試合・しあい： a game: baseball, checkers, football, chess.
競争・きょうそう： a contest where the winner is judged by a third party: a marathon, two guys fighting over a lady, the best score on the test, the salesman who sold the most this month.
競技 ：きょうぎ： games IN GENERAL. Like they don't say 'olympic games', they say 'olympic 競技’. or 'he really likes games' or ???