The Nakamura numbers for computable simple games

The Nakamura number of a simple game plays a critical role in preference aggregation (or multi-criterion ranking): the number of alternatives that the players can always deal with rationally is less than this number. We comprehensively study the restrictions that various properties for a simple game impose on its Nakamura number. We find that a computable game has a finite Nakamura number greater than three only if it is proper, nonstrong, and nonweak, regardless of whether it is monotonic or whether it has a finite carrier. The lack of strongness often results in alternatives that cannot be strictly ranked.

[1]  B. Peleg.  Game Theoretic Analysis of Voting in Committees , 1984 .

[2]  H. Reiju Mihara.  Arrow's Theorem and Turing computability , 1994 .

[3]  H. Reiju Mihara,et al.  Computability of Simple Games: A Characterization and Application to the Core , 2007, ArXiv.

[4]  K. Arrow,et al.  Social Choice and Individual Values , 1951 .

[5]  J. Banks,et al.  Positive Political Theory I: Collective Preference , 1998 .

[6]  A note on the core of voting games , 2000 .

[7]  M. Trick,et al.  Voting schemes for which it can be difficult to tell who won the election , 1989 .

[8]  Yasuhito Tanaka.  Type two computability of social choice functions and the Gibbard-Satterthwaite theorem in an infinite society , 2007, Appl. Math. Comput..

[9]  Kenjiro Nakamura,et al.  The vetoers in a simple game with ordinal preferences , 1979 .

[10]  Michel Truchon,et al.  Voting games and acyclic collective choice rules , 1995 .

[11]  M. Trick,et al.  The computational difficulty of manipulating an election , 1989 .

[12]  H. Reiju Mihara.  Nonanonymity and sensitivity of computable simple games , 2004, Math. Soc. Sci..

[13]  A. A. Lewis.  An infinite version of arrow's theorem in the effective setting , 1988 .

[14]  Ehud Kalai,et al.  Games in Coalitional Form , 2007 .

[15]  Rajat Deb,et al.  Rights as alternative game forms , 2004, Soc. Choice Welf..

[16]  R. Soare.  Recursively enumerable sets and degrees , 1987 .

[17]  H. Reiju Mihara,et al.  Arrow's theorem, countably many agents, and more visible invisible dictators 1 The main result of th , 1999 .

[18]  Robert J. Weber,et al.  Chapter 36 Games in coalitional form , 1994 .

[19]  Jeffrey S. Banks.  Acyclic social choice from finite sets , 1995 .

[20]  L. Shapley.  Simple games: an outline of the descriptive theory. , 2007, Behavioral science.

[21]  J. Kelly.  Social choice and computational complexity , 1988 .

[22]  Martin van Hees,et al.  Liberalism, efficiency, and stability : Some possibility results , 1999 .

[23]  M. Richter,et al.  Computable preference and utility , 1999 .

[24]  P. Odifreddi.  The theory of functions and sets of natural numbers , 1989 .

[25]  B. Peleg.  Chapter 8 Game-theoretic analysis of voting in committees , 2002 .

[26]  Equivalent game forms and coalitional power , 1990 .