For many years, the focus has been on games with perfect information. That is, in Chess and Go both of us are looking at the same board. In something like Poker, you have information that I don't have and I have information that you don't have, and so for either of us to make sense of each other's actions we need to model what hidden information the other player has, and also manage how we leak our hidden information. (A poker bot whose hand strength could be trivially determined from its bets will be easier to beat than a poker bot that doesn't.)
Current research is switching to tackling games with imperfect information. Deepmind, for example, has said they might approach Starcraft next.
I don't see too much difference between video games and board games, and there are several good reasons to switch to video games for games with imperfect information.
One is that if you want to beat the best human to be a major victory, there needs to be a pyramid of skill that human is atop of--it'll be harder to unseat the top Starcraft champion than the top Warcraft champion, even though the bots might be comparably difficult to code, just because humans have tried harder at Starcraft.
Another is that many games with imperfect information deal with reading faces and concealing information, which an AI would have an unnatural advantage at; for multiplayer video games, players normally interact with each other through a server as an intermediary and so the competition will be more normal.