The term "cooperative chess" includes chess variants that feature interpersonal cooperation rather than competition.
In such cooperative chess variants, two players play the game on either a classic or modified chess board, with each player striving to achieve a mutual, rather than solo, victory.
Variants
Two cooperative chess variants that encourage collaborative rather than competitive play are Hans Theo Frenkel's Metapontum and Aaron Lundstrom's DUO: A Social Chess.
References
Cooperative chess Wikipedia(Text) CC BY-SA