Will_Newsome comments on Searching for consequence-imagining games for children - Less Wrong Discussion
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For taking into account others' reasoning processes and how they interact with one's own, chess. Specifically the 'if I move here then he moves there then I move there' calculation algorithms. (Chess would probably also be good for general keeping track of consequences but other games might be able to teach that less discouragingly.)
I think that (for adults) poker is even better, because you have to think probabilistically (life isn't a perfect information game); but I'm not sure it is feasible for a young child to learn it.
Fights are messy but you practice kata precisely. (This applies to more than just chess vs. poker.)
It might be absolutely fantastic, except for the fact that you get it -wrong- so much of the time. You have to be adequately skilled at anticipating consequences in order for chess to teach you that you should. If you're pretty poor at guessing what somebody will do (or if your opponents are sufficiently unpredictable), it can easily teach you the opposite; don't bother wasting time planning things that won't happen.