Higher-order Uncertainty Research Articles

Impact of Higher-Order Uncertainty

In some games, the impact of higher-order uncertainty is very large, implying that present economic theories may be misleading as these theories assume common knowledge of the type structure after specifying the first or the second orders of beliefs. Focusing on normal-form games in which the players' strategy spaces are compact metric spaces, we show that our key condition, called stability under uncertainty, implies a variety of results to the effect that the impact of higher-order uncertainty is small. Our central result states that, under global stability, the maximum change in equilibrium strategies due to changes in players' beliefs at orders higher than k is exponentially decreasing in k. Therefore, given any need for precision, we can approximate equilibrium strategies by specifying only finitely many orders of beliefs.

Persuasion games with higher-order uncertainty

Shin (J. Econom. Theory 64 (1994) 253–264) showed that a perfectly revealing equilibrium fails to exist in persuasion games when the decision maker is uncertain about the interested party's payoff-relevant information. By explicitly integrating higher-order uncertainty into the information structure, this note shows that a perfectly revealing equilibrium does exist when disclosures are not restrained to intervals of the payoff-relevant state space.

Higher-order fluctuations and SU(1,1) higher-order squeezing in SU(1,1) generalized coherent states

The higher-order fluctuations in the SU(1,1) generalized coherent states are discussed. The definition of higher-order SU(1,1) Squeezing is introduced in terms of higher-order uncertainty relation. For two possible bosonic realizations of SU(1,1) Lie algebra, the second-, fourth- and sixth-order SU(1,1) squeezing are examined in detail. It is shown that the SU(1,1) generalized coherent states can be squeezed to not only second-order, but also fourth-and sixth-order. Hence, it follows that the higher-order squeezing will occur for the fluctuations of the square of amplitude in squeezed vacuum. SU(1,1) higher-order squeezing is a kind of non-classical property which is independent of second-order squeezing.

HIGHER-ORDER FLUCTUATIONS AND THEIR SQUEEZING OF ANGULAR MOMENTUM IN ATOMIC COHERENT STATES

The second-,fourth-and sixth-order fluctuations have been discussed by making use of- SU-(2)Lie algebra.On the basis of higher-order uncertainty relation,the definition of higher-order squeezing for the fluctuations of angular momentum has been put forward.In particular,the second-,fourth-and sixth-order squeezing in atomic coherent states are investigated.These methods and definition can be used for studying much higher-order squeezing.The higher-order squeezing can be generalized to the fluctuations of atomic variances thereby.

Higher-order uncertainty relations

The Schwartz inequality is used to obtain some generalized uncertainty relations among higher-order moments of the position and momentum operators.

An introduction to issues in higher order uncertainty

The specification of appropriate procedures for making inferences in the context of uncertain, incomplete or imprecise information is a key research issue in a variety of disciplines. Within and among these disciplines, great debates have occurred about the meaning and appropriateness of various proposals for reasoning under uncertainty. At the core of many of these debates are disagreements as to the appropriate meaning of the term the related term probability, and what it means to be unsure of one's own uncertainties. This latter issue, uncertainty about one's uncertainty, is commonly referred to as higher order uncertainty. This paper provides a brief overview of various perspectives in higher order uncertainty.

Is higher-order uncertainty needed?

In the artificial intelligence and decision-making communities, there seems to be agreement on the calculations used in most problems that are solved using uncertainty, but disagreement on the terminology used. The purpose here is to show that these problems can be modeled using traditional first-order beliefs, and that, when they are modeled this way, the calculations are the same as the calculations, The disagreements arise because researchers rephrase first-order beliefs in various guises of higher-order uncertainty. I argue that we should stay with the traditional first-order formulations of the problems.

Modeling and measuring the effects of vagueness in decision models

Because subjective probability assessments do not represent measurements of empirical quantities, they often exhibit vagueness, or ambiguity. We adopt a standard approach to modeling vagueness as a second-order probability (SOP) on a first-order probability. We define the expected value of including vagueness (EVIV) as the change in expected value due to explicit probabilistic representation of vagueness. We show that, under general conditions, the EVIV is always zero, regardless of the SOP or the loss or utility function. This shows that representing higher-order uncertainty as SOP never changes the output of a decision model. We analyze the value of reducing vagueness, and we provide a framework for measuring the expected value of reducing vagueness (EVRV) on a probability assessment using equivalent sample size and relative information multiple (RIM). We use influence diagrams to represent the flow of potential but unobserved evidence from a vagueness-reducing activity. We demonstrate our results for a simple problem implemented in Demos, a decision modeling software environment.

Representing partial ignorance

This paper advocates the use of nonpurely probabilistic approaches to higher-order uncertainty. One of the major arguments of Bayesian probability proponents is that representing uncertainty is always decision-driven and as a consequence, uncertainty should be represented by probability. Here we argue that representing partial ignorance is not always decision-driven. Other reasoning tasks such as belief revision for instance are more naturally carried out at the purely cognitive level. Conceiving knowledge representation and decision-making as separate concerns opens the way to nonpurely probabilistic representations of incomplete knowledge. It is pointed out that within a numerical framework, two numbers are needed to account for partial ignorance about events, because on top of truth and falsity, the state of total ignorance must be encoded independently of the number of underlying alternatives. The paper also points out that it is consistent to accept a Bayesian view of decision-making and a non-Bayesian view of knowledge representation because it is possible to map nonprobabilistic degrees of belief to betting probabilities when needed. Conditioning rules in non-Bayesian settings are reviewed, and the difference between focusing on a reference class and revising due to the arrival of new information is pointed out. A comparison of Bayesian and non-Bayesian revision modes is discussed on a classical example.

Comparing the Robustness of Trading Systems to Higher-Order Uncertainty

This paper compares the performance of a decentralized market with that of a dealership market when traders have differential information. Trade occurs as a result of equilibrium actions in a Bayesian game, where uncertainty is captured by a finite state space and information is represented by partitions on this space. In the benchmark case of trade with common knowledge of endowments, the two mechanisms deliver virtually identical outcomes. However, with differential information, the dealership market has strictly higher trading volume, and yields an efficient post-trade allocation in most states. In contrast, the decentralized market suffers from suboptimal trading volume. The reason for this poor performance is the vulnerability of the decentralized market to higher-order uncertainty concerning the fundamentals of the market. Traders may know that mutually beneficial trade is feasible, and perhaps know that they know, and yet a failure of common knowledge that this is so precludes efficient trade. The dealership market is robust to this type of uncertainty.

HIGHER-ORDER SQUEEZING OF ATOMIC DIPOLE

The higher-order fundamental quantum-mechanical fluctuation is given on the base of the higher-order uncertainty relation, from which the definition of higher-order squeezing for atomic dipole is introduced. As an example, we examine the second-, fourth-and sixth-order squeezing of atomic dipole in the two-photon Jaynes-Cummings model with the two-photon superposition state preparation.

A theoretical investigation into quantitative modal logic

Quantitative modal logic (QML) is a multi-modal formulation of possibilistic reasoning with the capacity of representing and reasoning about higher order uncertainty. In this paper, some results about QML and possibility theory are investigated. First, the concept of filtration in classical modal logic is introduced into QML and the fundamental filtration theorem is proved. Second, as a corollary, the finite model property of QML is provided. Finally, the theoretical and practical implications of finite model property are discussed.

Depth of knowledge and the effect of higher order uncertainty

Depth of knowledge and the effect of higher order uncertainty

Robust Linear-Optimal Control Laws for Active Suspension Systems

The stability robustness of linear-optimal control laws for quarter-car active suspension systems is evaluated using stochastic robustness analysis. Simultaneous parameters variations and neglected actuator and sensor dynamics are considered for LQ active suspension systems and for a single-measurement LQG system to determine the effects of uncertainty on system stability. The results indicate that neglected actuator and sensor dynamics have a small effect on stability robustness, while parameter uncertainty, particularly that of the “sprung mass” is of great concern. The effectiveness of Loop Transfer Recovery on active suspension systems with both parameter uncertainty and higher-order uncertainty is discerned. The analysis shows that when Loop Transfer Recovery is applied arbitrarily to uncertain systems, both estimator performance and system robustness can decrease. Nevertheless, it is concluded that the impact of the robustness recovery method is determined by stochastic stability robustness analysis, and the recovery design parameter that provides sufficient robustness with minimal performance degradation is readily identified. The effect of LQ design parameters on robustness also is considered. The paper presents robustness analysis and synthesis methods for a quarter-car model that can be applied to higher-order active suspension systems.