game-spaces

October 28th, 2025

The right spaces of games

There is a lot of fascinating work on studying the richness of behaviors in games (the main object of game theory) and in mechanism design

However in the age of games for AI agents, in particular, games start to play a role that is different from that typically envisioned by game theory or traditional game design:

Obviously, games are either given to us naturally or designed for a certain purpose (or, a combination of the two, e.g. selected among a given family as being the fittest towards achieving a certain purpose)

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There is a need for games that are valuable as benchmark and training environments, rather than entertainment or historical reasons

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There is a number of problems (think creativity, security, common sense) whose solutions are best understood to arise as solutions from some games

Designing a space of games for LLMs

Quite obviously, the most natural choiceo for games for LLMs are text-based games

What is a text-based game?

It is just a process with pre-defined rules that elicits moves from various players and awards them; players can be humans or LLMs or some other programs (e.g. a game solver)

For performance and reproducibility reasons, the games need to be able to run without humans in the loop, though it can be fun and useful to involve humans in these games

Why is it hard to write games?

The main reason seems to be that there is no good way to explore the space of games: there are so many possible choices around the scores, the possible algorithms that the environment should run, the move timing, the access to the information, the number of agents and their capabilities

It is easy to feel lost in that space, even if one restricts oneself to turn-by-turn text games, which seems to be quite appropriate to start working with LLMs

That's the first difficulty, and then there is the difficulty of spelling out what we expect from the games and how to measure one is making progress towards that goal

In the xent game paper, we propose an interesting mathematical definition of usefulness: the goal of a game is to be playable and to bring asymmetric transfer value to an existing corpus of games (labeled as 'interesting'), i.e. it should bring in new skills, while being relevant to pre-existing skills

We postulate that this choice of an interesting corpus of playable (i.e. learnable) games is in some sense some kind of axiom, in that it cannot be reduced to anything else (the upcoming "is-agi-unique" note is about this)

A canonical meta-game framework

What implementing this framework suggests

This framework suggests two kinds of ideas:

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Have a design that allows for automatic game generation (by e.g. genetic algorithms): this means in particular

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Have a reward structure that gives detailed score computation and partial credit as much as possible

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Try bypassing as much as possible the typical problems (semantic correctness, infinite loops, various errors) that can plague automatically-generated programs

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Focus on composability of game logic: make it easy to make games by putting together various subgames

These points lead one to focusing on a number of simple independent primitives, which lead to three axioms (composition, zero-sum stability, adversarial rescalings) from which to design the space of xent games

How the xent in xent games emerges

LLMs are already trained by playing a central game: predicting the next token; for which they get compensated (or rather, punished) in cross-entropy (xent) loss; this has led to us using the cross-entropy as a natural unit of reward for the xent games

What does one expect from games, concretely?

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Direct relevance to tasks that we care about, that LLMs are maybe not very good at yet

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Some good properties in terms of general usefulness: good games should teach us to play other good games (as recursive as it sounds)

Below, I will describe how answering the question of game usefulness via transfer led to a theoretical framework where xent games come in naturally

This in turn has led one to a simple game (the first xent game): given a text

s

, find a prefix

t

such that

xent(s|t)

is minimal

If we ask the question: if one starts from that example and asks what comes if we play the moves of the meta-game (the three axioms), the answer is simply that we get all the xent games

Another way to phrase the axioms is by saying that there should be three allowed types of 'moves' allowed in the space of games (where a 'move' is from one game to another game): these are are the moves of the meta-game

The cute name for this is "the Life of Games", an homage to the celebrated "Game of Life" which popularized the field of artificial life to the general public; George Walker Jr came up with the "Yudkowskian Nightmare" nickname, which is a little catchier at the end of this first quarter of century

This can then be iterated: new games bring new games, bring new models, which bring new games, and new models, and so on and so forth

And the miracle is that there are so many interesting games in that space!

... but this will be the topic of a future note