Over the years, I have figured out that the design of all (or at least most) games can be written as a combination of several paradigms: Fun, Balance, Narrative, Challenge, and Realism.
These principles aren't limited to video games, in fact they are just as applicable to tabletop games.
Important: All explanations below assume a video game with no errors. Multiplayer games assume a competent team acting in good faith (no griefers or noobs) and a relatively competent, balanced opposition team (in a competitive game).
Explanation of each paradigm and its tenets below.
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What makes a game fun?
-The player's brain should be engaged as much as possible, making meaningful choices that impact the game. For example, an MMORPG where you are constantly managing your energy reserves and skills on short cooldown fits this description. Alternately, this could describe an RTS where you are constantly scrambling to read what the other players are doing and counter, or an FPS where you have to focus on aiming, active ability use, and avoid hazards in your surroundings.
-Difficulty should match its advertised name. If you have the difficulty set to Narrative or Training Mode, it should be possible to play like a complete idiot and win anyway. If you have the difficulty set to Insanity or Torment, it should be as difficult as Dark Souls.
-Avoid making the player go through tedious/repetitive tasks, and avoid throwing in un-fun mechanics that add nothing to the game.
-Players shouldn't be forced to wait long and arbitrary amounts of time when the wait adds nothing to the game (for example, a game where crafting items takes 12 hours of real time). Even excessive respawn times make a game un-fun.
-Players shouldn't be punished for events outside of their control (examples: RNG-heavy mechanics, undodgeable/random/arbitrary instant death).
-Allow the player to save the game and exit anytime, anywhere.
-Add plenty of side areas and quests; open-world is a big plus.
-Mistakes should always be recoverable. This ranges from replenishable/regenerating health to talent/perk resets.
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What makes a game balanced?
-Tierwise balance: Mechanics (creatures, enemies, characters, items, abilities, skills) should be split into tiers/levels. All mechanics should be upgrade over similar mechanics in a lower tier, and a downgrade versus similar mechanics in a higher tier.
-Comparative balance: Mechanics (creatures, enemies, characters, items, abilities, skills) on the same tier should be roughly as good as other mechanics in the same category/tier.
-Competitive balance: Both sides of a fight should have the same opportunity to win if they are of the same tier. In particular, if both sides are of the same tier and equal skill, a fight should be a draw or a toss-up.
-Adaptive balance: The game should provide enough leeway for players to switch up their tactics on the fly to counter any reasonable opposition of similar tier (avoid degenerate rock-paper-scissors situations). A good example of this would be the original Starcraft, where you have 3 different factions with a wide variety of tactics to counter different situations.
-Multiple ways to win: There shouldn't be any single required strategy to win.
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What makes a good narrative-based game?
-Obviously, the storyline. Build the game around the storyline and lore, even if it makes certain parts imbalanced or linear.
-Well-designed characters.
-A rich amount of lore and side quests/content.
-Focus on the journey through the game rather than playing to get the most points or rushing to the end.
-Avoid using procedural generation to mask a lack of content.
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What makes a good challenge?
-Winning should be an achievement. Every mistake should have consequences, some of which are non-recoverable.
-Mistakes should require the player to start over. This principle is particularly popular in Japanese game design, particularly Mario clones and "bullet hell" games.
-Mistakes should have lasting consequences, including penalties on respawned characters, lack of save points, lack of save functionality, or even permadeath.
-The game should be balanced largely against the player.
-Avoid RNG. Losing because of bad RNG isn't a punishment for failing a challenge, it's just uncounterable abusive RNG. If you play the game perfectly, you shouldn't have any chance of losing.
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What makes for good realism?
-Fit game mechanics to the lore and environment, even if this compromises any of the other principles.
-Realistic survival requirements, including eating, drinking, resting, and temperature/environment control.
-Use of locational damage.
-Use of character skill instead of player skill (in a role-playing game). This rule is non-applicable in survival simulators where you "are" the character.
-Superior graphics aren't a guarantee of realism It's entirely possible for a game with ASCII art and vector graphics designed around realism to be more realistic than a top-tier MMORPG in glorious 4K resolution with HD textures.
-Implement balance and tiering based on real physics principles. For example, moving a boulder up a hill should take much more energy than moving a sack of wheat up the same hill. This principle can be applied even in sci-fi and fantasy settings. Summoning a flying battlecruiser through a warp gate should take more energy than bringing in a scouting drone.
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My personal preferences:
40% fun, 40% balance, 15% narrative, 5% realism. I mostly prefer single-player or PvE games that require a large amount of thought (particularly mental processing power per minute), yet are well balanced and give the player a large amount of latitude.
I consider designing specifically around challenge and excessive realism to be distasteful. Games shouldn't delete my character except if I tell them to, and they shouldn't prevent me from saving and exiting anywhere. I have little time or patience for repeatedly starting over (a new level, map, character, or whatever) - respawns and respecialization are almost a requirement.
Realism: On one hand, I have no interest in perpetually managing realistic character needs (food/water/temperature/rest) and IMO excessive realism damages game balance. On the other hand, a small amount of lore-wise realism is good (so the game mechanics make sense).