Computability of Simple Games: A Characterization and Application to the Core

The class of algorithmically computable simple games (i) includes the class of games that have finite carriers and (ii) is included in the class of games that have finite winning coalitions. This paper characterizes computable games, strengthens the earlier result that computable games violate anonymity, and gives examples showing that the above inclusions are strict. It also extends Nakamura’s theorem about the nonemptyness of the core and shows that computable games have a finite Nakamura number, implying that the number of alternatives that the players can deal with rationally is restricted.

[1]  H. Reiju Mihara,et al.  The Nakamura numbers for computable simple games , 2008, Soc. Choice Welf..

[2]  Y. Moschovakis.  Review: G. Kriesel, D. Lacombe, J. Shoenfield, Effective Operations and Partial Recursive Functionals; G. Kreisel, D. Lacombe, J. R. Shoenfield, Partial Recursive Functionals and Effective Operations , 1966 .

[3]  Mark Fey,et al.  May’s Theorem with an infinite population , 2004, Soc. Choice Welf..

[4]  Xiaotie Deng,et al.  On computational complexity of membership test in flow games and linear production games , 2002, Int. J. Game Theory.

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

[6]  H. Sabourian,et al.  COOPERATION AND EFFECTIVE COMPUTABILITY , 1995 .

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

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

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

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

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

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

[13]  M. Richter,et al.  Non-computability of competitive equilibrium , 1999 .

[14]  César Martinelli,et al.  Anonymity in large societies , 2005, Soc. Choice Welf..

[15]  Michael Boss.  Economic theory of democracy , 1974 .

[16]  John Duggan,et al.  Social choice and electoral competition in the general spatial model , 2006, J. Econ. Theory.

[17]  Thomas E. Armstrong,et al.  Arrow's theorem with restricted coalition algebras , 1980 .

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

[19]  R. Evans,et al.  Cooperation and Punishment , 2001 .

[20]  Vincent Conitzer,et al.  Coalitional Games in Open Anonymous Environments , 2005, IJCAI.

[21]  Stephen Spear,et al.  Learning Rational Expectations under Computability Constraints , 1989 .

[22]  L. A. Goodman,et al.  Social Choice and Individual Values , 1951 .

[23]  Barton L. Lipman.  Information Processing and Bounded Rationality: A Survey , 1995 .

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

[25]  Sabine Koppelberg,et al.  Handbook of Boolean Algebras , 1989 .

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

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

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

[29]  H. Reiju Mihara.  Anonymity and neutrality in Arrow’s Theorem with restricted coalition algebras , 1994 .

[30]  Xiaotie Deng,et al.  On the Complexity of Cooperative Solution Concepts , 1994, Math. Oper. Res..

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

[32]  A. Rubinstein.  Modeling Bounded Rationality , 1998 .

[33]  Robert I. Soare,et al.  Recursively enumerable sets and degrees - a study of computable functions and computability generated sets , 1987, Perspectives in mathematical logic.

[34]  Peter G. Hinman.  Review: Piergiorgio Odifreddi, Classical Recursion Theory. The Theory of Functions and Sets of Natural Numbers , 1990 .

[35]  K. Arrow.  Rationality of Self and Others in an Economic System , 1986 .

[36]  Thomas E. Armstrong,et al.  Precisely dictatorial social welfare functions: Erratum and Addendum to ‘arrows theorem with restricted coalition algebras’ , 1985 .

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

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

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

[40]  Kislaya Prasad.  On the computability of Nash equilibria , 1997 .

[41]  Luca Anderlini,et al.  Incomplete Written Contracts: Undescribable States of Nature , 1994 .

[42]  H. Reiju Mihara,et al.  Existence of a coalitionally strategyproof social choice function: A constructive proof , 2001, Soc. Choice Welf..

[43]  I. Gilboa.  Philosophical Applications of Kolmogorov's Complexity Measure , 1994 .

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

[45]  David Canning,et al.  Rationality, Computability, and Nash Equilibrium , 1992 .

[46]  Dieter Sondermann,et al.  Arrow's theorem, many agents, and invisible dictators☆ , 1972 .