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”.     

How to estimate the Value at Risk under incomplete information

A key problem in financial and actuarial research, and particularly in the field of risk management, is the choice of models so as to avoid systematic biases in the measurement of risk. An alternative consists of relaxing the assumption that the probability distribution is completely known, leading to interval estimates instead of point estimates. In the present contribution, we show how this is possible for the Value at Risk, by fixing only a small number of parameters of the underlying probability distribution. We start by deriving bounds on tail probabilities, and we show how a conversion leads to bounds for the Value at Risk. It will turn out that with a maximum of three given parameters, the best estimates are always realized in the case of a unimodal random variable for which two moments and the mode are given. It will also be shown that a lognormal model results in estimates for the Value at Risk that are much closer to the upper bound than to the lower bound.     

Closer estimators of a common mean in the sense of Pitman

Consider the problem of estimating the common mean of two normal populations with different unknown variances. Suppose a random sample of sizem is drawn from the first population and a random sample of sizen is drawn from the second population. The paper gives a family of estimators closer than the sample mean of the first population in the sense of Pitman (1937,Proc. Cambridge Phil. Soc.,33, 212–222). In particular, the Graybill-Deal estimator (1959,Biometrics,15, 543–550) is shown to be closer than each of the sample means ifm5 andn5.     

CONSISTENCY OF MLE OF THE PARAMETER OF EXPONENTIAL LIFETIME DISTRIBUTION FOR RANDOM CENSORING MODEL WITH INCOMPLETE INFORMATION

In this paper, we have discussed a random censoring test with incomplete information, and proved that the maximum likelihood estimator (MLE) of the parameter based on the randomly censored data with incomplete information in the case of the exponential distribution has the strong consistency.     

Communication, credible improvements and the core of an economy with asymmetric information

We analyze an economy with asymmetric information and endogenize the possibilities for information transmission between members of a coalition. We then define a concept of the Core that takes into account these communication possibilities. The internal consistency of the improvements is considered and an Internally Consistent Core, which requires credibility from the improvements is introduced. Received: September 1998/revised version: June 1999     

Note Equilibrium payoffs in stochastic games of incomplete information: the general symmetric case

We prove that the existence of equilibrium payoffs for stochastic games of incomplete symmetric information follows from the same result for stochastic games with complete information. Received January 1999/Revised October 2001     

The game for the speed of convergence in repeated games of incomplete information

We consider an infinitely repeated two-person zero-sum game with incomplete information on one side, in which the maximizer is the (more) informed player. Such games have value v _∞ (p) for all 0≤p≤1. The informed player can guarantee that all along the game the average payoff per stage will be greater than or equal to v _∞ (p) (and will converge from above to v _∞ (p) if the minimizer plays optimally). Thus there is a conflict of interest between the two players as to the speed of convergence of the average payoffs-to the value v _∞ (p). In the context of such repeated games, we define a game for the speed of convergence, denoted SG _∞ (p), and a value for this game. We prove that the value exists for games with the highest error term, i.e., games in which v _n (p)− v _∞ (p) is of the order of magnitude of . In that case the value of SG _∞ (p) is of the order of magnitude of . We then show a class of games for which the value does not exist. Given any infinite martingale 𝔛^∞={X _k }^∞ _k=1, one defines for each n : V _n (𝔛^∞) ≔E∑ⁿ _k=1 |X _k+1 − X _k|. For our first result we prove that for a uniformly bounded, infinite martingale 𝔛^∞, V _n (𝔛^∞) can be of the order of magnitude of n ^1/2−ε, for arbitrarily small ε>0. Received January 1999/Final version April 2002     

Recovery of signals under the condition on RIC and ROC via prior support information

In this paper, the sufficient condition in terms of the RIC and ROC for the stable and robust recovery of signals in both noiseless and noisy settings was established via weighted $l_1$ minimization when there is partial prior information on support of signals. An improved performance guarantee has been derived. We can obtain a less restricted sufficient condition for signal reconstruction and a tighter recovery error bound under some conditions via weighted $l_1$ minimization. When prior support estimate is at least 50% accurate, the sufficient condition is weaker than the analogous condition by standard $l_1$ minimization method, meanwhile the reconstruction error upper bound is provably to be smaller under additional conditions. Furthermore, the sufficient condition is also proved sharp.     

Tensor conditions for the existence of a common solution to the Lyapunov equation

Necessary and sufficient conditions for the existence of a common solution to the Lyapunov equation for two stable complex matrices are derived. These conditions are applied to the cases when a common weak solution to the Lyapunov equation exists. Conditions for the existence of a common solution to the Lyapunov equation for two complex 2 × 2 and two complex 3 × 3 matrices are derived.     

How to decrease the diameter of triangle-free graphs

Assume that G is a triangle-free graph. Let be the minimum number of edges one has to add to G to get a graph of diameter at most d which is still triangle-free. It is shown that for connected graphs of order n and of fixed maximum degree. The proof is based on relations of and the clique-cover number of edges of graphs. It is also shown that the maximum value of over (triangle-free) graphs of order n is . The behavior of is different, its maximum value is . We could not decide whether for connected (triangle-free) graphs of order n with a positive ε. Received: October 12, 1997     

不完全信息下考虑效率和规模的多时点并购决策

针对企业并购过程中存在不完全信息和多时点信息的问题,提出了一种考虑效率和规模的企业并购决策方法. 通过证据推理集结并购双方的不完全的、多时点的评价信息,利用\,DEA\,方法判断合并企业规模是否过大,并筛选出可行并购方案,再根据合并企业竞争型并购交叉效率高低决策最佳并购目标. 最后算例分析说明了方法的有效性和实用性.     

Computing tolerance limits for the lifetime of a k-out-of-n:F system based on prior information and censored data

Exact guaranteed-coverage and expected-coverage Bayesian tolerance limits for the lifetime distribution of a k-out-of-n:F system are computed by solving nonlinear equations. The bounds are based on exponential component test data and available prior information concerning the expected component lifetime which is described by an inverted gamma distribution. The Bayesian tolerance limits are valid for single (right or left), double and progressive (standard or general) censoring, and even have frequentist validity in the noninformative case. The derived results allow the reliability engineer to judge the quality of a system prior to assembly, which offers obvious practical and economic benefits. Minimum and expected percentages of conforming systems are assessed by constructing suitable tolerance limits. Even though the viewpoints are different, the Bayesian tolerance limits that adopt the natural diffuse prior coincide numerically with recently published conditional tolerance limits in the double censoring case. The proposed Bayesian approach may be deemed as an extension of the existing frequentist methodology under double censoring that also takes into account the presence of prior information and general progressive censoring. The perspective developed simplifies and unifies the computation of tolerance limits with both frequentist and Bayesian interpretations, and also provides a probabilistic way of updating the tolerance limits in the light of new, relevant data, which is especially important in the dynamic analysis of a sequence of data. Moreover, the Bayesian approach is shown to outperform the frequentist viewpoint in terms of accuracy. In most situations, the use of substantial prior information significantly increases the accuracy level and considerably reduces the required number of failures to attain a specified degree of accuracy. Two illustrative numerical examples are studied, including the analysis of a system of water pumps for cooling a reactor. The results developed are extended to the Weibull case with unknown scale parameter and other probability models.     

Bifurcating critical points of bending energy under constraints related to the shape of red blood cells

Considered is a variational problem for the bending energy of closed surfaces under the prescribed area and surrounding volume. Minimizers of this problem are interpreted as surfaces modeling the shape of red blood cells. We give a rigorous proof of the existence of a one-parameter family of critical points bifurcating from the sphere and study their stability/instability. In particular, for a few branches of critical points, we compute the exact values of the index and the nullity of critical points. Received: 8 September 2001 / Accepted: 25 October 2001 / Published online: 29 April 2002 Partly supported by Grant-in-Aid for Exploratory Research (Nos.09874026, 11874033) and for Scientific Research (No.12640200), Ministry of Education, Science, Sports, and Culture, Japan; and also by Sumitomo Foundation Dedicated to Professor Takaaki Nishida on his sixtieth birthday