# Cores of Many-Player Games; Nonemptiness and Equal Treatment

Working paper

Issue number:

. 09-W18

Publisher:

Vanderbilt University

Year:

2009

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 collec- tive 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 characteri- zation 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.