There are a few popular games where the object is to infer some unknown information, and in some cases there has been some nice mathematical analysis of advanced strategies.
Peter Winkler's classic paper on The Advent of Cryptology in the Game of Bridge shows how you can covertly communicate information to your partner even when your opponents are fully informed about your bidding system. You can think of Winkler as implementing a crude form of public-key cryptography in a bridge bidding system.
The game of Clue (a.k.a. Cluedo) is analyzed in the book One Hundred Prisoners and a Light Bulb by Hans van Ditmarsch and Barteld Kooi.
You may not have heard of the game Hanabi because it was invented pretty recently, but it won the prestigious Game of the Year award in 2013, and is still selling pretty well. There was an article called How to Make the Perfect Fireworks Display in the Mathematics Magazine which developed a powerful strategy for Hanabi using ideas from the theory of error-correcting codes.