Condorcet efficiency of two stage constant scoring rules

Previous research has developed a representation for the Condorcet efficiency of plurality and negative plurality rules on three candidates for large electorates under the impartial culture condition. This study considers the two stage voting rules on three candidates, plurality elimination and negative plurality elimination. A representation is obtained for the Condorcet efficiency of each for large electorates under impartial culture. The Condorcet efficiencies of both rules are equal and are shown to be substantially greater than the Condorcet efficiency of plurality and negative plurality.     

Are serial Condorcet rules strategy-proof?

Abstract. This paper explores the extent to which majority rule is invulnerable to manipulation by individuals and coalitions, even when majority rule is used to select more than one alternative. The resulting rule may or may not be strategy-proof, depending on the size of the coalitions that can form, and on the nature of the individual preferences over sets of alternatives. No individual can manipulate with respect to a wide family of preferences over sets. The only restriction on the domain of true and revealed individual preferences is that the selection rule is always well defined. Received: 1 November 1999 / Accepted: 7 May 2001 We thank two anonymous referees for suggestions that have significantly improved the paper. We are also grateful to l'Université de Caen for sponsoring a Workshop on Social Choice Theory, where a first draft of this paper was presented in May, 1999, and to the workshop participants for helpful observations. Work on the final version of the paper was done while one of the authors was a guest of the Project on Intergenerational Equity supported by the Ministry of Education, Science, and Technology of Japan. We are grateful to the Ministry and to the project leader, Professor Kotaro Suzumura, for their support.     

Condorcet efficiency of positional voting rules with single-peaked preferences

Consider a three-alternative election with n voters and assume that preferences are single-peaked. LetC(λ,n) be the Condorcet efficiency of the rule that assigns 1,λ and 0 points (respectively) to each first, second and third place vote. An exact representation is obtained forC(λ,∞). This relation shows that Borda rule (λ=1/2) is not the most efficient rule. In addition to this result, exact closed form relations are provided forC(0,n),C(1/2,n) andC(1,n). All these relations are obtained by assuming that every admissible configuration of preferences is equally likely to occur. This research was supported in part by the Swedish Insitute     

The construction of empirical credit scoring rules based on maximization principles

We examine the econometric implications of the decision problem faced by a profit/utility-maximizing lender operating in a simple “double-binary” environment, where the two actions available are “approve” or “reject”, and the two states of the world are “pay back” or “default”. In practice, such decisions are often made by applying a fixed cutoff to the maximum likelihood estimate of a parametric model of the default probability. Following (Elliott and Lieli, 2007), we argue that this practice might contradict the lender’s economic objective and, using German loan data, we illustrate the use of “context-specific” cutoffs and an estimation method derived directly from the lender’s problem. We also provide a brief discussion of how to incorporate legal constraints, such as the prohibition of disparate treatment of potential borrowers, into the lender’s problem.     

Combining density forecasts using focused scoring rules

We investigate the added value of combining density forecasts focused on a specific region of support. We develop forecast combination schemes that assign weights to individual predictive densities based on the censored likelihood scoring rule and the continuous ranked probability scoring rule (CRPS) and compare these to weighting schemes based on the log score and the equally weighted scheme. We apply this approach in the context of measuring downside risk in equity markets using recently developed volatility models, including HEAVY, realized GARCH and GAS models, applied to daily returns on the S&P 500, DJIA, FTSE and Nikkei indexes from 2000 until 2013. The results show that combined density forecasts based on optimizing the censored likelihood scoring rule significantly outperform pooling based on equal weights, optimizing the CRPS or log scoring rule. In addition, 99% Value‐at‐Risk estimates improve when weights are based on the censored likelihood scoring rule.     

Likelihood-based scoring rules for comparing density forecasts in tails

We propose new scoring rules based on conditional and censored likelihood for assessing the predictive accuracy of competing density forecasts over a specific region of interest, such as the left tail in financial risk management. These scoring rules can be interpreted in terms of Kullback-Leibler divergence between weighted versions of the density forecast and the true density. Existing scoring rules based on weighted likelihood favor density forecasts with more probability mass in the given region, rendering predictive accuracy tests biased toward such densities. Using our novel likelihood-based scoring rules avoids this problem.     

Constant scoring rules for choosing one among many alternatives

Quality & Quantity - When a group is to select a decision alternative from a finite set of m ⩾ 3 feasible alternatives, it is often desirable to choose the Condorect (majority)...     

Weighted scoring rules,the impartial culture condition,and homogeneity

The study considers the significance of weight selection for weighted scoring rules on three alternatives. Most research previously done in this area has been based on the assumption of the impartial culture condition. Intuition suggests that if preferences of individuals in a large group are at all distant or different from the impartial culture condition, then the weights selected for a weighted scoring rule become unimportant. It would seem likely in such a situation that one of the alternatives should be a clear front runner and win under almost any weighted scoring rule. A measure of social homogeneity is used in the study to represent the distance between a given set of individuals' preferences and the impartial culture condition. Based on computer simulation analysis, the results indicate that specific weight selection definitely becomes less important as the level of social homogeneity increases. However, the specific weight selection is still found to be of importance for situations with the greatest homogeneity that represent preferences the farthest distance from the condition of impartial culture.     

On the existence of consistent rules to adjudicate conflicting claims: a constructive geometric approach

For the problem of adjudicating conflicting claims, a rule is consistent if the choice it makes for each problem is always in agreement with the choice it makes for each “reduced problem” obtained by imagining that some claimants leave with their awards and reassessing the situation from the viewpoint of the remaining claimants. We develop a general technique to determine whether a given two-claimant rule admits a consistent extension to general populations, and to identify this extension if it exists. We apply the technique to a succession of examples. I gratefully acknowledge support from NSF under grant SBR-9731431 and SES 0214691, and the comments of Carmen Bevia, Rodrigo Velez, Chun-Hsien Yeh, and a referee.     

The invalidity of the Condorcet Jury Theorem under endogenous decisional skills

The current note clarifies that Condorcet Jury Theorem cannot be generalized to the extended setting where individual decisional skills are not assumed to be exogenous parameters even when these skills are homogeneous. This is true when skills are determined endogenously either by a central planner or, in a decentralized strategic setting, by the decision makers themselves.     

The power of bootstrap and asymptotic tests

The bootstrap discrepancy measures the difference in rejection probabilities between a bootstrap test and one based on the true distribution. The order of magnitude of the bootstrap discrepancy is the same under the null hypothesis and under non-null processes described by Pitman drift. If the test statistic is not an exact pivot, critical values depend on which data-generating process (DGP) is used to determine the null distribution. We propose using the DGP which minimizes the bootstrap discrepancy. We also show that, under an asymptotic independence condition, the power of both bootstrap and asymptotic tests can be estimated cheaply by simulation.     

The asymptotic codifference and covariation of log-fractional stable noise

Many econometric quantities such as long-term risk can be modeled by Pareto-like distributions and may also display long-range dependence. If Pareto is replaced by Gaussian, then one can consider fractional Brownian motion whose increments, called fractional Gaussian noise, exhibit long-range dependence. There are many extensions of that process in the infinite variance stable case. Log-fractional stable noise (log-FSN) is a particularly interesting one. It is a stationary mean-zero stable process with infinite variance, parametrized by a tail index α$\alpha$ between 1 and 2, and hence with heavy tails. The lower the value of α$\alpha$, the heavier the tail of the marginal distributions. The fact that α$\alpha$ is less than 2 renders the variance infinite. Thus dependence between past and future cannot be measured using the correlation. There are other dependence measures that one can use, for instance the “codifference” or the “covariation”. Since log-FSN is a moving average and hence “mixing”, these dependence measures converge to zero as the lags between past and future become very large. The codifference, in particular, decreases to zero like a power function as the lag goes to infinity. Two parameters play an important role: (a) the value of the exponent, which depends on α$\alpha$ and measures the speed of the decay; (b) a multiplicative constant of asymptoticity c$c$ which depends also on α$\alpha$. In this paper, it is shown that for symmetric α$\alpha$-stable log-FSN, the constant c$c$ is positive and that the rate of decay of the codifference is such that one has long-range dependence. It is also proved that the same conclusion holds for the second measure of dependence, the covariation, which converges to zero with the same intensity and with a constant of asymptoticity which is positive as well.     

The primal route to the turnpike and asymptotic stability

Asymptotic turnpike theorems in the case where the future is discounted are traditionally derived under the assumptions that a utility function is twice differentiable and that is Hessian is negative definite. Replacing these assumptions by a weaker one. I give a proof of the theorem simpler than existing ones.     

The structure of exactly strongly consistent social choice functions

For ? an anonymous exactly strongly consistent social choice function (ESC SCF) and x an alternative, define $\text{b}{}_{\mathrm{x}}\text{=b(}{}^{\mathrm{f}}\text{)}{}_{\mathrm{x}}$ to be the size of a minimal blocking coalition for x. By studying the correspondence between ? and $\text{{b(}{}^{\mathrm{f}}\text{)}{}_{\mathrm{x}}\text{}}$, we establish the existence, uniqueness and monotonicity of ESC SCF's. We also prove the following conjecture of B. Peleg: A non–constant anonymous ESC SCF depends on the knowledge of every player's full preference profile.     

The evolution of monetary policy rules

This article examines the evolutionary forces involved in the development of monetary policy rules over time, considering in particular how these relate to proposals made by Milton Friedman and Walter Bagehot. The lines between money and credit and between monetary and fiscal policy are given special focus.     

A Note on the Probability of Having a Strong Condorcet Winner

In an election, a strong Condorcet winner is a candidate who is top-ranked by more than 50% of the voters. The purpose of this note is to provide some algebraic representations for the probability of having a strong Condorcet winner in three-candidate elections. Three alternative procedures for generating voting situations are considered: the Impartial Culture condition, the Impartial Anonymous Culture condition and the Maximal Culture condition. It turns out that the conclusions we obtain strongly depend on the way for generating voting situations.     

地产市场的"情"和"义"

又到春暖花开时.清冷已近一年的中国地产市场在牛年春天的暖阳里开始显露复苏的迹象,并似有"重拾昨日辉煌"的气势.……