The first is London:
Anatoly: I'm a chess player, Mr. Molokov - you go and play these other games!The line is traditionally delivered with a dramatic disgust as Anatoly sends Molokov off.
The Broadway line is sort of its opposite, Molokov ordering Anatoly to play an ambassador:
Molokov: He went to University with the First Secretary. You have no choice.I'm bringing up these lines because neither of them would have been said by a Russian, particularly not in the Soviet days.
The Russian language doesn't have a word meaning "mister." Today there is occasional use of the extremely formal господин (gahspuh-DEEN) - which was a way to address a nobleman in the old Imperial Russia, closer to "master/Lord" than "mister". It was totally out of use in the Soviet period. It was far more common to use товарищ (tah-VAR-ish), which in English is translated as "comrade." This was a common form of address in the period, and it's frequently enough used by film and stage Russians that it's something of a cliché, but it is much more accurate to have Anatoly say "comrade Molokov" than "Mr. Molokov." Now, and then as well, it would also be acceptable to have him simply call him "Aleksandr" or "Molokov," the latter being a bit overly formal.
Broadway's line is really odd. The leader of the USSR was the General Secretary of the Communist Party of the Soviet Union; for the period described by Chess there are only three candidates for the post. Yuri Andropov was General Secretary from November 1982 to February 1984; Konstantin Chernenko followed until March 1985; and Mikhail Gorbachev until August 1991. Given the timing of the London and Broadway scripts, Gorbachev was General Secretary.
"First Secretary" was a peculiar form of the title used by Nikita Khrushchev (in power 1953-1964), who was consciously distancing himself from Stalin. Brezhnev restored it to "General Secretary" in 1966, and this form was used until the CPSU was outlawed in 1991. But being General Secretary was not like being President in the US. Two Russians in discussion would have referred to "Gorbachev," and definitely not to "the First Secretary." This is actually really helpful for staging Chess over two decades after the fall of the Soviet Union. It immediately contextualizes the piece for Molokov to say "He went to university with Gorbachev."
Happily, Gorbachev did go to university. There's no record of whether he actually knew the only ambassador to Thailand that was appointed while he was General Secretary, Anatoly Valkov, but photographs of the candidates can be seen here. Valkov is the image of the bureaucrat that Anatoly didn't want to play chess against. Sadly there are no detailed biographies for these Russian ambassadors, but Valkov was born in 1927, and his predecessor Kasatkin in 1930, so it's quite possible for either of them to have been at university at the same time as Gorbachev (born in 1931).
So in our revised form, these lines go (London):
Anatoly: I'm a chess player, Comrade Molokov - you go and play these other games!And then Broadway:
Molokov: He went to University with Gorbachev. You have no choice.Both of these help set the tone for a modern audience, placing the characters more firmly in their Cold War settings.
There is also some value in talking about Russian names, at least briefly. Russians tend to have a middle name (patronymic) based on their father's first name, with ovich added for men and ovna for women, so Viktor's children Ivan and Marina would be Ivan Viktorovich and Marina Viktorovna. Giving the characters patronymics helps make them feel more authentically Russian at key moments. I would suggest Anatoly Viktorovich (a reference to Korchnoi), Aleksandr Grigorevich, and Svetlana Petrovna.
There are also stereotyped nicknames for most common Russian names. For Anatoly it's Tolya, for Aleksandr there is Sasha (it can be a man's name in Russian), and for Svetlana there is Sveta and the less common Lana, the latter of which would work more smoothly in English. These are much warmer in usage than first names or patronymics, and give another variant to make the dialogue feel more authentically Russian.
I'm going to go through some more dialogue for other ways to add context that may not have been obviously needed in 1986/1988. It's one of the little things about making Chess more of a period piece that I think is a dramaturgical necessity.