Diminishing Returns for Balance

It's a common problem in game design. Your stat system works perfectly at level 5, but not at level 50 or, God forbid, 500. If a player spreads their points out between each of their statistics evenly or accrues small bonuses in one stat or the other, the effect is perfect... but min/maxers who jam all of their starting points into Dexterity completely ruin the game and can defeat the Final Boss solo at level 3.

What to do?

There are three general solutions:
 * Do nothing. Leave the game broken and hope nobody notices. You can get away with this if the brokenness is reasonably esoteric; who cares if 4% of the game's diehard fandom discovers how to beat the bonus boss with a naked level 1 character after the game's been out for five years?
 * Hard cap the stats. Restrict player creativity arbitrarily for the balance of the game. If your game mechanics in general fall prey to this, then capping them all equally can be a good option. If it's one specific mechanic, however, players are going to be miffed that they can raise their Strength to 10, but can only raise their Vitality to 6.
 * Employ Diminishing Returns for Balance. Every point of Charisma over a certain point has half the effect of the last one. You can alter the coefficients as necessary for each attribute, and if you do a smooth enough job nobody will notice. The game works as it should and everyone's happy.

Of course, the third option has its troubles. First, it's a lot more work, and takes a lot longer time to playtest and get just right. Second, if it's done poorly it just makes the game mechanics confusing; explaining to the player that, above a certain point, each point of Dexterity only provides 36.74% of the benefit of each previous point makes your game sound arcane and confusing. And if you don't explain it to them, astute players will feel cheated when they realize that they pushed all of their points into Strength and are getting no appreciable benefit from half of those stat points.

Fighting Game

 * A lot of fighting games use this to keep combos under control, causing each hit of the same type to cause less damage than the last until a certain amount of time has passed.
 * Some games will cause the first hit after an arbitrary number to automatically whiff even if the enemy is still well within its hit box, giving the opponent enough time to recover and counter.

First-Person Shooter

 * This blog entry analyzes that while Team Fortress 2's sniper and spy classes can be useful in small numbers, a team mostly consisting of these two classes is going to be wiped out pretty easily.

Hack and Slash

 * Diablo II uses this for many skill bonuses, such as Dodge giving you an 18% chance to dodge with the first point, but quickly tapering down to less than 1% bonus per point by level 20. This type of balance wound up turning many skills into "one point wonders." Just put a single point in the skill, and the " x all skills" bonuses on your equipment end up giving you just as much of a bonus as actually maxing the skill would have in the first place.

Mecha Game

 * To encourage flat out tanking rather than dodge tanking in Super Robot Wars. Diminishing returns made that if the unit is successful in dodging the attack, the other foe gets an innate plus-10% to hit. Some units just plain refuse to die with diminishing returns, however, as if they were equipped with sufficient barriers and defenses. By the time the enemy has a 100% hit rate, your unit will only be able to taken down by boss units. And woe to the Diminishing Return if the unit has counter...

MMORPGs

 * Dragon Saga's archer classes have a skill that lets them juggle enemies in the air by holding down the 'Z' key. Predictably it quickly became one of the most hated exploits even after more broken ones were found. One attempt to fix it saw its damage gradually decrease to 30% over consecutive hits... with no other change, meaning that the only 'improvement' was that the victim survived longer in a helpless state.
 * City of Heroes originally let you add as many of a type of 'Enhancements' to a power as you liked, leading to builds that were far and away so much better than anything else that there was no reason to use any different kind of enhancement build. This led, eventually, to The Great Diversification, when Diminishing Returns for Balance was instituted, making every enhancement of the same kind give less return. The returns diminished so harshly that anything over three of a kind was useless. The net effect was to make the game overall more difficult and to weaken linear, straightforward powers that only benefit from one type of enhancement. Certain powers and builds became useless overnight.
 * It was however very necessary as the Inventions System that came out a few years later and gave many newer ways to enhance powers would have been absolutely gamebreaking if there were no diminishing returns. Inventions also had the side-effect of making many builds that were previously unfeasible under the old system into very effective ones that can rival the most powerful of the pre-diversification builds. If anything Min-Maxing was made even better.
 * Multiple damage or defense upgrades in EVE Online employ Diminishing Returns. The first such module has full effect, the second approximately 80%, and after the third a fourth becomes near-pointless.
 * World of Warcraft uses this for many cases, including the duration of stuns and similar effects on other players to prevent "stunlocks". Defensive stats also use them to encourage tanks to invest in all forms (if available to them) rather than just one. Hard caps are also in effect in some cases, but Blizzard is trying to get rid of those or change things around so they cant be reached.
 * For instance, there is a hard cap for armor. At some point druid tanks, helped by a generous armor multiplier to compensate leather equipment and lack of shields, could easily push past it. Many healers also had a talent to increase a players armor for a short time after healing them with a critical effect. This was then addressed by several means such as changing the armor multiplier to only work on specific types of items and the healer talent changing to a plain damage reduction effect.
 * Trade skill leveling works in a similar fashion. Early on, it's ridiculously easy to cook spiced bread with inexpensive flour and spice you can get anywhere or make cheap potions with flowers common as dirt but as increase your skill it becomes harder to increase your skill. Ingredients needed to craft become rare and expensive and it often takes multiple attempts to actually gain a skill point, each attempt consuming the reagents regardless of a possible lack of skill gain.
 * Champions Online implemented this feature fairly early on to balance out the downright insane returns people were getting on Dex/Ego builds (things like critting on almost half your attacks for almost double damage). They instituted a "softcap" partially determined by the character's level that seems to be working well as a compromise; low level players see better returns early on (Dex became a feasible defensive stat BEFORE the late teen levels). but the higher level min/maxers can still push their stats high enough to get noticeably better performance from their characters.
 * eRepublik skill training works like this, in the case of strength when you start out you gain .5 every time you train, by the time you hit 4 strength this is down to .04 every time you train.
 * Star Trek Online has a skill point-based system with diminishing returns the more you rank it up. The only reason for maxing out some skill trees is to get the ability to train high-level, class-specific powers on Bridge Officers.
 * Ragnarok Online inverts this in before its renewal patch for its offensive stats; while the cost to increase a stat went up as it got higher, you'd get bonus damage when you reach certain points (for example, every 10 str points gave a melee attack bonus). This bonus grows as you reached higher and higher stats, to the point that oftentimes that it was worth paying more for the higher stat (so you'd have less total stats), if it meant reaching a damage bonus. The renewal patch changes this behavior.

Puzzle Game

 * Used in Puzzle Quest, where putting more and more points into each stat gave less and less reward per point.

Real Time Strategy

 * League of Legends uses hard and soft caps for some stats. Attack Speed and Cooldown Reduction hardcap at 2.5 attacks per second and 40% reduction, respectively, with no diminishing returns. Movement Speed has no hard cap, but two "soft caps" make it hard to increase past a certain point-raising your movement speed past one soft cap causes increases to have their effectiveness cut, and going past the second causes a steeper cut.
 * Rise of Legends (no relation) increases the price of a given unit as their population increases, to make it more expensive to mass single unit types.

Role Playing Game

 * In older Final Fantasy games (I, II and III mostly) by endgame evasion would outgrow your quickly diminishing armor stat, meaning it was often better by game's end to wear no armor.
 * The Fable games give you three sets of stats and let you choose how to level. Unfortunately, each level of a stat costs about 5 times the cost of the previous level. Since leveling a low stat is so cheap, almost everyone advances roughly equally.
 * The Elder Scrolls III: Morrowind is a treasure trove of this trope:
 * The most direct example is the patentable method The Elder Scrolls has used for leveling for a long time - you are awarded experience for misses and critical failures as opposed to successes. Your level is proportionately related to your ability to use a skill successfully, hence you will be able to learn something you have no skill in quickly, but master something you have a lot of skill in only through devoted use (since you will stop failing as often). There are only a few exceptions, such as some implementations of magic.
 * Alchemy, meanwhile, is broken beyond all belief and is the perfect example of what can happen when this trope is not enforced. Not only can mass herb mixing give you lots of Alchemy skill (snowball effect), it will also give your base stats lots of free level-up points AND eventually absurdly powerful and expensive potions as your skill increases. Lots of money, lots of experience, powerful stackable buffs, all while doing very little. Additionally, this leads to the legendary Fortify Intelligence Stacking trick and can ultimately result in a situation where the game crashes from mathematical overflow.
 * When you buy from trainers, it costs increasing value for each level. (The actual increase in price, however, is typically considered trivial compared to one's ability to make money.)
 * In what could be a direct call-out to the introduction to this trope, the Merchant skill is broken at 50. Up to 50, the price decrease of all merchants' goods makes sense as a better level should indicate you are better at buying and selling. After 50, things start getting more expensive again. Many mods have attempted to correct this issue.
 * The Mario & Luigi RPGs award points into every stat automatically on each level up, and then let the player apply a bonus to one stat by timing a roulette spin. If one stat is getting too high, or you've used the roulette on the same stat too many times, the numbers on the roulette get smaller. There's always a 3 on the wheel, however, so with good timing you can minmax any of your stats.
 * The Might and Magic games had an interesting take on this trope, where the stat system only offered bonuses at certain intervals, which exponentially decreased in frequency as the number got higher. You received bonuses to relevant skills, for example, at 13, 15, 17, 19, 21, 25, 30, 35, 40, 50, 75, 100, 125, 150, etc., until finally stopping offering bonuses past 500. So, a stat of 15 and 16, for example, was functionally the same. This tended to make raising most stats higher than around 50 or so rather pointless, and rendered most minor stat gains worthless well before that.
 * The first Geneforge used the Might and Magic version of this trope--going from 10 to 11 in a skill, for instance, does nothing, whereas going from 10 to 12 provides a bonus equivalent to going from 9 to 10. From 10 to 20, the bonuses came from even numbers only, and above 20 bonuses were only gained at 23, 26, and 29. This incidentally meant that getting one point of a skill from a quest could just mean that you'd be capable of getting a bonus if you were willing to buy another level of the skill. Later games replaced this with the Dungeons and Dragons system of having higher levels cost more points--in that case, the only problem is that it encourages minmaxers to get their free quest-related bonuses as late in the game as possible, so as to maximize the number of skill points saved.
 * In Demon's Souls stats tend to start giving diminishing returns when brought up to somewhere around level 30-50; Endurance is the most extreme, as at level 40 it outright stops giving stamina boosts.
 * Its spiritual successor Dark Souls have a soft cap of 40, but diminishing returns really hits hard at 50, in which the only reason you want 50 in that stat is for spell requirements (such as 50 Intelligence for White Dragon Breath and 50 Faith for Sunlight Spear). Subverted for Carrying Capacity, which is tied with Endurance stat; while Endurance stops giving any stamina boost after 40, increasing it beyond 40 gives you one point of carry capacity per level, that is, a flat rate increase.

Tabletop Games

 * The Role Playing Game Role Master has this for the skill system. Every ten points in a skill up to +50 gives you +5. The next ten give you +2, up to +70. Then you get +1, then +1/2. Magic items and stat bonuses, on the other hand, are linear, so +10 is +10 whether you're adding to 10, 100 or 1000.
 * Dungeons and Dragons (both 3e and 4e) uses a variant of this in their point-buy system for stats - The modifier value derived from your stats remains constant for easy math, but the cost of each stat grows faster the higher you get.
 * This only applies for new characters - points gained from leveling (and from items or spells) have no diminishing returns, although the cost of most magic items grows quadratically with the benefit.
 * Older editions also increased the Wish cost for raising stats: if your stat was anything between 1 and 15, then a Wish spell would improve it by +1, but if your stat was already 16 then each Wish would only raise it by 1/10 of a point; if it was 20 or more, each Wish would increase the stat by 1/20 of a point! This was back in the day when the Wish spell aged the caster by five years, and required a System Shock roll (basically a Fortitude save) to avoid instant death (with success merely leaving you bedridden for a week).
 * Similarly, the New World of Darkness tends to encourage evening out one's portfolio past a certain level -- each new dot in a category typically costs its new rating multiplied by a flat rate, so buying the fourth dot in an Attribute from the third costs twice as much as buying the second dot in an Attribute up from the first. Similarly, the character-creation rules make the fifth dot in a Trait (out of a normal maximum of five dots) cost twice as many of starting points to buy.
 * Averted in character creation, where costs are linear, except for the highest level. This leads to the problem where it is mathematically superior to start with an extremely minmaxed character, and spend xp to cheaply flesh out secondary abilities, than make a balanced character and pay a premium to increase your abilities later. Shadowrun 4th Edition has the same setup and suffers from the same effect.
 * Skills in GURPS start to give noticeably diminishing returns around level 14 (90% chance on normal tasks) and additional levels become a total waste of points at level 24 (90% chance on "impossible" tasks) except in the most extreme settings where techniques can have penalties of -30.
 * Using the SPECIAL system based on GURPS, the Fallout games require increasing numbers of skill points to improve skills once they're up to 100% (provided the installment you're playing allows skills past 100%).
 * Earlier editions of GURPS had the point cost of any skill double with each level, so increasing anything beyond +7 or so (depending on skill difficulty) would be ridiculous. Later editions made the cost linear with level, allowing for easy minmaxing with opposed skills (like swordfighting).
 * In some games the basic mechanics are inherently designed to create this. For example, any Tabletop RPG that uses a bell curve (e.g. 3d6) as opposed to a flat statistical probability (e.g. 1d20) for its core mechanic will get a version of diminishing returns: in a bell curve the numbers in the middle of the probability space are created by more of the possible permutations of the dice, while the numbers on the edge of the probability space are created by relatively few permutations. This means that changing a very low or very high number grants a smaller benefit than raising a number in the middle of the range.
 * Swashbucklers of the 7 Skies employs similar ability improvement to the aforementioned The Elder Scrolls video game series: you only gain points to improve your abilities when you fail. Characters who max out an ability and then focus exclusively on it are going to advance very slowly, if at all, while those who dabble in many things or throw themselves into scenarios where they've got no real skill are going to develop faster.