Strong and coalition-proof political equilibria under plurality and runoff rule

We analyze the effects of voter coordination on the equilibrium of voting games. Specifically, we analyze the concepts of strong equilibrium and coalition-proof equilibrium for plurality rule and runoff rule elections. We characterize these equilibria in the three candidate case, and provide results for important special cases in the general multicandidate case.      

On simplex method with most-obtuse-angle rule and cosine rule

A class of simplex methods for solving linear programming (LP) problems, with cosine pivot rule, have been presented in some recent papers. In this paper we show that the cosine rule used in this class is equivalent to the most-obtuse-angle pivot rule, proposed by Pan (1990) [6]. The relation between the direct method for LP and the most-obtuse-angle rule is discussed.     

Dempster's rule of combination

The theory of belief functions is a generalization of probability theory; a belief function is a set function more general than a probability measure but whose values can still be interpreted as degrees of belief. Dempster's rule of combination is a rule for combining two or more belief functions; when the belief functions combined are based on distinct or “independent” sources of evidence, the rule corresponds intuitively to the pooling of evidence. As a special case, the rule yields a rule of conditioning which generalizes the usual rule for conditioning probability measures. The rule of combination was studied extensively, but only in the case of finite sets of possibilities, in the author's monograph A Mathematical Theory of Evidence. The present paper describes the rule for general, possibly infinite, sets of possibilities. We show that the rule preserves the regularity conditions of continuity and condensability, and we investigate the two distinct generalizations of probabilistic independence which the rule suggests.     

A NOTE ON THE GRADIENT PROJECTION METHOD WITH EXACT STEPSIZE RULE

In this paper, we give some convergence results on the gradient projection method with exact stepsize rule for solving the minimization problem with convex constraints. Especially, we show that if the objective function is convex and its gradient is Lipschitz continuous, then the whole sequence of iterations produced by this method with bounded exact stepsizes converges to a solution of the concerned problem.     

A Pieri rule for skew shapes

The Pieri rule expresses the product of a Schur function and a single row Schur function in terms of Schur functions. We extend the classical Pieri rule by expressing the product of a skew Schur function and a single row Schur function in terms of skew Schur functions. Like the classical rule, our rule involves simple additions of boxes to the original skew shape. Our proof is purely combinatorial and extends the combinatorial proof of the classical case.     

A multiplier rule on a metric space

A multiplier rule is proved for constrained minimization problems defined on a metric spaces. The proof requires a generalization of the values of a derivative in the classical case that the metric space is a normed space.     

A Littlewood-Richardson rule for factorial Schur functions

We give a combinatorial rule for calculating the coefficients in the expansion of a product of two factorial Schur functions. It is a special case of a more general rule which also gives the coefficients in the expansion of a skew factorial Schur function. Applications to Capelli operators and quantum immanants are also given.

     

On the convergence of the random arrival rule in large claims problems

We study the behavior of the random arrival rule in claims problems with a large number of claimants. We assume that the amount to divide is not too “small” compared with total claims, and each claim is not too “large” compared with any other positive claims. Then, the random arrival rule behaves like the proportional rule as the number of claimants increases. We are grateful to Yuki Funaki, William Thomson, and three referees for their comments. This work was supported by the Brain Korea 21 Project in 2003     

Heterogeneous cost sharing, the directional serial rule

The directional serial rule is introduced as a natural serial extension, generalizing the Moulin–Shenker cost sharing rule to heterogeneous cost sharing models. It is the unique regular rule compatible with the radial serial principle. In particular, this shows the incompatibility of the serial principle with differentiability of a cost sharing rule as a function of the individual demands.I would like to thank the editor and the referee for their comments which have been most useful.     

An interpretability improvement for fuzzy rule bases obtained by the iterative rule learning approach

Interpretability is one of the key concepts in many of the applications using the fuzzy rule-based approach. It is well known that there are many different criteria around this concept, the complexity being one of them. In this paper, we focus our efforts in reducing the complexity of the fuzzy rule sets. One of the most interesting approaches for learning fuzzy rules is the iterative rule learning approach. It is mainly characterized by obtaining rules covering few examples in final stages, being in most cases useless to represent the knowledge. This behavior is due to the specificity of the extracted rules, which eventually creates more complex set of rules. Thus, we propose a modified version of the iterative rule learning algorithm in order to extract simple rules relaxing this natural trend. The main idea is to change the rule extraction process to be able to obtain more general rules, using pruned searching spaces together with a knowledge simplification scheme able to replace learned rules. The experimental results prove that this purpose is achieved. The new proposal reduces the complexity at both, the rule and rule base levels, maintaining the accuracy regarding to previous versions of the algorithm.     

A largest-distance pivot rule for the simplex algorithm

We propose a new pivot rule for the simplex algorithm, which is demonstrative in the dual space intuitively. Although it is based on normalized reduced costs, like the steepest-edge rule and its variants, the rule is much simpler and cheaper than the latter. We report computational results obtained with the 47 largest Netlib problems in terms of the number of rows and columns, all of the 16 Kennington problems, and the 17 largest BPMPD problems. Over the total 80 problems, a variant of the rule outperformed the Devex rule with iterations and time ratio 1.43 and 3.24, respectively.     

A short proof of a general multiplier rule

A general multiplier rule obtained by Hestenes is shown to follow directly from the Lagrange multiplier rule by a simple compactness argument. A similar simplification is effected in the proof of Gittleman's extension of Hestenes' rule.     

Independent rule in classification of multivariate binary data

We consider the performance of the independent rule in classification of multivariate binary data. In this article, broad studies are presented including the performance of the independent rule when the number of variables, d, is fixed or increased with the sample size, n. The latter situation includes the case of d=O(n^τ) for τ>0 which cover “the small sample and the large dimension”, namely dn when τ>1. Park and Ghosh [J. Park, J.K. Ghosh, Persistence of plug-in rule in classification of high dimensional binary data, Journal of Statistical Planning and Inference 137 (2007) 3687–3707] studied the independent rule in terms of the consistency of misclassification error rate which is called persistence under growing numbers of dimensions, but they did not investigate the convergence rate. We present asymptotic results in view of the convergence rate under some structured parameter space and highlight that variable selection is necessary to improve the performance of the independent rule. We also extend the applications of the independent rule to the case of correlated binary data such as the Bahadur representation and the logit model. It is emphasized that variable selection is also needed in correlated binary data for the improvement of the performance of the independent rule.     

On the fuzzy intersection rule

The paper provides a complete exposition of the fuzzy intersection rule in variational analysis and its applications to generalized differentiation theory. New forms of the rule and further applications are presented.     

Combination of partially non-distinct beliefs: The cautious-adaptive rule

The combination rule is critical in an evidence based fusion process. The conjunctive rule is most common eventhough when the cognitive independence – distinctness – assumption is often questionable. A new combination rule is tested here in both discrete and continuous cases, accounting for a partial non-distinctness between evidences. It is based on ‘generalized discounting’, that we define for separable basic belief assignments (bbas) or basic belief densities (bbds), to be applied to the source correlation derived from the cautious rule. This correlation can be specified in both considered cases of consonant bbas/bbds (as proposed by Dubois et al.) and separable bbas/bbds (as proposed by Denœux). Then, the so-called ‘cautious-adaptive’ rule varies between the conjunctive rule and the cautious one, depending on the discounting level. In the Gaussian case with standard deviation σ, the evidence non-distinctness will be parameterized by a factor [0,1] dividing σ. It leads to the generalized discounting needed in the cautious-adaptive formulation.     

Computation of the random arrival rule for bankruptcy problems

Bankruptcy problems are a fundamental class of fair division problems in microeconomics. Among the various solution concepts proposed for the problem, the random arrival rule is one of the most prominent. In this paper, we conduct a computational analysis of the rule. It is shown that the allocation returned by the rule is #P-complete to compute. The general complexity result is complemented by a pseudo-polynomial-time dynamic programming algorithm for the random arrival rule.     

A note on a generalized Cramer''s rule

A short proof of a generalized Cramer's rule is presented.     

Exploring the VCG mechanism in combinatorial auctions: The threshold revenue and the threshold-price rule

We explore interesting potential extensions of the Vickrey–Clarke–Groves (VCG) rule under the assumption of players with independent and private valuations and no budget constraints. First, we apply the VCG rule to a coalition of bidders in order to compute the second price of the coalition. Then, we introduce and formulate the problem of determining that partition of players into coalitions which maximize the auctioneer’s revenue in the case whereby such coalitions take part to a VCG auction each one as a single agent; in particular, we provide an integer linear formulation of this problem. We also generalize this issue by allowing players to simultaneously belong to distinct coalitions in the case that players’ valuation functions are separable. Finally, we propose some applications of these theoretical results. For instance, we exploit them to provide a class of new payment rules and to decide which bids should be defined as the highest losing ones in combinatorial auctions.     

An exponential example for Terlaky's pivoting rule for the criss-cross simplex method

Recently T. Terlaky has proposed a new pivoting rule for the criss-cross simplex method for linear programming and he proved that his rule is convergent. In this note we show that the required number of iterations may be exponential in the number of variables and constraints of the problem.     

Continuity properties of majority rule with intermediate preferences

This paper shows that certain conditions which have previously been shown (by Grandmont) to be sufficient for a society's majority rule relation to be transitive or acyclic are also sufficient for the map from distributions of voter preferences to indices identified with their majority rule relations to be continuous. Applications of this result to societies with certain classical assumptions on preferences reveal that, in such societies, the map from distributions of voter preferences to their majority rule equilibria is also continuous.