Someone asked Rav Shmuel Eliyahu is it is permitted to play the game "Catan" on Shabbos, considering the game involves doing business among the players.

Rav Shmuel Eliyahu, Chief Rabbi of Tzfat, responded that it is a problem and the game cannot be played. The Shulchan Aruch says that children who are playing such games on Shabbos involving business dealings should be allowed to continue and not stopped, but adults playing Catan should be told to stop and not play on Shabbos.
source: Kipa

Rav Avraham Stav, on the other hand, disagrees and says that every Shabbos outside of shul one can see children playing all sorts of games accumulating cards and candies or pits or other things, and also we see many families spending time together Shabbos afternoon playing games like Catan and Monopoly.

Rav Stav mentions the Rama who says such games should not be played, but if one is playing, he shouldnt be told to stop, as it is better to let him violate the prohibition by accident rather than intentionally after being told. So really such games should not be played.

Yet, the Aruch Hashulchan suggests that perhaps the games that are prohibited are games that would obligate the loser to pay the winner at the end with some object not part of the game but if the game is just cards or nuts there is clearly no legal contract obligating the two sides and it is just a game. The Aruch Hashulchan says this does not violate the prohibition of business dealings on Shabbos. The Piskei Teshuvos paskens lhalacha that the Aruch Hashulchan is to be relied upon to allow games, especially for children who anyways have no full legal standing for what they do. Rav Dov Lior adds that one should have in mind to start that any kinyan will only happen after Shabbos, as a precaution.

According to Rav Stav, the prohibition is only when there is actually winners and losers with actual winnings being paid by the loser. A boxed game that has no actual business happening has no prohibition and can be played. While some say it is worth being strict to not use fake money in such games on Shabbos, Rav Shlomo Zalman Auerbach paskened that there is no reason to be machmir regarding the use of fake money in games on Shabos.
source: Kipa