. Not every possible state of the world can be unfolded in a hierarchy of beliefs. It will be shown by means of a simple argument, based in Tarski’s indefinability theorem, that there exist states of the world that are not expressible in that way. Moreover, this result implies that there is no way to represent those states of the world in a consistent language. However, if we assume agents do not have negative self-referential beliefs, the unfolding of beliefs suffices. D 2004 Elsevier B.V. All rights reserved. Keywords: States of the world; Circularity; Non-well-founded sets; Definability JEL classification: B40; C79

1. Introduction The main goal of economic theory has been to explain how the interaction of many selfinterested agents leads to the emergence of orderly social patterns. Game Theory has shown to be a powerful tool in this respect. Despite the fact that in many cases, the use of *Tel.: +54-291-4595138; fax: +54-291-4595139. E-mail address: [email protected] (F. Tohme´). 0165-4896/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.mathsocsci.2004.03.008

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Nash equilibrium or its refinements does not pinpoint a unique outcome, Game Theory has solved the problem of the origin of social order in the case of certainty about the characteristics of the agents and the environment. However, the presence of uncertainty makes it harder to explain how many agents can coordinate their actions or plans to lead to a (frequently unintended) single state of the world. Uncertainty can exist about either the characteristics of the game or about the decisions of the agents. Following Brandenburger (1996), we call the former case a situation of structural uncertainty while in the latter case, the players face strategic uncertainty. In either case, it means that any player (or an external observer) has to complete with beliefs her picture of the situation. Any of the many possible complete characterizations of the game and the beliefs of the agents constitute a state of the world, which can be represented as an ordered pair: x ¼ hs; BðxÞi In this ‘‘equation’’, we have that: 

s is a state of nature, which represents a possible realization of the objective features of the game: the actions available to the agents, the payoff functions, the order of play, etc.  B(x) is the structure of beliefs held by the agents about s and about the beliefs of the other agents.  x is the state of the world, i.e., the actual state of nature and the beliefs held by the agents. Notice that there exists a circularity between x and B(x) which transcends a mere notational issue. On one hand, the latter determines the former. But also, and this is crucial, since x is the full characterization of the actual game, the beliefs of the agents have to include a description of how they envision the whole situation. In other words, x defines B(x). We capture in the notation this circularity by denoting with B(x) the structure of beliefs at x. Due to this circularity, an x that verifies the equation above leads to a well-defined game. Therefore, to answer how and if such a x may arise seems critical for the discussion of how order may emerge from the interaction of many intentional agents. In the literature on applied economics, states of the world are described by both the objective features of the game (our state of nature) and the profile of types of the agents. A type is basically a shorthand to describe all the beliefs of an agent as well as what is believed about him. While this accounts for most of B(x), to reach a full equivalence with our definition, an additional feature is needed, namely the beliefs about the state of the world itself. This is solved by assuming a commonly known probability distribution of types. So, each type knows that distribution and can entertain a belief about the entire situation. A circularity appears here again, this time in the form of this commonly known distribution. While this is more than enough in applications, the procedure of disregarding the process by which states of the world appear seems appropriate only as a first approximation. In fact, since the characterization of incomplete information games by Harsanyi

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(1967), the problem of justifying the existence of states of the world has been seen as a serious challenge for game theory as a whole. As we will discuss a bit more in Section 2, this question seems to be settled, at least under certain conditions. All the proofs that show the existence of a x share a basic characterization of inductive processes of formation of beliefs. The sequences of beliefs are then shown to have limits that coincide with the corresponding states of the world. An example may illustrate the way in which states of the world can be defined by induction (Vassilakis, 1991): Example 1. Consider two firms, A and B competing a` la Cournot, but ignoring each other’s reaction functions. Let S be the set of all such reaction functions. Any sa S is such that s = h fA, fB, qA, qBi, where fi is i’s reaction function and qi is i’s initial output level. Suppose that the real state of nature is s*= h f, f, q, qi, i.e., both firms have the same reaction functions and the same initial output levels. Since each firm ignores the other’s reaction function, it has to form beliefs in order to decide its output level. The set of all beliefs of A about B’s reaction function will be BA0 (S) = D(S), while B’s beliefs about A’s reaction function will be BB0(S) = D(S), where D(S) is the set of probability measures on S. But A cannot make a decision unless it forms beliefs about B’s beliefs about A’s reaction function; these beliefs can be represented as BA1 (S) = D(S  BB0 (S)). Proceeding in this way, we consider the limits in this construction: BA ¼ DðS  BB Þ BB ¼ DðS  BA Þ The state of the world is now x* ¼ hs*; BA ; BB i since the context in which firms A and B have to make their decisions on output level is fully described by the actual parameters (reaction functions and initial output levels) and their beliefs about those parameters and about the other’s belief about those parameters, etc. It is rather evident that the limit beliefs are beliefs about the true state of the world, i.e., beliefs about the context in which the firms have to decide their output levels. That is, BA = def BA(x*) and BB = defBB(x*). Calling B(x*) the joint vector hBA(x*), BA(x*)i, it follows that x* ¼ hs*; Bðx*Þi The proofs that show the convergence to the limit depend on very stringent conditions on the topological and measure-theoretical properties of both the set of states of nature and the belief-formation process. In this paper, instead, we want to keep a high degree of generality. This implies that we will abstract from topological or measure-theoretical considerations as much as it is possible and use only tools drawn from logic and set theory. Even at this level of generality, we can prove the existence of x. The difference with the usual approaches is based on how we handle the circularity in the definition of states of

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the world. While our strategy of proof is still based on the ‘‘unfolding’’ of B(x) in a hierarchy of beliefs by means of a inductive characterization, we will not use topological or measure-theoretical arguments to ensure its convergence. To drop these conditions is not completely unproblematic: in the most general case, the sequence may become transfinitely long. In that case, no argument of convergence to a limit can be given. But then, the fundamental circularity of states of the world is no longer a liability. While circular definitions are not admissible in the standard set theory known as ZFC (Zermelo – Frenkel + Axiom of Choice), an alternative, ZFA, which preserves most of the standard axioms, ensures the existence of a set equivalent to x. This in turn is the basis of our proof of existence of states of the world. But even with this turn, the question of the existence of states of the world cannot be so easily assumed to be settled. As we will show, if agents can form negative selfreferential beliefs, there will states of the world that cannot arise as limits of inductive processes. Notice that this is not a drawback for the usual treatment of states of the world in applied economics where beliefs are not formed step by step. It is, instead, a real limitation of inductive characterizations of beliefs. Furthermore, this shows that the question of the right justification for the existence of states of the world is still open. The plan of the paper is as follows: in Section 2, we present the inductive characterization of states of the world and show that some conditions, required usually in the proofs of existence of states of the world, leave out of the picture interesting cases, in which genuine states of the world arise as a result of a discontinuous process of belief formation. In Section 3, we will see that the circularity and the possibility of discontinuities in the process of belief formation may lead generically to an unending sequence of beliefs. But in Section 4, we introduce a way out from this problem, by means of the alternative set theory ZFA in which circular definitions are seen as representing legitimate sets. In Section 5, we will show that the results in Section 4 depend on a strict hierarchical ordering of the sequence of beliefs, such that no negative self-referentiality is allowed. Finally, in Section 6, we sum up our arguments.

2. Belief-formation processes Despite the clear perception of the role of circularity in the behavior of real world economies, the usual modeling strategy has been to represent economies as seen by an omniscient external agent, leaving as ‘‘common knowledge’’ the set of alternative values of the features (rules, preferences and beliefs) which constrain and condition real agents’ decisions (Binmore, 1990). Although several authors emphasized the existence of circularity, mediated through expectations, between individual and aggregate behavior, these insights were not fully conveyed into formal models until John Harsanyi characterized Bayesian games. Although his goal was to asses the equilibrium outcomes in a game played by agents with incomplete information, Harsanyi’s approach just assumed that agents had common knowledge of the distribution of the features that were matter of circular definition.

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Moreover, by means of his definition of the agents’ types, he eliminated the circularities involved in reasoning about the others’ reasoning. In other words, instead of formalizing the process that leads to the decision-making procedures, Harsanyi’s (1967) characterization emphasized its final outcomes. In terms of the characterization we made of a state of the world in Section 1, we can easily translate Harsanyi’s approach as giving a definition of B(x) supported by an underlying state of nature s. Harsanyi’s construction is based on the characterization of two collections: 

the basic set S of possible states of nature. Its elements are precise descriptions of the objective features of a game, namely the actions available to the agents, the payo. functions, the order of play, etc.  the final class of states of the world, X. It includes all the possible complete descriptions of games (completed with beliefs) with objective features drawn from S. In Harsanyi’s construction, each saS may support a sequence of beliefs. The zeroth element in the sequence is the belief held about s (which we denote B0(x)). The next element consists of the beliefs held about this belief, B1(x). Next come the beliefs about the beliefs about the beliefs about s of the other agents, etc. By an inductive characterization, we have that each n-level belief is a belief about the (n  1)-level beliefs. If the limit of this sequence exists, it is called B(x). The resulting state of the world xaV is defined as being s plus the structure of beliefs engendered by s, B(x). Formally: Definition 1. Given the sequence (B 0(x),B1(x),. . .) of beliefs generated by a set I of agents, such that B 0 ðxÞaS B1 ðxÞ ¼ B 0 ðxÞ  belðB 0 ðxÞÞAIA ::: Bn ðxÞ ¼ B n1 ðxÞ  belðB n1 ðxÞÞAI A B(x) is the limit of the sequence: BðxÞ ¼ Ba ðxÞ ¼ limn
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If we assume a suitable—in a sense that will be made precise in a moment—bel, the state of the world x can be constructed up from s by means of the hierarchy of beliefs. ¯ a complete On the other hand, any analysis in applied game theory begins with a x, description of a game including the beliefs of the agents. The question that Harsanyi intended to answer was: What are the conditions that ensure that any given description of ¯ a game x¯ coincides with an inductively defined x? His conjecture was that for each x, there exists a state of nature s, such that by a construction as in Definition 1, the resulting ¯ Although the construction is inductive, notice that the goal of x verifies that x = x. Harsanyi was in fact to provide a coinductive characterization. That is, given the beliefs B(x), the goal is to find an appropriate unfolding in terms of a countable hierarchy of beliefs. Harsanyi did not give any proof of his conjecture. Notice that a critical element in the conjecture is that the unfolding of B(x) ends at the first infinite ordinal. This claim was proven by Armbruster and Bo¨ge (1979), Bo¨ge and Eisele (1979), Mertens and Zamir (1984) and Brandenburger and Dekel (1993), among others. The definition of bel as well as of an adequate topology over S are fundamental for these results. The least restrictive set of such assumptions that ensures the convergence to a limit in that literature is the following (Fagin et al., 1999).2 

   

Let S, the space of states of nature, be a compact metric space and bel(S) the set of Borel probability measures on S. It is endowed with the topology of weak convergence of measures. bel(S) is also a compact metric space. Let B0 = S, B1 = B0  bel(B0)jIj,. . .,Bn = Bn  1  bel(Bn  1)jIj,. . . For all n>1, the actual belief of agent i is Bin(x)abel(Bn  1). For all n>1, Bin(x) assigns probability 1 to the subspace of B n  1 consisting of sequences hx0, x1,. . ., x n  1i such that xi n  1 = Bin  1(x). For all n>1, the belief of agent i is such that Bin  1(x) is the marginal of Bin(x) on B n  2.

The most demanding of these assumptions are the last two, that imply that the verification of the consistency of the beliefs involves to check that any (n  1)th stage belief is the marginal of the (n)th. Kolmogorov’s Existence Theorem (Billingsley, 1968) n n states that there exists a unique BðxÞa jl n¼1 B such that B (x) is the projection on the n n + 1 coordinate of B (x) for every n. This ensures that the sequence of beliefs leads to a unique limit probability distribution. However, a natural question here is whether the hierarchical sequence of beliefs converges or not if we drop those assumptions. In the literature, it is customary to represent B(x) as a distribution of probabilities over 6 Moreover, it is assumed that the sequence of higher order probability distributions (distributions over distributions. . .) 2

It must be noted that a topology either over S or over X formalizes the notion of ‘‘closeness’’ among the descriptions that constitute the elements of either collections. Sometimes, abusing a bit of language, we will follow the literature in equating the descriptions (syntactical structures) with the entities that they represent (i.e. the semantics of the description). We will make the distinction explicit only if necessary for the clarity of the argument.

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converges to Bn(x). In this Bayesian setting, the consistency requirement on sequences of beliefs arises naturally (Chuaqui, 1991). On the other hand, it is easy to imagine situations where consistency leads to absurd results (Kyburg, 1974). Let us consider one which exhibits this phenomenon (Dekel and Gul, 1997; Nozick, 1993): Example 2. (Newcomb’s paradox)3. A human agent has to play against an all-knowing Genie who can predict the human’s choices. There are two boxes A and B, the first translucent, the second opaque. The Genie offers the human to take either both boxes or only box B. The agent can see that box A has $1000 inside, but the Genie tells him that if he chooses B, he will leave $1 000 000 inside, otherwise (if the agent chooses both boxes) he will leave B empty. The final decision made by the agent depends on what he believes about the powers of the Genie. A possibility is to think that the Genie may be able to predict correctly the agent’s choice. According to that, the agent should choose only box B and the Genie should leave the million dollars inside the box. But then, at the moment of grabbing only box B, the agent reflects that either his belief was right and therefore there are $1 000 000 inside the box or he was wrong and therefore the box is empty. Then, he looses nothing grabbing both boxes. In other words, the belief that constitutes part of the state of the world (because it determines the agent’s actual behavior) is inconsistent with the beliefs he held during his internal deliberation. In fact, the analysis of beliefs in Cognitive Science, Philosophical Logic and Artificial Intelligence, usually disregards the Bayesian conception.4 In that literature, beliefs are conceived as meta-linguistic statements about statements in an object language. For example, if agent i believes that ‘‘snow is white’’, this constitutes a statement in a language about statements in the language used to refer to the real world. The semantics of this metalinguistic assertion is that i has a model of the real world, in which the sentence ‘‘snow is white’’ is true. We will focus on this particular notion of beliefs, expressed in a language with which we can ‘‘say something’’ about states of nature as well as about sentences in the language. Each sentence will have an interpretation (the set of states of the world of which the proposition is believed to be true). In this setting, bel can be quite different from a probability distribution. It is represented by a correspondence from sentences in level n to sentences in level n + 1. More formally, if we abuse a bit of language by calling B to this particular bel, we have that: Definition 2. A language-based hierarchy of beliefs: assume that an agent iaI has a formal language, Li0, expressive enough to make assertions about states of nature in S. For this, it is enough to assume that Li0 is a first-order language, with an intended interpretation over S. Each admissible formula will have the form / u bx1bx2. . .bxn f (x1, x2,. . ., xn; a1,. . ., am), where n is an arbitrary number of bounded variables,5 m the number of constants

3

This puzzle has been applied to the analysis of the Prisoner’s Dilemma (Sobel, 1991). See, for example, Hintikka (1962), Loui (1998), Perlis (1986) and Gupta and Belnap (1993). 5 Recall that bxf (. . .,x,. . .) = Iax If (. . .,x,. . .). 4

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involved, and f( ) a propositional function with n + m arguments. The interpretation of / is a structure6 over S, q/a, such that / is true (in Tarski’s sense) over q/a . Then, we can define inductively a hierarchy of beliefs:  Any formula /0 in the language Li0 is called a 0-level belief. Its interpretation, defined

above, is q/0a. Its domain is the set of states of the world of level zero, V0 = S.  Given a n-level belief /n, a (n + 1)-level belief is expressed as a formula /n + 1 = B(/n) in a first-order language Lin + 1. The intended interpretation of expressions in Lin + 1 Lin + 1 has as its domain Vn = S  (2Vn  1)jIj. Then q/n + 1a is a structure over Vn. The entire hierarchy of languages is hL0 ; . . . ; Ln ; . . . ; La ; Laþ1 ; . . .i , such that L ¼ [bz0 Lb is the universal language for states of the world. This construction, which simply recasts the hierarchical construction of beliefs given in Definition 1, is basically an adaptation of Hao Wang’s (1963) system R, intended to provide a constructive framework for mathematical reasoning. We assume for the analysis of the convergence of hierarchies of beliefs that every possible belief can be represented in L.7 In fact, the Bayesian approach can be seen as a particular instance of this representation. If every Vn is endowed with a r-field ln, the meaning of a belief sentence /n + 1 is a probabilistic structure q/n + 1a over hVn, lni. The belief operator B verifies: Definition 3.  B(Lb  1)pLb if b is a successor ordinal.

But we do not require .

ðContinuityÞ : limb
While the condition in Definition 3 seems natural, to drop the continuity of the process responds to the arguments illustrated in Example 2. In fact, it implies that there exist beliefs held about limb
3. Circularity is unavoidable As discussed in the last section, we have to find a characterization of an extended belief hierarchy ample enough to represent cases where the reasoning processes of the agents allow for discontinuous jumps. Since such a hierarchy will in general not have a countable 6 7

A structure is a domain with certain relations defined among its elements. See (Boolos and Jeffrey, 1989). We will analyze the problem of the definability of beliefs in Section 5.

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number of stages, it opens the possibility of an endless process of belief formation. If this were true, we would be facing the fact that a state of the world can not be unfolded as a hierarchy of beliefs upon a state of nature. That is, circularity would be essential. In fact this is exactly what we want to show here. As a first step in our argument, let us consider a generalized hierarchy of beliefs which may be called Harsanyi’s structure: Definition 4. Construction of a transfinite hierarchy of beliefs and its limit  Step 1: Consider the collection X={b: b is a belief about x}, such that saX. That is,

each possible belief about x should be an element of X.  Step 2: Given a belief function B: X ! X, build a sequence of elements in X:

BEL ¼ hB0 ðxÞ; B1 ðxÞ; B2 ðxÞ; . . . ; Ba ðxÞ; Baþ1 ðxÞ; . . .i where B0(x) = B0(s) and Bn(x) is the n-th iteration of B(s). This is called a net or a Moore –Smith sequence (Kelley, 1955), indexed by the ordinal numbers (the class Ord) instead as by the natural numbers.  Step 3: Assume that BEL has a limit B(x). Notice that the notion of limit applied in Step 3 depends on the context. If we adhere to the characterization of beliefs of Definition 2, we have that lim Bb ðxÞ ¼ BðxÞ;

baOrd

while, as discussed in Section 2, when beliefs are represented as Bayesian probability distributions, B(x) = Ba(x), where a is the first limit ordinal. That is, in the Bayesian case the belief-formation process jumps only once to a limit ordinal. Under the general characterization of beliefs given in Definition 2, instead, the hierarchy can be transfinite, since the belief-formation process is not necessarily continuous.8 The results obtained restricting the attention only to the (a-)infinite hierarchy seem to show that it is relatively easy (and convenient) to get rid of the circularity involved in Harsanyi’s notion of state. However, as said, there are no good reasons why the process should stop at the first infinite ordinal. We, instead, define the hierarchy as indexed by the entire class of ordinals, Ord. But then the resulting object is no longer an admissible entity for Zermelo– Frenkel’s set theory. To see this, let us give a previous definition and a formal result which will be applied to support our argument: Definition 5. Given a collection X and a binary relation R over X we say that vaX is founded w.r.t. R, if there does not exist F (defined by transfinite recursion) such that for each ordinal h, Fh = w({F(b)}b < h) with F(0) = v, where w({F(b)}b < h) = xaX such that xRFb for all b < h. 8

A similar assumption can be found in the work of Lipman (1991).

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Then, we have the following result (which is based on the so called Hypergame paradox) (Bernardi and D’Agostino, 1996): Proposition 1. Given a relation R on a set X, there does not exist iaX such that for all vaX: vRi if and only if v is founded w:r:t: R Proof of Proposition 1. Suppose that such i exists. This means that vRi if and only if v is founded w:r:t: R Then i is founded w.r.t. R. The proof of this claim is as follows: Suppose i is not founded. Then, there exists a F such that F(0) =i and for every ordinal h, F(h)RF(b) for every b < h. In particular F(1)RF(0). That is, F(1)Ri. Therefore, F(1) is founded and therefore i must also be founded. Absurd. Since i is founded, iRi. But then, just defining a recursive function F as F(h) = i for every ordinal h we see that i is not founded. Absurd. 5 Before we apply this result to the construction of limit beliefs, we will try to give an intuition of its meaning. An object is said founded with respect to a relation if it cannot generate an endless transfinite descending chain (according to the order defined by the relation). Proposition 1 says that there cannot exist a ‘‘common ancestor’’ (in the relation) for all the founded elements. The proof is based on a paradox: if i is a common ancestor for all founded elements, then i itself is founded. Therefore, i is an ancestor of itself and so, i generates an infinite descending chain of is. That is, i is not founded. In other words i is founded ! i is not founded This expression is actually a variant of the Hypergame paradox ( p ! I p).9 An obvious requirement for the validity of Proposition 1 is that R must be an irreflexive relation. That means that for no xaX, it is true that xRx. With this proviso at hand, we can apply these notions to the construction of limit beliefs: Definition 6. Consider the Harsanyi hierarchy BEL ¼ hB0 ðxÞ; B1 ðxÞ; B2 ðxÞ; . . . ; Ba ðxÞ; Baþ1 ðxÞ; . . .i This sequence defines implicitly an ordering on the collection of beliefs about x. Given ¯ b2 if and only if there exist two ordinal two possible beliefs b1, b2aX, we say that b1 R numbers, k1, k2, k1>k2 and b1 = Bk1(x), b2 = Bk2(x). Therefore, there exists a transfinite recursive function F¯ such that for each ordinal h, F¯ = w({F¯ (b)}b < h) with F¯ (0) = s, ¯ F¯ (b) for all b < h. where w({F¯ (b)}b < h) = xaBEL such that R 9 Call a game (in common usage) standard if every play ends in a finite number of steps. Hypergame is a game in which player 1 chooses a standard game, which is played from then on. Is Hypergame standard or not? If it is, then player 1 can choose Hypergame, then player 2, can choose also Hypergame, and so on without end. So, if Hypergame is standard, it implies that it is not standard (Zwicker, 1987).

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Now we can apply Proposition 1 to prove the following: Proposition 2. Either B(x) is a fixed point or for every ordinal b BðxÞ p Bb ðxÞ Proof of Proposition 2. There are two cases: ¯ is an irreflexive ordering. Suppose that there exists b* such that B(x) = Bb*(x).  R

Consider the entire hierarchy BEL={Bb(x)}b>0, which is equivalent to the bounded hierarchy BELV= hB0(x),B1(x),. . .,Bb*(x)i. Therefore, B0(x) = s (the state of nature) is ¯ for every b z 1, each Bb(x) is founded. By ¯ Thus, since Bb(x)Rs, founded w.r.t. R. Proposition 1, there does not exist an s verifying this. Absurd. ¯ is not an irreflexive ordering. Therefore there exists a belief b* such that b* Rb*. ¯  R ¯ ¯ Then, there exists an ordinal b* such that F (b*) = b* and for every b>b*, F (b) = b*. ¯ )}bV< b = w({. . .,b*,b*. . .}). That is, b* is a fixed ¯ = w({F(bV This means that b*=F(b) point for w. 5 In other words, B(x) must be a fixed point or else it lacks a characterization as a mathematical object, since it cannot be seen as a limit of a sequence.10 This argument applies also for the Bayesian approach, since every belief Bn(x) is defined for a finitely generated space. The limit belief is defined on an infinitely generated space and therefore, it lies outside the domain of the elements of the sequence. In fact, it is isomorphic to its image under the belief operator, and not a limit in the topological sense. In the framework of Category Theory, it constitutes a fixed point (Vassilakis, 1991). In any case, as said, the belief operator does not necessarily preserve limits (in either a topological ¯ cannot or category-theoretical sense). In other words, the order relation that it generates, R be assumed to be reflexive. To see why this is a serious problem, it can be useful to recall the example in Section 1 to convey the full implications of Proposition 2. It shows that agents build their beliefs in a systematic form. So, the Cournot competitors have to ponder about the possibilities of their rivals in order to reach a stage where they have no further reasons to change their minds. Our result shows that, using the language of dynamic systems, this ‘‘steady state’’ must constitute an unstable state, since otherwise, the agents would be always able to get to it in their reasoning process. That is, either the agents reach that stable state of mind just because of the form of their beliefs or they will have to keep thinking forever! So, to provide a generic foundation for states of the world, we have to look for an alternative set-theoretical characterization of B(x) in order to show that the agents’ 10

A related argument can be found in the work of Fagin et al. (1999). The goal there is to compare the structure of the hierarchy—when beliefs are represented as probability distributions—with the structure obtained when beliefs are represented as sets of (believed) sentences. The authors find that in the last case, the hierarchy must also be transfinite, but it stops at a countable ordinal. In both cases, they assume continuity.

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reasoning processes, no matter how many discontinuities they may exhibit, will inevitably lead to the state of the world.

4. States of the world as non-well-founded objects Since the strategy of unfolding constructively the beliefs held at a state of the world seems to lead to infinite regresses, unless they constitute fixed points, we have to contemplate their circular characterization as inherent. That is, to accept their circularity as primitive, instead of seeing them as limits of hierarchies of well-behaved beliefs. The discussion on the foundations of mathematics at the beginning of the last century was haunted by the presence of circularities (recall, for example, Russell’s paradox). Therefore, a good deal of e.ort was devoted to eliminate them from mathematics, and particularly from its foundations. On one hand, predicative (non-circular) definitions were sought for every mathematical notion and replaced, when possible, characterizations generated by applications of what Russell called the Vicious Circle Principle: no totality can contain members definable only in terms of this totality (Wang, 1963). This was accompanied, as well, by the creation of new theories of sets, which all had axioms explicitly designed to avoid the presence of circularities. The most important theory of sets in contemporary mathematics is Zermelo – Frenkel’s. It was intended as a form to characterize proper sets defined by means of a bounded11 number of iterated application of operations (like pairing, union, powerset or taking images under set-operations). It includes the following axiom, called the Foundation (Fundierungs) Axiom or Regularity Axiom (Devlin, 1993): axFðxÞ ! ay½FðyÞ ^ bz Iðzay ^ FðzÞÞ Here the variables x, y, z represent sets12 and F a property of sets. The axiom means that if there is a set x verifying a property F, there must exist another set y verifying the same, such that it does not contain any element that verifies F as well. In other words, if F is verified at all, there is no unending descendent chain of sets verifying the property. Circular characterizations usually violate this Axiom. This is the case of our definition of state of the world: Proposition 3. The definition of x as hs, B(x)i is not definable in Zermelo – Frenkel’s set theory (ZFC). Proof of Proposition 3. Trivial. Just define F(x) u ‘‘x = hs,B(x)i’’. Then, being x a state of the world, it must verify Proposition 2. If B(x) does not coincide with any element of BEL—a sequence indexed by Ord—it cannot be identified with any bounded subsequence of BEL, and therefore, it is not a set but a proper class (see Devlin, 1993). Since ZFC characterizes only proper sets, x is not definable there. If, instead, B(x) is a fixed point, then B(B(x)) = B(x). In fact, call bB(x) the result of applying b times B(S) over B(x). It is immediate that bB(x) = B(x). It follows that for every baOrd, b>1, F(bB(x)) and 11 12

That is, there exists a caOrd that indexes the sequence. Objects in mathematical practice can be seen as sets.

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F(b  1B(x)) are valid (0B(x) = x), while bB(x) a b  1B(x) (because b  1B(x) = hs, b B(x)i). Therefore, it violates the Foundation axiom and thus is not definable in ZFC. 5 ¯ in the Proof of Proposition 2 is Corollary. The Foundation axiom implies that R irreflexive. ¯ is not irreflexive. This means that there exists at least one baOrd Proof. Suppose that R such that B(x) = Bb(x). That is, B(x) is a fixed point. But this violates the Foundation axiom, according to the Proof of Proposition 3. 5 This last result shows that the elimination of circularities prescribed by ZFC can be the source of new problems in the set-theoretical foundations of mathematics. This realization lead, in the last 20 years, to a renewed interest on circular (also called non-well-founded) phenomena. A consequence of this line of inquiry has been the development of an alternative theory of sets, which keeps all other axioms of ZFC but drops the Foundation axiom (Aczel, 1988; Devlin, 1993; Barwise and Moss, 1996). Another axiom is added, called in general the Anti-Foundation Axiom (AFA), which has been presented in several versions. Before we introduce the formulation of AFA that better fits for our analysis, we define two new entities: Definition 7.  A general system of equations is e = hX, A, ei, where X is a set of indeterminates, A a set

of ‘‘constants’’, X \ A = h and e: X ! V(X [ A), provides the equations (where V(X \ A) is the class of sets build up from elements in A). Equations have the following form: x = e(x)aV(X \ A).  A solution to e is a function so¯l on X, such that so¯l(x) = so¯l(e(x)). This so¯l is a substitution function, which assigns to each indeterminate a set in V(A) (i.e., without indeterminates and constructed entirely on elements of A). In the case of states of the world, we take X = V, A = S, i.e., our indeterminates are states of the world, and the constants are states of nature. Then, we have the following version of the Anti-Foundation Axiom: Definition 8. Solution Lemma: for every general system of equations e, there exists a unique solution so¯l. Notice that the replacement of the Foundation Axiom for the Solution Lemma constitutes a very conservative move. The new set theory (called sometimes ZFA) is such that every set A definable in ZFC is also definable in ZFA (Barwise and Moss, 1996). In particular, this is true for the ordinal numbers, since they admit a definition which does not require the Foundation Axiom.13 Therefore, all the constructions that can be made over the ordinal numbers, like our characterization of BEL, are valid in ZFA.

13

A set A a ZFA is an ordinal if and only if it is a transitive, linearly ordered by a and no member of A is reflexive (Barwise and Moss, 1996, pp. 86). In this characterization of the set A, transitive means that if B a A, for all C a B, C a A. To say that A is reflexive means that AaA.

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On the other hand, the role of privilege that ordinal numbers enjoy in ZFC can still be kept in ZFA. This is because it is possible to write down a model for ZFA based on Ord. This model obtains as an instance of a more general construction of a model for ZFA. Let us first consider a model M of ZFC in which the sets of solutions to bisimilar equations are deemed identical.14 In words: if two equations are bisimilar, their solutions are considered identical. Then M can be extended to a model MAFA of ZFA. The crucial step consists in considering that each definition of a set in ZFA is the solution to a general system of equations. This solution corresponds, in turn, to a class of equivalence in M, where the equivalence is among all the sets that verify the general system of equations. The set of equivalence classes of M is called MAFA (Devlin, 1993; Barwise and Moss, 1996). To build up a model based on Ord, it is possible to take as M the inner model of ZFC, i.e., the minimal transitive class that includes Ord. For a set A defined in ZFA, its corresponding equivalence class over this M will be defined as all the sets of ordinals that verify the equations for which A is a solution. If A is already definable in ZFC, its corresponding equivalence class is a singleton set. In the case of sets definable only in ZFA, the Axiom of Choice indicates that there exists a systematic way to select one representative element from each equivalence class. In our case, the Solution Lemma indicates that the equation x ¼ hs*; BðxÞi has a unique solution, x* = O(s*), a set entirely constructed upon s*.15 This solution is, by construction, based on the endless transfinite hierarchy of Proposition 2: Proposition 4. The solution to equation x ¼ hs*; BðxÞi is Oðs*Þ ¼ hs*; BELc ðxÞi, for a caOrd. Proof of Proposition 4. We know that B(x) can be unfolded as the sequence BEL. By Proposition 2, either B(x) is a fixed point or for every ordinal Bb(x) a BEL, B(x) p Bb(x). Bb(x). By Proposition 3, we know that either case violates the assumptions of ZFC, but both violations stem—as shown by the Corollary to Proposition 3—from the Foundation Axiom. In ZFA, instead, by the Solution Lemma, there exists a solution O(s*), i.e., based on the ‘‘constant’’ s*. In either case, the solution must be an equivalence class in a model of ZFC, and any of its elements can be taken as the representative. However, each element in a model of ZFC corresponds to a proper set. In our case, this implies that there must exist a caOrd such that B(x) is a fixed point for the bounded sequence hs*, B1(x), B2(x),. . ., Ba(x),. . .Bc(x)i. More precisely, B(x) = Bc(x). 5 In other words: in an Anti-Founded Set Theory, a state of the world obtains as a proper subsequence of the endless unfolding postulated in Section 3. That means that B(S) always has a fixed point, even if it is not continuous. The following result resumes this conclusion.

14

Two equations are said bisimilar if for every indeterminate and every constant in one, it corresponds an indeterminate and a constant in the other. 15 See Heifetz (1996) for an application of Non-Well-Founded Set Theory to the related notion of type of an agent.

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Theorem 1. A state of the world x, given its underlying state of nature s* obtains as a fixed point of the belief-formation operator B(S). Proof of Theorem 1. Follows immediately from Propositions 2 and 4.

5

5. Non-definability of states of the world The results reported in the previous section depend on a very critical feature of the construction of a language-based hierarchy of beliefs as presented in Definition 2. The critical assumption made implicitly in Section 2 is that, for every stage n and for every /n  1, there exists a /n = B(/n  1)aLn. Without this assumption, it cannot be ensured that at each stage n, every belief about stage n  1 is expressible in Ln. This leaves open the possibility of having a sequence that stops abruptly at a non-representable belief which in no way is a fixed point. That is, without that assumption, there may not exist a welldefined state of the world.16 This issue is also crucial for the existence of a state of the world supporting a rationalizable outcome. If there are beliefs that are not expressible, the agents cannot have a complete set of beliefs to choose from to rationalize every possible action they see performed by the others (Stalnaker, 1998). The following example exhibits a very simple case of a state of the world that cannot be represented in any consistent language (Brandenburger and Keisler, 1999): Example 3. Consider a simple situation with two players 1 and 2. Assume that 1 has the following structure of beliefs B1: /0: Whatever 1 believes about the state of nature s. ... /b: 2 is mistaken about B1. /b + 1: 2 believes that 1 believes /b. The question is that if 1 thinks that 2 is right then 1 believes that 2 is wrong about Bi. On the other hand, assume that 1 thinks that 2 is wrong. Then 1 does not believe that 2 is mistaken about B1. That is, that 2 is right. In other words: If 1 believes that 2 is right ! 1 believes that 2 is wrong and If 1 believes that 2 is wrong ! 1 believes that 2 is right Therefore, any /b + 2 leads to a contradiction. However, this means that B1 cannot be represented in a consistent system.

16

An interesting question to be answered is whether a belief that almost represents the true state of affairs does not lead to the definition of a state of the world that is close to the true one (Anderson, 1986).

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This is again a problem related to the circularity of the characterization of states of the world. But this time with the added complication of the existence of self-reference in the definition: a state of the world is such only if the agents believe that it is a possible state of the world. It is not surprising, then, that negative results arise in such a setting.17 Example 3 shows that negative beliefs in a self-referential expression may lead to contradictions. This means that a consistent reasoner should not held some structures of beliefs, in particular those that combine self-reference with negation. But if she does, the resulting state of the world cannot be derived as in Definition 2. This paradox is of another type than Newcomb’s, discussed in Example 2. It is more akin to the famous Liar paradox that postulates a sentence / defined as ‘/ is false’, which if true is false or if it is assumed false is true. This paradox is an intuitive background for technical results of impossibility. In what follows, we will introduce a variant of Tarski’s indefinability theorem (Boolos and Jeffrey, 1989) to prove the impossibility of deriving each possible state of the world. To see that this problem is of concern for the definition of a Harsanyi structure of beliefs, consider the following result: Lemma 1. For each Harsanyi structure BEL = hB0(x), B1(x), B2(x),. . ., Ba(x), Ba + 1(x),. . .i, there exists a hierarchy of languages hL0,. . ., Ln, . . ., La, La + 1, . . .i, such that X={b: b is a belief about x}pLX, the universal language for the description of states of the world. Moreover, Bb a Lb, for each b = 0,. . .,Length(BEL)  1. Proof of Lemma 1. Immediate from the fact that a belief about s, B0(x)) can be expressed as /0, even in the case of Bayesian beliefs, and that for n>0, Bn + 1(x), a belief about Bn(x), can be expressed as /n + 1 a Ln + 1. On the other hand, the definition of language-based hierarchies of beliefs does not require the continuity of beliefs. Therefore, this representation of beliefs is still valid for each baOrd. Since BEL has a limit by definition and in ZFA, this limit is reached for a caOrd, the languages from Lc + 1 do not matter for the equivalence.18 That is, each Bb(x) can be expressed as a /b, for b V c, where c = Length(BEL)  1. 5 According to this result, the properties of LV matter for the proper characterization of any Harsanyi structure. A first property, of metamathematical import, is related to the consistency of beliefs:19 Definition 9. Given LX, the universal language for the description of states of the world, a subset L¯ is said to be belief-consistent if  It verifies all the (classical logic) (in particular the principle of non-contradiction:

¯ I(/^ I /), for every / aL). 17

Just consider the abundance of negative results for systems with the potential for self-reference (Boolos and Jeffrey, 1989). 18 In ZFC, this limit may not be reached, and therefore BEL may be indexed by the entire class Ord. But even so, there exists a Lb for each baOrd. 19 In what follows, we will use B for the more appropriate Bi and in general L for Li to simplify a bit the notation.

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 If /aL is derived according to Definition 2 from saS, B(/)a L¯ while if / is not

¯ derived, IB(/)a L. ¯ but not both of them. ¯  For every / aL, either B(/) a L¯ or IB(/) a L,

This means that a belief-consistent set is a theory (either propositional or of higher order) in which each expression is either believed or not. Another metamathematical property of importance is completeness: ¯ we say that V is definable complete in L if for every Definition 10. Given a language L, ¯ such that q/a = A, then B(/)aL. structure A over V, there exists /aL In words: the universe of states of the world is definable complete if for every possible structure of states of the world, a belief can be defined that coincides with it. Now we have the following result by Smullyan (1992): Proposition 5. Let L¯ be a belief-consistent set of expressions defining states of the world and X the space of states of the world. Then, there exists an expression / such that q/a is a ¯ structure over X but / g L. Proof of Proposition 5. Trivial. Let / a LV (the general language for the description of states of the world), / u IB(/). Since there exists states of the world in which agents ¯ Then, if / may have such beliefs, we have that q/a is defined over V. Assume that /aL. is derivable from a hierarchical construction (as defined in Definition 2), it follows that, on one hand B(/) (by the characterization of a consistent set) and IB(/) (by the definition of /) on the other. If instead we assume that / is not derivable, if follows that B(/) (by the definition of /) while IB(/) (by the definition of a consistent set). ¯¯ Therefore, in either case, B(/) and IB(/) are in L. Absurd. Therefore, / is not in L while q/a is a structure over V. 5 As an immediate corollary we have that ¯ Theorem 2. If L¯ is belief-consistent then X is not definable-complete in L. Proof of Theorem 2. Take A = q/a, a structure over V, where / = I B(/). According to the ¯ 5 Proof of Proposition 5, / g L. This result holds to the extent in which we allow (negative) self-referentiality in the definition of beliefs. In the case in which / u IB(/), if we assume /aLn, since its equivalent expression IB(/) is also in Ln, we have that /aLn  1. This feature can be avoided if in the hierarchical construction in Definition 2 we require that if /aLn, /gLk for k < n. This clear distinction between object and meta-levels seems to be fundamental in general for the avoidance of the negative results that arise in the languages such that their meta-languages are representable in their interpretation (Paulos, 1976). Therefore, we may restrict our language to include only those expressions constructed hierarchically. We have that: ¯ be the space of states of the world that can be represented in Theorem 3. Let X L = vn z 0Ln, constructed according to Definition 2 with the proviso that for each / a Ln, / g Lk for k < n. Then L is belief-consistent and X¯ is definable-complete in L.

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Proof of Theorem 3. Trivial. L is belief-consistent since there is no /aL, derived according to Definition 2 that verifies both B(/) and IB(/). On the other hand, by ¯ is representable by a /aL and since L is beliefdefinition, each structure over V ¯ , there exists a consistent, B(/)aL. By the same token, we have that for each A over V /Va L (for example, /V=w*aU V, where w* indicates the real state of the world) and since /V is derivable, we have that B (/V) a L. 5 Of course this leaves some states of the world out of the hierarchical construction. This is a rather negative result since it makes the agents incapable of rationalizing each possible action of the others.20 On the other hand, it seems extremely unlikely that negative selfreferential statements may arise in interactions among rational agents, but this is matter of further work. 6. Conclusions In this paper, we revisited the notion of state of the world, showing that it can be applied in various economic and game-theoretic contexts. Its essential circular nature arises from the fact that it is defined in terms of the beliefs held by the agents, which in turn depend on the state of the world. We expanded the notion of ‘‘belief’’ in order to encompass all possible cases of belief generation. Bayesian beliefs, represented as consistent probability distributions, seem too narrow as a representation of processes of rational deliberation in decision-making, particularly when other parties are involved. We examined the procedure of unfolding the beliefs held at a given state of the world. We saw that, when the belief operator is not in general continuous, the only case in which this procedure is feasible occurs when the beliefs to be unfolded constitute already a fixed point for the operator. Otherwise, the sequence of beliefs is (transfinitely) endless. In the framework of Non-Well-Founded Set Theory, however, we can show that there must always exist a fixed point for the belief operator. Therefore, states of the world are defined in terms of that operator. The classical results for Bayesian belief generation become particular cases of this one. The caveat is that the unfolding of beliefs works only if no negative self-referential beliefs are allowed in the definition of states of the world. With this proviso, each possible state of the world constructed hierarchically obtains as a fixed point. Instead, if we do not restrict the kind of beliefs that can be held, there will exist states of the world that cannot be unfolded. This fact shows that Harsanyi’s procedure does not allow for a full account of states of the world. This has also important consequences for the use of solution concepts like rationalizability, which requires that each agent should be able to rationalize each possible situation. It is possible to think of situations that arise from self-defeating beliefs and in which no rational (consistent) agent can rationalize them. On the other hand, it is immediate that this kind of beliefs cannot be unfolded hierarchically. If we focus only on 20

A quite similar result has been derived by Brandenburger and Keisler (1999).

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‘‘unfoldable’’ states of the world, the results are instead positive: rational reasoners will be able to rationalize every possible action of the others. The question that remains to be answered is whether negative self-referential beliefs may appear in a game-theoretical setting in a meaningful form. If true, this realization may erode the foundations of Game Theory.

Acknowledgements Thanks are due to Professors Adam Brandenburger, Larry Moss, John Nachbar, and Julio Olivera for the useful conversations and helpful comments. Of course, errors and missinterpretations are my own responsability. I wish also to thank the Group of Logic and Methodology of the Department of Mathematics, University of California, Berkeley for their hospitality and support during my stay as a Fulbright Scholar during the winter of 2003. This research was partially supported by Secretari´a de Ciencia y Te´cnica of the Universidad Nacional del Sur and by a grant of the National Council of Scientific Research of Argentina (CONICET).

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