I realized while playing a game of Wargame that Soviet and Warsaw Pact tanks have anti-tank missiles far more than their NATO counterparts. I'm aware of the advantages and disadvantages of gun-launched ATGMs, but why exactly did the Soviets decide to pursue them far more than NATO's brief and problematic flirtations with them (e.g. the M-551 Sheridan and its Shillelagh missile)?
With regards to MBTs having missiles, the short of it is that the Khrushchev era defense industry had a mania for missiles. The Soviet premier had already tried to prohibit the development of heavy tanks in 1960, arguing that they were death traps in an age of guided missiles (ATGM). The three main tank design bureaus responded by incorporating ATGMs into their designs for a next generation Soviet tank. By Khrushchev's fall, these designs were well underway. The T-64 was the byproduct of these design efforts and it included gun tube launched missiles. The development of the M-551 Sheridan and its Shillelagh missile also spurred interest and development within Soviet defense circles.
The advantages of the ATGMs looked great on paper. ATGMs promised greater accuracy and range along with a greater chance of a kill. In practice, the 9K112 Kobra (NATO codename AT-8 Songster) was highly problematic. The Kobra was quite expensive (R5000 in the 70s, when a diesel tank engine cost R9000) which prohibited extensive training with the weapon. Because of the costs, each tank in wartime would be allocated 4 rounds (there's something for you to keep tabs of while playing the game!). As is often the case with high-technology weapons systems, the high cost inhibited extensive testing so that the Kobra was unreliable.
source
Zaloga, Steve. T-80 Standard Tank: The Soviet Army's Last Armored Champion. Oxford: Osprey, 2009.
It appears that you are asking how historically accurate a certain game is, specifically concerning armament in the game. This is fine, but could you include which game you are talking about and the era in which the game is set?