ISRAEL JOURNAL OF MATHEMATICS 224 (2018), 133–158 DOI: 10.1007/s11856-018-1639-7 TIGHTNESS GAMES WITH BOUNDED FINITE SELECTIONS BY Leandro F. Aurichi ∗ Instituto de Ciˆ encias Matem´ aticas e de Computa¸ c˜ao, Universidade de S˜ao Paulo Caixa Postal 668, S˜ ao Carlos, SP, 13560-970, Brazil e-mail: aurichi@icmc.usp.br AND Angelo Bella ∗∗ Department of Mathematics, University of Catania, Citt` a Universitaria Viale A. Doria 6, 95125 Catania, Italy e-mail: bella@dmi.unict.it AND Rodrigo R. Dias † Centro de Matem´atica, Computa¸ c˜ao e Cogni¸c˜ao, Universidade Federal do ABC Avenida dos Estados, 5001, Santo Andr´ e, SP, 09210-580, Brazil e-mail: rodrigo.dias@ufabc.edu.br ∗ Partially supported by FAPESP (2013/05469-7 and 2015/25725) and by GNSAGA. ∗∗ The research that led to the present paper was partially supported by a grant of the group GNSAGA of INdAM. † Partially supported by FAPESP (2012/09214-0). Received May 12, 2016 and in revised form September 21, 2016 133