On strong representations of games by social choice functions

Games in characteristic function form are used to model an allocation of decision power among individuals involved in a voting situation. The problem of strong representation is to find a strategically acceptable social choice function that entails the allocation of power prescribed by a given game. Within the class of non-weak characteristic function games, we fully characterize the games that admit a strong representation. We apply this result to Peleg's problem of strong representation of simple games. Our results indicate that a strong representation requires significantly more than has been recognized in the literature.     

The structure of decision schemes with cardinal preferences

This paper replacesGibbard’s (Econometrica 45:665-681, 1977) assumption of strict ordinal preferences by themore natural assumption of cardinal preferences on the set pure social alternatives and we also admit indifferences among the alternatives. By following a similar line of reasoning to the Gibbard-Satterthwaite theoremin the deterministic framework, we first show that if a decision scheme satisfies strategy proofness and unanimity, then there is an underlying probabilistic neutrality result which generates an additive coalitional power function. This result is then used to prove that a decision scheme which satisfies strategy proofness and unanimity can be represented as a weak random dictatorship. A weak random dictatorship assigns each individual a chance to be a weak dictator. An individual has weak dictatorial power if the support of the social choice lottery is always a subset of his/her maximal utility set. In contrast to Gibbard’s complete characterization of randomdictatorship, we also demonstrate with an example that strategy proofness and unanimity are sufficient but not necessary conditions for a weak random dictatorship.     

An alternative direct proof of Gibbard’s random dictatorship theorem

We present an alternative proof of the Gibbards random dictatorship theorem with ex post Pareto optimality. Gibbard(1977) showed that when the number of alternatives is finite and larger than two, and individual preferences are linear (strict), a strategy-proof decision scheme (a probabilistic analogue of a social choice function or a voting rule) is a convex combination of decision schemes which are, in his terms, either unilateral or duple. As a corollary of this theorem (credited to H. Sonnenschein) he showed that a decision scheme which is strategy-proof and satisfies ex post Pareto optimality is randomly dictatorial. We call this corollary the Gibbards random dictatorship theorem. We present a proof of this theorem which is direct and follows closely the original Gibbards approach. Focusing attention to the case with ex post Pareto optimality our proof is more simple and intuitive than the original Gibbards proof.Received: 15 October 2001, Accepted: 23 May 2003, JEL Classification: D71, D72Yasuhito Tanaka: The author is grateful to an anonymous referee and the Associate editor of this journal for very helpful comments and suggestions. And this research has been supported by a grant from the Zengin Foundation for Studies on Economics and Finance in Japan.     

Testing block‐diagonal covariance structure for high‐dimensional data

A test statistic is developed for making inference about a block‐diagonal structure of the covariance matrix when the dimensionality p exceeds n, where n = N ? 1 and N denotes the sample size. The suggested procedure extends the complete independence results. Because the classical hypothesis testing methods based on the likelihood ratio degenerate when p > n, the main idea is to turn instead to a distance function between the null and alternative hypotheses. The test statistic is then constructed using a consistent estimator of this function, where consistency is considered in an asymptotic framework that allows p to grow together with n. The suggested statistic is also shown to have an asymptotic normality under the null hypothesis. Some auxiliary results on the moments of products of multivariate normal random vectors and higher‐order moments of the Wishart matrices, which are important for our evaluation of the test statistic, are derived. We perform empirical power analysis for a number of alternative covariance structures.     

Sample size and the Binomial CUSUM Control Chart: the case of 100% inspection

The Binomial CUSUM is used to monitor the fraction defective (p) of a repetitive process, particularly for detecting small to moderate shifts. The number of defectives from each sample is used to update the monitoring CUSUM. When 100% inspection is in progress, the question arises as to how many sequential observations should be grouped together in forming successive samples. The tabular form of the CUSUM has three parameters: the sample size n, the reference value k, and the decision interval h, and these parameters are usually chosen using statistical or economic-statistical criteria, which are based on Average Run Length (ARL). Unlike earlier studies, this investigation uses steady-state ARL rather than zero-state ARL, and the occurrence of the shift can be anywhere within a sample. The principal finding is that there is a significant gain in the performance of the CUSUM when the sample size (n) is set at one, and this CUSUM might be termed the Bernoulli CUSUM. The advantage of using n=1 is greater for larger shifts and for smaller values of in-control ARL. First version: September 1998/Third revision: September 2000     

Party loyalty as habit formation

In most democracies, at least two out of any three individuals vote for the same party in sequential elections. This paper presents a model in which vote‐persistence is partly due to the dependence of the utility on the previous voting decision. This dependence is termed ‘habit formation’. The model and its implications are supported by individual‐level panel data on the presidential elections in the USA in 1972 and 1976. For example, it is found that the voting probability is a function of the lagged choice variable, even when the endogeneity of the lagged variable is accounted for, and that the tendency to vote for different parties in sequential elections decreased with the age of the voter. Furthermore, using structural estimation the effect of habit is estimated, while allowing unobserved differences among respondents. The structural habit parameter implies that the effect of previous votes on the current decision is quite strong. The habit model fits the data better than the traditional ‘party identification’ model. Copyright © 2003 John Wiley & Sons, Ltd.     

Action embedded in social structure: A systematic theory Part I,theory and evidence

This research develops a systematic theory, which premises a responsible and reasonable actor, of how social structure can affect rational social action, applies this general theory to explain 29 specific empirical findings about voting, and illustrates how deductive theories can synthesize prior empirical findings and provide a focus for subsequent empirical research. Part II, Explanations applies the systematic theory to account for specific empirical propositions that relate social statuses to the decisions voters make. Voters are classified by such graduated social statuses as socioeconomic class, age, and education; and by such nominal social statuses as religion, urban or rural residence, region, gender, race, and ethnicity. Positions on two key issues affect voting choice, the desire for economic equity and the desire for social equality. The members of the various social groups interpret these issues as they affect their own self-interest, and align with the party they perceive as furthering their interests. To corroborate aspects of the theory and to quantify the effects of the variables, survey data are analyzed.     

Asymmetrical Factorial Designs Containing Clear Effects

The asymmetrical or mixed-level factorial design is a kind of important design in practice. There is a natural problem on how to choose an optimal (s ²)s ⁿ design for the practical need, where s is any prime or prime power. This paper considers the clear effects criterion for selecting good designs. It answers the questions of when an (s ²)s ⁿ design with fixed number of runs contains clear two-factor interaction (in brief 2fi) components and when it contains clear main effects or clear 2fis. It further gives the complete classification of (s ²)s ⁿ designs according to the clear 2fi components, main effects and 2fis they have.     

A conditional distribution approach to uniform sampling on spheres and balls in Lp spaces

Liang and Ng (Metrika 68:83–98, 2008) proposed a componentwise conditional distribution method for L _p -uniform sampling on L _p -norm n-spheres. On the basis of properties of a special family of L _p -norm spherical distributions we suggest a wide class of algorithms for sampling uniformly distributed points on n-spheres and n-balls in L _p spaces, generalizing the approach of Harman and Lacko (J Multivar Anal 101:2297–2304, 2010), and including the method of Liang and Ng as a special case. We also present results of a numerical study proving that the choice of the best algorithm from the class significantly depends on the value of p.     

Kosten-optimale Prüfpläne für die Gut-Schlecht-Prüfung

Summary A lot is accepted if the number of defective units in a sample of sizen does not exceed the acceptance numberc. The usefulness of the sampling plan (n, c) is described by the regret function. This regret functionR(p), depending on the proportionp of defective units in the lot, is the expectation of the avoidable costs. There always exists an optimum sampling plan which minimizes the maximum ofR(p). The dependence of the maxima ofR(p) onn andc is studied and some theorems are given which are useful for calculating the minimax solution, that is the optimum sampling plan.      

The Median Voter Model Fails an Empirical Test: The Procedure, Useful in the Absence of a Better One, Is Not Valid for Multidimensional Issues

Abstract . The median voter model is a frequently used tool in analyzing local public sector issues. However, its use in analyzing multidimensional issues has been criticized. This study examines the criticism in the context of the decision by a municipality to establish a local constitution by voting for a home rule charter. A model is developed and tested using a variety of income measures. This model reveals that median income is not a better explanatory variable than other income measures. Based on this evidence, the authors conclude that the median voter model is not appropriate for analyzing multidimensional issues.     

A social choice function implementable via backward induction with values in the ultimate uncovered set

We prove the existence of a social choice function implementable via backward induction which always selects within the ultimate uncovered set. Whereas the uncovered set is the set of maximal elements of the covering relation, the ultimate uncovered set is the set obtained by iterative application of this covering operation. Dutta and Sen (1993) showed that any social choice function which is the solution of a generalized binary voting procedure is implementable via backward induction. Our result follows from Dutta and Sen's theorem, in that we construct a binary voting procedure always selecting within the ultimate uncovered set. We use the classical multistage elimination procedure, which always selects an alternative within the uncovered set. When this procedure is also used to select among all of the possible agendas or orderings of alternatives within the procedure, the alternative selected (from the agenda selected) will be within the uncovered set of the uncovered set. Our result follows from repeated application of this construction. Intuitively, the procedure constructed consists of requiring agents to vote on how they should vote and so on. Received: 7 April 1997 / Accepted: 15 October 1998     

Local monotonicity of power: Axiom or just a property?

This paper discusses whether Local Monotonicity (LM) should be regarded as a property of the power distribution of a specific voting game under consideration, indicated by a power measure, or as a characteristic of power per se. The latter would require reasonable power measures to satisfy a corresponding LM axiom. The former suggests that measures which do not allow for a violation of LM fail to account for dimensions of power which can cause nonmonotonicity in voting weight. Only if a measure is able to indicate nonmonotonicity, it can help design voting games for which power turns out to be monotonic. The argument is discussed in the light of recent extensions of traditional power indices.     

On the structure of voting systems between two alternatives

Voting systems between two alternatives have been widely studied in the literature of Social Choice. One of the results given by Fishburn (The theory of social choice. Princeton University Press, Princeton, 1973) allows us to characterize anonymous, neutral and monotonic voting systems by means of functions satisfying adequate conditions. From among all kinds of functions, the class of affine functions is highly interesting because from them it is possible to obtain the voting systems most used in practice. In this paper we analyze the structure of the set of these functions and we show that this set is convex and its extreme points are the functions that generate the following voting systems: simple majority, absolute majority, unanimous majority and Pareto majority. Moreover, we suggest a simple method for choosing a voting system when two alternatives are under consideration.     

Do congressional leaders detect and deter cartel cheating? Evidence from committee assignments

Abstract. The present study extends recent work in the public choice literature that suggests that monopoly legislators form cartel-like organizations in an effort to extract greater benefits in the political process. With any cartel organization cheating is expected to arise. If any political cartel (or business cartel) is to be successful, cheating must be detected and punished. Previous work examining the voting behavior and committee placement of Congressional Black Caucus members presents observable evidence of cartel success, but it fails to examine how cheating by individual Congressional Black Caucus members is detected and deterred. This study points out that (1) use of ADA voting records by Democratic House leaders renders “cartel cheating” easily detectable, (2) any observed cheating is punished by Democratic leaders through placement on less desirable House committees, and (3) such a punishment diminishes the “legislative” level of representation that an individual Representative can provide. Received: 29 May 2001 / Accepted: 29 July 2002 The authors thank two anonymous referees of this journal, Amihai Glazer, David Hobson and Troy Gibson for helpful comments. This work was supported through a Business Advisory Council Faculty Scholars Grant.     

The endowment game when n = 2

This paper considers a well defined problem in the implementation literature: A planner wants to re-allocate goods among two agents but she does not know each agent’s initial endowment of goods. We completely characterize social choice functions that can be implemented by feasible mechanisms in Nash equilibria in this n = 2 case.     

Increasing sample sizes do not always increase the power of UMPU-tests for 2×2 tables

We consider the uniformly most powerful unbiased (UMPU) one-sided test for the comparison of two proportions based on sample sizes m and n, i.e., the randomized version of Fisher's exact one-sided test. It will be shown that the power function of the one-sided UMPU-test based on sample sizes m and n can coincide with the power function of the UMPU-test based on sample sizes m+1 and n for certain levels on the entire parameter space. A characterization of all such cases with identical power functions is derived. Finally, this characterization is closely related to number theoretical problems concerning Fermat-like binomial equations. Some consequences for Fisher's original exact test will be discussed, too.     

Single-stage election procedures for large electorates

The Condorcet efficiency of single-stage election procedures is considered under the assumption of impartial culture for large electorates. The most efficient ranked voting rule is either Borda rule or a truncated scoring rule. A decision rule is established to determine the number of candidates, k, that individuals should be required to vote for, whether or not ranking should be required, and the scoring rule that should be used if ranking is required. This decision depends upon the number of candidates available and the probabilities that individuals will vote if they must rank k candidates or simply report k candidates.     

Measurement of power,probabilities, and alternative models of man

Starting from Max Weber’s definitions of power we discuss the meaning of his concept Chance and its relationships to the probabilistic notions that play a central role in definitions of indices of measuring voting power. Using Martin Hollis’s distinction between two models of man—plastic and autonomous—we argue that the common measures of voting power when interpreted in terms of probabilities seem to be in better conformity with the model of plastic man than with the model of autonomous man. The paper elaborates on the probability interpretation with applications to the modelling of power measures with a priori unions and to the concept of “real voting power” based on relative frequencies of historical events. Power as potential—which in our view is what Weber very likely meant—remains an elusive concept, but one that should be amenable to game-theoretic analysis.