Chess originated in India, and evolved on its spread to the Middle East and southern Europe. The 20th century chess scene was totally dominated by the Soviet Union. But these days, thanks to Viswanathan Anand, one might say that India has finally become the best at its own game.
The game Reversi or Othello was invented in England, but its modern reinvention was in Japan, and as seen in the link, a number of World Champions in the game have been Japanese. This might be attributed to its similarity with the Asian game Go.
Mahjong is originally a Chinese game, but in America has most of its association with (now) old Jews.