Arrow's Theorem

Social choice and telling the truth

Perhaps the most influential result in social choice theory is Arrow’s impossibility theorem, which states that a seemingly modest set of desiderata cannot be satisfied when aggregating preferences [1]. While Arrow’s theorem might appear rather negative, it can also be interpreted in a positive way by identifying what can be achieved in preference aggregation.In this talk, I present a number of variations of Arrow’s theorem–such as those due to Mas-Colell and Sonnenschein [8] and Blau and Deb [2]–in their choice-theoretic version. The critical condition in all these theorems is the assumption of a rationalizing binary relation or equivalent notions of choice-consistency. The bulk of my presentation contains three escape routes from these results. The first one is to ignore consistency with respect to a variable set of alternatives altogether and require consistency with respect to a variable electorate instead. As Smith [12] and Young [14] have famously shown, this essentially characterizes the class of scoring rules, which contains plurality and Borda’s rule. For the second escape route, we factorize choice-consistency into two parts, contraction-consistency and expansions-consistency [11]. While even the mildest dose of the former has severe consequences on the possibility of choice, varying degrees of the latter characterize a number of appealing social choice functions, namely the top cycle, the uncovered set, and the Banks set [3,9,4]. Finally, I suggest to redefine choice-consistency by making reference to the set of chosen alternatives rather than individual chosen alternatives [6]. It turns out that the resulting condition is a weakening of transitive rationalizability and can be used to characterize the minimal covering set and the bipartisan set. Based on a two decades-old conjecture due to Schwartz [10], the tournament equilibrium set can be characterized by the same condition or, alternatively, by a weak expansion-consistency condition from the second escape route. Whether Schwartz’s conjecture actually holds remains a challenging combinatorial problem as well as one of the enigmatic open problems of social choice theory.Throughout the presentation I will discuss the algorithmic aspects of all considered social choice functions. While some of the mentioned functions can be easily computed, other ones do not admit an efficient algorithm unless P equals NP [13,5,7].

Extension of Arrow's theorem to symmetric sets of tournaments

This paper considers an aggregation rule when each alternative consists of elements of multiple issues. I assume that each issue-specific aggregation rule can be applied to each issue and explore whether various different issue-specific aggregation rules are compatible. When the set of alternatives has some structure (which we call a set connected by loops), there exists a powerful individual whose preference always coincides with the social ordering in every issue. As a corollary, Arrow's General Possibility Theorem is obtained.

Non-Dictatorial Social Choice Through Delegation

Social choice theory without the Pareto Principle

This paper provides two brief proofs of Arrow’s general possibility theorem. The second one is simple and short. Our proofs are inspired by the pioneering work by Inada (Ann. Inst. Stat. Math. 6:115–122, 1954).

The “Big-Four” accounting firms dominate the global accounting/auditing industry, and the big-seven consulting firms (Bain; McKinsey; Booz; Deloitte; PwC; KPMG and E&Y) dominate the global business/management consulting industry and stifle competition. During 1990-2017, the global auditing/accounting industry and the global management consulting industry experienced significant structural changes that have implications for Financial Stability, systemic risk and the proper functioning of capital markets. Some of the results included the collapses of stock prices and bond prices of firms suspected of earnings management; and substantial litigation against auditing firms, CRAs and board of directors. Accounting/audit firms and consulting firms have always been key elements in the fight against earnings management, securities fraud, corruption and asset quality management because of their unique position as external auditors and advisors. This chapter introduces some efficient Auditor allocation and Compensation Mechanisms. These new “Learning Business Models” and contracts can solve the conflicts of interest, Antitrust, greed, Regret, Deadweight-Losses, complexity and industrial organization problems inherent in the Auditing/consulting industry; and each such model contravenes Myerson-Satterthwaite Impossibility Theorem, Arrow's Impossibility Theorem, Sen's Impossibility Theorem, Gibbard's Theorem, the Gibbard-Satterthwaite Impossibility Theorem, and the Green-Laffont Impossibility Theorem. These issues have implications for international trade and international capital flows given the prevalence of accounting and management consulting in almost all aspects of modern business.

In this note I consider a simple proof of Arrow's Impossibility Theorem (Arrow 1963). I start with the case of three individuals who have preferences on three alternatives. In this special case there are 133=2197 possible combinations of the three individuals' rational preferences. However, by considering the subset of linear preferences, and employing the full strength of the IIA axiom, I reduce the number of cases necessary to completely describe the SWF to a small number, allowing an elementary proof suitable for most undergraduate students.

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.

It is the purpose of this article to give an exact formulation of Arrow's famous impossibility theorem and its proof. Although the results presented here are not new in the sense that the theorem is refuted or questionned, it may be of some interest especially for the reader who wants to get a more or less complete view of the argumentation leading up to the theorem. Special attention is paid to the logical structure of the argument in the conviction that a lot of misunderstanding exists about what Arrow and subsequent writers tried to say. By way of introduction to the central theme some remarks are made concerning the relation of the Arrow problem to welfare economics.

Kenneth J. Arrow was one of the most important intellectuals of the twentieth century, and his “impossibility theorem” is arguably the starting point of modern, axiomatic social choice theory. In this review, we begin with a brief discussion of Arrow's theorem and subsequent work that extended the result. We then discuss its implications for voting and constitutional systems, including a number of seminal results—both positive and negative—that characterize what such systems can accomplish and why. We then depart from this narrow interpretation of the result to consider more varied institutional design questions such as apportionment and geographical districting. Following this, we address the theorem's implications for measurement of concepts of fundamental interest to political science such as justice and inequality. Finally, we address current work applying social choice concepts and the axiomatic method to data analysis more generally.

Chapter 1 Impossibility theorems in the arrovian framework

4 - SIMPLICITY: IMPOSSIBILITY THEOREMS

Kenneth Arrow recently told us of an interesting episode in the formative era of social choice theory. It was in the session chaired by Lawrence Klein at the December 1948 Meeting of the Econometric Society that the general possibility theorem was first presented. Arrow recalls: ‘in the audience was this contentious Canadian, David McCord Wright, who objected because among the objectives, [Arrow] hadn’t mentioned freedom as one of the essential values in social choice and apparently he went out of the room saying that Klein and Arrow were communists’ (Kelly, 1987, p. 56).