He shared first with Evgeny Sveshnikov at Sochi in 1983, tied for 2nd and 3rd with Viswanathan Anand, behind István Csom, at New Delhi in 1987, and took second, behind Vladimir Malaniuk, at Budapest in 1989.
Vaisser played twice for France in Chess Olympiads:
- In 1998, at fourth board in 33rd Chess Olympiad in Elista (+2 −1 =4);
- In 2002, at second reserve board in 35th Chess Olympiad in Bled (+2 −3 =1).
Vaisser was awarded the International Master (IM) title in 1982 and the Grandmaster (GM) title in 1985.
In 2010 Vaisser won the 20th World Senior Chess Championship and the 23rd in 2013.
- Anthony Miles vs Anatoli Vaisser, ol (men) 1998, Horwitz Defense: General (A80), 1/2-1/2
- Anatoli Vaisser vs A Mutzner, Mendrisio open 1988, Dutch Defense: Raphael Variation (A80), 1–0
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