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I am looking for a game (at least by some definition) that:
1. Is a 0-player game; that means that, at least once the game has started, no player is able to influence the course of the game.
2. Has good replay value; in particular, the game should not be the same every time.

There are a couple games that fit this criteria that I can think of: The card game of War, and John Conway's Game of Life. Any other suggestions?
I think, Tetris could fit your description, as long as you do not touch any buttons...
The Settlers have 0-player mode.
Ring King, if you have an NES.
You can watch AI vs AI in Rocket League, that could probably fall under the definition of 0-player game where you can't influence the course of the game.
Pachinko or Slots?
Counterstrike (or anything similar) with bots only.

Letting two chess engines play against each other. (Or Go or any other board game engine)

Any simulation of sufficiently complex systems.
Progress Quest, of course.
The [games] in my mind are almost always better -- The Maxx
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Lifthrasil: Letting two chess engines play against each other. (Or Go or any other board game engine)
To be honest, I always wondered how matches of that nature would pan out.

Pitting programs like Sargon, Fritz, Hoyle & Zillions Of Games against each other sounds almost Street Fighter-esque in nature.

(Could Deep Blue be the competition's Vega/M.Bison?)
Post edited April 01, 2019 by ChainsawGenie
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ChainsawGenie: The [games] in my mind are almost always better -- The Maxx
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Lifthrasil: Letting two chess engines play against each other. (Or Go or any other board game engine)
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ChainsawGenie: To be honest, I always wondered how matches of that nature would pan out.

Pitting programs like Sargon, Fritz, Hoyle & Zillions Of Games against each other sounds almost Street Fighter-esque in nature.

(Could Deep Blue be the competition's Vega/M.Bison?)
Are you aware that The Max was the name of the diner in Saved By the Bell? :D
Watching the Planet Earth series. 0-player, good replay value.
Also The Matrix.
Candyland was the board game that taught 7 year old me that there are situations in life that have predetermined winners and losers
Many 4x/territory control games let you set this up, and then let AI go.

Civ4 can.
SNES emulate "Romance of the Three Kingdoms 2" for this.
FLOSS game "Fracas" can be configured this way.
Tree Simulator 2020
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dtgreene: I am looking for a game (at least by some definition) that:
1. Is a 0-player game; that means that, at least once the game has started, no player is able to influence the course of the game.
2. Has good replay value; in particular, the game should not be the same every time.

There are a couple games that fit this criteria that I can think of: The card game of War, and John Conway's Game of Life. Any other suggestions?
Distant Worlds Universe (fully automated everything setting) or Dwarf Fortress's world generation.
Distant Worlds Universe is pretty interesting, you can legitimately spend the entire game following around one ship/character, and experience totally different things happening each time you generate a new universe in-game.