How are Fighters Linear but Wizards Quadratic?

Question

The phrase "Linear Fighters, Quadratic Wizards" gets bandied about a lot, but I've found I don't have a good way to explain it to newer players.

The tier system post has some examples of how wizards are better than fighters in specific situations, but I don't find the examples very satisfactory: the wizards in the examples seem to mostly rely on cheesy abuses that wouldn't happen in an actual game. For example the post says a wizard can kill a dragon using shivering touch from Frostburn, or using mindrape and love's pain from the Book of Vile Darkness, but many games won't allow those books.

In an actual play scenario, with no access to any expansion books, and assuming a group of characters that aren't grossly evil: what sorts of trends make wizards (or, more generally, full spellcasters) more powerful than non-spellcasting classes? At what character level does this start to happen, and what spells available at that level are responsible for the change?

I'm interested in responses pertaining to both 3.5e and Pathfinder; if there are important differences between the two, I'd be interested in hearing about those as well.

Answer 1

Breadth of Option

Unexpected monster rears out of the darkness, clearly well beyond the battered party's ability to handle?

Wizard teleports home. Fighter manages to kill the thing half to death before he gets eaten.

Ambuscade! The earl's men have the party cornered, and demand they surrender - only execution awaits if they do.

Wizard casts glitterdust - blinding most of the enemies and allowing his comrades to cut them to pieces. Or he turns invisible and leaves them to their fate. Maybe he uses a colour spray, and stuns half of the enemies, or burning hands to kill a different half. Maybe he runs away, and leaps off a cliff while casting featherfall. Fighter faces down 15 halberds and hopes to god his HP and AC hold out long enough for his Cleave feat to cut through them, because other than hoofing it, that's his only option.

It's less true, but still prevalent, at low levels, that Wizards can escape or defeat situations that kill fighters. At levels past 7, though, once Wizards get vastly more powerful options, foes become deadlier in melee than the Fighter, and Wizards can prepare enough spells to prepare for many different circumstances, the gap widens enough that the two classes are, barring self-nerfing/DM hand on the scales, playing different games.

The arguments against this generally incorporate self-nerfing. I've personally had first-time players pick the better spells on the wizard spell list, reliably. Animate Rope is indeed a wizard spell - but new players can easily see that it's utility compared to Burning Hands is very low. Fire Trap, Detect Scrying, Dimensional Anchor, Arcane Eye - all have uses, but new players pick Polymorph, and some of them even turn into hydras. Literally first-time players. There is a wide-spread culture of casters intentionally playing weaker than their class options offers them, especially amongst 'traditionalist' DnD players. This is to avoid GM banhammer/fudging the rules to smash them, generally, and has dated back to the earliest days of DnD.

Wizard spells are stronger than fighter options for less action economy (standard vs Full-Round), they can use them from range instead of their best tricks requiring melee, they can target groups, single targets, saving throws, hp, AC, no save at all,line of sight/effect blocking, creating minions, destroying enemy weapons/shields, altering the battlefield, and decide which to do on the fly instead of needing feat chains (that are still inferior to spells even if you have them), they can layer buffs to be tougher in every way than a fighter, they can use spells to be faster in every mode of movement than a fighter, they can literally see things before they happen and set up the perfect counter, spy on people from far away, turn enemies into allies, read minds, solve every single kind of encounter, create armies of minions (even without cheese, the literal lesser planar binding spell, the literal Dominate Person spell), turn into a gas and slip through the keyhole of doors, you know, whatever they like.

The sole 'downside' for this real ultimate power is that they can only use so much of it before needing a naptime. And since they can teleport home, or use spells to secure a safe place to nap, and since killing defenseless parties is kind of a dick DM move, in effect this is not a big deal without time pressures. And even if you can consistently use time pressures without it negatively impacting the story, you still have to effectively read the wizard's mind - as without his spells being used, an 'easy' encounter can swiftly lead to a TPK (at least, of the non-teleport/invisibility/gaseous form capable members of the party). Which is again bad DMing.

Individual wizards might self-nerf into being fighter-level, but the wizard class is a powerhouse of might. It becomes godlike, odin or zeus while the fighter stays firmly in hercules territory. But even at low levels, the wizard has options the fighter doesn't, and at everything but 'random damage comes out of nowhere! you take hp loss' has choices that will keep him alive instead of choices that lead to him relying on dice rolls against AC in order to not die.

Answer 2

Strictly speaking, it’s not actually true

The terms “linear” and “quadratic” come from mathematics.

• Linear growth is one in which the rate of change is constant. Notably, this means that no one level can be particularly special, each level would involve the same bonuses as the one before it, and growth only happens because of the accumulation.

• For quadratic, it is instead the rate of change that grows linearly, causing the value itself to grow faster than linearly.

Neither of these terms apply to fighter or wizard. The fighter does “accelerate” as he levels – most notably when he gets new iterative attacks – so his growth is better than linear, though probably not as good as quadratic. The wizard, on the other hand, more than doubles in power with each new spell level – that is, his growth is exponential, or better, and exponential growth moves a lot faster than quadratic.

And I bring this up because it starts to shed some light on how this trope is being misapplied.

The typical meaning of the trope is also not actually true

What people mean when they say “linear fighters, quadratic wizards” is that the fighter starts in a better position, but grows more slowly, so that by higher levels the wizard overtakes him and is better. This is true in a great many games, including AD&D.

Note that the definitions of linear and quadratic, even if they did apply, do not require that the initial conditions favor the fighter. The fighter could start out weaker, and then only see the gap between them widen as the wizard grows more quickly.

In fact, this is actually what we see with 3.5. Even if you have a 1st-level arena deathmatch where the fighter and wizard start adjacent to each other – a situation which plays to all of a fighter’s strengths – the wizard can have a significant statistical advantage.¹

Why? It’s all about the spells

Spells are, at every single level, the best things you can do with a standard action in the game. Swinging a greatsword can average 13 damage at level 1, which can put down a lot of things, but sleep and color spray can do that as well. Not so much at 1st level, but by say 3rd or so, grease becomes a massive problem for the enemy, and glitterdust just completely sandbags a lot of opponents. Ray of enfeeblement is hyper-reliable, if dealing with a single tough target. Meanwhile, charm person, comprehend languages, identify, and feather fall solve problems the fighter just has zero answer to.

And as the wizard levels, these things get massively better. Haste adds a disgusting amount of damage to the party’s total output (though it does require a fighter or similar to put it on). Fly completely changes the game. Solid fog, at character level 7th, is an excellent example of something that a fighter literally has no response to. The wizard can drop solid fog on an enemy warrior, and forget about him for several rounds while dealing with his allies. And then you start getting into complete game-changers, like teleport and overland flight.

The issue of preparation is one that is often brought up at this point. There is some merit to it – particularly at lower levels, where spell slots are actually in short supply – but it ignores a very fundamental fact about spells in 3.5: there are always options that are very broadly applicable. At low levels, you won’t go wrong preparing grease or glitterdust if you expect any kind of fighting. Haste and fly and solid fog are similarly hyper-reliable, broadly-applicable spells. Once you get high enough to have extra 4th-level spell slots, yes I would expect any spellcaster worth his salt to prepare freedom of movement or dimension door to handle grapples – and prior to that, grease can be used for that too, in a pinch.

On the other hand, stuff like comprehend languages or contact other plane are spells that I probably would not expect a wizard to just happen to have prepared (though I’ve been known to prepare the former when exploring unknown, but not necessarily hostile, lands). But a key thing to remember here is that the wizard can change his prepared spells every day. Yes, a wizard can get caught without a spell that he needs today, not tomorrow. It does happen. But it also happens that tomorrow can work. And in that case, the wizard can swap. The fighter cannot.

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1. ^{The wizard has an advantage on initiative (likely a higher Dexterity thanks to no armor, and could very conceivably take Improved Initiative), and if he wins initiative can open with color spray or sleep to put the fighter down immediately. If necessary, a coup de grace with a scythe is basically guaranteed to kill even a fighter. The DC should be 15-17, targeting Will; a fighter with an unusual focus on Wisdom and Will saves might have a bonus as high as +5, maybe (Iron Will and 16 Wisdom), but that’s an extreme investment and still only gives him a 50-50 shot in the best of circumstances. More likely, though, he only has +1 or +2, giving him a 75% chance of failure. This is not enough to guarantee victory over the fighter – the wizard cannot survive even one hit from the fighter – but in simulation the wizard wins more often than not unless the fighter is very purpose-built to counter the wizard.}

Answer 3

Three reasons, in order of increasing importance: flexibility, narrative agency, and expectations.

1. Flexibility: A well-built fighter can compete with a wizard in pretty much any combat-specific area. Wizards can be very good at grappling, so can fighters. Wizards can deal very large amounts of damage, so can fighters. Wizards can be good at ranged attacks, etc. To really get optimized for a given tactic, the fighter has to dedicate his entire character build to that tactic. To switch to a different tactic (e.g. from "guy who throws boulders" to "guy who grapples"), the fighter's player tears up the character sheet and makes a new character. The wizard optimizes by memorizing spells, and to switch to a new tactic he memorizes different spells.
2. Narrative agency: imagine that a wizard and a fighter each want to sass a demon. The fighter goes on a quest to find a portal to drop him on the right plane (assume the DM takes pity and places a portal; otherwise the fighter can't go on the demon-sassing adventure), then investigates for a while to find where the demon lives, then trudges over to to his house, then pesters his butler until he gets in to see the demon, and by then the fighter has forgotten why he even cared. The wizard takes a nap, casts Gate, delivers a cutting barb, then goes back to his nap. The same applies to underwater adventures (cast water breathing and go), aerial adventures (overland flight and go), hanging out on the plane of fire (fire resistance), building a castle (wall of stone), digging a tunnel (polymorph into a thoqqua or umber hulk), industrial revolution (animate dead), etc.
3. Expectations: the fighter is essentially a mundane guy doing well-defined things. No-one thinks that a fighter can punch a curse away. "I don't know how it works; it's punching!" is not a convincing argument. The wizard is a magical guy doing poorly-defined things. A wizard might well be able to magic a curse away. "I don't know how it works; it's magic!" is a very plausible argument. Fundamentally, wizards have a huge benefit-of-the-doubt advantage on top of their already-significant rules advantage.

Answer 4

D&D creatures may have evasion abilities such as flying, invisibility, damage reduction, darkness, illusions, ensnarement, walls, and teleportation. Spellcasters have spells which can defeat monsters that have these evasion abilities, and they have spells which can give themselves these evasion abilities. Fighters don't have any intrinsic way to deal with these evasion abilities; they can compensate by buying an array of magic items, but it's very expensive to get a perfect set of defenses. As characters increase in level, these abilities become more common, and it gets harder and harder for the fighter to keep up.

Low-level wizard spells related to evasion abilities include invisibility, and glitterdust to pierce invisibility, at character level 3, and fly at character level 5. These effects become more pronounced at higher levels.

Spellcasters have save or lose spells which will defeat a given opponent unless they can make a saving throw. Low-level spells include color spray, cause fear, entangle, web, blindness, hold person, suggestion, resilient sphere, black tentacles, et cetera. At higher levels, most opponents will have at least one bad save, and a spell targeting that save is very likely to instantly defeat the opponent. This allows spellcasters to fight without targeting an opponent's hit points at all.

Spellcasters have utility spells which are much more powerful than the corresponding mundane abilities. For example mundane classes might invest lots of skill points in skills like Climb and Jump and Swim and Balance, but the wizard can just cast fly which makes those skills irrelevant. Mundane classes might invest in Stealth and Perception, but the wizard can cast invisibility which gives a +20 bonus to stealth checks. Of course the spells only last for a few minutes at a time, but the effect on the game of these spells goes up as the wizard gains levels and has more spell slots to devote to them.

Spellcasters can, if necessary, use all their spells in one battle. One term for this is a "nova". The game is balanced assuming that spellcasters will split up their spells among several encounters each day, but the DM may have difficulty forcing the party to fight that many encounters between rests. If the spellcaster can limit the number of battles per day, they become much more powerful.

In 3.5e, buff spells such as polymorph deserve special mention, because they can make the wizard into a more powerful combatant than the fighter. In Pathfinder, polymorph spells are less powerful. Clerics have their own set of buff spells including divine power and divine favor and righteous might, but I don't have practical experience with a cleric that used these spells, so I can't comment on how effective they are.

Spellcasters have other important abilities as well, such as summoned monsters and direct damage. These abilities are powerful, but not all spellcasters use them, and they're not as important to the "quadratic wizards" issue as the abilities listed above.

Overall I'd say that spellcasters start to pull ahead of fighter-types somewhere around character levels five to seven, as they start getting spells such as fly and greater invisibility. The effect is stronger if the party is doing just one battle per day, and weaker if the party is doing many battles per day.

Answer 5

This is an attempt to describe where linear vs quadratic comes from, not talk about the power level differences between fighters and wizards in general.

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Fighters gain HP as they gain level. It takes (very) roughly as many rounds for a Fighter to kill a Fighter at level 1 as at level 20.

The Fighter's damage output goes up linearly against an expected defence level of an even-level foe.

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Each of a Wizard's spells scale. At level 1, they can shoot off their top-tier spells a few times per day, and each change the situation significantly.

As they gain levels, they keep roughly as many top-tier spells; but almost all of their spells scale. The ability for their spells to "hit" scale with their attributes and any feats/items they invest in boosting it (and with the spell level).

Some spells clearly scale, like Fireball, which does (level)d6 damage for the same 3rd level slot. Some spells scale in less obvious ways, like glitterdust, which at low levels cripples low level foes, and at higher levels ... cripples higher level foes.

As the wizard gains levels, the number of such spells they can cast grows linearly, and the power of each spell also grows roughly linearly. The product of two linear values ... is quadratic.

Now, one might object, eventually the wizard's spell load reaches "infinite", in that they can cast a spell every round and never run out.

But the wizard's spells don't always work. Sleep won't work on undead, for example. A wizard has to both handle "X game changers" per day, and the chance that they can change a given situation.

But as your number of spells goes up, if you pick spells well, the chance that you don't have the perfect spell for this situation goes down. This is another reflection of how the "depth" of the wizard's spell list is a source of power (which we describe as linear).

There are reasons to argue that a Wizard is super-quadratic; but the core is that the wizard has Linear Scaling number of Linear Scaling choices. The fighter mainly keeps one Linear Scaling ability (hit things hard while soakinga linear amount of damage).

Answer 6

This concept has been around since the early days and modern RPGs (for example, 3.5e) are less susceptible to this problem.

In AD&D, wizards started off with 1d4hp, a weak attack and not much in the way of spells. Magic missile was barely as good as an average warrior's attack. They couldn't use any of the cool weapons or wear any of the good armors. They were extremely squishy and basically useless at level 1.

A fighter would start off with 1d10hp, a strong attack, to-hit and damage bonuses from STR, extra hp from CON, extra AC from dex and they got all the best equipment. A level one fighter was actually quite an ass kicker in early D&D.

As levels progressed, fighters gained the same hp per level, their attack damage went up slightly (increasingly fractionally every few levels) and their saving throws got marginally better. A level 10 fighter did maybe 2-3 times the damage of a level 1 fighter, had maybe 10x the hp and probably better equipment. That's it.

Wizards, on the other hand, got much stronger spells after only a few levels. IIRC, fireballs were something ridiculous like 6D6 damage with area of effect and you could memorize a fairly large number of them. And this was a relatively low level spell. With decent INT (which every wizard has, that being their core stat), wizards got ridiculous levels of power at the higher levels. While the warrior is doing 2-3 times the damage he did at level 1, the wizard is throwing around huge amounts of damage, gating in powerful demons/elementals/monsters, casting wish and generally making the elements his bitch. And the wizard had spells that replaced all the utility skills of the thief, bard, etc, only they were better. And that's without getting into all the stuff wizards could do that no one else could like polymorph, etc.

Answer 7

One view of the problem is: put simply, in D&D 3.5e, starting at medium levels, you cannot survive getting attacked by monsters.

The place I usually notice this is around level seven, when fighting a Tyrannosaurus becomes a reasonable thing that can happen. The Tyrannosaurus is CR8; it attacks with +20 to-hit for average 23.5 damage, grapples at +30, and can swallow you whole.

The party's fighter cannot tank this monster. Even if someone invested most of their money in armor class, they'd probably be AC 26 at best, which means a 75% chance to get hit. Whoever tries to stand in front of it will get eaten and quite possibly killed.

While the fighter is getting eaten, the wizard can cast fly and be totally fine. Summon monster also works pretty well, or tiny hut, or greater invisibility.

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Tyrannosaurs are an unusually bad case, but this happens with other monsters as well. Fighters are designed with the assumption that they can wear heavy armor and survive fights with monsters, but monster attack bonuses and attack damage scale so quickly that armor does not work.

In this view, the problem isn't even that the fighter can't contribute -- the fighter's sword is a perfectly viable means of damaging the monster. The problem is that everyone expects the fighter to be standing next to the monster (if only because the fighter uses melee attacks), and nobody in the party is capable of doing that and surviving.

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All that other stuff, about wizards having more options out-of-combat: that's true too, and that's part of the problem. We should also keep in mind that spellcasters are intended to be played in an adventure where they have many combats per day, but most DMs don't enforce that very well, so all the spellcaster's daily powers get concentrated in one or two combats, making them much more powerful than intended.

But I think a large part of it is based on the fighter's combat role, and the fact that the fighter is entirely unsuited to surviving it.

Answer 8

I don't really think it's true.

Wizards are definitely weaker and lower levels, and more powerful at higher levels, and they are certainly more flexible. But there are distinct limitations and caveats about that.

Knowing the right wizard spells means planning ahead. You get a set number of spells and then you run out. Saving throws for powerful enemies tend to be quite good, and theirs spell resistance as well. By the time they get to the point where they are powerful, almost everything in the game is designed to counter it.

Ie, big single powerful monsters with great saves and spell resistance. Deep challenging dungeons, and planar adventures where rest is rare. And monster summons will or henchmen will universally be deployed against the wizard.

The vast majority of wizard builds rely on saving throws. It's true, you can make a wizard a better DPS, by focusing on single enemy touch attacks, with things like spell specialisation and bonus damage per die, but almost no one builds that way.

Fighters are consistently ahead of wizards in hit points, by using things like power attack, their damage scales because of high BAB - but it gets no saving throw, and adds things like magic weapon effects, and ability increases, belts of giant strength, potions of enlarge and additional attacks which makes it fairly on par, if not ahead for single big boss enemies.

Disabling attacks can be replicated with combat maneuvers, utility spells can be replicated with skills or plain creativity, and nearly everything else, you can get a magical item for (like a ring of invisibility) if you have the inclination. Magical items can be brought, and most of the key tactical advantages can be replicated with a magical item, particularly if you have UMD (which you can buy with a trait).

I think the power difference is basically situational, until wizards get their highest 7, 8, 9ths level spells. There are combats or situations where the wizard have a sheer advantage, like enemies almost immune to attacks, or a bunch of enemies standing in clusters, and then there are situations, where the wizard is next to useless (like a high save opponent with decent spell resistance), or a create with high reach or mobility and attacks that can basically take an entire wizard characters hit points in a few hits.

Played cleverly, I think they are not wildly apart. Sure, if you look at in the abstract outside of an actual game, they look like they are. But in practice, virtually everything in the game is designed to provide challenge at all levels, and that includes countering high level wizards strengths.