In certain judgmental situations where a “correct” decision is presumed to exist, optimal decision making requires evaluation of the decision-makers’ capabilities and the selection of the appropriate aggregation rule. The major and so far unresolved difficulty is the former necessity. This article presents the optimal aggregation rule that simultaneously satisfies these two interdependent necessary requirements. In our setting, some record of the voters’ past decisions is available, but the correct decisions are not known. (...) We observe that any arbitrary evaluation of the decision-makers’ capabilities as probabilities yields some optimal aggregation rule that, in turn, yields a maximum-likelihood estimation of decisional skills. Thus, a skill-evaluation equilibrium can be defined as an evaluation of decisional skills that yields itself as a maximum-likelihood estimation of decisional skills. We show that such equilibrium exists and offer a procedure for finding one. The obtained equilibrium is locally optimal and is shown empirically to generally be globally optimal in terms of the correctness of the resulting collective decisions. Interestingly, under minimally competent (almost symmetric) skill distributions that allow unskilled decision makers, the optimal rule considerably outperforms the common simple majority rule (SMR). Furthermore, a sufficient record of past decisions ensures that the collective probability of making a correct decision converges to 1, as opposed to accuracy of about 0.7 under SMR. Our proposed optimal voting procedure relaxes the fundamental (and sometimes unrealistic) assumptions in Condorcet’s celebrated theorem and its extensions, such as sufficiently high decision-making quality, skill homogeneity or existence of a sufficiently large group of decision makers. (shrink)

The paper introduces the concept of polar decision rules and establishes that majority rules are polar rules. We identify second best rules and penultimate rules in cases that majority rules are optimal or the most inferior, respectively. We especially specify the almost expert rule and the almost majority rule as the secondary rules of the expert and majority rules, respectively.

We focus on the dichotomous choice model, which goes back as far as Condorcet (1785; Essai sur l'application de l'analyse a la probabilité des décisions rendues a la pluralité des voix, Paris). A group of experts is required to select one of two alternatives, of which exactly one is regarded as correct. The alternatives may be related to a wide variety of areas. A decision rule translates the individual opinions of the members into a group decision. A (...) decision rule is optimal if it maximizes the probability of the group to make a correct choice. In this paper we assume the correctness probabilities of the experts to be independent random variables, selected from some given distribution. Moreover, the ranking of the members in the team is (at least partly) known. Thus, one can follow rules based on this ranking. The polar different rules are the expert and the majority rules. The probabilities of the two polar rules being optimal were compared in a series of papers. The main purpose of this paper is to outline the results, providing exact formulas or estimates for these probabilities. We consider a variety of distributions and show that for all of these distributions the asymptotic behaviour of the probabilities of the two polar rules follows the same patterns. (shrink)

We study the uncertain dichotomous choice model. In this model a set of decision makers is required to select one of two alternatives, say ‘support’ or ‘reject’ a certain proposal. Applications of this model are relevant to many areas, such as political science, economics, business and management. The purpose of this paper is to estimate and compare the probabilities that different decision rules may be optimal. We consider the expert rule, the majority rule and a (...) few in-between rules. The information on the decisional skills is incomplete, and these skills arise from an exponential distribution. It turns out that the probability that the expert rule is optimal far exceeds the probability that the majority rule is optimal, especially as the number of the decision makers becomes large. (shrink)

Animals depleting one patch of resources must decide when to leave and switch to a fresh patch. Foraging theory has predicted various decision mechanisms; which is best depends on environmental variation in patch quality. Previously we tested whether these mechanisms underlie human decision making when foraging for external resources; here we test whether humans behave similarly in a cognitive task seeking internally generated solutions. Subjects searched for meaningful words made from random letter sequences, and as their success rate (...) declined, they could opt to switch to a fresh sequence. As in the external foraging context, time since the previous success and the interval preceding it had a major influence on when subjects switched. Subjects also used the commonness of sequence letters as a proximal cue to patch quality that influenced when to switch. Contrary to optimality predictions, switching decisions were independent of whether sequences differed little or widely in quality. (shrink)

Decision theory aims at a general account of rationality covering humans but to begin makes idealizations about decision problems and agents' resources and circumstances. It treats inerrant agents with unlimited cognitive power facing tractable decision problems. This book systematically rolls back idealizations and without loss of precision treats errant agents with limited cognitive abilities facing decision problems without a stable top option. It recommends choices that maximize utility using quantizations of beliefs and desires in cases where (...) probabilities and utilities are indeterminate and using higher-order utility analysis in cases of limited access to probabilities and utilities. For agents burdened with mistakes, it advocates reasonable attempts to correct unacceptable mistakes before deciding. In decision problems without a stable top option, a topic of game theory, it proposes maximizing self-conditional utility among self-supporting options. In games of strategy, the new principles lead to solutions that are Pareto optimal among equilibria composed of jointly self-supporting strategies. Offering an account of bounded rationality, the bookmakes large strides toward realism in decision theory. (shrink)

The current note clarifies why, in committees, the prior probability of a correct collective choice might be of particular significance and possibly should sometimes even be the sole appropriate basis for making the collective decision. In particular, we present sufficient conditions for the superiority of a rule based solely on the prior relative to the simple majority rule, even when the decisional skills of the committee members are assumed to be homogeneous.

A dichotomous decision-making context in committees is considered where potential partisan members with predetermined votes can generate inefficient decisions and buy neutral votes. The optimal voting rule minimizing the expected costs of inefficient decisions for the case of a three-member committee is analyzed. It is shown that the optimal voting rule can be non-monotonic with respect to side-transfers: in the symmetric case, majority voting is optimal under either zero, mild or full side-transfer possibilities, whereas (...) unanimity voting may be optimal under an intermediate side-transfer possibility. The side-transfer possibilities depend on the power of partisans (their ability or willingness to pay for neutral votes) relative to the corruptibility of neutral members (personal cost of deliberately casting a `wrong' vote). (shrink)

Modest difference in individual decisional skills may warrant substantial inequality in power. This claim has been illustrated in Ben Yashar and Nitzan (Economics Letters 174:93–95, 2019), applying the symmetric uncertain dichotomous choice setting and focusing on the skill-dependent (s-d) power of the decision-makers under the optimal decision rule. The same claim is valid when one focuses on the relationship between skill heterogeneity and the distribution of the second type of power, viz., the group members' marginal contribution (...) (mc). This has been shown in Ben Yashar et al. (Journal of Theoretical Politics 33(2):225–235, 2021), where mc has been formally related to s-d power. A preliminary study of the relationship between inequality of the distributions of the two measures has also been presented. The current paper considerably expands this investigation by applying simulations and sheds light on the reason for the increased inequality of the two types of power. Since reward (in terms of status, payment or both) depends on s-d power and mc, inequality of the distribution of rewards depends on the inequality of the two types of power. Comparison of the two types of power inequality is therefore interesting if one wishes to shed light on the role of the two types of power on reward inequality. In particular, the results of the comparison may clarify the effectiveness of the incentives to invest in skills, dependent on the prevailing norms and the institutional characteristics that determine the relationship between rewards and the two types of power. For this reason, the current study focuses on the comparison between inequality of s-d power and marginal contributions showing that, in small (large) groups, on average, the latter is more (less) unequal than the former and both are more unequal than the optimal weights of the decision-makers and much more unequal than their skills. The robustness of these findings is shown by applying alternative symmetric probability distribution functions. Another novel task undertaken in the current study is the sensitivity analysis of inequality of the power measures with respect to changes in the size of the group and the skills of the group members. (shrink)

In a homogeneous jury, in which each vote is correct with the same probability, and each pair of votes correlates with the same correlation coefficient, there exists a correlation-robust voting quota, such that the probability of a correct verdict is independent of the correlation coefficient. For positive correlation, an increase in the correlation coefficient decreases the probability of a correct verdict for any voting rule below the correlation-robust quota, and increases that probability for any above the correlation-robust quota. The (...) jury may be less competent under the correlation-robust rule than under simple majority rule and less competent under simple majority rule than a single juror alone. The jury is always less competent than a single juror under unanimity rule. (shrink)

Some decisions result in cognitive consequences such as information gained and information lost. The focus of this study, however, is decisions with consequences that are partly or completely noncognitive. These decisions are typically referred to as ‘real-life decisions’. According to a common complaint, the challenges of real-life decision making cannot be met by decision theory. This complaint has at least two principal motives. One is the maximizing objection that to require agents to determine the optimal act under (...) real-world constraints is unrealistic. The other is the precision objection that the numeric requirements for applying decision theory are overly demanding for real-life decisions. Responses to both objections are aired in the History section of this chapter. The maximizing objection is addressed with reference to work by Weirich and Pollock, while the precision objection is countered via a proposal by Kyburg and another by Gärdenfors and Sahlin. However, the Current Research section urges a different response to the precision objection by introducing a comparative version of decision theory. Drawing on Chu and Halpern’s notion of generalized expected utility, this version of decision theory permits many choices to be based on merely comparative plausibilities and utilities. Finally, the Further Research section undertakes an open-ended exploration of three of the assumptions upon which this form of decision theory (and many others) is based: transitivity, independence, and plausibilistic decision rules. (shrink)

A broad range of intellectual perspectives may be brought to bear on any important social institution. To this general rule, the institution of private property is no exception. The desirability of private property has been endlessly debated across the disciplines: philosophical, historical, economic, and legal. Yet there is very little consensus over its proper social role and limitations. Is it possible to find a unique solution to questions of property and private ownership, good for all resources and for all (...) times? The famous defense of private property that is found in chapter 5 of John Locke'sSecond Treatise of Governmentanswers this question in the affirmative, for Locke writes as though all property was given to mankind in common, and then seeks to find the quickest and most expeditious way to convert all common property into private property. His implicit, but undefended, assumption is that common forms of property are both undesirable and unstable, while private forms of ownership are always just the opposite. Yet his preferred method for moving from a commons to a regime of private property—the unilateral decision to appropriate by each actor—is one that has been frequently condemned as a sop to unbridled egotism, even though it is subject to two constraints: the first against waste, and the second requiring the appropria-tor to leave “enough, and as good” for others. (shrink)

In a three-candidate election, a scoring rule s (s in [0,1]) assigns 1, s, and 0 points (respectively) to each first, second and third place in the individual preference rankings. The Condorcet efficiency of a scoring rule is defined as the conditional probability that this rule selects the winner in accordance with Condorcet criteria (three Condorcet criteria are considered in the paper). We are interested in the following question: What rule s has the greatest Condorcet efficiency? (...) After recalling the known answer to this question, we investigate the impact of social homogeneity on the optimal value of ?. One of the most salient results we obtain is that the optimality of the Borda rule (s=1/2) holds only if the voters act in an independent way. (shrink)

According to rule-consequentialism’s moral criterion, a given action is morally right if and only if it complies with an ideal code of rules, regardless of the consequences of that action. Rules are to be assessed by their consequences, not actions. This being so, one of the many accusations that have been made against rule-consequentialism is that it can turn suboptimal decisions into morally right decisions and optimal decisions into morally wrong decisions. After all, in certain circumstances, a (...) rule that has the best consequences overall can require an action that does less good or forbid an action that does more good. This is the core of the original rule-worship objection. However, recently, different versions of rule-consequentialism have been developed in order to address that challenge. This paper focuses on three of them, those offered by Brad Hooker, Susan Wolf, and David Copp. My claim is that their arguments, however ingenious, are unsuccessful, because they are innocuous if the charge is reformulated as it should be. (shrink)

In this paper, we derive necessary and sufficient conditions on team based tasks in order for a selection criterion applied to individuals to produce optimal teams. We assume only that individuals have types and that a team’s performance depends on its size and the type composition of its members. We first derive the selection principle which states that if a selection criterion exists, it must rank types by homogeneous team performance, the performance of a team consisting only of that (...) type. We then prove that a selection criterion exists if and only if replacing the team’s lowest ranked type, as measured by homogeneous team performance, with a higher ranked type increases team performance. Finally, we show that the replace the lowest ranked property rules out most common types of team complementarities, including benefits to diverse types and types that fill structural holes. (shrink)

Information is a peculiar quantity. Unlike matter and energy, which are conserved by the laws of physics, the aggregation of knowledge from many sources can in fact produce more information (synergy) or less (redundancy) than the sum of its parts. This feature can endow groups with problem‐solving strategies that are superior to those possible among noninteracting individuals and, in turn, may provide a selection drive toward collective cooperation and coordination. Here we explore the formal properties of information aggregation as a (...) general principle for explaining features of social organization. We quantify information in terms of the general formalism of information theory, which also prescribes the rules of how different pieces of evidence inform the solution of a given problem. We then show how several canonical examples of collective cognition and coordination can be understood through principles of minimization of uncertainty (maximization of predictability) under information pooling over many individuals. We discuss in some detail how collective coordination in swarms, markets, natural language processing, and collaborative filtering may be guided by the optimal aggregation of information in social collectives. We also identify circumstances when these processes fail, leading, for example, to inefficient markets. The contrast to approaches to understand coordination and collaboration via decision and game theory is also briefly discussed. (shrink)

Katarzyna de Lazari-Radek and Peter Singer’s wonderful book, The Point of View of the Universe: Sidgwick and Contemporary Ethics, contains a wealth of intriguing arguments and compelling ideas. The present paper focuses on areas of continuing dispute. The paper first attacks LazariRadek’s and Singer’s evolutionary debunking arguments against both egoism and parts of common-sense morality. The paper then addresses their discussion of the role of rules in utilitarianism. De Lazari-Radek and Singer concede that rules should constitute our moral decision (...) procedure and our public morality. This paper argues that, if no one should be blamed for complying with the optimal decision procedure and optimal public rules, there are strong reasons to accept that these same rules determine what is morally permissible from what is morally wrong. (shrink)

Katarzyna de Lazari-Radek and Peter Singer’s wonderful book, The Point of View of the Universe: Sidgwick and Contemporary Ethics, contains a wealth of intriguing arguments and compelling ideas. The present paper focuses on areas of continuing dispute. The paper first attacks LazariRadek’s and Singer’s evolutionary debunking arguments against both egoism and parts of common-sense morality. The paper then addresses their discussion of the role of rules in utilitarianism. De Lazari-Radek and Singer concede that rules should constitute our moral decision (...) procedure and our public morality. This paper argues that, if no one should be blamed for complying with the optimal decision procedure and optimal public rules, there are strong reasons to accept that these same rules determine what is morally permissible from what is morally wrong. (shrink)

Given free information and unlimited processing power, should decision algorithms use as much information as possible? A formal model of the decision-making environment is developed to address this question and provide conditions under which informationally frugal algorithms, without any information or processing costs whatsoever, are optimal. One cause of compression that allows optimal algorithms to rationally ignore information is inverse movement of payoffs and probabilities (e.g., high payoffs occur with low probably and low payoffs occur with (...) high probability). If inversely related payoffs and probabilities cancel out, then predictors that correlate with payoffs and consequently condition the probabilities associated with different payoffs will drop out of the expected-payoff objective function, severing the link between information and optimal action rules. Stochastic payoff processes in which rational ignoring occurs are referred to as compressed environments, because optimal action depends on a reduced-dimension subset of the environmental parameters. This paper considers benefits and limitations of economic models versus other methods for studying links between environmental structure and the real-world success of simple decision procedures. Different methods converge on the normative proposition of ecological rationality, as opposed to axiomatic rationality based on informational efficiency and internal consistency axioms, as a superior framework for comparing the effectiveness of decision strategies and prescribing decision algorithms in application. (shrink)

This paper considers whether the category of sovereign “decision,” as it is used in Carl Schmitt’s Political Theology, has analogies with the paradigm of “choice,” as it is theorized in neoliberalism. Both decision and choice belong to that mode of judgement that “cuts” the field of alternatives into two, into two “extreme” alternatives. This mode of judgment not only presupposes the subject of the decision, but also sets up the terms of the choice, clearly indicating the (...) class='Hi'>optimal option. For Schmitt, a political decision arises in a “case of necessity”: in the “serious case” of war. And yet, it is precisely the neoliberal depoliticization of war that reduces it to a rule, and every choice fundamentally to a stance for or against. To escape from the neoliberal reduction of political decision in the form of a choice, I examine another mode of decision, absent from Schmitt’s texts: “disclosedness.”. (shrink)

How, and how well, do people switch between exploration and exploitation to search for and accumulate resources? We study the decision processes underlying such exploration/exploitation trade‐offs using a novel card selection task that captures the common situation of searching among multiple resources (e.g., jobs) that can be exploited without depleting. With experience, participants learn to switch appropriately between exploration and exploitation and approach optimal performance. We model participants' behavior on this task with random, threshold, and sampling strategies, and (...) find that a linear decreasing threshold rule best fits participants' results. Further evidence that participants use decreasing threshold‐based strategies comes from reaction time differences between exploration and exploitation; however, participants themselves report non‐decreasing thresholds. Decreasing threshold strategies that “front‐load” exploration and switch quickly to exploitation are particularly effective in resource accumulation tasks, in contrast to optimal stopping problems like the Secretary Problem requiring longer exploration. (shrink)

In a three-candidate election, a scoring rule λ, λ∈[0,1], assigns 1,λ and 0 points (respectively) to each first, second and third place in the individual preference rankings. The Condorcet efficiency of a scoring rule is defined as the conditional probability that this rule selects the winner in accordance with Condorcet criteria (three Condorcet criteria are considered in the paper). We are interested in the following question: What rule λ has the greatest Condorcet efficiency? After recalling the (...) known answer to this question, we investigate the impact of social homogeneity on the optimal value of λ. One of the most salient results we obtain is that the optimality of the Borda rule (λ=1/2) holds only if the voters act in an independent way. (shrink)

In many voting systems, voters’ preferences on a set of candidates are represented by linear orderings. In this context, scoring rules are well-known procedures to aggregate the preferences of the voters. Under these rules, each candidate obtains a fixed number of points, sk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_k$$\end{document}, each time he/she is ranked k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}th by one voter and the candidates are ordered according to the total number of (...) points they receive. In order to identify the best scoring rule to use in each situation, we need to know which properties are met by each of these procedures. Although some properties have been analyzed extensively, there are other properties that have not been studied for all scoring rules. In this paper, we consider two desirable social choice properties, the Pareto-optimality and the immunity to the absolute loser paradox, and establish characterizations of the scoring rules that satisfy each of these specific axioms. Moreover, we also provide a proof of a result given by Saari and Barney, where the scoring rules meeting reversal symmetry are characterized. From the results of characterization, we establish some relationships among these properties. Finally, we give a characterization of the scoring rules satisfying the three properties. (shrink)

What is the optimal size of a democratic society? While not taking an explicit stand on this issue, Hélène Landemore's model of democracy in Democratic Reason suggests that democracies ought to be small, certainly smaller than many existing states. If, as Landemore argues, we must rely on the random selection of representatives, then we should be concerned about both the size of the population and the way cognitive diversity is distributed within it. Given the realities of party politics and (...) media framing, this means that smaller political societies will yield wiser decisions than very large ones. (shrink)

In this study I analyse the performance of a democratic decision-making rule: the weighted majority rule. It assigns to each voter a number of votes that is proportional to her stakes in the decision. It has been shown that, for collective decisions with two options, the weighted majority rule in combination with self-interested voters maximises the common good when the latter is understood in terms of either the sum-total or prioritarian sum of the voters’ well-being. (...) The main result of my study is that this argument for the weighted majority rule — that it maximises the common good — can be improved along the following three main lines. (1) The argument can be adapted to other criteria of the common good, such as sufficientarian, maximin, leximin or non-welfarist criteria. I propose a generic argument for the collective optimality of the weighted majority rule that works for all of these criteria. (2) The assumption of self-interested voters can be relaxed. First, common-interest voters can be accommodated. Second, even if voters are less than fully competent in judging their self-interest or the common interest, the weighted majority rule is weakly collectively optimal, that is, it almost certainly maximises the common good given large numbers of voters. Third, even for smaller groups of voters, the weighted majority rule still has some attractive features. (3) The scope of the argument can be extended to decisions with more than two options. I state the conditions under which the weighted majority rule maximises the common good even in multi-option contexts. I also analyse the possibility and the detrimental effects of strategic voting. Furthermore, I argue that self-interested voters have reason to accept the weighted majority rule. (shrink)

This article explores links between the modern theory of rational choice and ethics. Ethics allows us to answer an unsolved question in modern decision theory: the structuring problem in decisions. Such a problem cannot be solved coming from the principle of expected utility. This principle can solve the problem of ‚choosing’ among given alternatives, but does not establish which alternatives should be taken into account in decision. In order to understand the structuring problem, the act of ‚choosing’ has (...) to be completed with three human acts: operating, predicting and evaluating. At the same time, the subjective criteria of rational choice – probability and utility – have to be completed with three objective criteria: efficiency, truth and goodness. Utility is a subjective value of an alternative considered in relation to another. Efficiency, truth and goodness are objective qualities of the joint alternatives in relation to the reality: the real operating possibilities of the agent, the real possibilities of his environment and the true good produced. So, the rule that guarantees a decision to be optimal would be: “do that which most certainly maximises our preferences, according to real operative resources, true knowledge and the right will”. However, this rule cannot be applied without the development of virtues, which could be seen as the main moral competences in decision-making. (shrink)

We consider several natural fragments of the alternating-time temporal logics ATL* and ATL with restrictions on the nesting between temporal operators and strategic quantifiers. We develop optimal decision procedures for satisfiability in these fragments, showing that they have much lower complexities than the full languages. In particular, we prove that the satisfiability problem for state formulae in the full `strategically flat' fragment of ATL* is PSPACE-complete, whereas the satisfiability problems in the flat fragments of ATL and ATL$^{+}$ are (...) $\Sigma^P_3$-complete. We note that the nesting hierarchies for fragments of ATL* collapse in terms of expressiveness above nesting depth 1, hence our results cover all such fragments with lower complexities. (shrink)

Florida International University, USA edwardj{at}fiu.edu ' + u + '@ ' + d + ' '/ /- -> Several authors have analyzed the optimal k -majority rule based on a variety of criteria. Buchanan and Tullock argued that, in constitutional settings, the criterion should be that all changes meet the Pareto criterion; otherwise the status quo should be preferred. They then asserted that unanimity rule would be the preferred voting rule in this setting. In parliamentary settings, (...) they claimed that a near majority rule would be preferred because it minimizes the sum of decision costs and external costs. This article investigates both claims in an N -voter, two-alternative setting. We show the conditions under which unanimity rule is less likely to select BT preferred alternatives than other k -majority rules and prove that the difference in performance can be negligible when N is large and certain weak conditions are met. Furthermore, if we define external costs as the expected number of losers from a BT-inferior vote, then external costs become negligible for a range of supermajority rules. This implies that unanimity rule and a range of supermajority rules should be equally preferred when decision costs are added. Finally, we show that the external cost function can actually increase for certain populations. Many of the broader conclusions should also hold for multiple alternatives. Key Words: constitutional design • Pareto criterion • external costs • social contract. (shrink)

In many learning or inference tasks human behavior approximates that of a Bayesian ideal observer, suggesting that, at some level, cognition can be described as Bayesian inference. However, a number of findings have highlighted an intriguing mismatch between human behavior and standard assumptions about optimality: People often appear to make decisions based on just one or a few samples from the appropriate posterior probability distribution, rather than using the full distribution. Although sampling-based approximations are a common way to implement Bayesian (...) inference, the very limited numbers of samples often used by humans seem insufficient to approximate the required probability distributions very accurately. Here, we consider this discrepancy in the broader framework of statistical decision theory, and ask: If people are making decisions based on samples—but as samples are costly—how many samples should people use to optimize their total expected or worst-case reward over a large number of decisions? We find that under reasonable assumptions about the time costs of sampling, making many quick but locally suboptimal decisions based on very few samples may be the globally optimal strategy over long periods. These results help to reconcile a large body of work showing sampling-based or probability matching behavior with the hypothesis that human cognition can be understood in Bayesian terms, and they suggest promising future directions for studies of resource-constrained cognition. (shrink)

A transformative decision rule transforms a given decision probleminto another by altering the structure of the initial problem,either by changing the framing or by modifying the probability orvalue assignments. Examples of decision rules belonging to thisclass are the principle of insufficient reason, Isaac Levi'scondition of E-admissibility, the de minimis rule, andthe precautionary principle. In this paper some foundationalissues concerning transformative decision rules are investigated,and a couple of formal properties of this class of rules areproved.

Virtual enterprise is a dynamic alliance of businesses, in which multiple members undertake joint research, development, manufacturing, operation, etc. The complexity of the relationship between business members, coupled with many new technologies or methods applied in the alliance operation, leads to more uncertain factors and difficulties in the operation and risk management of the virtual enterprise. The distributed decision-making model is a fast and effective decision-making model, in which dispersed intellectual resources and information resources are dynamically integrated through (...) virtual organization forms and operating mechanisms, and the dynamic reorganization, result evaluation, and synthesis of the decision-making system are realized by certain control rules. On the basis of summarizing and analyzing previous research works, this paper expounded the research status and significance of the risk control of virtual enterprise; elaborated the development background, current status, and future challenges of distributed decision-making model; introduced the related works of decision-making variable calculation and constraint determination; constructed a risk control model for virtual enterprise based on distributed decision-making model; analyzed the multiobjective model and interval programming model of risk control; established a risk control approach for virtual enterprise based on distributed decision-making model; performed the optimal allocation of risk control funds and the selective optimization of backup allies; and finally conducted a case analysis. The study results show that, in view of the diversity of relationships between alliance businesses and the complexity of the distribution of cooperative areas, the distributed decision-making model can more reasonably describe the relationship between alliance members and accurately control the risks of virtual enterprise from multiple angles, which can effectively manage the risks of virtual enterprise. The study results of this paper provide a reference for further research on the risk control of virtual enterprise based on distributed decision-making model. (shrink)

We characterize seniority rules, also known as lexical dictatorships, under weak consistency constraints on the group’s choice function. These constraints are base triple-acyclicity in the case of binary choices and rationalizability (although not rationality) in the case of choices between an arbitrary number of alternatives. Existing results on these weakened constraints remain silent on the treatment of the group’s most junior individuals and therefore do not yield a complete characterization of seniority rules. We also impose a universal domain, binary strict (...) Pareto optimality, binary Pareto indifference, binary independence of irrelevant alternatives, and the newly introduced condition of conflict resolution. The latter condition requires a social choice rules not to remain indecisive between alternatives for which individuals have conflicting preferences. (shrink)

The concept of transformative decision rules provides auseful tool for analyzing what is often referred to as the`framing', or `problem specification', or `editing' phase ofdecision making. In the present study we analyze a fundamentalaspect of transformative decision rules, viz. permutability. A setof transformative decision rules is, roughly put, permutable justin case it does not matter in which order the rules are applied.It is argued that in order to be normatively reasonable, sets oftransformative decision rules have to (...) satisfy a number ofstructural conditions that together imply permutability. Thisformal result gives support to a non-sequential theory of framing,i.e., a theory which prescribes no uniform order in which differentsteps in the framing process have to be performed. (shrink)

This study was conducted to corroborate findings that females invoke a decision rule that is significantly different from that of their male counterparts when making ethical value judgements. In addition, the study examines whether the same decision rule is used by men and women for all types of ethical situations. The results show that males and females use different decision rules when making ethical evaluations, although there are types of situations where there are no significant (...) differences in decision rules used by men and women. The results do not suggest that any one particular decision rule is used by the majority of either males or females in different types of ethical judgements. There is a greater diversity in decision rules used by females than by males. (shrink)

Gigerenzer, Todd, and the ABC Research Group give an interesting account of simple decision rules in a variety of contexts. I agree with their basic idea that animals use simple rules. In my commentary I concentrate on some aspects of their treatment of decision rules in behavioural ecology.

Adam Elga (Philosophers’ Imprint, 10(5), 1–11, 2010) presents a diachronic puzzle to supporters of imprecise credences and argues that no acceptable decision rule for imprecise credences can deliver the intuitively correct result. Elga concludes that agents should not hold imprecise credences. In this paper, I argue for a two-part thesis. First, I show that Elga’s argument is incomplete: there is an acceptable decision rule that delivers the intuitive result. Next, I repair the argument by offering a (...) more elaborate diachronic puzzle that is more difficult for imprecise Bayesians to avoid. (shrink)

We consider a decision board with representatives who vote on proposals on behalf of their constituencies. We look for decision rules that realize utilitarian and (welfarist) egalitarian ideals. We set up a simple model and obtain roughly the following results. If the interests of people from the same constituency are uncorrelated, then a weighted rule with square root weights does best in terms of both ideals. If there are perfect correlations, then the utilitarian ideal requires proportional weights, (...) whereas the egalitarian ideal requires equal weights. We investigate correlations that are in between these extremes and provide analytic arguments to connect our results to Barberà and Jackson (J Polit Econ 114(2):317–339, 2006) and to Banzhaf voting power. (shrink)