Faction Calculus

In Strategy Games whether turn-based, real-time, or 4X, when there are different factions, these will (usually) have different advantages, disadvantages, and play styles to entertain the player. With a low amount of factions, there are clear, fundamental differences, but as the factions grow, the differences subside, until eventually the sides are not polar opposites as much as points on a gradient or two.

Two Factions:
With two factions, it will be the USSR vs. NATO, Allies vs. Nazis, Humans vs. Orcs, Elves Versus Dwarves, or the USA vs. an unspecified terrorist organization (who are not Middle-Eastern). One faction--"Powerhouse" for convenience--will employ the Elite Army trope, relying on a small number of powerful units; the other faction--"Subversive"--will use faster, weaker, and somewhat cheaper units (depending on gameplay mechanics), but will usually not have to resort to the Zerg Rush any more than the Powerhouse will have to resort to the Tank Rush. Historically, the Evil Empire is typically the Powerhouse faction, but in keeping with current real-world trends, it's becoming more common for the US or NATO to be the Powerhouse while weaker, decentralized non-state bad guys play as the Subversives.
 * Example: Allies (Subversive) vs. Russia (Powerhouse) in Command & Conquer: Red Alert.

Three Factions:
With three factions, the previous Subversive faction becomes "The Balanced" by moving up the power scale, while a new Subversive faction emerges. The new Subversives rely on weak, swift, massed units and special powers/dirty tactics such as stealth, deception or Mind Control. The Balanced, as suggested by its name, has become a balance between the new Subversive and the Powerhouse, not quite as powerful but not quite as weak either. Typical Subversives include The Undead, terrorist organizations, WWII Russians, and the morally ambiguous.
 * Examples: Atreides (Balanced), Harkonnen (Powerhouse), and Ordos (Subversives) from Westwood's Dune games. Starcraft's Zerg (Subversive), Terrans (Balanced), and Protoss (Powerhouse) are another Trope Codifier.

Four Factions:
When a fourth faction is introduced, the Balanced splits in two; The new Balanced and the Cannons. The Balanced moves towards the Powerhouse in a Magic Knight fashion, while the Cannons specialize with more Squishy Wizard and Glass Cannon units. With this split, the power field has become close, and factions grow alike.
 * Examples: Warcraft III''s Human Alliance (Balance), Orcish Horde (Powerhouse), Undead Scourge (Subversive), and Night Elf Sentinels (Cannons).

Five Factions:
The fifth faction, The Horde, completes the circle by being a mix of the Powerhouse's power and the Subversive's numbers. The Horde doesn't have very powerful units. They're probably better than the Subversives' and worse than the Cannons', and it's only through sheer numbers that the Horde will win. The Horde takes the Zerg Rush aspect of the Subversive as its only tactic, but they won't need any other tactics. Numbers will suffice. The Subversive, having lost their numerical superiority, will compensate with further special powers, making basic Subversive units weaker than basic Horde units, but no longer used as much for Zerg Rushing.
 * Example: The Rise of the Reds mod for Generals: Zero Hour has the USA (Balanced), the Russian Federation (Powerhouse), the European Union (Cannons), the Global Liberation Army (Subversive), and China (Horde).

Six or More Factions:
At this point, there are no established conventions. See A Commander Is You for the many variations that occur at this point.

Two

 * Command & Conquer. Tiberian Dawn: GDI vs. Nod - A rare example of the good guys being the Powerhouse.
 * Command & Conquer. Red Alert: Allies vs. Soviets
 * Company of Heroes: USA (Subversive) vs. Wehrmacht (Powerhouse)
 * Star Wars Empire At War: Rebel Alliance vs. the Empire
 * Total Annihilation: ARM (Subversive) vs CORE (Powerhouse), although the differences are fairly small.
 * Somtaaw vs Beast in Homeworld: Cataclysm; Hiigarans vs Vaygr in Homeworld 2. Two more examples of Powerhouse good guys.
 * SWINE: Subversive Rabbits, powerhouse Pigs.
 * Dark Reign: JDA/Imperium vs Sprawlers/Freedom Guard
 * Warcraft II: As Humans and Orcs are Cosmetically Different Sides at the lower end of the Tech Tree, advanced units differ enough to require different strategies. Orcs are more of a powerhouse, relying on Bloodlusted Ogre-Magi, while Humans rely on subversive but powerful spells such as Invisibility, Blizzard and Exorcism.
 * Tomorrow's War: In "asymetric warfare" scenarios the professional army is the powerhouse while the insurgents have to be subversive.

Three

 * Dune II, Dune 2000 and Emperor: Battle for Dune: Artreides, Harkonnen and Ordos
 * Command & Conquer. Red Alert 2: Yuri's Revenge: Allies (Subversive), Soviets (Powerhouse), and Yuri (Balanced, to the point of becoming a Game Breaker).
 * Command & Conquer. Tiberium Wars: GDI (Balanced), Nod (Subversive) & Scrin (Powerhouse)
 * Command & Conquer: Red Alert 3: Allies, Soviets, and Empire of the Rising Sun
 * Command & Conquer: Generals: USA, GLA, and China. Breaks the mold a bit, in that the USA is somewhere between Balanced and Powerhouse, and China is much closer to a Horde mentality than Powerhouse, but the GLA plays the Subversive role to a T.
 * Act of War: Direct Action: US Army, Task Force Talon, Consortium
 * Star Wars Empire At War Forces of Corruption: Rebel Alliance, the Empire, and the Zann Consortium
 * Starcraft: Terrans, Protoss and Zerg
 * Rise of Legends: Vinci (powerhouse), Alin (subversive), and Cuotl (balanced). That being said the units themselves are largely the same, with the traits of each race mostly coming into play through the Hero Units and differences in tech options.
 * World in Conflict: USA, Soviets, NATO. Here, though, the differences are much smaller then in other games - most units and supports are exactly the same on either side, and many more only have subtle differences. What little differences there are makes Soviets the Powerhouse, US Subversive, and Nato Balanced (Nato uses a mix of units and supports from the other two factions).
 * Para World: Tries to break this by having two "typical" factions, the Norsemen (powerhouse) and Dustriders (subversive), but then try to alter the mix by having a "stealthy" team, the Dragon Clan. However, stealth doesn't work too well when you're leading a big honking army.
 * Supreme Commander, though all sides tactics boil down to Zerg Rush eventually.
 * Sins of a Solar Empire starts out this way, with TEC (Balanced), Advent (Subversive) and Vasari (Powerhouse), but due to research topics all the factions start sliding subtly in other directions: TEC goes Powerhouse, Vasari go Subversive and the Advent go Balanced.
 * Conquest: Frontier Wars: Terrans, Mantis, Celareons
 * Ground Control: Crayven Corporation vs Order of the New Dawn. Phoenix Mercenaries were added in Dark Conspiracy. The sequel featured the NSA and the Virons. The Terran Empire was unplayable.
 * Earth 2150: United Civilised Sates, Eurasian Dynasty, Lunar Corporation
 * Metal Fatigue: Rimtech (Balanced), Mil-Agro (Powerhouse) and Neuropa (Subversive)
 * Brutal Legend: Tainted Coil (Powerhouse), Ironheade (Balanced), and Drowning Doom (Subversive)
 * A non-video game example in Babylon 5: The Vorlons are the Powerhouse, the Shadows are Balanced, and the Sherridan-led Army of Light is Subversive.
 * Rise of Legends: Vinci (Balanced), Cuotl (Powerhouse), and Alin (Subversive), though the Vinci will out-Powerhouse the Cuotl if allowed to tech up enough.
 * End War: USA (Balanced), European Federation (Subversive), and Russia (Powerhouse)
 * Achron's 3 factions are difficult to place since they don't follow any one category. For example, Grekim relies the most on trickery and special abilities, but unlike Subversive, it focuses on few high powered units. CESO is most like The Horde, while Vecgir are somewhere in between.
 * End of Nations: Order of Nations, Liberation Front, Shadow Revolution.

Four

 * Warcraft III: Human Alliance, Orcish Horde, Undead Scourge, Night Elf Sentinels
 * Dawn of War (Basic): Space Marines, Orks, Eldar, and Chaos.
 * Age of Mythology has four with the Titans expansion: Greeks (Balanced), Egyptians (Subversive), Norse (Powerhouse), and Atlanteans (Cannon)
 * Supreme Commander's Forged Alliance expansion adds the Seraphim.
 * Star Trek Armada: Federation, Klingons, Romulans, Borg. The Romulans do not appear in the sequel.
 * The Romulans just don't get a campaign in Armada 2. They're still there as enemies/multiplayer factions, along with the Cardassians and Species 8472.
 * Battle Realms: The 4 Clans; Dragon (Balanced), Serpent (Subversive), Wolf (Powerhouse), Lotus (Cannon).
 * Company of Heroes' expansion, Opposing Fronts, adds in two more factions - the Panzer Elite and British. By comparison, the Panzer Elite are Cannons, the British are Balanced (but in a very defensive sense), the Americans are Subversive, and the Wehrmacht are the Powerhouse.
 * Sword of the Stars base game: Humans (Balanced), Liir (Subversive), Hivers (Powerhouse), and Tarka (Cannon).
 * The first expansion added the Zuul (Horde). The second one added the Morrigi (Arguably took over Subversive from the Liir).

Five

 * Magic: The Gathering's five colors are divided like this. Green is Powerhouse, large, expensive and powerful units. Red is Subversive, quick, hard hitting but weak units. Blue is a mix of Balanced and Cannon: lots of very specialized Squishy Wizard units, with a few expensive powerful attackers. White is Horde, with the "White Weenie" deck coming into play. A mix of protection and flying keep it from being a sheer zerg faction. Last but not Least, Black is another Balanced/Cannon mix, being a "do anything at a price" faction, they have a very Jack of All Stats feel, but pay outrageous costs for their flexibility.
 * Dawn of War Winter Assault adds the Imperial Guard.
 * Theoretically, there are five armies in the original Advance Wars storyline: Orange Star (USA), Blue Moon (Russia/Canada), Yellow Comet (Asia), Green Earth (Western Europe), and Black Hole (Big Bad). In practice, it tends more towards "Allied Nations vs. Black Hole".
 * And there are no differences between any of the armies, which CO you use makes all the difference
 * Also theoretically, Sacrifice has five factions if one plays with one god straight - Persephone (Balanced), James (Powerhouse), Stratos (Subversive), Pyro (Cannon) and Charnel (Horde). Technically speaking, a player can mix and match with the various powers, meaning his unit combinations can come anywhere inbetween.

Six or more

 * Lords of Magic has EIGHT races. Fire (Cannon-subversive-ish), Chaos (Cannon-horde-ish), Air (Subversive-horde-ish), Life (Subversive-Balanced), Water (Balanced), Order (Powerhouse-Cannon), Earth (Powerhouse-Horde-VERY SLOW-ish), and Death (Balanced, in the way that having a Horde of Nukes is balanced).
 * Dawn of War added the Tau and Necrons in Dark Crusade, and the Sisters of Battle and Dark Eldar in Soulstorm for a final total of 9 races. This reflects almost all the major playable factions of Warhammer 40000
 * Warhammer40000 itself has sixteen different factions (five different Space Marine variants, Eldar, Dark Eldar, Tau, Chaos Space Marines, Chaos Daemons, Ordo Malleus, Ordo Hereticus, Orks, Necrons, Imperial Guard, and Tyranids). They're generally organized in accordance of how focused on shooting versus close combat they are and by whether they tend to field hordes versus elite units.
 * Rise of Nations has over 20 different nations to choose from, but most of the differences are small.
 * Age of Empires series have lots of factions to play with (at least in skirmish/multiplayer) and they always add more for each expansions. There aren't a lot of differences in the first two games though, generally defining each race by their unique unit.
 * Although some factions will specialise in other aspects not previously mentioned, such as economic boosts, or particular unit types (say, infantry, over cavalry or archers).
 * Age of Wonders 2 has twelve races, alphabetically: Archons, Dark Elves, Draconians, Dwarves, Elves, Frostlings, Goblins, Halflings, Humans, Orcs, Tigran, and Undead. Additionally, there are two Elvish wizard factions, two Human wizard factions (one is you), and two Undead wizard factions. Humans are the most balanced straightforward units by race, but everyone else is a bit mixed.
 * The sequel/expansion "Age of Wonder: Shadow Magic" brought that number up to fifteen.
 * Battle for Wesnoth has six factions in the Default era: Rebels (elves), Knalgan Alliance (dwarves and outlaws), Loyalists (humans), Northerners (orcs), Undead, and Drakes (dragonmen and lizardmen).
 * Master of Orion has ten races who actually play fairly differently. Each has a special racial bonus (and a few have handicaps to go along with that if their bonus is too powerful - most races can only colonize half the planets in the galaxy until they've researched the proper tech, but Silicoids can colonize them all right away. But their population grows HALF as fast as everyone else's!). They also have certain tech fields that they are better or worse at, and tendencies toward good or bad relations with certain other races. This results in quite a bit of difference in how each race plays, plus a rather extreme case of unbalance. The Psilons are excellent at all research AND are liked by most other races, making them one of the clearly best races, while the Klackons have an equally overpowered ability (bonuses to industrial production) but are widely disliked (they're still one of the top races though). Mrrshans and Alkaris have awful abilities (small combat bonuses) and are generally disliked, making them the hardest to play, while the Darloks (bonuses to all spying activities) are hated by all, but can often become the leaders in technology by stealing everyone else's tech, and can keep themselves alive by inciting wars between other empires.
 * Warlord's Battlecry 2 has 12 races that play differently, and Warlord's Battlecry 3 has SIXTEEN! Though for the most part it is a sliding gradient scale, there are several races that have unique combat gimmicks which make them difficult to categorize - the undead, for example, can upgrade their units on the field, simultaneously full-healing them. The Swarm in WBC 3 can resource starve their opponents while hammering them with weak - but very cheap - troops. Daemons can use their units to summon whole new units onto the field for free, effectively replacing lost troops mid-battle or freely building up the ranks beforehand, but the base units are hideously expensive. And then there's the Plaguelords in WBC 3, who are just totally overpowered; with access to cheap hordes of units with powerful upgrades "and" powerful area of effect monsters, their only real weakness is a lack of effective ranged units.
 * Sid Meier's Alpha Centauri has seven factions: Gaea's Stepdaughters, Human Hive, University of Planet, Morgan Industries, Spartan Federation, The Lord's Believers, and U.N. Peacekeepers. The expansion, Alien Crossfire, adds seven more: Cybernetic Consciousness, Nautilus Pirates, Free Drones, Data Angels, Cult of Planet, Manifold Caretakers, and Manifold Usurpers. There may still only be seven factions in the game. Also, if one alien faction (i.e. Caretakers/Usurpers) is in the game, the other must be as well. The printed manual for Alien Crossfire went into detail describing the process of autoplaying games with all 1,584 combinations of factions just to make sure everything is balanced juuuuust right.
 * And then there's the unlockable faction in Alien Crossfire - the Firaxians (named after Firaxis Games, the developer). They're intentionally a Lethal Joke Character, for people who just want to kick inordinate amounts of butt inordinately early, so they don't factor into the balancing.
 * Iron Kingdoms is two games (two sides of a coin really) warmachine and Hordes; Warmachine has six factions while hordes has five. Has even more if you count the sub groups in the mercenaries (3) and Minions (2) factions.
 * Wizard 101 has seven schools of magic. Storm is a Glass Cannon, Fire is a Gradual Grinder mixed with some Glass Cannon, Ice is the Stone Wall and later on learns Gradual Grinder spells, Life is Combat Medic, Myth is The Beast Master, Death is based off Life Drain based spells, and balance is a mix of Jack of All Stats and status buffing. All of them are actually very well balanced.
 * Monsterpocalypse has six factions each are one of six agendas, G.U.A.R.D. (Protectors), Terrasaurs (Radicals), Planet Eaters (Destroyers), Lords of Cthul (Fiends), Martian Menace (Invaders), and Shadowsun Syndicate (Collaborators). The later sets add six more factions to each agenda, Elemental Champions (Protectors), Empire of the Apes (Radicals), Savage Swarm (Destroyers), Subterran Uprising (Fiends), Tritons (Invaders), Ubercorp International (Collaborators).
 * Game Mod Red Alert 3 Paradox has 8 factions which are fairly deverse. Out of those, 4 play in typical Command & Conquer fashion while the other 4 do NOT (they play like an RPG, Sim City and both sides of a Tower Defense game, respectively). The orientations are equally deverse, the Allies posses air power and "working together", the Soviets have tanks ad absurdum, bringing raw unmatched firepower in, the Empire are versatile and very fast in addition to a powerful navy, the Confederates have great infantry and all sorts of stealthy mean tricks, the Talons are even more about working together, creating impenetrable shields with their strong defense values, the Chinese are the towers of a Tower Defense game, short-ranged but powerful, the Electrical Protectorate are the creeps of tower defense games with Zerg Rush tactics that would make the actual Zerg proud and the Syndicate are all about range and speed while bringing entire cities on the battlefield.
 * The 4X space simulation X Universe series has ten different starship-manufacturing factions, each of which uses a different design philosophy best described in terms of the armor triangle (speed-defense-offense). The core factions are as follows: the Argon and Paranid (variations on Balanced), the Terrans and Teladi (Powerhouse), the Split (Cannons), and the Boron (poor Balanced, being a pacifist race forced into warfare). The Kha'ak and Xenon are The Horde, while the Yaki are somewhere between Powerhouse and Cannons, and the Pirates are Subversives.
 * Dark Souls has a total of nine factions the player can join, each of them offering different, spells, weapons, armor, and how you interact with other players in PvP.