# Arrow’s impossibility theorem as a special case of Nash equilibrium: a cognitive approach to the theory of collective decision-making

*Edgardo Bucciarelli* () and
*Andrea Oliva* ()

Additional contact information

Edgardo Bucciarelli: University of Chieti-Pescara

Andrea Oliva: Abruzzo Academy Foundation, Legal Entity of the Prefecture of Pescara

*Mind & Society: Cognitive Studies in Economics and Social Sciences*, 2020, vol. 19, issue 1, No 4, 15-41

**Abstract:**
Abstract Metalogic is an open-ended cognitive, formal methodology pertaining to semantics and information processing. The language that mathematizes metalogic is known as metalanguage and deals with metafunctions purely by extension on patterns. A metalogical process involves an effective enrichment in knowledge as logical statements, and, since human cognition is an inherently logic–based representation of knowledge, a metalogical process will always be aimed at developing the scope of cognition by exploring possible cognitive implications reflected on successive levels of abstraction. Indeed, it is basically impracticable to maintain logic–and–metalogic without paying heed to cognitive theorizing. For it is cognitively irrelevant whether possible conclusions are deduced from some premises before the premises are determined to be true or whether the premises themselves are determined to be true first and, then, the conclusions are deduced from them. In this paper we consider the term metalogic as inherently embodied under the framework referred to as cognitive science and mathematics. We propose a metalogical interpretation of Arrow’s impossibility theorem and, to that end, choice theory is understood as a mental course of action dealing with logic and metalogic issues, in which a possible mathematical approach to model a mental course of action is adopted as a systematic operating method. Nevertheless, if we look closely at the core of Arrow’s impossibility theorem in terms of metalogic, a second fundamental contribution to this framework is represented by the Nash equilibrium. As a result of the foregoing, therefore, we prove the metalogical equivalence between Arrow’s impossibility theorem and the existence of the Nash equilibrium. More specifically, Arrow’s requirements correspond to the Nash equilibrium for finite mixed strategies with no symmetry conditions. To demonstrate this proof, we first verify that Arrow’s set and Nash’s set are isomorphic to each other, both sets being under stated conditions of non–symmetry. Then, the proof is completed by virtue of category theory. Indeed, the two sets are categories that correspond biuniquely to one another and, thus, it is possible to define a covariant functor that preserves their mutual structures. According to this, we show the proof–dedicated metalanguage as a precursor to a special equivalence theorem.

**Keywords:** Social choice theory; Metalogic; Cognition; Abductive semantics; Theoretical isomorphisms; Non-cooperative game theory; Arrow–Nash equivalence (search for similar items in EconPapers)

**Date:** 2020

**References:** View references in EconPapers View complete reference list from CitEc

**Citations:** Track citations by RSS feed

**Downloads:** (external link)

http://link.springer.com/10.1007/s11299-019-00222-3 Abstract (text/html)

Access to the full text of the articles in this series is restricted.

**Related works:**

This item may be available elsewhere in EconPapers: Search for items with the same title.

**Export reference:** BibTeX
RIS (EndNote, ProCite, RefMan)
HTML/Text

**Persistent link:** https://EconPapers.repec.org/RePEc:spr:minsoc:v:19:y:2020:i:1:d:10.1007_s11299-019-00222-3

**Ordering information:** This journal article can be ordered from

http://www.springer.com/economics/journal/11299

**DOI:** 10.1007/s11299-019-00222-3

Access Statistics for this article

Mind & Society: Cognitive Studies in Economics and Social Sciences is currently edited by *Riccardo Viale*

More articles in Mind & Society: Cognitive Studies in Economics and Social Sciences from Springer, Fondazione Rosselli Contact information at EDIRC.

Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().