Review of Economic Design

, Volume 14, Issue 1–2, pp 131–162

Cores of many-player games; nonemptiness and equal treatment

Open Access
Original Paper

DOI: 10.1007/s10058-009-0095-6

Cite this article as:
Wooders, M. Rev Econ Design (2010) 14: 131. doi:10.1007/s10058-009-0095-6


This paper provides sufficient conditions to ensure nonemptiness of approximate cores of many-player games and symmetry of approximate core payoffs (the equal treatment property). The conditions are: (a) essential superadditivity—an option open to a group of players is to partition into smaller groups and realize the worths of these groups and (b) small group effectiveness (SGE)—almost all gains to collective activities can be realized by cooperation only within members of some partition of players into relatively small groups. Another condition, small group negligibility (SGN), is introduced and shown to be equivalent to SGE. SGN dictates that small groups of players cannot have significant effects on average (i.e., per capita) payoffs of large populations; thus, SGN is a analogue, for games with a finite player set, of the condition built into models with a continuum of player that sets of measure zero can be ignored. SGE implies per capita boundedness (PCB), that the supremum of average or per capita payoffs is uniformly bounded above. Further characterization of SGE in terms of its relationship to PCB is provided. It is known that if SGE does not hold, then approximate cores of many-player games may be empty. Examples are developed to show that if SGE does not hold and if there are players of “scarce types” (in other works, players with scarce attributes) then even if there is only a finite number of types of players and approximate cores are non-empty, symmetry may be lost; moreover, even players of abundant types may be treated asymmetrically by the core.


Core Market games Approximate cores Equal treatment Core convergence Small group effectiveness Symmetry Per capita boundedness Small group negligibility Games with a continuum of players 

JEL Classification

C71 D71 C78 H41 D40 D50 

Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.Department of EconomicsVanderbilt UniversityNashvilleUSA

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