He gained the International Master and International Grandmaster titles in 1980 and 1982 respectively, either side of two momentous Soviet Championship victories in 1980 (Vilnius - shared with Alexander Beliavsky) and 1981 (Frunze - shared with Garry Kasparov, whom he defeated in round 2).

In international tournaments, he has had many fine results, including outright or shared first place at Naleczow 1980, Sarajevo 1981, Cienfuegos 1983, Troon 1984, Sverdlovsk 1984, Szirak 1986, Sarajevo 1986, Sevastopol 1986, Lugano Open 1988, Tel Aviv 1990 (and again in 1999), London MSO 1999 and Andorra 2002. There were creditable second place finishes at Tallinn 1983, Sochi 1985, Trnava 1988, Calcutta 1988, Erevan 1988 and Herzliya 1998.

In the World Championship cycle, he was a runner-up at the Erevan Zonal of 1982 and qualified for the Interzonal at Las Palmas later the same year. Posting only a modest score however, he failed to progress to the Candidates stage of the competition.

It is the suggestion of his fellow professionals that his physical condition and occasional low energy levels were the only barrier to an even brighter career. Boris Gelfand (an Elo rated 2700 grandmaster) remarked that he has never beaten Psakhis at normal speed chess. Psakhis himself is more modest about his abilities, claiming that he can offer good advice on defensive technique due to the many dreadful positions he has had to play his way out of.

He was champion of Israel in 1997 and shared the title in 1999. He has also represented his adopted country at the Chess Olympiad seven times between 1990-2002. At the European Team Championship, he was first a member of the Soviet team at Plovdiv in 1983, when he won individual and team gold medals. Representing Israel thereafter, he took the board 4 individual gold medal at Batumi in 1999.

Over the years, he has assisted in many training programs, dating back to the late 1980s when he worked with Kasparov and Artur Yusupov. He played a training match with Kasparov in 1990 and lost heavily (1-5). Other students and/or chess stars whom he has seconded include Susan Polgar, Judit Polgár, and Emil Sutovsky.

In terms of chess style, he was in his youth a player of sharp, complex positions, but nowadays prefers to play in a positional sense. Consequently, he has developed an affinity with the French Defence and is a noted expert on it, beginning with writing The Complete French (and The Complete Benoni) for B.T. Batsford in the early nineties. His latest Batsford effort is a four-volume treatise on the French in 2003/4, titled Vol.1 French Defence: 3. Nd2 (Tarrasch), Vol.2 French Defence: Advance and Anti-French Variations, Vol.3 French Defence: 3. Nc3 Bb4 (Winawer) and Vol.4 French Defence: Steinitz, Classical and Other Variations.