In a game of diplomacy I am playing, a Turkish army in Moscow is attacking Warsaw with support from Ukraine, Galicia, and Silesia. A Russian Army in Warsaw is attacking Moscow with support from Livonia and St. Petersburg. Finally, A Russian Army in Prussia is attempting to move into Warsaw. What happens?
Moscow is moving to Warsaw with strength 4 (itself + 3 supports). Warsaw is moving to Moscow with strength 3 (itself + 2 support). The Army in Prussia moving into Warsaw is irrelevant (It's of strength 1, which is also less than 4). The MOS-WAR move has greater strength, so succeeds and the Army in Warsaw is dislodged. The Prussian army remains in Prussia.
The Turkish Army in Moscow moves to Warsaw. The Russian Army in Warsaw is dislodged and must retreat (although in this situation, it will have nowhere to retreat to, since all adjacent territories are occupied, so it will be disbanded). The Russian Army in Prussia doesn't move.
Rulebook can be found here: https://media.wizards.com/2015/downloads/ah/diplomacy_rules.pdf
The key rule is:
A unit moves with its own strength combined with all of its valid supports. It can complete its move unless it’s opposed by a unit that is supported equally or better. One unit supporting another provides a combined strength of two and will defeat an opponent’s unsupported unit. Likewise, a unit with two supporting units (strength of 3) will defeat an opponent’s unit with only one support (strength of 2).
Units can’t trade places without the use of a convoy.
There are three separate moves that are in conflict:
The Turkish Army in Moscow has strength 4 (3 supports).
The Russian Army in Warsaw has strength 3 (2 supports).
The Russian Army in Prussia has strength 1 (no support).
Because of this rule, Turkish Army in Moscow moves to Warsaw because it has 4 strength, which is greater than either of the opposing moves of 3 strength and 1 strength. Meanwhile, the Russian Army in Warsaw and the Russian Army in Prussia are each opposed by an order of more strength (3 vs 4 and 1 vs 4 respectively), so given that and that the Russian Army in Warsaw cannot trade places with the unit in Moscow, both of the Russian orders fail.
If the unit that was attacked had no orders of its own to move elsewhere, it’s defeated and dislodged from the province. The dislodged unit must retreat or be disbanded.
Given that the Russian army in Moscow cannot move (it's move order fails), and it is being defeated (the Turkish Army in Moscow is successfully moving to Warsaw), the Russian Army in Warsaw is dislodged.
A unit can’t retreat to:
• a province that is occupied;
• the province from which the attacker came; or
• a province that was left vacant by a standoff during the same turn.
If there is no available province to retreat to, the dislodged unit is immediately disbanded and removed from the game board.
There is no possible retreat for the Russian Army in Warsaw since all surrounding territories are either occupied or the province from which the attacker came, so it is forcibly disbanded.
What Russia could do better
The fundamental problem is that Russia moving Prussia to Warsaw doesn't contribute to the defense of Warsaw. Because it's a separate maneuver rather than supporting or attacking one of the supporting units in the main conflict, it doesn't help in the strength comparison in the Warsaw - Moscow conflict. Instead, it's a follow-on move assuming that Warsaw completes its attack successfully (which does not happen in this case).
Russia could prevent the loss of Warsaw in this situation by moving Army Prussia to Silesia. This would cut the support of Silesia, which would make the opposing moves between Warsaw and Moscow each 3 strength, thus causing a standoff where neither unit moves.