While explaining the appeal of rules-light games to a friend who generally prefers a crunchier game, my friend asked me "then why have rules at all?"
I think the answer basically comes down to this:
A game is made more fun than just imagining through the limited introduction of rule. We use rules when we want to resolve things that are ambiguous, or when we want to abstract something, or when it is more fun to have a rule or constraint of some kind.
Some things are just ambiguous and dice mechanics are useful for those situations. Are you strong enough to knock the door down? It is hard to know, the player could say: "well I hit the door really hard." And we could try to work with that, but fundamentally this is an ambiguous situation and it is better to abstract it using a die roll.
While in most cases in a rules-light roleplaying game descriptive exploration and activity is encouraged, there are areas where description is either too ambiguous (as above) or otherwise finicky or difficult adjudicate so we abstract that decision with a die roll. Combat, saving throws, and thief skills all fall into this category.
Rules are fun:
Rules ARE fun. They are fun to think about, they are fun to experiment with. Constraints can create challenge. Niche protection and spell memorization rules create interesting constraints. The many tables for spells, criticals, and fumbles in Dungeon Crawl Classics are a good example of rules being fun. They certainly slow the game a bit compared to what the game would play like in their absence. However, they provide a lot of fun without creating many constraints on what is possible in the game. What is problematic is that in many RPGs is that there are many rules because it seems to be that the "rules are fun" reason has been taken to extremes. Because we like rules for a few things or because a few rules are useful it doesn't mean rules are good or necessary when created to cover all situations.
A key to an excellent game is making sure that the rules are serving a specific purpose. The rules covering every eventuality is not useful to a good game. Common sense can serve a lot of the need that rules purport to. The referee was introduced into wargaming as a way to obviate the need for encyclopedic rules and to make it possible for anything to be attempted (see Playing at the World). Referees can still do this job without a lot of policing! That is because the referee is not the players' opponent.
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