Impossibility Theorem Research Articles

The Death of Welfare Economics: History of a Controversy

The death of welfare economics has been declared several times. One of the reasons cited for these plural obituaries is that Kenneth Arrow’s impossibility theorem, as set out in his path-breaking Social Choice and Individual Values in 1951, has shown that the social welfare function – one of the main concepts of the new welfare economics as defined by Abram Bergson (Burk) in 1938 and clarified by Paul Samuelson in the Foundations of Economic Analysis (1947, ch. VIII) – does not exist under reasonable conditions. Indeed, from the very start, Arrow kept asserting that his famous impossibility result has direct and devastating consequences for the Bergson-Samuelson Social Welfare Function (1948, 1950, 1951a, 1963), though he seemed to soften his position in the early eighties. On his side, especially from the seventies on, Samuelson remained active on this issue and continued to defend the concept he had devised with Bergson, tooth and nail, against Arrow’s attacks (1967, 1977, 1981, 1987, 2005). The aim of this paper is precisely to examine this rather strange controversy, which is almost unknown in the scientific community, even though it lasted more than fifty years and saw a conflict between two economic giants, Arrow and Samuelson, and behind them two distinct communities – the fading welfare economics against the emerging social choice theory –, two conflicting ways of dealing with mathematical tools in welfare economics and, above all, two different conceptions of social welfare. By relying on different kinds of material, I attempt to grasp what the exchanges between Arrow, Samuelson and others, both overtly and behind the scenes, reveal regarding the motivations of my two main actors.

Visualization of Clusters in an Educational Data Set Based on Convex-Hull Shape Preservation Algorithm

We study the problem of visualization of clusters in an educational data set based on convex-hull shape preservation algorithm. This problem considers multidimensional data with pre-established classes with the requirement of elements of different classes must be presented at distinctive regions. Such problem is commonly found on economic and social data, where visualization is important to understand a phenomenon before further analysis. In this paper, we propose an algorithm that uses a nonlinear transformation to preserve some data distance properties and display in a convenient format to interpretation. The proposed visualization algorithm is a partition-conforming projection, as defined by Kleinberg [An impossibility theorem for clustering, Adv. Neural Inform. Processing Syst. 15: Proc. 2002 Conf., 2003, The MIT Press, p. 463.], and completely separates the convex hull of data classes by applying locally linear operations. We applied this algorithm to visualize data from an important exam applied for over four million students of the Brazilian educational system Exame Nacional do Ensino Médio (ENEM). Results show that the proposed algorithm successfully separates unintelligible data and presents it more accessible to further visual analysis.

Characterizing and Adapting the Consistency-Latency Tradeoff in Distributed Key-Value Stores

The CAP theorem is a fundamental result that applies to distributed storage systems. In this article, we first present and prove two CAP-like impossibility theorems. To state these theorems, we present probabilistic models to characterize the three important elements of the CAP theorem: consistency (C), availability or latency (A), and partition tolerance (P). The theorems show the un-achievable envelope, that is, which combinations of the parameters of the three models make them impossible to achieve together. Next, we present the design of a class of systems called Probabilistic CAP (PCAP) that perform close to the envelope described by our theorems. In addition, these systems allow applications running on a single data center to specify either a latency Service Level Agreement (SLA) or a consistency SLA. The PCAP systems automatically adapt, in real time and under changing network conditions, to meet the SLA while optimizing the other C/A metric. We incorporate PCAP into two popular key-value stores: Apache Cassandra and Riak. Our experiments with these two deployments, under realistic workloads, reveal that the PCAP systems satisfactorily meets SLAs and perform close to the achievable envelope. We also extend PCAP from a single data center to multiple geo-distributed data centers.

How well does theory explain reality? - A case of Arrow's impossibility theorem

In this paper, we investigate how well Arrow's impossibility theorem, a typical no-existence proof, can explain real human decision-making. Since the conclusion of a no-existence proof is universal, it may also be useful to examine instances of illogical decision-making which are actually handled in reality if an Arrovian approach is seriously considered. Then, we try to modify the framework to reflect the actual decision-making situation as much as possible by allowing individuals to have their own sets of alternatives. Since individual sets of alternatives are varied in nature, Arrow's framework is not satisfied in general because a social ordering of all alternatives does not normally exist. Therefore, it may be concluded that Arrow's impossibility result holds not only in theory but also in reality. This conclusion may open up a way to manage an illogical social decision such as cyclical decision-making and strategic negotiation, among others which are popular and effective in actual decision-making.

How well does theory explain reality? - A case of Arrow's impossibility theorem

In this paper, we investigate how well Arrow's impossibility theorem, a typical no-existence proof, can explain real human decision-making. Since the conclusion of a no-existence proof is universal, it may also be useful to examine instances of illogical decision-making which are actually handled in reality if an Arrovian approach is seriously considered. Then, we try to modify the framework to reflect the actual decision-making situation as much as possible by allowing individuals to have their own sets of alternatives. Since individual sets of alternatives are varied in nature, Arrow's framework is not satisfied in general because a social ordering of all alternatives does not normally exist. Therefore, it may be concluded that Arrow's impossibility result holds not only in theory but also in reality. This conclusion may open up a way to manage an illogical social decision such as cyclical decision-making and strategic negotiation, among others which are popular and effective in actual decision-making.

Impossible by design? Fairness, strategy, and Arrow’s impossibility theorem

The design process often requires work by teams, rather than individuals. During team based design it is likely that situations will arise in which individual members of the team have different opinions, yet a group decision must still be made. Unfortunately, Arrow’s impossibility theorem indicates that there is no method for aggregating group preferences that will always satisfy a small number of ‘fair’ conditions. This work seeks to identify methods of combining individual preferences that can come close to satisfying Arrow’s conditions, enabling decisions that are fairer in practice. First, experiential conjoint analysis was used to obtain individual empirical utility functions for drinking mug designs. Each empirical utility function represented individual members who were part of a design team. Then, a number of functions for constructing group preference were analysed using both randomly generated preferences and empirical preferences derived from the experiential conjoint survey. The analysis involved checking each of Arrow’s conditions, as well as assessing the potential impact of strategic voting. Based on the results, methods that should be used to aggregate group preference within a design team in practice were identified and recommended.

The “Win-Win-Win Papakonstantinidis Model”: from Social Welfare’s Philosophy towards a Rural Development Concept by Rural Tourism Approach: The WERT Case Study

The article is dealing with two interconnected problems based on the conjectures: a) social welfare is a condition for rural development and not the prerequisite for it; b) shape a new landscape (the “win-win-win”) based on critique of the “Impossibility Theorem (Kenneth Arrow 1951) through the Nash Bargaining Solution (Nash, John 1950). Specifically, this article discusses and analyses social welfare and rural development objectives integrating elements from the impossibility theorem, the bargaining theory, and the theory of agency by (a) reviewing the literature on coordination “social welfare” and “rural development” (b) reversing the focus from “voting” to “bargaining” and (c) underlining that Social choice is the perquisite of social welfare, using the “win-win-win Papakonstantinidis model’s solution as the bridge between “voting”(Arrow) and “bargaining”(proposal). This solution highlights the Role of Rural Community as an “Aggregation” corresponding to its “sensitization process”.

The Austrians and the Arrow Impossibility Theorem

Some economists in the Austrian tradition interpret the Arrow Impossibility Theorem (AIT) as support for their laissez-faire standpoint. They presume that the AIT cannot be applied in market settings, but in the context of voting,it adds weight to their arguments against the possibility of socialism and planning.The examples of Boettke (2001) and Boettke and Leeson (2002)are taken as examples. It is shown that their stance is founded on a misinterpretation of Arrow, and no compelling case for exempting markets from the AITis found. The Austrian writers draw on Buchanan’s critique of Arrow, which critique is in turn shown to be faulty and unreliable. Finally a possible avenue for future Austrian research is identified, showing that the AIT might not apply to markets.

Cross-Currency Consistency, Three-Part SDF Factorizations, and an Impossibility Theorem for the Stationarity of Exchange Rates in International Economies

Cross-Currency Consistency, Three-Part SDF Factorizations, and an Impossibility Theorem for the Stationarity of Exchange Rates in International Economies

Complexity, and New Dynamic-Pricing 'Learning' Models that Reduce Antitrust Liability, Deadweight Losses, Regret and GPS-Manipulation In the 'Sharing Economy'

During 2005-2018, the “Sharing Economy” (which includes Airbnb; Apple; Alibaba, Uber; WeWork; Liquidspace, Lyft; Ebay; Didi Chuxing; Taxify; Amazon; HomeAway; etc.) blossomed across the world, triggered structural changes in industries and significantly affected international capital flows primarily by disobeying a wide variety of statutes laws in many countries and illegally reducing and changing the nature of competition in many industries often to the detriment of social welfare. Furthermore, Uber’s and Airbnb’s pricing systems are not efficient and can generate deadweight-losses and Regret. Other “sharing economy companies” (“SEOs”) also face similar pricing, antitrust, deadweight-loss and Regret issues (eg. WeWork; Lyft; Taxify; LiquidSpace; ShareDesk; Homeaway; etc.). This article: i) develops new dynamic pricing models for the Sharing Economy companies; ii) derives some stability properties of mixed games and dynamic algorithms that are inherent in new dynamic pricing models introduced herein which eliminate antitrust liability for SEOs and also reduce deadweight losses, greed, Regret and GPS manipulation - that is, the new dynamic pricing models exhibit “Multi-Stability”. The new dynamic pricing models have desirable properties such as individual rationality; exact and approximate welfare maximization; strong budget balance; dominant-strategy incentive compatibility of sellers and buyers; pareto optimality and feasibility – and thus, each model contravenes the Myerson-Satterthwaite Impossibility Theorem. The pricing models include multi-sided auctions wherein at any time t, several auctions can simultaneously occur, and the payoff functions of any buyer-seller pair in any auction depends on the bidding done by at least another buyer-seller pair either at the same time, or at a different time (this “long-memory” bias of buyers and sellers is new in the literature). More importantly the issues and models discussed herein have or can have significant implications for inequality, labor mobility and income dynamics (for employees of SEOs).

An Impossibility Theorem in Repeated Games

We prove a negative result in repeated games which shows that a sizable part of the set of feasible individually rational payoffs can never be supported by strategies that are at all robust to players’ discount factors. We find the cutoff defining this region and interpret it as a limit on the ability to punish deviations when future rewards for randomization cannot be finely calibrated. Furthermore, we present a robust folk theorem to support payoffs in the complementary region with “Blackwell-Nash” strategies that remain SPNE at all greater discount factors.

The Competitive Paradigm in Spatial Economics

This paper discusses the reasons for the spatial impossibility theorem, which states that the competitive paradigm cannot explain the formation of large urban agglomerations and trade flows. This result is especially meaningful insofar as it is internal to the theory itself. We then briefly explores different solutions to remedy to this methodological failure.

Aggregating extended preferences

An important objection to preference-satisfaction theories of well-being is that they cannot make sense of interpersonal comparisons. A tradition dating back to Harsanyi (J Political Econ 61(5):434, 1953) attempts to solve this problem by appeal to people’s so-called extended preferences. This paper presents a new problem for the extended preferences program, related to Arrow’s celebrated impossibility theorem. We consider three ways in which the extended-preference theorist might avoid this problem, and recommend that she pursue one: developing aggregation rules (for extended preferences) that violate Arrow’s Independence of Irrelevant Alternatives condition.

Rawls’s maximin rule and Arrow’s impossibility theorem

Rawls’s maximin rule is closely related to Arrow’s impossibility theorem. When there are at least three social states and three social members, a social welfare functional with full domain is the maximin rule if and only if it satisfies anonymity, binary independence, weak ordinal non-comparability, the weak Pareto principle, and the Pareto indifference principle.

The possibility of Arrovian social choice with the process of nomination

In this paper, we introduce an Arrovian social choice framework with the process of nomination. We consider a two-stage social choice procedure in which some alternatives are first nominated by aggregating the opinions of nominators, and then the society makes a collective choice from the nominated alternatives by aggregating the preferences of voters. Each nominator’s opinion is a positive, negative, or neutral view as to whether each alternative deserves to be eligible for collective decision making. If a voter is a nominator, his preference space is restricted by his opinion as follows: he always prefers positive alternatives to neutral ones and neutral alternatives to negative ones, according to his opinion. When each nominating voter has such a preference space, we first characterize Arrow-consistent preference domains at the second stage of the social choice framework. Second, we find a resolution of Arrow’s impossibility theorem when at least one nominating voter exists.

An impossibility theorem for parameter independent hidden variable theories

Recently, Roger Colbeck and Renato Renner (C&R) have claimed that ‘[n]o extension of quantum theory can have improved predictive power’ (Colbeck & Renner, 2011, 2012b). If correct, this is a spectacular impossibility theorem for hidden variable theories, which is more general than the theorems of Bell (1964) and Leggett (2003). Also, C&R have used their claim in attempt to prove that a system׳s quantum-mechanical wave function is in a one-to-one correspondence with its ‘ontic’ state (Colbeck & Renner, 2012a). C&R׳s claim essentially means that in any hidden variable theory that is compatible with quantum-mechanical predictions, probabilities of measurement outcomes are independent of these hidden variables. This makes such variables otiose. On closer inspection, however, the generality and validity of the claim can be contested. First, it is based on an assumption called ‘Freedom of Choice’. As the name suggests, this assumption involves the independence of an experimenter׳s choice of measurement settings. But in the way C&R define this assumption, a no-signalling condition is surreptitiously presupposed, making the assumption less innocent than it sounds. When using this definition, any hidden variable theory violating parameter independence, such as Bohmian Mechanics, is immediately shown to be incompatible with quantum-mechanical predictions. Also, the argument of C&R is hard to follow and their mathematical derivation contains several gaps, some of which cannot be closed in the way they suggest. We shall show that these gaps can be filled. The issue with the ‘Freedom of Choice’ assumption can be circumvented by explicitly assuming parameter independence. This makes the result less general, but better founded. We then obtain an impossibility theorem for hidden variable theories satisfying parameter independence only. As stated above, such hidden variable theories are impossible in the sense that any supplemental variables have no bearing on outcome probabilities, and are therefore trivial. So, while quantum mechanics itself satisfies parameter independence, if a variable is added that changes the outcome probabilities, however slightly, parameter independence must be violated.

A general impossibility theorem and its application to individual rights

In this paper, we generalize Green and Laffont’s (1979) impossibility theorem to the following form: in quasi-linear environments, when the set of each agent’s types is sufficiently rich, we cannot find mechanisms that allow bounded deviations from the decisive efficiency, incentive compatibility and budget-balance at the same time. Hence, it is impossible to find an incentive compatible mechanism with minimum social welfare losses. Furthermore, we discuss the compatibility problems between incentive and individual rights in a quasi-linear environment (see Sen, 1970a,b; Deb et al., 1997). Specifically, some new impossibility results are established.

Deflecting Arrow by Aggregating Rankings of Multiple Correlated Criteria According to Borda

AbstractThis paper has two objectives. The first is to review and address concerns raised by Hazelrigg that Arrow's impossibility theorem prevents the selection of rational aggregation methods for use in engineering trade studies. In addressing these concerns, the work of Saari is cited to establish the fact that the Borda count is the only ‘non‐dictatorial’ positional voting method that satisfies the criteria for a rational decision procedure while using complete information. Hence, the resulting rank ordering of the alternatives is the most reliable outcome. Several previous studies that use other aggregation methods are critiqued, and Borda is applied to examples to illustrate the differences in the outcomes. The second objective is to extend the applicability of Borda to include attributes such as cost, schedule duration, and certain technical and performance measures that are generally more reasonably described as correlated random variables. Exact Probabilities by Simulation with Borda, a method introduced by Hulkower that improves a technique by Book for determining which candidate in a trade study is the probable lowest‐cost alternative, is generalized to include multiple correlated criteria, each of which is expressed as a random variable and thus incorporates probabilistic uncertainty. Copyright © 2016 John Wiley & Sons, Ltd.

An Axiomatic Framework for Ex-Ante Dynamic Pricing Mechanisms in Smart Grid

In electricity markets, the choice of the right pricing regime is crucial for the utilities because the price they charge to their consumers, in anticipation of their demand in real-time, is a key determinant of their profits and ultimately their survival in competitive energy markets. Among the existing pricing regimes, in this paper, we consider ex-ante dynamic pricing schemes as (i) they help to address the peak demand problem (a crucial problem in smart grids), and (ii) they are transparent and fair to consumers as the cost of electricity can be calculated before the actual consumption. In particular, we propose an axiomatic framework that establishes the conceptual underpinnings of the class of ex-ante dynamic pricing schemes. We first propose five key axioms that reflect the criteria that are vital for energy utilities and their relationship with consumers. We then prove an impossibility theorem to show that there is no pricing regime that satisfies all the five axioms simultaneously. We also study multiple cost functions arising from various pricing regimes to examine the subset of axioms that they satisfy. We believe that our proposed framework in this paper is first of its kind to evaluate the class of ex-ante dynamic pricing schemes in a manner that can be operationalised by energy utilities.

Finding Strategyproof Social Choice Functions via SAT Solving

A promising direction in computational social choice is to address research problems using computer-aided proving techniques. In particular with SAT solvers, this approach has been shown to be viable not only for proving classic impossibility theorems such as Arrow's Theorem but also for finding new impossibilities in the context of preference extensions. In this paper, we demonstrate that these computer-aided techniques can also be applied to improve our understanding of strategyproof irresolute social choice functions. These functions, however, requires a more evolved encoding as otherwise the search space rapidly becomes much too large. Our contribution is two-fold: We present an efficient encoding for translating such problems to SAT and leverage this encoding to prove new results about strategyproofness with respect to Kelly's and Fishburn's preference extensions. For example, we show that no Pareto-optimal majoritarian social choice function satisfies Fishburn-strategyproofness. Furthermore, we explain how human-readable proofs of such results can be extracted from minimal unsatisfiable cores of the corresponding SAT formulas.