I see you didn't link to empirically validated claims. That is because I didn't find any directly applicable references. So I settled for the LW links. Probably I should have stated this negative find. That might prompted someone to supply them should they be there.

I do have quite a few references about games and teaching concepts - but all of these address games in parenting, computer games, motor skills and literacy/numeracy. None of these are applicable here.

At best these would apply:

• Playing linear numerical board games promotes low-income children’s numerical development, 2008, Robert S. Siegler and Geetha B. Ramani, http://www.psy.cmu.edu/~siegler/sieg-ram08.pdf

Quote from the study: “Providing children from low-income backgrounds with an hour of experience playing board games with consecutively numbered, linearly arranged, equal-size squares improved their knowledge of numerical magnitudes to the point where it was indistinguishable from that of children from uppermiddle-income backgrounds who did not play the games. Playing otherwise identical non-numerical board games did not have this effect.”

This supports the claim that games with clear concepts can quickly convey these concepts (albeit possibly only very simple concepts).

• Home Numeracy Experiences and Children’s Math Performance in the Early School Years, 2009, Jo-Anne LeFevre et al, http://www.psy.cmu.edu/~siegler/fall10-lefevre-bisanz09.pdf

Deals with “the importance of home experiences in children’s acquisition of mathematics”. This supports the claim that concepts present in games significantly improve acquisition of complex concepts (albeit again simple ones).

Look at a statement like "Classical games with dice or shuffled cards surely build some intuition for probability theory which is present in most games in so far as some most games need some controlled random variables".

I wanted to draw the connection between the fact that probabilty theory is neccessary to explain those games theoretically (the "controlled random variable" part) and the ability to actually infer this in a game (the "intuition" part), But rereading it I agree that it comes across as either trivial or meaningless.