Agent's Beliefs Research Articles

The hierarchical approach to modeling knowledge and common knowledge

One approach to representing knowledge or belief of agents, used by economists and computer scientists, 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 world, his beliefs about other agents' beliefs about other agents' beliefs about the world, and so on. (Economists have typically modeled belief in terms of a probability distribution on the uncertainty space. In contrast, computer scientists have modeled belief in terms of a set of worlds, intuitively, the ones the agent considers possible.) We consider the question of when a countably 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 a countable hierarchy suffices in this case. These conditions also show that whether a countable hierarchy suffices may depend on the “richness” of the states in the underlying state space. We also consider the question of whether a countable hierarchy suffices for “interesting” sets of events, and show that the answer depends on the definition of “interesting”.

Imprecise Beliefs in a Principal Agent Model

This paper presents a principal-agent model where the agent has multiple, or imprecise, beliefs. We model this situation formally by assuming the agent's preferences are incomplete. One can interpret this multiplicity as an agent's limited knowledge of the surrounding environment. In this setting, incentives need to be robust to the agent's different beliefs. We study whether robustness implies simplicity. Under mild conditions, we show the unique optimal contract has a two-wage structure; a flat payment and bonus. That is, all output levels are divided into two groups, and the optimal incentive scheme pays the same amount for all output levels in each group. We also show that a two-state two-action framework can be thought of as a reduced form of the original model. We solve explicitly the principal's problem in this case, and discuss some implications of our model for firm ownership.

Endogenous Uncertainty and Market Volatility

Endogenous Uncertainty is that component of economic risk and market volatility which is propagated within the economy by the beliefs and actions of agents. The theory of Rational Belief (see Kurz [1994]) permits rational agents to hold diverse beliefs and consequently, a Rational Belief Equilibrium (in short, RBE) may exhibit diverse patterns of Endogenous Uncertainty. This paper shows that most of the observed volatility in financial markets is generated by the beliefs of the agents and the diverse market puzzles which are examined in this paper, such as the equity premium puzzle, are all driven by the structure of market expectations. To make the case for this theory we present a single RBE model, which builds on developments in Kurz and Beltratti [1997] and Kurz and Schneider [1996], with which we study a list of phenomena that have been viewed as anomalies in financial markets. The model is able to predict the correct order of magnitude of: (i) the long term mean and standard deviation of the price\dividend ratio;(ii) the long term mean and standard deviation of the risky rate of return on equities;(iii) the long term mean and standard deviation of the riskless rate;(iv) the long term mean equity premium.In addition, the model predicts (v) the GARCH property of risky asset returns;(vi) the Forward Discount Bias in foreign exchange markets.We also conjecture that an adaptation of the same model to markets with derivative assets will predict the appearance of smile curves in derivative prices.The common economic explanation for these phenomena is the existence of heterogenous agents with diverse but correlated beliefs. Given such diversity, some agents are optimistic and some pessimistic. We develop a simple model which allows agents to be in these two states of belief but the identity of the optimists and the pessimists fluctuates over time since at any date any agent may be in these two states of belief. In this model there is a unique parameterization under which the model makes all the above predictions simultaneously. That is, although the parameter space of the RBE is large, all parameterizations outside a small neighborhood of the parameter space fail significantly to reproduce some subset of variables under consideration. Any parameter choice in this small neighborhood requires the optimists to be in the majority but the rationality of belief conditions of the RBE require the pessimists to have a higher intensity level. This higher intensity has a decisive effect on the market: it increases the demand for riskless assets, decreases the equilibrium riskless rate and increases the equity premium. In simple terms, the large equity premium and the lower equilibrium riskless rate are the result of the fact that at any moment of time there are agents who hold extreme pessimistic beliefs and they have a relatively stronger impact on the market. The relative impact of these two groups of agents who are, at any moment of time, in the two states of belief is a direct consequence of the rationality of belief conditions and in that sense it is unique to an RBE. As for the correlation among the beliefs of agents, the paper shows that the dynamics of asset prices are strongly affected by such correlation. The pattern of correlation which was used in the model can be explained intuitively in terms of its effect on the dynamics of prices. The model correlation causes periods of price rises (i.e. bull markets) to develop slower than periods of decline (i.e. bear markets) hence the model dynamics do not permit prices to shoot directly from the bottom to the top but the opposite is possible and takes the form of market crashes.

Coherence and truth conducive justification

Since Klein and Warfield (1996) insist that it is coherentists that they are addressing, and since the coherence theory of justification advanced by Laurence BonJour (1985) seems to be the theory that most influences their own discussion, one would expect their argument against the truth conduciveness of coherentist justification to apply, if at all, to BonJour's theory. In what follows I will show that this is not the case. I will also show that an important premiss is omitted from Klein's and Warfield's argument against the truth conduciveness of justification as understood in the version of coherentism they explicitly consider, which I will call the Naive Coherence Theory. According to the Naive Coherence Theory, an agent's belief set is justified iff that set is epistemically coherent, the degree of justification of the belief set being defined as the degree of its coherence. Fortunately, the premiss that Klein and Warfield omit can be defended without begging the question against the Naive Coherence Theory. The same cannot be said of an analogous premiss that would make Klein's and Warfield's argument effective against BonJour's theory. In BonJour's version of coherentism, as BonJour takes pains to point out, the coherence of an agent's beliefs at a particular moment is not a sufficient condition for their being justified. BonJour writes:

A modal logic for fusing partial belief of multiple reasoners

We present PLn, a multi-agent epistemic logic in which each agent can perform uncertain (possibilistic) reasoning. The original feature of this logic is the presence of a distributed belief operator, with the purpose of merging the belief of different agents. Unlike the corresponding operator in the categorical (non-uncertain) case, our distributed belief operator accumulates support for the same fact coming from different agents. This means that opinions shared by different agents can be combined into a stronger distributed belief. This feature is useful in problems like pooling expert opinions and combining information from multiple unreliable sources. We provide a possible worlds semantics and an axiomatic calculus for our logic, and prove soundness, completeness and decidability results. We hint at some possible applications of PLn in the conclusions

Resource Bounded Belief revision

The AGM paradigm for belief revision provides a very elegant and powerful framework for reasoning about idealized agents. The paradigm assumes that the modeled agent is a perfect reasoner with infinite memory. In this paper we propose a framework to reason about non-ideal agents that generalizes the AGM paradigm. We first introduce a structure to represent an agent's belief states that distinguishes different status of beliefs according to whether or not they are explicitly represented, whether they are currently active and whether they are fully accepted or provisional. Then we define a set of basic operations that change the status of beliefs and show how these operations can be used to model agents with different capacities. We also show how different operations of belief change described in the literature can be seen as special cases of our theory.

Iterated belief revision, reliability, and inductive amnesia

Belief revision theory concerns methods for reformulating an agent's epistemic state when the agent's beliefs are refuted by new information. The usual guiding principle in the design of such methods is to preserve as much of the agent's epistemic state as possible when the state is revised. Learning theoretic research focuses, instead, on a learning method's reliability or ability to converge to true, informative beliefs over a wide range of possible environments. This paper bridges the two perspectives by assessing the reliability of several proposed belief revision operators. Stringent conceptions of “minimal change” are shown to occasion a limitation called inductive amnesia: they can predict the future only if they cannot remember the past. Avoidance of inductive amnesia can therefore function as a plausible and hitherto unrecognized constraint on the design of belief revision operators.

Three Conceptions of Rational Agency

Rational agency may be thought of as intentional activity that is guided by the agent's conception of what they have reason to do. The paper identifies and assesses three approaches to this phenomenon, which I call internalism, meta-internalism, and volitionalism. Internalism accounts for rational motivation by appeal to substantive desires of the agent's that are conceived as merely given; I argue that it fails to do full justice to the phenomenon of guidance by one's conception of one's reasons. Meta-internalism explains this phenomenon by postulating higher-order dispositions, consitutive of (rational) agency itself, which causally interact with the agent's normative beliefs to produce corresponding motivations to action. I show that meta-internalism comes to grief over cases of akrasia, insofar as it leaves no room for the capacity for rational guidance when agents voluntarily act at variance with their judgments about what they have reason to do. Volitionalism, I contend, improves on both internalism and meta-internalism. Its distinctive feature is the postulation of a kind of motivation that is directly subject to the agent's control, and independent of the dispositions and desires to which the agent is passively subject.

Mapping Moral Motivation

In this paper we defend a version of moral internalism and a cognitivist account of motivation against recent criticisms. The internalist thesis we espouse claims that, if an agent believes she has reason to A, then she is motivated to A. Discussion of counter-examples has been clouded by the absence of a clear account of the nature of motivation. While we can only begin to provide such an account in this paper, we do enough to show that our version of internalism can be defended against putative counter-examples. All theories of motivation which take what motivates to be a psychological state run foul of the following plausible constraint: the reason why you ought to do an action and the reason why you do it can be the same. In our view, however, while what motivates is a reason (which is a fact) the state of being motivated is a cognitive stage, viz. the belief that one has reason to act. In cases where the agent's relevant beliefs are false, then she has no reason to act, but nontheless her action can be explained in other ways.

Endogenous Uncertainty: A Unified View of Market Volatility

Endogenous Uncertainty: A Unified View of Market Volatility

Social States of Belief and the Determinants of the Equity Risk Premium in A Rational Belief Equilibrium

September 4, 1997 We review the issues related to the formulation of endogenous uncertainty in rational belief equilibria(RBE). In all previous models of RBE, individual states of belief were the foundation for the construction of the endogenous state space where individual states of belief were described with the method of assessment variables. This approach leads to a lack of anonymity where the belief of each individual agent has an impact on equilibrium prices but as a competitor he ignores it. The solution is to study a replica economy with a finite number of types but with a large number of agents of each type. This gives rise to type-states which are distributions of beliefs within each type. The state space for this economy is then constructed as the set of products of the exogenous states and the social states of belief which are vectors of distributions of all the types. Such an economy leads to RBE which do indeed solve the problem of anonymity. We then study via simulations the implications of the model of RBE with social states for market volatility and for the determinants of the equity risk premium in an RBE. Under i.i.d. assessments one uses the law of large numbers to induce a single social state of belief and we show that the RBE of such economies have the same number of prices as in rational expectations equilibrium (REE). However, the RBE may exhibit large fluctuations if agents are allowed to hold extreme beliefs. Establishing 5% boundary restrictions on beliefs we show that the model with a single social state of belief cannot explain all the moments of the observed distribution of returns. We then introduce correlation among beliefs and this leads to the creation of new social states. We next show that under correlation among beliefs the model simulations reproduce the values of four key moments of the empirical distribution of returns. The observed equity premium is then explained by two factors. First, investors demand a higher risk premium to compensate them for the endogenous increase in the volatility of returns. Second, at any moment of time there are both rational optimists as well as rational pessimists in our financial markets and such a distribution leads automatically to a decrease in the riskless rate and to an increase of the risk premium. We show that correlation among beliefs of agents leads to fluctuations over time in the social distribution of beliefs and such fluctuations add to endogenous volatility and to a higher equilibrium equity risk premium. JEL Classification Numbers: D58, D84, G12.

A Tableau-Based Proof Method for Temporal Logics of Knowledge and Belief

ABSTRACT In this paper we define two logics, KLn and BLn, and present tableau-based decision procedures for both. KLn is a temporal logic of knowledge. Thus, in addition to the usual connectives of linear discrete temporal logic, it contains a set of unary modal connectives for representing the knowledge possessed by agents. The logic BLn is somewhat similar; it is a temporal logic that contains connectives for representing the beliefs of agents. In addition to a complete formal definition of the two logics and their decision procedures, the paper includes a brief review of their applications in AI and mainstream computer science, correctness proofs for the decision procedures, a number of worked examples illustrating the decision procedures, and some pointers to further work.

AAAI-96 Workshop on Agent Modeling

Agent modeling -- the ability to model and reason other agents' knowledge, beliefs, goals, and actions -- is central to intelligent interaction. The Workshop on Agent Modeling, held as part of the Thirteenth National Conference on Artificial Intelligence, was organized to bring together researchers working in these areas to assess the state of the art and discuss the common issues in representation and reasoning with models of agents.

An Epistemic Dimension of Blameworthiness

The author first argues against the view that an agent is morally blameworthy for performing an action only if it is morally wrong for that agent to perform that action. The author then proposes a replacement for this view whose gist is summarized in the principle: an agent S is morally blameworthy for performing action A only if S has the belief that it is wrong for her to do A and this belief plays an appropriate role in S's Aing. He focuses on explicating the role an agent's belief that a prospective action, A, of hers is wrong must play in the production of her A-ing in order that she be blameworthy for A-ing. Towards this end, the author makes use of cases involving akrasia and selfdeception.

Convergence in Economic Models with Bayesian Hierarchies of Beliefs

I study a model where hierarchies of beliefs (the beliefs about the beliefs of other agents, etc.) are important. I provide conditions under which optimal actions of agents will converge to the Nash equilibrium of the model characterized by the true, previously unknown “fundamentals.” The conditions are (i) a contraction property on the best-response mappings and (ii) a mutual absolute continuity condition on beliefs. Violation of (i) may result in an “anything is possible” result: any stochastic process of actions is consistent with maximizing behavior and Bayesian updating. Violation of (ii) may result in cyclical behavior of actions on each sample path.Journal of Economic LiteratureClassification Numbers: C70, C73, D81, D82, D83, D84.

The dynamics of default reasoning

In this paper we study default reasoning from a dynamic, agent-oriented, semantics-based point of view. In a formal framework used to specify and to reason about rational agents, we introduce actions that model the (attempted) jumping to conclusions that is a fundamental part of reasoning by default. Application of such an action consists of three parts. First it is checked whether the formula that the agent tries to jump to is a default, thereafter it is checked whether the default formula can consistently be incorporated by the agent, and if this is the case the formula is included in the agent's beliefs. As for all actions in our framework, we define the ability and opportunity of agents to apply these actions, and the states of affairs following application. To formalise formulae being defaults, we introduce the modality of common possibility. This modality is related to, but not reducible to, the notions of common knowledge and ‘everybody knows’-knowledge. To model the qualitative difference that exists between hard, factual knowledge and beliefs derived by default, we employ different modalities to represent these concepts, thus combining knowledge, beliefs, and defaults in one framework. Based on the concepts used to model the default reasoning of agents, we look into the dynamics of the supernormal fragment of default logic. We show in particular that by sequences of jumps to conclusions agents can end up with extensions in the sense of default logic of their belief.

Coordination and correlation in Markov rational belief equilibria

This paper studies the effect of correlation in the rational beliefs of agents on the volatility of asset prices. We use the technique of generating variables to study stable and non-stationary processes needed to characterize rational beliefs. We then examine how the stochastic interaction among such variables affects the behavior of a wide class of Rational Belief Equilibria (RBE). The paper demonstrates how to construct a consistent price state space and then shows the existence of RBE for any economy for which such price state space is constructed. Next, the results are used to study the volatility of asset prices via numerical simulation of a two agents model. If beliefs of agents are uniformly dispersed and independent, we would expect heterogeneity of beliefs to have a limited impact on the fluctuations of asset prices. On the other hand, our results show that correlation across agents can have a complex and dramatic effect on the volatility of prices and thus can be the dominant factor in the fluctuation of asset prices. The mechanism generating this effect works through the clustering of beliefs in states of different levels of agreement. In states of agreement the conditional forecasts of the agents tend to fluctuatetogether inducing more volatile asset prices. In states of disagreement the conditional forecasts fluctuatein diverse directions tending to cancel each other's effect on market demand and resulting in reduced price volatility.

Rational belief structures and rational belief equilibria

The paper introduces some simplifying tools and methods for studying Rational Beliefs and for proving existence of Rational Belief Equilibria. We identify a set of stable non-stationary stochastic processes, named SIDS processes. Furthermore we introduce the concept of a Rational Belief Structure, which formulates the Rational Beliefs of the agents as beliefs about the distribution of exogenous variables and the beliefs of other agents. The use of the developed apparatus is demonstrated by showing existence of a set of Rational Belief Equilibria in an Overlapping Generations Model with money and one commodity.

A semantics for reasoning consistently in the presence of inconsistency

A great deal of research has been devoted to nontrivial reasoning in the presence of inconsistency. However, previous formalisms on this account do not permit consistent reasoning in the presence of inconsistency—they may conclude a statement on one hand and the negation of the statement on the other. In this paper, we propose a logic that allows an agent to reason consistently, even though there are inconsistencies in the agent's beliefs. We first give the semantics of the logic and then present a simple, sound and complete axiomatization for the logic, thus forming the formal basis of reasoning consistently in the presence of inconsistency.

Intransitive choices based on transitive preferences: The case of menu-dependent information

The paper considers decision contexts in which the set of alternatives from which choices have to be made (the ‘menu’) may convey information about the desirability of these alternatives. Our analysis is motivated by the fact that in specifying an appropriate description of a choice situation an outside observer always has to neglect some ‘dimensions’ of the decision problem. The central claim is that properties of observed behaviour may depend crucially on the neglected dimensions even when their influence is arbitrarily small. Specifically, we prove the following result. Suppose that in a discrete choice model an agent's beliefs about the ‘quality’ of the available alternatives depend on the specific menu to choose from. Then, even when the difference in beliefs given different menus is arbitrarily small (but positive), any practicable description of the decision situation necessarily implies that cyclic choices will be observed. The result suggests that transitivity - as a condition on observable behaviour - is rather questionable in any context where one cannot completely exclude the possibility that menu-dependent information may play some role.