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I said a while ago I'm working on this... and here it is, just finished in the old year (well, where I live at least). I certainly didn't expect it to be this long.
Group 1: High Men, Nomads and Orcs.
These races have the weakest basic units with no bonuses at all, and before they unlock their race-specific ones the best that can be said about their military is that it's cheap. Advanced military infrastructure and unit production may be restricted to cities near mithril and adamantium deposits.
This isn't much of a problem, because with access to (almost) the full building tree they lend themselves well to economic support.
High Men may lack raw power in the early game, but they make up for it with speed and cost-effectiveness as the only race to get fully functional cavalry for the list price. Pikemen are slow and fragile for their cost and requirements, but 8 figures with armour piercing attacks makes them extremely deadly, especially with a few bonuses. On a completely different line, priests without a mark-up are attractive as combatants or supporters, even without fancy minerals or wasteful military infrastructure.
At the high end, High men lack the flying units of their competitors and have steep infrastructure requirements spread across different lines. However, 6-figure magicians are a bargain for the price and paladins are well worth the wait. Formidable combat units in their own right, magic immunity makes them ignore threats that would make an attack futile with anything else. Their holiness bonus also means one of them can increase the effectiveness of any ground-based stack - an endgame empire without at least one well-developed High Men city is definitely missing something.
Last not least, there's the mighty warship for ruling the seas... or, better, the skies with a flight spell.
High Men are generally liked as a master race: 5 races don't mind them at all, only Klackons and Dark Elves truly hate them... and those hate everyone who isn't them.
Nomads balance out their slower growth with a nifty gold bonus. While they aren't conventionally powerful in the early game, they can still conquer their home continent quickly: Horsebowmen can shoot on the move to avoid to-hit penalties, and are quite capable of finishing wounded units in melee. Rangers are rather expensive for a very modest increase in combat strength, but a single one of them can double a stacks overland speed.
With Pikemen and Priests Nomads have solid medium-tier alternatives to Team Blitzkrieg.
Fantastic Stables are expensive, but at least all Nomad unique units are unlocked by buildings from the same branch... and Griffins are extremely capable. Flying and first strike is an excellent combination, armour piercing makes it even better. Unlike other fliers, they are good enough combatants that they wouldn't be laughable as a ground unit (compare War Mammoths... and those are more expensive).
Diplomacy is another of their strong features, only halflings are better liked.
Orcs seem deliberately made as bland as possible - only they have access to all buildings, but some of their units are plainer than plain. The cavalry doesn't have first strike, and the halberdiers don't negate it. This means that even with a focus on earlier, cheaper units they don't really get an early-game advantage over Nomads or High Men, even though shamans for the list price are nice to have.
With no priests, plain 4-figure magicians that are hardly worth building unless needed against ranged garrisons and no high-powered combat unit, they don't have much going for them in the late game. By the time they can build Wyvern Riders, nothing that's susceptible to poison should be a problem, making them just an overpriced, underperforming flyer. Their most impressive asset, provided they can abuse them, are warships... but those aren't unique to Orcs.
Surprisingly, Orcs get along very well with many races, being slightly better liked by Gnolls, Barbarians and Trolls Than High Men. Unfortunately, they are despised by High Elves and especially Dwarves.
Group 2: Gnolls, Klackons and Lizardmen.
No research base, no magic weapons, limited power base, no animist guilds to support a large number of low-end units... for any staying power, it pays to branch out and conquer someone else.
On the plus side, their basic units get sizeable advantages for a head start to conquest. Their race-specific units are balanced to be competitive without magic weapons, allowing one to crank out serviceable troops in every location. Especially attractive with the Alchemy perk.
Gnolls need to subjugate some more advanced races for power and research, but apart from that they give up very little for their military head start, being able to build all the basic infrastructure and not generating crippling unrest.
Their Wolf Riders are fast, tough and hit hard. Unfortunately, their lack of first strike and low defense means they require outside help to keep any momentum instead of spending a lot of time recuperating.
Moreover, Wolf Riders are priced fairly while Gnolls get fantastic deals on anything else with 2 additional swords at no extra cost. Their most impressive unit is in fact the humble Spearman - outfighting vanilla Halberdiers at a quarter of the cost.
Interracial tensions are quite light - 5 races with no additional unrest, and even the normally antisocial Myrran races like furries.
Klackons are a bit of an oddity. Rebellious when conquered by another race, their cities are unwelcome news to many players unless they're in need of Engineers. As a starting race, better workers and reduced unrest are very pleasant, but expansion carries a dilemma: keeping other races in line is going to be difficult, razing and resettling is inefficient and leaves one short of power/research. Neither problem is unsolvable, but being forced to compensate for an inherent weakness makes it more difficult to exploit opportunities.
On the military side, +2 shields are nice, but paying twice as much for regular units seems steep in comparison to gnolls and lizardmen, who get comparable goodies for free. Stag beetles have enough shields to shrug off weak attacks, take many hits to kill, can attack fliers, deal full damage to the end and have an attack powerful enough to get through high defences. Unfortunately, they are quite expensive, demand two buildings that serve no other purpose and are only modest damage dealers. Not bad and probably he best deal considering everything else is overpriced... but hardly spectacular.
Everyone else hates bugs.
Lizardmen may grow quickly, but their utter lack of production multipliers can only be described as crippling. Worse, without Foresters or Animist Guilds they also struggle to feed themselves, let alone support a decent low-tech army. Klackons and Gnolls may lack economic clout, but Lizardmen don't even have the option to build up the basics before starting a sustainable offense. Their movement type is a blessing, otherwise starting on a small island without victims would be a death sentence.
The military side of things looks much better. With a free shield and heart, they get the toughest basic units for the list price, and their Halberdiers in particular are excellent. Shamans and Javelineers are neither as cheap nor particularly damaging, but their sturdiness keeps them from being direct damage fodder and lets them win many shoot-outs, even against Longbowmen and Slingers. With the melee prowess of halberdiers, Javelineers also remain an asset after emptying their quivers. Dragon Turtles sport the highest defence in the game (tied with Golems) for an incredible price, share the basic strengths of Stag Beetles and can swim. The only question is whether they are worth the infrastructure when lizardmen get equally good deals on basic units.
All in all, Lizardmen are under the greatest pressure to conquer other races quickly, but they have the tools to do so. They cause average rebellion in their subjects: nobody apart from Klackons and Dark Elves truly hates them, but nobody apart from Halflings likes them either.
Group 3: Barbarians, Halflings and High Elves.
These races combine the best of both worlds, at least superficially. They get some natural economic advantage that is in effect independent of infrastructure, enjoy bonuses to their basic units and can improve those even more with alchemists guilds.
Barbarians may not have the greatest lategame potential for their economy, but they have the basics and their fast growth rate - which also applies to outposts – gives them a decent head start.
Their thrown weapons are useful against fliers and opponents with few shields. More importantly, they make attack bonuses scale extremely well - even their spearmen can become powerful, and their cavalry has very respectable first strike potential.
Berserkers are the ultimate expression of this, and a contender for the most deadly (mostly) melee unit when given a little love from minerals and enchantments. Unlike the competition, they are neither fragile nor all that expensive nor flummoxed by fliers.
The ranged units are effective but unspectacular, unless one is able to abuse flying warships.
Barbarians don't cause extreme unrest, not even in Dark Elves, but they are also short of true friends - Nomads and Orcs only.
Halflings also get a very respectable head start thanks to their better farmers and benefit from the same quick outpost growth. They can also make a respectable economic contribution, despite their limited infrastructure: Nobody else feeds an army as efficiently, with no need to waste hammers and maintenance on stables and animists guilds.
Similarly, their selection of units is limited but what they can build is excellent. Despite the reduced strength, their swordsmen outfight some halberdiers (thanks to more figures and the lucky bonus), are cheaper and far less susceptible to being picked off from afar.
Their bowmen and shamans are quite good already but pale in comparison to slingers, which are generally regarded as the finest ranged units in the game and have excellent upgrading potential.
Halflings are the most loved race in the game, 8 out of the other 13 don't mind them at all.
High Elves can definitely shine as a builder race, having access to all multiplier and research buildings while generating power by their mere existence. Reaching their potential is another matter though: First they are hindered by a a low growth rate, later on by unrest concerns. They also have no engineers or units capable of purifying putrified tiles.
The accuracy bonus is generally worth the increased cost for spear- and swordsmen, less so for halberdiers which start to look like a very poor man's pikemen. Cavalry benefit from it nicely enough because more damage means less counterattacks, but again 60 vs. 40 hammers is noticeable... dubious whether it's worth the increased upkeep.
Their magicians have two movement points and get the usual accuracy bonus, making their regular attacks comparable to those of warlocks. Against high-defence targets, this works better than High Men's strength in numbers.
Longbowmen are a contender for the best ranged unit: although they are a lot less robust than Slingers, they are even better damage dealers overall and cheaper. Elven Lords seem overshadowed by Paladins, lacking the shields, hearts or abilities... but they cost less, don't require 1000 hammers worth of religious buildings and deliver the hardest first strike of all Normal units.
Pegasi certainly have their use as ranged fliers, and continue the racial theme of being fragile yet very efficient at killing things that can't fight back. They don't compare favourably at all to Airships.
Unrest is still acceptable on Arcanus considering the friendly relations with High Men, Halflings and Nomads. However, High Elves are the most hated race on Myrror and disliked overall.
Group 4: Beastmen, Dark Elves, Draconians, Dwarves and Trolls
The Myrran races would fit in the above categories well enough - Trolls in group 2, everyone else in group 3. However, they have several things in common beyond being tied to the Myrran trait: being more powerful, more expensive and more antisocial than their counterparts from Arcanus. And, sadly, not benefiting as much from Myrror's mineral wealth as the group 3 entries.
Beastmen have a very solid economy - all religious, research and production buildings, only missing Merchants Guilds. Hardly a reason to complain for a race that also generates mana naturally and doesn't grow that slowly.
Their basic units are powerful enough with an additional sword, cross and heart, but at twice the hammer cost they ought to be. The priests are probably the best deal - they pay a smaller mark-up and can hold their own in melee.
Centaurs and Manticores are rather underwhelming. Centaurs may be preferable to Horsebowmen but not by enough to justify the substantial cost difference, and the unit type is just not as useful on Myrror where all basic units are either sturdier than standard or able to shoot back. Manticores don't even compare well to Wyvern riders and are downright embarrassing compared to Draconian units. Minotaurs are beefy fighters, able to challenge most regular unit in melee combat. Large shields and their ability to soak up a lot of damage before damage output drops makes them quite capable of dealing with ranged units, and as usual fewer but more powerful attacks work better against high-defence opponents. All in all, a very capable unit that's slightly less impressive than other high-end units at its best (fewer figures also mean it doesn't gain as much from stacking multiple bonuses, even though the to-hit bonus helps) but exceptionally solid at its worst.
Beastmen are well-liked for a Myrran race, which is not at all.
Dark Elves have the ultimate economy in the game with 1 mana per population and all economic buildings... but slow to get there with serious growth and unrest issues.
Their normal units are more than a little dubious. Free ranged hits are usually nice, especially in the large-ish spearman garrisons one will likely need to quell the omnipresent unrest. At other times, it's exasperating to see adamantium-armed cavalry fling matches at unicorns that are vigorously impaling one's casters from behind.
Even when this feature isn't a liability, paying 250% for 4 crosses and some weak magic attacks is steep... 'if I wanted Shamans, I'd rather be able to build Shamans'. Or whatever else the closest analogy is - Dark Elves usually end up paying more for less.
Their higher-end units are a lot more interesting: Priests get ranged attacks surpassing other races' magicians while retaining their fighting skills and healing spell. Nightblades can keep whole armies at bay. Warlocks are extremely deadly and far superior to any magicians. Nightmares tick all the right boxes, being ranged fliers with a still respectable combat strength and not even that expensive... but often enough, they will be ignored for more casters.
Dark Elves are ridiculously unpopular even by Myrran standards. Barbarians are the only race to only dislike Dark Elves, everyone else hates them.
Draconians are the last Myrran race able to build up their cities well - all research and religioius buildings, but doing so may take a while with limited production (no miners guild) and food (no animists guild).
Being able to defend one's cities from most threats with a single spearman is quite an advantage, as is easy exploration with 2-speed fliers. Firebreathing for all melee units complements flying very well since it will be applied in every bout rather than being skipped when a fast enemy attacks. A shield and two crosses round this out very nicely.
For situations where risking one's still fragile and expensive melee units (250% normal cost!) would be unwise, Draconians have access to Shamans, Bowmen and Magicians which all lack the fire breath but cost only 50% extra. Bowmen especially benefit from flying if they don't need to fear retaliation and can empty their quivers from point blank range.
The unique units continue the theme of an all-flying race and fix holes in the line-up: Doom Drakes remain able to fight after taking a few hits and don't need to rack up bonuses to be competitive (with enough bonuses, Halberdiers tend to be better). Air Ships allow allies from other races to travel at the same pace, and their single high-powered attack will get through high defenses or magic immunity.
Like all Myrran races, they will face some significant unrest problems.
Dwarves are short, sturdy creatures fond of drink and industry. With more productive workers and a substantial bonus to taxes and mineral yields, they get an incredible economic start despite a slow growth rate. In the end they can still keep up with races who spent substantially more on infrastructure, with access to pacifying magic allowing a high tax rate they can be far ahead of everyone else.
They should definitely branch out though, because while they are excellent at generating gold there is no way to convert this directly into casting skill, research or food. Taking off a 3-production-worker for a 2-food-farmer when it could be the other way round is inherently wasteful.
They have access to few units, and even less that are worth building. While 50 hammers are a lot for the most basic combat unit, swordsmen with no gold ukeep and the usual dwarven bonuses (+2 hit points, +4 resistance) are very solid. Halberdiers aren't bad as such, but between not negating first strikes and being lodged between no-upkeep swordsmen and the much more powerful hammerhands there is little reason to build them. Engineers are an absolute bargain: At no additional cost they are the cheapest dwarven unit, still get 3 hearts meaning they can beat vanilla swordsmen in a fight as regulars, and they build roads at twice the normal speed.
Steam Cannons are rather expensive for what they do: almost twice the price and twice the gold upkeep of catapults and not clearly better, their main advantage is easier availability. Hammerhands will most likely form the backbone of one's army as one of the hardest hitters in the game that also benefit from rock solid defense and resistance. Golems are more of a niche unit: Their extremely high resistance (which renders their special abilities irrelevant) lets them ignore many threats entirely and their high defense is an asset against ranged attacks or multitudes of weak opponents. Otherwise, they usually end up taking more damage than Hammerhands because they don't finish the fight as quickly.
Dwarves get along fine with Halflings, Nomads and High Men. However, because Orcs, Elves and Trolls utterly despise them the average doesn't look good on either plane: only Klackons and Dark Elves face more of a rebel problem.
Trolls are the only Myrran race without a convincing economy, the lack of Miners Guilds is especially annoying considering the abundant mineral wealth. With Animists guilds and all religious buildings for food-neutral riot suppression, they can still pull their weight providing food for the rest of the empire. Their very low growth rate actually helps here: farmers eventually cease to be productive and more growth eats into the food surplus.
Price tag and seemingly large bonuses (+2 swords, +3 crosses, +3 hearts!) are misleading, Trolls do not hold their own in a fair fight past the early game. The advantages do not compensate for fewer figures and no magic weapons. Their main benefit is their regeneration ability - if not killed outright they can disengage and heal, and won battles don't result in any real casualties. The result is effortless conquest until they run into a brick wall.
Spearmen are very useful fodder/garrison units, swordsmen and halberdiers are fairly expensive for what what they do, Shamans are hilariously uneconomical and mostly unneeded. Going directly from spearmen to War Trolls makes sense - they are fast, considerably stronger than Halberdiers at little extra cost and a Fighters Guild is a good investment before focusing on an army anyway.
War Mammoths are hideously overpriced and require plenty of otherwise useless infrastructure... not good in a production-poor race for a unit that won't become a staple. A powerful first striker is a great complement to regenerating units that thrive on attrition though, and while other races can fill the niche better none of them are native to Myrror.
Trolls are unpopular as a master race, especially on Myrror itself. Halflings, Orcs and Gnolls consider them friends, they don't benefit from military resources and and they should have little trouble with the squishy garrisons on Arcanus. All of these make switching planes asap worth considering.
Closing thoughts
I spend entirely too much time on this, but organising what I knew and testing things in ILSe's excellent combat simulator gave me a few surprises. I've always considered Master of Magic the sort of game so full of content and possibilities that that proper balance is neither needed nor possible.
Now I'm convinced the designers came closer than I gave them credit for, with 11 or 12 of the 14 races seeming reasonably well-balanced against one another.
The good: Halflings. Economically, the best diplomacy and easy food are the most flexible advantages to have and among the most powerful. Even with a focus on other races, having Halflings in charge is probably ideal. Their basic units are among the best on Arcanus, Slingers are a candidate for the most practical racial unit in the game.
The bad: Orcs. Perfectly useful as economic support, but the only race to be almost totally dominated by a competitor.
The ugly: Klackons. There are ways of making them work and in the right circumstances they have their advantages... but the vast majority of times, they are just awkward.
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