In 1982 he won the championship of Soviet Russia. Vaisser 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. He tied for first in the Cappelle-la-Grande Open twice: in 1987 with Anthony Kosten and Jonny Hector, and in 1991 with Matthew Sadler.
Vaisser played twice for France in the Chess Olympiads of:
- 1998, on the fourth board at the 33rd Chess Olympiad in Elista (+2 −1 =4);
- 2002, on the second reserve board at the 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. In the following year, he won the over-65 section of the 24th World Senior Championship, which was split for the first time in two divisions (over-50 and over-65).
- 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
- Anatoly Vaisser rating card at FIDE
- Anatoly Vaisser chess games at 365Chess.com
- Anatoli Vasser player profile and games at Chessgames.com
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