A key benefit of Bayesian reasoning is that it stipulates how to optimally integrate unreliable sources of information. The authors present evidence that humans use Bayesian inference to determine how much to trust advice from another person, based on information about that person's knowledge and strategy.

We relax the assumption that priors are common knowledge, in the standard model of games of incomplete information. We make the realistic assumption that the players are boundedly rational: they base their actions on finite-order belief hierarchies. When the different layers of beliefs are independent of each other, we can retain Harsányi's type-space, and we can define straightforward generalizations of Bayesian Nash Equilibrium and Rationalizability in our context. Since neither of these concepts is quite satisfactory, we propose a (...) hybrid concept, Mirage Equilibrium, providing us with a practical tool to work with inconsistent belief hierarchies. When the different layers of beliefs are correlated, we must enlarge the type-space to include the parametric beliefs. This presents us with the difficulty of the inherent openness of finite belief subspaces. Appealing to bounded rationality once more, we posit that the players believe that their opponent holds a belief hierarchy one layer shorter than they do and we provide alternative generalizations of BNE and Rationalizability. Finally, we show that, when beliefs are degenerate point beliefs, the definition of Mirage Equilibrium coincides with that of the generalized BNE. (shrink)

Within this paper we consider a model of Nash bargaining with incomplete information. In particular, we focus on fee games, which are a natural generalization of side payment games in the context of incomplete information. For a specific class of fee games we provide two axiomatic approaches in order to establish the Expected Contract Value, which is a version of the Nash bargaining solution.

Quasi-truth is a formal approach to a pragmatically-oriented view of truth. The basic plan motivating the framework consists in providing for a more realistic account of truth, accommodating situations where there is incomplete information, as typically happens in the practice of science. The historical sciences are a case in hand, where incomplete information is the rule. It would seem, then, that the quasi-truth approach would be the most appropriate one to deal with historical sciences, then. In (...) this paper, we explore this possibility and use the historical sciences as a test case for the approach of quasi-truth. Our claim is that, on what concerns historical sciences, the quasi-truth approach fails in two basic senses; first, by misrepresenting some cases concerning incomplete information, and second, by falling short of accounting for many features of incomplete information peculiar to historical sciences. We conclude that, despite its stated goals, quasi-truth must be either amended or substituted if the goal of a more faithful representation of scientific practice is to be achieved. (shrink)

A mistake in “Five legitimate definitions of correlated equilibrium (CE) in games with incomplete information” motivates a re-examination of some extensions of the solution concept that Aumann introduced.

This article presents four analyses of an interaction between the middle-Bronze Age Pharaoh Nibmuarea and the Babylonian king Kadashman-Enlil as described in the Amarna letters (Moran [1992] The Amarna Letters, The Johns Hopkins Universiy Press, Baltimore, Maryland). Intent on denying the Pharaoh his daughter in marriage, the Babylonian king was faced with the choice of sending messengers who could (''dignitaries'') or could not identify (''non-dignitaries'') his missing sister in the Pharaoh's court. Intent on marrying the king's daughter, the Pharaoh was (...) faced with the choice of showing the sister or showing someone else. Based on the assumption of complete information (game 1), the analysis revealed a dominant-strategy equilibrium: Nibmuarea shows the sister and Kadashman-Enlil sends non-dignitaries. Based on the assumption of one-sided incomplete information (Pharaoh's misperception; game 2), the analysis revealed that the Pharaoh had a dominant strategy of showing the sister irrespective of whether the king is keen or reluctant to learn about his sister's fate. Based on the assumption of one-sided incomplete information (Kadashman-Enlil's misperception; game 3), the analysis revealed that if non-dignitaries are sent, the Pharaoh prefers showing someone other than his sister. Based on the assumption of two-sided incomplete information (game 4), the Pharaoh finds it more beneficial to present the sister irrespective of whether his intentions are genuine or feigned. With incomplete information, it is difficult to judge the other's intentions; the cost of being caught cheating by not showing the sister to knowledgeable messengers was quite high. These analyses highlight the strategic uncertainty that characterized this Bronze-Age interaction. (shrink)

The papers collected in this volume exemplify some of the trends in current approaches to logic, language and computation. Written by authors with varied academic backgrounds, the contributions are intended for an interdisciplinary audience. The first part of this volume addresses issues relevant for multi-agent systems: reasoning with incomplete information, reasoning about knowledge and beliefs, and reasoning about games. Proofs as formal objects form the subject of Part II. Topics covered include: contributions on logical frameworks, linear logic, and (...) different approaches to formalized reasoning. Part III focuses on representations and formal methods in linguistic theory, addressing the areas of comparative and temporal expressions, modal subordination, and compositionality. (shrink)

The Common Prior Assumption (CPA) plays an important role in game theory and the economics of information. It is the basic assumption behind decision-theoretic justifications of equilibrium reasoning in games (Aumann, 1987, Aumann and Brandenburger, 1995) and no-trade results with asymmetric information (Milgrom and Stokey, 1982). Recently several authors (Dekel and Gul, 1997, Gul, 1996, Lipman, 1995) have questioned whether the CPA is meaningful in situations of incomplete information, where there is no ex ante stage and (...) where the primitives of the model are the individuals' beliefs about the external world (their first-order beliefs), their beliefs about the other individuals' beliefs (second-order beliefs), etc., i.e. their hierarchies of beliefs. In this context, the CPA is a mathematical property whose conceptual content is not clear. The main results of this paper (Theorems 1 and 2) provide a characterization of Harsanyi consistency in terms of properties of the belief hierarchies that are entirely unrelated to the idea of an ex ante stage. (shrink)

We study a dynamic auction mechanism in the context of private provision of a discrete public good under incomplete information. The bidders have private valuations, and the cost of the public good is common knowledge. No bidder is willing to provide the good on her own. We show that a natural application of open ascending auctions in such environments fails dramatically: The probability of provision is zero in any equilibrium. The mechanism effectively auctions off the ‘right’ to be (...) the last one to contribute, but intuition suggests that neither player wishes to be the last one to contribute. Since the player who contributes first has the advantage of being able to free ride on the contributions of the other players, no player wants to ‘win’ the auction. (shrink)

Knowledge reduction of information systems is one of the most important parts of rough set theory in real-world applications. Based on the connections between the rough set theory and the theory of topology, a kind of topological reduction of incomplete information systems is discussed. In this study, the topological reduction of incomplete information systems is characterized by belief and plausibility functions from evidence theory. First, we present that a topological space induced by a pair of (...) approximation operators in an incomplete information system is pseudo-discrete, which deduces a partition. Then, the topological reduction is characterized by the belief and plausibility function values of the sets in the partition. A topological reduction algorithm for computing the topological reducts in incomplete information systems is also proposed based on evidence theory, and its efficiency is examined by an example. Moreover, relationships among the concepts of topological reduct, classical reduct, belief reduct, and plausibility reduct of an incomplete information system are presented. (shrink)

There are several three-valued logical systems that form a scattered landscape, even if all reasonable connectives in three-valued logics can be derived from a few of them. Most papers on this subject neglect the issue of the relevance of such logics in relation with the intended meaning of the third truth-value. Here, we focus on the case where the third truth-value means unknown, as suggested by Kleene. Under such an understanding, we show that any truth-qualified formula in a large range (...) of three-valued logics can be translated into KD as a modal formula of depth 1, with modalities in front of literals only, while preserving all tautologies and inference rules of the original three-valued logic. This simple information logic is a two-tiered classical propositional logic with simple semantics in terms of epistemic states understood as subsets of classical interpretations. We study in particular the translations of Kleene, Gödel, ᴌukasiewicz and Nelson logics. We show that Priest’s logic of paradox, closely connected to Kleene’s, can also be translated into our modal setting, simply by exchanging the modalities possible and necessary. Our work enables the precise expressive power of three-valued logics to be laid bare for the purpose of uncertainty management. (shrink)

We investigate a speci c model of knowledge and beliefs and their dynamics. The model is inspired by public announcement logic and the approach to puzzles concerning knowledge using that logic. In the model epistemic considerations are based on ontology. The main notion that constitutes a bridge between these two disciplines is the notion of epistemic capacities. Within the model we study scenarios in which agents can receive false announcements and can have incomplete or improper views about other agent's (...) epistemic capacities. Moreover, we try to express the description of problem speci cation using the tools from applied ontology { RDF format for information and the Protege editor. (shrink)

By a counter-example, we show that the set of the three spin components Sx, Sy, Sz of spin-1 systems is not informationally complete.

Tononi has proposed a fundamental theory of consciousness he terms Integrated Information Theory (IIT). IIT purports to explain the quantity of conscious experience by linking it with integrated information: information shared by the system as a whole and quantified by adopting a modified version of Shannon's definition of information. Since the fundamental aspect of IIT is information the theory allows for the multiple realizability of consciousness. While there are several concepts within IIT that need further (...) theoretical development, the main failings of the theory are an absence of a link between conscious experience and awareness and the use of Shannon's limited data based definition of information. These limitations prevent the theory from satisfying Chalmers' principles of structural coherence and organizational invariance which any functionalist theory should obey. It is not clear whether IIT can be repaired without simply reducing the theory to a general statement of computational functionalism. (shrink)

We consider the procedure for small-sample estimation of reliability parameters. The main shortcomings of the classical methods and the Bayesian approach are analyzed. Models that find robust Bayesian estimates are proposed. The sensitivity of the Bayesian estimates to the choice of the prior distribution functions is investigated using models that find upper and lower bounds. The proposed models reduce to optimization problems in the space of distribution functions.

This article considers an asymmetric contest with incomplete information. There are two types of players: informed and uninformed. Each player has a different ability to translate effort into performance in terms of the contest success function. While one player’s type is known to both players, the other is private information and known only to the player himself. We compare the Bayesian Nash equilibrium outcome of a one-sided private information contest to the Nash equilibrium with no private (...) information, in which both players know the type of the other player. We show conditions under which uncertainty increases the investment of the uninformed player and the rent dissipation of the contest, while decreasing the expected net payoff of the informed player. In addition, we consider conditions under which the informed player—before knowing his own type—prefers that the uninformed player knows his type. Moreover, we show conditions for the existence/non-existence of equilibrium in a two-stage contest in which the informed player declares his type (or does not declare) in the first stage and in the second stage the two players play according to the information available to them. (shrink)

We give some information about new proofs of the incompleteness theorems, found in 1990s. Some of them do not require the diagonal lemma as a method of construction of an independent statement.

In spite of incomplete knowledge we are permanently forced to act in complex real-life situations. First, a modern concept of information, the non-trivial transition from information to knowledge, patterns of missing knowledge, and the concept of perspective notions are studied. The main sections review some guidelines for action under incomplete information. A modern view of the concepts of holism and wholeness reveals that (in contrast to some critics) general system theory does not require any metaphysical (...) assumption or previously accepted worldview. The concepts of holism and wholeness, as well as general system theory, are well-founded, even under strict criteria. (shrink)

Gödel's Incompleteness Theorems have the same scientific status as Einstein's principle of relativity, Heisenberg's uncertainty principle, and Watson and Crick's double helix model of DNA. Our aim is to discuss some new faces of the incompleteness phenomenon unveiled by an information-theoretic approach to randomness and recent developments in quantum computing.

Graham Priest's In Contradiction (Dordrecht: Martinus Nijhoff Publishers, 1987, chapter 3) contains an argument concerning the intuitive, or ‘naïve’ notion of (arithmetic) proof, or provability. He argues that the intuitively provable arithmetic sentences constitute a recursively enumerable set, which has a Gödel sentence which is itself intuitively provable. The incompleteness theorem does not apply, since the set of provable arithmetic sentences is not consistent. The purpose of this article is to sharpen Priest's argument, avoiding reference to informal notions, consensus, or (...) Church's thesis. We add Priest's dialetheic semantics to ordinary Peano arithmetic PA, to produce a recursively axiomatized formal system PA★ that contains its own truth predicate. Whether one is a dialetheist or not, PA★ is a legitimate, rigorously defined formal system, and one can explore its proof‐theoretic properties. The system is inconsistent (but presumably non‐trivial), and it proves its own Gödel sentence as well as its own soundness. Although this much is perhaps welcome to the dialetheist, it has some untoward consequences. There are purely arithmetic (indeed, Π0) sentences that are both provable and refutable in PA★. So if the dialetheist maintains that PA★ is sound, then he must hold that there are true contradictions in the most elementary language of arithmetic. Moreover, the thorough dialetheist must hold that there is a number g which both is and is not the code of a derivation of the indicated Gödel sentence of PA★. For the thorough dialetheist, it follows ordinary PA and even Robinson arithmetic are themselves inconsistent theories. I argue that this is a bitter pill for the dialetheist to swallow. (shrink)

The aim of this paper is to comprehensively question the validity of the standard way of interpreting Chaitin's famous incompleteness theorem, which says that for every formalized theory of arithmetic there is a finite constant c such that the theory in question cannot prove any particular number to have Kolmogorov complexity larger than c. The received interpretation of theorem claims that the limiting constant is determined by the complexity of the theory itself, which is assumed to be good measure of (...) the strength of the theory. I exhibit certain strong counterexamples and establish conclusively that the received view is false. Moreover, I show that the limiting constants provided by the theorem do not in any way reflect the power of formalized theories, but that the values of these constants are actually determined by the chosen coding of Turing machines, and are thus quite accidental. (shrink)

Gödel’s incompleteness applies to any system with recursively enumerable axioms and rules of inference. Chaitin’s approach to Gödel’s incompleteness relates the incompleteness to the amount of information contained in the axioms. Zurek’s quantum Darwinism attempts the physical description of the universe using information as one of its major components. The capacity of quantum Darwinism to describe quantum measurement in great detail without requiring ad-hoc non-unitary evolution makes it a good candidate for describing the transition from quantum to classical. (...) A baby-universe diffusion model of cosmic inflation is analyzed using quantum Darwinism. In this model cosmic inflation can be approximated as Brownian motion of a quantum field, and quantum Darwinism implies that molecular interaction during Brownian motion will make the quantum field decohere. The quantum Darwinism approach to decoherence in the baby-universe cosmic-inflation model yields the decoherence times of the baby-universes. The result is the equation relating the baby-universe’s decoherence time with the Hubble parameter, and that the decoherence time is considerably shorter than the cosmic inflation period. (shrink)

Annual Reports on Medical Assistance in Dying (MAID) issued by Health CanadaCanada on behalf of the Government of Canada strive to make essential information public, on this large national program that over five years has provided death to over 30,000 Canadians. The source of the information is self-reporting by physicians and nurse practitioners who offer to act as MAID assessors and providers, as well as self-reporting by cooperating pharmacists. Focus of this discussion is the latest available document in (...) this series, the Third Annual Report on MAID published in July 2022 and providing data on 10,064 deaths by MAID in 2021. In addition to reviewing essential information about MAID recipients and MAID providers, the discussion is centered on the method of data collection, the rapid and unexpected annual increase of cases, the underlying medical conditions, the nature of suffering reported by MAID applicants, the use of palliative care, the decision-making about one’s eligibility, the role of the reflection period, necessary requirement to obtain informed consent and the denial of assisted dying. In each case, the attention is drawn to missing or indeterminate elements, and essential but unavailable analysis of data, in order to identify further information that ought to be collected and used to generate future annual reports and to refine the delivery of MAID program and its oversight. (shrink)

This paper shows that some classes of multimodal paraconsistent logics endowed with weak forms of negation are incompletable with respect to Kripke semantics. The reach of such incompleteness is discussed, and we argue that this shortcoming, more than just a logical predicament, may be relevant for attempts to characterize quantum logics and to handle quantum information and quantum computation.

In lieu of an abstract, here is a brief excerpt of the content:Thomas G. Pavel INCOMPLETE WORLDS, RITUAL EMOTIONS' IN recent years, the notion of "fictional world" has enjoyed a considerable rise in fortune. The expression, however, is not entirely new. To refer to the world of a literary work, of a novel or of a play, has always been a favorite way of speaking for literary critics and aestheticians. In most cases, these were informal worlds. A discussion of (...) the world of Balzac, Dostoyevsky, HenryJames, or Proust often amounts to a description of each author's preferences in die selection and rearrangement of his material. Such discussion primarily fits authors who purposefully attempt to build comprehensive representational artifacts. One speaks quite naturally about l'univers balzacien, or about the universe of Musil. But is one entitled to refer with equal ease to the world ofBenjamin Constant, meagerly perceptible in his Adolphe, a short novel built in the spirit of economy, displaying just a few characters and a minimum of worldly information? Perhaps density and comprehensiveness are not indispensable characteristics of literary worlds. Aestheticians and narratologists using the "fictional worlds" terminology seem to believe mat any kind of artistic text projects a world, be it rich or poor. Such worlds are imagined in the fashion of modern philosophy of logic as states of affairs, more or less comprehensive. Some writers believe that the states of affairs comprised in fictional worlds must be fairly sizable, since diey should include "not only what the author indicates, but also whatever is required (entailed) by that which he indicates." 1 The laws of nature which are not specifically contradicted by the text would belong to its world. Thus, a few notorious cases aside, there is no reason to doubt that every child born in fiction has been engendered by a human fadier as long as the text does not report an exception. And if it does? Then, assuming that the laws of logic still govern if not fiction itself at least our understanding of it, we may appeal to what "possible world" philosophers have called "small miracles." * Editor's note: This article, as well as the contributions by Gerald Prince, Peter van Inwagen, and Kendall Walton which follow it, grew out ofa symposium on "Styles ofFictionality," organized by Professor Pavel and held at Harvard University in 1981. The Fall, 1983 issue of PHILOSOPHY AND LITERATURE will contain additional papers presented at the symposium. 48 Thomas G. Pavel49 Imagine, for instance, the world in which Notre Dame de Paris would be covered with blue paint. While everything else would preserve the properties it possesses in the actual world, with the laws of nature and the caused chains maintained, a small local change, somewhat miraculous, would modify only the color of the venerable building. Similarly, the world of Balzac may be understood as the world of early and mid-nineteenth-century France, slightly modified by the addition of a few hundred imagined characters. That diese conform surprisingly well to the general laws of French society of the time is so much the goal of Balzac's successful project, that for the sociologically-minded reader die' actual nonexistence of Rastignac, Rubempré, Vautrin, or Birotteau may appear a mere accident. These characters, one may surmise, display every trait that actual counterparts of theirs would possess, except existence. Hence, why would Rastignac's or Vautrin's emergence in a world omerwise identical to nineteenth-century France amount to anydiing more than a "small miracle"? On the other hand, Balzac's world being blessed wim hundreds of such miracles, the very magnitude of the project vindicates the attribute of "visionary," ascribed to Balzac by Béguin, or, if metaphorically undercut, mat of"God the Father" given him by Thibaudet. For, would not such magnitude entail more than just local changes in an otherwise faithful representation of the "actual" nineteenthcentury France? Similarly, would we interpret a fictional world in which a child has been engendered not by a human father, but by a divinity, as identical in all respects to the actual world, except for the limited modification attributed to a small miracle? 2 If we grant for a moment the improbable assumption that nineteenth-century France was a given and that Balzac... (shrink)

ABSTRACT What characterizes human reasoning is the ability of dealing with incomplete information. Incomplete logic is developed for modeling incomplete knowledge. The most distinctive feature of incomplete logic is its semantics. This is an alternative presentation of partial semantics. In this paper, we will introduce the general notion of incomplete logic (ICL), compare it with partial logic, and give the resolution method for it. We will also show how ICL can be applied to nonmonotonic (...) reasoning. We define nonmonotonic derivation as monotonic derivation in ICL from the database and some consistent assumptions. The mechanism of ICL makes it easy to assert the consistency of an assumption without asserting the assumption itself. (shrink)

I discuss the design of the method of entropic inference as a general framework for reasoning under conditions of uncertainty. The main contribution of this discussion is to emphasize the pragmatic elements in the derivation. More specifically: (1) Probability theory is designed as the uniquely natural tool for representing states of incomplete information. (2) An epistemic notion of information is defined in terms of its relation to the Bayesian beliefs of ideally rational agents. (3) The method of (...) updating from a prior to a posterior probability distribution is designed through an eliminative induction process that singles out the logarithmic relative entropy as the unique tool for inference. The resulting framework includes as special cases both MaxEnt and Bayes’ rule. It therefore unifies entropic and Bayesian methods into a single general inference scheme. I find that similar pragmatic elements are an integral part of Putnam’s internal realism, of Floridi’s informational structural realism, and also of van Fraasen’s empiricist structuralism. I conclude with the conjecture that their valuable insights can be incorporated into a single coherent doctrine—an informational pragmatic realism. (shrink)