The Bet Yosef (Siman 308) cites the Shiboleh Haleket who rules that a ball used for games is Mukse on Shabbat. The fact that it is used for playing does not render it a vessel. Moreover, it is not even fit to cover a pot, since generally it is dirty. In the Shulhan Aruch, Maran states that it is prohibited to play ball on Shabbat.
Most Poskim understand that Maran is ruling in accordance with the Shiboleh Haleket, that the ball is Mukse. That is how the Mishna Berura, the Aruch HaShulhan, Kaf Hahaim, Gaon Rabbi Zalman, the Levush and MaHaritatz understand the Shulhan Aruch. The Rambam in a Teshuva (Responsa 114) also ruled that a certain game played in his time, using ivory pieces, was Mukse, presumably because the function of entertainment does not render it a vessel. If so, Maran's opinion is not limited to a ball, but to all games. Therefore, Sepharadim should avoid playing any type of game on Shabbat, including backgammon, chess or ball. Games which use an hourglass to time the moves are even more problematic, because it is prohibited to measure on Shabbat.
The Rema argues with Maran and rules that the custom of Ashkenazim is to be lenient regarding using a ball on Shabbat. It is important to know this ruling of the Ashkenazim, because Hacham Bension has a rule (Or Lesion Vol. 2) that anything permitted to an Ashkenazi adult is permitted to a Sepharadic child. Therefore, he allows Sepharadic children under Bar Misva age to play ball on Shabbat. Hacham Moshe Halevy (Israel, 1961-2001) rules that although women are also included in the prohibition, if they will not listen to rebuke, it is better that they transgress inadvertently.
However, Hacham Bension suggests a different approach to understanding the Shulhan Aruch's prohibition of playing ball on Shabbat. It may not be related to Mukse and the Shiboleh Haleket's opinion at all, but to an incident recorded in the Talmud Yerushalmi. The Talmud relates that a certain city in Israel, Tur Shimon, was destroyed because its inhabitants were accustomed to playing ball on Shabbat, instead of learning Torah. If so, the prohibition would be limited only to ball and not other games. Based on this, he rules that those adults that are lenient to play games, besides ball, on Shabbat have what to rely on.
This is difficult to accept because the Rambam explicitly prohibited certain games on Shabbat, and all the Poskim mentioned above understood Maran as referring to all games, based on the Shiboleh Haleket. Nevertheless, even the strict opinions permit handling a child's toy that will help calm a child and stop him from crying.
SUMMARY
Sepharadic adults may not play ball or any games on Shabbat, but children under the age of Bar Misva may do so. It is permitted to handle a child's toy to stop him from crying.
