A Comment on Arrow’s Impossibility Theorem

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Alec Sandroni ◽  
Alvaro Sandroni
Keyword(s):  
Social Choice ◽  
Choice Function ◽  
Social Preference ◽  
Social Choice Function ◽  
Individual Preference ◽  
Impossibility Theorem ◽  
Preference Order ◽  
Individual Choice ◽  
Choice Functions ◽  
Preference Orders

AbstractArrow (1950) famously showed the impossibility of aggregating individual preference orders into a social preference order (together with basic desiderata). This paper shows that it is possible to aggregate individual choice functions, that satisfy almost any condition weaker than WARP, into a social choice function that satisfy the same condition (and also Arrow’s desiderata).

scholarly journals Local‐global equivalence in voting models: A characterization and applications

2021 ◽  
Vol 16 (4) ◽  
pp. 1195-1220
Author(s):  
Ujjwal Kumar ◽  
Souvik Roy ◽  
Arunava Sen ◽  
Sonal Yadav ◽  
Huaxia Zeng
Keyword(s):  
Social Choice ◽  
Choice Function ◽  
Social Choice Function ◽  
Choice Functions ◽  
Local Strategy ◽  
Social Choice Functions ◽  
True Type ◽  
Strategy Proof ◽  
Voting Models ◽  
Strategy Proofness

The paper considers a voting model where each voter's type is her preference. The type graph for a voter is a graph whose vertices are the possible types of the voter. Two vertices are connected by an edge in the graph if the associated types are “neighbors.” A social choice function is locally strategy‐proof if no type of a voter can gain by misrepresentation to a type that is a neighbor of her true type. A social choice function is strategy‐proof if no type of a voter can gain by misrepresentation to an arbitrary type. Local‐global equivalence (LGE) is satisfied if local strategy‐proofness implies strategy‐proofness. The paper identifies a condition on the graph that characterizes LGE. Our notion of “localness” is perfectly general. We use this feature of our model to identify notions of localness according to which various models of multidimensional voting satisfy LGE. Finally, we show that LGE for deterministic social choice functions does not imply LGE for random social choice functions.

scholarly journals Finding Strategyproof Social Choice Functions via SAT Solving

2016 ◽  
Vol 55 ◽  
pp. 565-602 ◽  
Author(s):  
Felix Brandt ◽  
Christian Geist
Keyword(s):  
Social Choice ◽  
Choice Function ◽  
Social Choice Function ◽  
Search Space ◽  
Computational Social Choice ◽  
Choice Functions ◽  
Sat Solving ◽  
Computer Aided ◽  
Research Problems ◽  
Social Choice Functions

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.

scholarly journals Partial ex-post verifiability and unique implementation of social choice functions

2020 ◽  
Author(s):  
Hitoshi Matsushima
Keyword(s):  
Social Choice ◽  
Choice Function ◽  
Partial Information ◽  
Social Choice Function ◽  
The State ◽  
Equilibrium Path ◽  
Choice Functions ◽  
Central Planner ◽  
Ex Post ◽  
Social Choice Functions

Abstract This study investigates the unique implementation of a social choice function in iterative dominance in the ex-post term. We assume partial ex-post verifiability; that is, after determining an allocation, the central planner can observe partial information about the state as verifiable. We demonstrate a condition of the state space, termed “full detection,” and show that with full detection, any social choice function is uniquely implementable even if the information that can be verified ex-post is very limited. To prove this, we construct a dynamic mechanism according to which each player announces his (or her) private signal, before the other players observe this signal, at an earlier stage, and each player also announces the state at a later stage. In this construction, we can impose several severe restrictions such as boundedness, permission of only tiny transfers off the equilibrium path, and no permission of transfers on the equilibrium path. This study does not assume either expected utility or quasi-linearity.

Arrow’s Theorem

2020 ◽  
Author(s):  
Conal Duddy ◽  
Ashley Piggins
Keyword(s):  
Social Welfare ◽  
Social Choice ◽  
Social Welfare Function ◽  
Social Preference ◽  
Social Choice Theory ◽  
Impossibility Theorem ◽  
Preference Ordering ◽  
Individual Values ◽  
Arrow’S Theorem ◽  
Arrow's Theorem

Kenneth Arrow’s “impossibility” theorem is rightly considered to be a landmark result in economic theory. It is a far-reaching result with implications not just for economics but for political science, philosophy, and many other fields. It has inspired an enormous literature, “social choice theory,” which lies on the interface of economics, politics, and philosophy. Arrow first proved the impossibility theorem in his doctoral dissertation—Social Choice and Individual Values—published in 1951. It is a remarkable result, and had Arrow not proved it, it is unlikely that the theorem would be known today. A social choice is simply a choice made by, or on behalf of, a group of people. Arrow’s theorem is concerned more specifically with the following problem. Suppose that we have a given set of options to choose from and that each member of a group of individuals has his or her own preference over these options. By what method should we construct a single ranking of the options for the group as a whole? Any such method may be represented mathematically by a “social welfare function.” This is a function that receives as its input the preference ordering of each individual and then generates as its output a social preference ordering. Arrow defined some properties that would seem to be essential to any reasonable social welfare function. These properties are called “unrestricted domain,” “weak Pareto,” “independence of irrelevant alternatives,” and “non-dictatorship.” Each of these properties, when taken alone, does appear to be very necessary indeed. Yet, Arrow proved that these properties are in fact mutually incompatible. This troubling fact has been central to the study of social choice ever since.

scholarly journals SOCIAL CHOICE AND JUST INSTITUTIONS: NEW PERSPECTIVES

2007 ◽  
Vol 23 (1) ◽  
pp. 15-43 ◽  
Author(s):  
MARC FLEURBAEY
Keyword(s):  
Social Choice ◽  
Social Preferences ◽  
Well Being ◽  
Impossibility Theorem ◽  
Choice Functions ◽  
New Approach ◽  
Second Best ◽  
Interpersonal Comparisons ◽  
Theories Of Justice ◽  
Social Choice Functions

It has become accepted that social choice is impossible in the absence of interpersonal comparisons of well-being. This view is challenged here. Arrow obtained an impossibility theorem only by making unreasonable demands on social choice functions. With reasonable requirements, one can get very attractive possibilities and derive social preferences on the basis of non-comparable individual preferences. This new approach makes it possible to design optimal second-best institutions inspired by principles of fairness, while traditionally the analysis of optimal second-best institutions was thought to require interpersonal comparisons of well-being. In particular, this approach turns out to be especially suitable for the application of recent philosophical theories of justice formulated in terms of fairness, such as equality of resources.

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