Karpov, Kasparov play 25th anniversary chess rematch
The grandmasters resumed their legendary rivalry in two so-called semi-rapid games, in which each player starts off with 25 minutes on the clock and has five seconds added after each move.
In both games Karpov, 58, struggled to manage his time and the clock ran out on him.
The three-day match is being played in Valencia in eastern Spain. Ten more games remain: two more in this same format and eight blitz games, in which each player starts with five minutes on the clock, with two seconds added after each move.
The two men once waged one of the sporting world's greatest rivalries. From 1984 to 1990 they met five times for the world championship and pretty much drew even: Kasparov won 21 games, Karpov took 19 and they drew 104 times.
On Tuesday, Kasparov, 46, who had the black pieces in the first game and white in the second, told reporters that the day's wins give him a clear cushion — but that Karpov remains a formidable opponent.
The first title bout between the two started in September 1984 in Moscow and lasted nearly five months before it was halted with no winner declared on the grounds that both were exhausted. Kasparov won a rematch in 1985 and captured his first world title, at age 22, becoming the world's youngest champion.
Kasparov is considered by some to have been the best player in chess history. He retired from top-level professional play in 2005, saying that after dominating the game for two decades he had little left to achieve.
He is now an active political opponent of Russian Prime Minister Vladimir Putin.
Karpov was the reigning champion when Kasparov challenged him in 1984. He remains active and until recently was ranked among the world's top 100 players.