Since you mentioned bridge in the question, but nobody has said anything about it, I'll take a stab. Interestingly, bridge has several more-or-less orthogonal mathematical aspects to it.
The play of the hand necessarily involves calculating or estimating probabilities. These are not so difficult as to be mathematically interesting, but I do think they can be slightly more challenging that counting your outs in poker. In bridge there are often multiple possible ways of combining chances to make your contract, some highly dependent for their success upon the order in which the chances are taken.
Coming up with efficient communication schemes is central to both bidding and defense. I don't really know enough of the theory behind designing bidding systems to comment. But designing an efficient "relay" system probably involves a smidgen of math.
Finally there's more esoteric stuff. For instance, since bridge is not a game of complete information, one doesn't usually expect combinatorial game theory structures to arise. However it can happen that the bidding and play reveal enough information so that everyone knows what cards everyone else has, in which case there is of course complete information. Sometimes this actually brings added complexity though! One manifestation of this is higher order throw-ins, which can be analyzed via nimbers, etc.
