Isomorphism is a concept in Graph Theory, where 2 graphs, if they have the same nodes, connected by the same edges, are the same graph, no matter how they’re shaped. Basically this means that if two seemingly different systems have the same shape, they’re actually the same system.

One of the most popular examples of this is the Rest system in World of Warcraft. MMOs are known for being addictive. You pay a subscription to them, so there’s only so much time before your subscription runs out, and you want to get the most out of it. To avoid encouraging players to play constantly, many MMOs implemented penalties for playing continuously, to incentivize players to log off. Naturally players didn’t like having their xp gains drop to 50% as they continued to play during the WoW beta, so the developers tweaked the interface so that the “unrested” penalty became a rested bonus, granting 200% xp gains. The actual numbers didn’t change at all, but player reception to them did.

Another example I’ve described on this blog is the Hollowing systems in Demon’s Souls, Dark Souls 2, and Dark Souls 3. In the Souls series, your character has 2 states, Human and Hollow (Body and Soul for Demon’s Souls). Human form lets you engage with online functions, and has access to your full health bar, hollow form locks off online functions and has a reduced health bar. You can spend uncommon, finite supply, consumable items (Stone of Ephemeral Eyes, Humanity, Human Effigy, Ember) to regain your Human form, and thus your full healthbar. Naturally, the Souls games are all tuned such that you still take 3-4 hits to die from common enemies even if your healthbar only has half its max value, but the idea of being punished doubly for dying is offensive to a lot of people.

Perhaps in response to this, Dark Souls 3 framed its hollowing system as a bonus rather than a penalty. Your max health in Dark Souls 3 is smaller proportionally to your min health than in the other games, but it’s not displayed as a chunk of your health bar that is missing all the time, and you’re not hideous looking without an ember active. The actual numbers aren’t any different, but people seem to like it better.

In Breath of the Wild, weapons have a low durability and cannot be repaired. Almost all weapons of a category (sword, spear, heavy blunt) have the same moveset with different attack values, durability, and properties, so any given weapon is nearly 100% interchangeable for any other weapon in a given category. This means that rather than having an inventory of weapons like most fantasy RPGs, you instead have more of a loadout of whatever you’ve picked up recently, making the inventory system more similar to a modern First Person Shooter, where weapons are disposable, interchangeable, and very few are really situational. So rather than thinking of your inventory of weapons as your babies that you love, it’s more practical to think of them as a pool of damage to be unloaded into enemies and refilled with whatever they drop. Your total pool of damage is a weapon’s attack power times the number of hits it can do, plus the next weapon down the list until you get to the end. Thus in a manner rather similar to a FPS, you might have a higher damage weapon that’s good for finishing off strong foes quickly, but once the ammo (durability) empties out, you’re gonna have to find another one. So you may want to conserve ammo with your strong weapons, instead of seeing your weapons as persistent upgrades to your character like they would be in another game.

BotW cleverly stacks the deck in your favor in a subtle way, by having a “leveling” system chugging along in the background as you kill enemies. Every enemy has an xp value for defeating them, proportional to their strength. As you gain more xp, the enemies across the world get scaled up to stronger versions of themselves, and their equipment gets scaled up too, and becomes more likely to have advanced bonuses. This means that the pool of weapons available in the world gets stronger at a faster rate than the enemies do, and of course many enemies, such as bosses, remain unscaled regardless of your level. Since weapons always knock enemies down when they break, forcing them to drop their weapon, you’re constantly getting fed a supply of weapons that progressively grow in strength, so you can’t be left behind on the power curve unless you deliberately throw your weapons away (and Eventide Isle is designed to show how Link can still succeed, even starting with an empty inventory).

And of course, while this system in BotW might be really fun for gameplay enthusiasts who grok how the system is supposed to work and how they’re supposed to interact with it, it outright contradicts the more common framing of weapons being tools instead of ammo. This means that many players who can’t dissociate the theme and function of the weapon system just see the whole thing as a frustrating mess. This is amplified by cases like the Lightscale Trident, which is thematically the most powerful Zora Spear ever forged, and a sacred heirloom of the Zora Royal Family, but it breaks about as easily as most other weapons. Most people expect weapons to be semi-permanent upgrades that can be selected from at any time, with a durability system as some mild flavor that adds a sense of realism. So when a game breaks that framing and treats durability as an ammo counter, it’s aggravating to a lot of people.

In the pursuit of depth, it’s important to avoid having isomorphism within the same game; meaning, there shouldn’t be systems whose relationships are identical to one another. This means avoiding things that are proportionally the same in scale, rotation, mirroring, etc. 10 hp with 1 damage is proportionally the same as 30hp with 3 damage, unless there’s a 3rd factor that doesn’t scale along with those. Most RPGs have self-similar proportionality between the characters’ health, damage output, and enemy health and damage output across a game, meaning that the experience at the beginning frequently feels the same as the experience at any other point in the game.

In a blank endless room, the absolute (relative to the world, not each other) rotation of 2 characters locked onto each other is self similar, regardless of how much to the right they move, the only thing that can vary in proportion is how far away they are. The same is true of their absolute world position. If the two of them are translated any number of units away from the world’s origin point, in a blank endless room, there’s no real difference between any possible positions, except as relative to each other. If the characters are locked onto each other perfectly, then they’re not allowed to have different rotations relative to each other, therefore removing 3, and even 2d, space entirely. Avoiding isomorphism means avoiding symmetry, perfect mirroring, perfect rotation, perfect scaling, where nothing really changes.

Tekken attempts to avoid this mirroring/rotation problem by having characters be more dominant on their left or right side, able to catch people who sidestep to one side or the other, so even though you might be playing on an endless stage, sidestepping around your opponent in either direction is still important and different per-matchup. Not to mention that they otherwise avoid the problem by introducing terrain, which characters can be closer to or further from, and the rotation of the characters can orient them against the wall, or into corners, which might be harder to escape from. This makes the absolute position/rotation of the characters important.

Games with movement that’s locked to 4 or 8 directions (2d Legend of Zelda, except Link Between Worlds; Princess Remedy) have a weak form of this baked into the movement system, where relative positions of each character become more important to line up attacks, since they can’t be aimed at an arbitrary angle. In these games it’s really important to approach your opponents from either the cardinal or diagonal lines of the movement axes, depending on where your attacks are oriented relative to your character, and where the enemy’s attacks are relative to them. A Link to the Past and the gameboy zelda games accentuate this more by having Link’s attack originate from his side, rather than appearing straight from his center, as in Zelda 1. This has a weak effect in most cases, but it’s there. Since sword wielding enemies frequently have their sword a little off to the side, moving carefully can sometimes allow you to poke around their sword, which is interesting.

Most fighting games have what’s called “Guts Scaling”, where characters take less damage as their health gets lower. Since this is applied uniformly to all attacks, the effect is that characters have a little more health concentrated at the bottom of their health bar than they appear to. This doesn’t make it easier to come back from behind, but it does mean that matches spend more time in the stage where players look like their health is low. In other words, this is isomorphic in most fighting games, it’s purely a psychological trick. In Guilty Gear, the guts scaling is more extreme than other games, but chip damage and throws are exempt from this scaling, dealing their full damage, meaning that as someone gets lower in health, chip damage and throws become proportionally stronger, which means blocking is proportionally stronger at the beginning of the match, and weaker at the end.

Symmetry can frequently be useful however, for guaranteeing consistency and fairness. Multiplayer games are frequently symmetric in order to maintain fairness between teams. Asymmetric multiplayer games (Counter Strike, Overwatch, TF2, who serves in Tennis) are frequently played with the sides reversed to create symmetry, to give each team a chance to play each side. Many hitboxes for competitive games are designed to standardize oddities to create consistent outcomes (such as projecting 3d hitboxes orthographically). In boxing, left handed boxers (called southpaw) are frequently said to have an advantage, because they fight with their strong hand on the same side as their opponent. This breaks the mirroring of strong right arm versus strong right arm, closed stance, or open stance. The general rule is that asymmetry creates interest and breaks up redundancy, but symmetry creates fairness and consistency.

Developers should recognize framing and be careful with whether they label something a bonus or a penalty. If it’s more common to be “penalized” than to not be, players would probably prefer if that lack of penalty were framed as a bonus instead. Skilled players should recognize isomorphic game systems and not assign meaning based on whether the developer calls something a bonus or penalty. Developers should attempt at every stage of development to try to have relationships between different elements change in proportion, rather than retain the same proportion at a different scale, rotation, or so on. This means introducing reasons why the characters might want to rotate or not, or move around relative to each other or not. It means making levels, attacks, scaling, asymmetric relative to rotation, position, distance, and time.