Noname manuscript No. (will be inserted by the editor) Convexity and positivity in partially defined cooperative games Jan Bok · Martin ˇ Cern´ y · David Hartman · Milan Hlad´ ık Received: date / Accepted: date Abstract Partially defined cooperative games are a generalisation of classical cooperative games in which payoffs for some of the coalitions are not known. In this paper we perform a systematic study of partially defined games, focusing on two important classes of cooperative games: convex games and positive games. In the first part, we focus on convexity and give a polynomially decidable condition for extendability and a full description of the set of symmetric convex extensions. The extreme games of this set, together with the lower game and Jan Bok Computer Science Institute, Faculty of Mathematics and Physics, Charles University, Prague, Malostransk´ e n´am. 25, 118 00 Praha 1, Czech Republic. ORCID: 0000-0002-7973-1361 E-mail: bok@iuuk.mff.cuni.cz This author is the corresponding author. Martin ˇ Cern´ y Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles Univer- sity, Prague, Malostransk´ e n´am. 25, 118 00 Praha 1, Czech Republic. ORCID: 0000-0002-0619-4737 E-mail: cerny@kam.mff.cuni.cz Milan Hlad´ ık Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles Univer- sity, Prague, Malostransk´ e n´am. 25, 118 00 Praha 1, Czech Republic. ORCID: 0000-0002-7340-8491 E-mail: hladik@kam.mff.cuni.cz David Hartman Computer Science Institute, Faculty of Mathematics and Physics, Charles University, Prague, Malostransk´ e n´am. 25, 118 00 Praha 1, Czech Republic and Institute of Com- puter Science of the Czech Academy of Sciences, Pod Vod´arenskou vˇ eˇ z´ ı 271/2, Prague 8, Czech Republic ORCID: 0000-0003-3566-8214 E-mail: hartman@iuuk.mff.cuni.cz arXiv:2010.08578v1 [cs.GT] 16 Oct 2020