## Abstract

In his classic monograph, *Social Choice and Individual Values*, Arrow introduced the notion of a decisive coalition of voters as part of his mathematical framework for social choice theory. The subsequent literature on Arrow’s Impossibility Theorem has shown the importance for social choice theory of reasoning about coalitions of voters with different grades of decisiveness. The goal of this paper is a fine-grained analysis of reasoning about decisive coalitions, formalizing how the concept of a decisive coalition gives rise to a social choice theoretic language and logic all of its own. We show that given Arrow’s axioms of the Independence of Irrelevant Alternatives and Universal Domain, rationality postulates for social preference correspond to strong axioms about decisive coalitions. We demonstrate this correspondence with results of a kind familiar in economics—representation theorems—as well as results of a kind coming from mathematical logic—completeness theorems. We present a complete logic for reasoning about decisive coalitions, along with formal proofs of Arrow’s and Wilson’s theorems. In addition, we prove the correctness of an algorithm for calculating, given any social rationality postulate of a certain form in the language of binary preference, the corresponding axiom in the language of decisive coalitions. These results suggest for social choice theory new perspectives and tools from logic.

## References

- Abramsky S (2015) Arrow’s theorem by arrow theory. In: Villaveces A, Kossak R, Kontinen J, Hirvonen Å (eds) Logic without borders: essays on set theory, model theory, philosophical logic and philosophy of mathematics. De Gruyter, Berlin, pp 15–30Google Scholar
- Agotnes T, van der Hoek W, Wooldridge M (2009) On the logic of preference and judgment aggregation. Auton Agents Multi-Agent Syst 22(1):4–30Google Scholar
- Arrow KJ (1959) Rational choice functions and orderings. Economica 26(102):121–127Google Scholar
- Arrow KJ (2012) Social choice and individual values, 3rd edn. Yale University Press, New HavenGoogle Scholar
- Arrow KJ (2014) Origins of the impossibility theorem. In: Maskin E, Sen A (eds) The Arrow impossibility theorem. Columbia University Press, New York, pp 143–148Google Scholar
- Baigent N, Arrow KJ, Sen A, Suzumura K (2010) Topological theories of social choice. Handbook of social choice and welfare, vol 2. North-Holland, Amsterdam, pp 301–334Google Scholar
- Bao N, Halpern NY (2017) Quantum voting and violation of Arrow’s impossibility theorem. Phys Rev A 95:062306. https://doi.org/10.1103/PhysRevA.95.062306 Google Scholar
- Barberá S (1980) Pivotal voters: a new proof of Arrow’s theorem. Econ Lett 6(1):13–16Google Scholar
- Blair DH, Pollak RA (1979) Collective rationality and dictatorship: the scope of the Arrow theorem. J Econ Theory 21:186–194Google Scholar
- Blair DH, Bordes G, Kelly JS, Suzumura K (1976) Impossibility theorems without collective rationality. J Econ Theory 13:361–379Google Scholar
- Blau JH (1979) Semiorders and collective choice. J Econ Theory 21:195–206Google Scholar
- Buchanan J (1954) Social choice, democracy and free markets. J Polit Econ 62(2):114–123Google Scholar
- Campbell D (1992) Equity, efficiency and social choice. Clarendon Press, OxfordGoogle Scholar
- Campbell DE, Kelly JS (2002) Impossibility theorems in the Arrovian framework. In: Arrow KJ, Sen AK, Suzumura K (eds) Handbook of social choice and welfare, vol 1. North-Holland, Amsterdam, pp 35–94Google Scholar
- Chagrov A, Zakharyaschev M (1997) Modal Logic. Clarendon Press, OxfordGoogle Scholar
- Ciná G, Endriss U (2016) Proving classical theorems of social choice theory in modal logic. Auton Agents Multi-Agent Syst 30(5):963–989Google Scholar
- Dryzek J, List C (2003) Social choice theory and deliberative democracy: a reconciliation. Br J Polit Sci 33(1):1–28Google Scholar
- Endriss U (2011) Logic and social choice theory. In: Gupta A, Benthem J (eds) Logic and philosophy today. College Publications, London, pp 333–377Google Scholar
- Fishburn P (1970) Arrow’s impossibility theorem: concise proof and infinitely many voters. J Econ Theory 2:103–106Google Scholar
- Fishburn P, Rubinstein A (1986) Algebraic aggregation theory. J Econ Theory 38:63–77Google Scholar
- Fleurbaey M, Mongin P (2005) The news of the death of welfare economics is greatly exaggerated. Soc Choice Welf 25:381–418Google Scholar
- Geanakoplos J (2005) Three brief proofs of Arrow’s theorem. Econ Theory 26(1):211–215Google Scholar
- Gibbard AF (2014) Intransitive social indifference and the Arrow dilemma. Rev Econ Des 18:3–10Google Scholar
- Givant S, Halmos P (2009) Introduction to Boolean algebras. Springer, New YorkGoogle Scholar
- Grandi U, Endriss U (2013) First-order logic formalisation of impossibility theorems in preference aggregation. J Philos Logic 42(4):595–618Google Scholar
- Guha AS (1972) Neutrality, monotonicity, and the right of veto. Econometrica 40(5):821–826Google Scholar
- Hammond P (1976) Equity, Arrow’s conditions, and Rawl’s difference principle. Econometrica 44:793–804Google Scholar
- Hansson B (1976) The existence of group preference functions. Public Choice 28:89–98Google Scholar
- Herzberg F, Eckert D (2012) The model-theoretic approach to aggregation: impossibility results for finite and infinite electorates. Math Soc Sci 64:41–47Google Scholar
- Hurley S (1985) Supervenience and the possibility of coherence. Mind 94(276):501–525Google Scholar
- Kalai E, Muller E, Satterthwaite M (1979) Social welfare functions when preferences are convex and continuous: impossibility results. Public Choice 34:87–97Google Scholar
- Kelly JS (1988) Social Choice theory: an introduction. Springer, BerlinGoogle Scholar
- Kirman AP, Sondermann D (1972) Arrow’s theorem, many agents, and invisible dictators. J Econ Theory 5:267–277Google Scholar
- Kraft CH, Pratt JW, Seidenberg A (1959) Intuitive probability on finite sets. Ann Math Stat 30(2):408–419Google Scholar
- Kroedel T, Huber F (2013) Counterfactual dependence and Arrow. Nous 47(3):453–466Google Scholar
- Lauwers L, Van Liedekerke L (1995) Ultraproducts and aggregation. J Math Econ 24:217–237Google Scholar
- Le Breton M, Weymark J (2010) Arrovian social choice theory on economic domains. In: Arrow KJ, Sen A, Suzumura K (eds) Handbook of social choice and welfare, vol 2. North-Holland, Amsterdam, pp 191–299Google Scholar
- List C, Polak B (2010) Introduction to judgment aggregation. J Econ Theory 145(2):441–466Google Scholar
- MacAskill W (2016) Normative uncertainty as a voting problem. Mind 124:967–1004Google Scholar
- Mackie G (2003) Democracy defended. Cambridge University Press, CambridgeGoogle Scholar
- Makinson D (1996) Combinatorial versus decision-theoretic components of impossibility theorems. Theory Decis 40:181–190Google Scholar
- Mas-Colell A, Sonnenschein H (1972) General possibility theorems for group decisions. Rev Econ Stud 39(2):185–192Google Scholar
- Monjardet B (1967) Remarques sur une classe de procédures de votes et les théorèmes de possibilité. In: La Décision. Colloque du CNRS, Aix-en-Provence, pp 117–184Google Scholar
- Monjardet B (1978) Une autre preuve du théorème d’Arrow. R.A.I.R.O. Recherche Opérationnelle 12:291–296Google Scholar
- Monjardet B (1983) On the use of ultrafilters in social choice theory. In: Pattanaik PK, Salles M (eds) Social choice and welfare. North-Holland, Amsterdam, pp 73–78Google Scholar
- Morreau M (2015) Theory choice and social choice: Kuhn vindicated. Mind 123(493):239–262Google Scholar
- Murakami Y (1968) Logic and social choice. Dover, New YorkGoogle Scholar
- Nipkow T (2009) Social choice theory in HOL. J Autom Reason 43(3):289–304Google Scholar
- Okasha S (2011) Theory choice and social choice: Kuhn versus Arrow. Mind 120(477):83–115Google Scholar
- Pacuit E, Yang F (2016) Dependence and independence in social choice: Arrow’s theorem. In: Abramsky S, Kontinen J, Väänänen J, Vollmer H (eds) Dependence logic. Springer, Berlin, pp 235–260Google Scholar
- Patty J, Penn EM (2014) Social choice and legitimacy: the possibilities of impossibility. Cambridge University Press, CambridgeGoogle Scholar
- Reny PJ (2001) Arrow’s theorem and the Gibbard–Satterthwaite theorem: a unified approach. Econ Lett 70:99–105Google Scholar
- Riker WH (1982) Liberalism against populism: a confrontation between the theory of democracy and the theory of social choice. William H. Freeman, San FranciscoGoogle Scholar
- Rubinstein A (1984) The single profile analogues to multi profile theorems: mathematical logic’s approach. Int Econ Rev 25(3):719–730Google Scholar
- Saari DG (2008) Disposing dictators, demystifying voting paradoxes: social choice analysis. Cambridge University Press, CambridgeGoogle Scholar
- Salles M (2017) On quine on Arrow. Soc Choice Welf 48:877–886Google Scholar
- Sen A (1969) Quasi-transitivity, rational choice and collective decisions. Rev Econ Stud 36(2):381–393Google Scholar
- Sen A (1970) The impossibility of a Paretian liberal. J Polit Econ 78(1):152–157Google Scholar
- Sen A (1979) Personal utilities and public judgements: or what’s wrong with welfare economics. Econ J 89(335):537–558Google Scholar
- Sen A (1993) Internal consistency of choice. Econometrica 61(3):495–521Google Scholar
- Sen A (2014) Arrow and the impossibility theorem. In: Maskin E, Sen A (eds) The Arrow impossibility theorem. Columbia University Press, New York, pp 29–42Google Scholar
- Sen A (2017) Collective choice and social welfare: an expanded edition. Harvard University Press, CambridgeGoogle Scholar
- Shelah S (2005) On the Arrow property. Adv Appl Math 34:217–251Google Scholar
- Stegenga J (2013) An impossibility theorem for amalgamating evidence. Synthese 190(12):2391–2411Google Scholar
- Suzumura K (1983) Rational choice, collective decisions, and social welfare. Cambridge University Press, CambridgeGoogle Scholar
- Tang P, Lin F (2009) Computer-aided proofs of Arrow’s and other impossibility theorems. Artif Intell 173(11):1041–1053Google Scholar
- Taylor A, Zwicker W (1992) A characterization of weighted voting. Proc Am Math Soc 115(4):1089–1094Google Scholar
- Taylor AD, Zwicker WS (1999) Simple games: desirability relations, trading, pseudoweightings. Princeton University Press, PrincetonGoogle Scholar
- Troquard N, van der Hoek W, Wooldridge M (2011) Reasoning about social choice functions. J Philos Logic 40(4):473–498Google Scholar
- von Neumann J, Morgenstern O (1947) Theory of games and economic behavior, 2nd edn. Princeton University Press, New HavenGoogle Scholar
- Weymark JA (1984) Arrow’s theorem with social quasi-orderings. Public Choice 42:235–246Google Scholar
- Wiedijk F (2007) Arrow’s impossibility theorem. Formaliz Math 15(4):171–174Google Scholar
- Wilson R (1972) Social choice theory without the Pareto principle. J Econ Theory 5:478–486Google Scholar
- Yu NN (2012) A one-shot proof of Arrow’s impossibility theorem. Econ Theory 50(2):523–525Google Scholar