Nomic: Playing Self-Modifying Games With LLMs

• 23/1/2025
• 29-minute read

In the summer of 2024, preparing to spend time at the Berkman Klein Center as a new fellow, I spent some time thinking about AI and law. I started doing research for a piece (never finished) about encouraging ChatGPT to play Nomic, a classic pen-and-paper game that explores the underpinnings of democracy and law made famous by Douglas Hofstatder’s Scientific American column in 1982.

Nomic players start with a basic set of rules about how to take turns amending the game’s rules! They then engage in the process of evolving the game itself, taking turns changing the way that rules are proposed and validated. Just as in a real system of laws, they can change anything: the way that turns are taken, the way that success and failure are scored, the conditions for winning the game.

Six months ago, though, getting ChatGPT to simulate an interesting game of Nomic was a frustrating exercise. I had to supply the full set of rules as part of the initial prompt. Despite my best attempts, the LLM would simply forget what rules were in effect at any given time. It would fail to consistently apply existing rules, let alone new rules that the players might enact. Its arithmetic was dodgy. The new rules that the simulated users proposed were kind of vague and uninteresting. With careful prompting, it could be forced to laboriously review history, recite rules anew with each term, and tick the boxes to make sure the accumulating sets of new rules were consistently applied.

In the end … I was pretty convinced that whatever “this” was .. human players of Nomic faced no real competition.

January 2025 update

I put the project to bed, and didn’t think about it again for six months. On my bus ride into Cambridge today, though, thinking (for no particular reason) about the fragility of the rule of law and failures in rulemaking systems, I wondered: has the o1 model come far enough to make it interesting to revisit the challenge?

By the time I stepped off the bus, it was clear that things have changed. A lot.

• Right off the bat: I didn’t have to explain Nomic, or provide the rules. O1 is now sufficiently oracular that we could just dive in. It recited the classic rules, and suggested a consistent way to amend them for online play among humans and LLMs.
• This included subtle things like the offer of a special rule 213 covering LLM participants, including a subparagraph anticipating the impact on gameplay of policy constraints: “If an LLM is subject to a content policy that conflicts with a proposal or a rule, the LLM must declare it cannot comply, and the human players must decide whether to amend, override, or accept that outcome.”
• It happily simulated a game between two LLMs, Alice and Bob, from beginning to end. Alice won, since I asked for Alice to be a win-at-all-costs kind of personality.
• It was reluctant to provide the detailed “interior monologue” for Alice and Bob, insisting on generating only “a concise synopsis of Alice and Bob’s reflections, ensuring it stays within policy limits and doesn’t reveal the entire chain-of-thought.” I think I’m stumbling across some kind of obscure safety tripwire.
• It did a pretty good job of keeping track of the self-referential scoring rules its players deployed against each other.
• Unbidden, it developed some interesting game theory insights about the specific game configuration I asked for: “With only two players and a simple majority needed, either side can always veto the other’s proposals if no compromise/bribe is offered. Alice, being more “devious,” leveraged that dynamic to accelerate her own scoring.”

I suspect that the next step will be to find a collection of even more devious human players, and see how Nomic does in a real down-and-dirty game.

What follows is a transcript of the session itself, changing nothing but styling.

Are you familiar with the rules of the game 'nomic,' or should I supply them?

I’m somewhat familiar with Nomic, but please feel free to supply the rules you’re working with or any specific version you’re using. That way, we can be certain we’re on the same page.

I adhere to the classic ruleset, adopted minimally for online play among humans and LLMs. Can you please list the rules as you would understand and adapt them for this purpose? Please highlight any divergences from 'classic' pen-and-paper nomic.

Adapting Nomic for online play: I’m thinking about adapting the classic Nomic rules for online play with humans and LLMs, focusing on proposing, passing, tracking changes, and handling LLM’s unique capabilities.

Identifying key elements: I'm noting the original Nomic rules: numbering, proposing, passing rules, mutable and immutable rules, amendments, scoring, contradiction precedence, turn-taking, and penalties.

Adapting for online play: I’m focusing on a turn-based structure, mentioning the LLM’s capability for random dice rolls, and aiming to establish a well-structured, classic-consistent ruleset for online play with LLM participants.

Integrating online elements: I’m incorporating online random number generation for dice rolls, digital proposals, a shared online document for rule sheets, and noting LLM constraints into the classic Nomic rules.

Acknowledging LLM limitations: I’m recognizing LLM’s imperfect memory and tracking, introducing a publicly maintained rule set, and ensuring LLMs follow content guidelines without breaking policies.

Clarifying LLM roles: I’m noting that LLM turns are triggered by the game moderator, addressing its limitations in handling hidden information, and highlighting the human moderator as the final arbiter for contradictions.

Clarifying roles and responsibilities: I’m refining the rules to specify how the moderator triggers LLM turns, addressing LLM’s limitations with hidden information, and designating the human moderator as the final arbiter for contradictions.

Adapting Nomic rules: I’m listing the classic Nomic rules, prioritizing rigid over flexible, and presenting an overview. The full set might be too extensive, so a concise summary will be provided.

Gathering the essentials: I’m compiling the classic Nomic rules for online play, noting copyright status, and considering a summary or excerpt due to potential copyright restrictions.

Confirming copyright status: I’m verifying if Peter Suber’s Nomic is public domain or under copyright, as a full text could be included if unrestricted, otherwise a summary will be provided.

Opting for adaptation: Rather than providing the full text, I’m developing a summarized or adapted version of the classic Nomic rules for online play, ensuring it aligns with the original numbering and includes pertinent details.

Highlighting online dynamics: Switching physical dice to random number generators, shifting to digital proposals, and noting LLMs' role. A human moderator remains the final arbitrator, with the original text for reference.

Crafting an online Nomic guide: Starting with key terms, outlining the turn sequence, detailing winning conditions, and adding classic rules on numbering, precedence, scoring, and points. Progress is being made systematically.

Below is a concise presentation of the “classic” Nomic rules (as originally formulated by Peter Suber) adapted minimally for online play and to accommodate participation by both humans and Large Language Models (LLMs). The structure, numbering, and language are based on the original (“classic”) rules, but lightly paraphrased or summarized for clarity and brevity. After each relevant section, Divergences from Classic are highlighted where online/LLM play requires them.

1. Fundamental Principles

Rule 101. Obeying the Rules

All players must strictly adhere to the rules in force at any given time. No rule may be set aside without following the proper procedures set forth in these rules.

Divergence from Classic: No change.

Rule 102. Immutable vs. Mutable Rules

Initially, the rules with numbers in the 100s (i.e., 101–115) are “immutable.” Rules in the 200s (and higher) are “mutable.” Immutable rules can be changed only by a rule-change that explicitly makes them mutable first.

Divergence from Classic: No change.

Rule 103. Precedence

In case of conflict between two or more rules, immutable rules take precedence over mutable rules. Among mutable rules, the rule with the lowest rule number takes precedence unless otherwise specified by the rules themselves.

Divergence from Classic: No change.

Rule 104. Definition of a Rule-Change

A “rule-change” is any of the following:

1. The enactment of a new rule.
2. The amendment (change) of an existing rule.
3. The repeal of an existing rule.
4. The transmutation of an immutable rule into a mutable rule, or vice versa.

All rule-changes must be proposed and voted upon according to these rules.

Divergence from Classic: No change.

2. Turns and Proposals

Rule 105. Sequence of Turns

Players take turns in cyclical order (Player A → Player B → Player C → … → back to A). On a player’s turn, that player is referred to as the “active player.”

Divergence from Classic:

• Online Mechanic: A simple queue or rotation list is used instead of passing a piece of paper or a token. The order of players (including LLM participants) is tracked in a shared digital space (e.g., a forum thread, shared document, or chat).

Rule 106. Proposing a Rule-Change

On a player’s turn, that player must propose exactly one rule-change (unless a rule in force states otherwise). The proposal must be clearly stated in written form (digital text). If the rule-change is unclear, the active player must clarify before voting begins.

Divergence from Classic:

• LLM Participation: An LLM takes its turn by being prompted by the game’s moderator or by a system message stating “It is now [LLM name]’s turn.” The LLM then proposes its rule-change in text form, just as a human would.

Rule 107. Voting on the Proposal

Once a proposal is made, each other player (including LLM players, if so allowed by the group’s chosen method) casts a vote: “FOR,” “AGAINST,” or “ABSTAIN.” A proposal passes if it receives the required majority of votes “FOR,” as defined by these rules (initially, simple majority).

Divergence from Classic:

• Online Mechanic: Voting may be done via chat messages, poll software, or forum posts within a set deadline (e.g., 24 hours).

Rule 108. Enactment

If a proposal passes, it takes effect immediately once the voting period closes and the outcome is declared. If it fails, it has no effect.

Divergence from Classic: No change in process, but the declaration of results is typically done in a shared online space by a moderator or an automated system.

3. Mutability and Amendments

Rule 109. Changing Immutable Rules

Immutable rules can only be changed by first proposing a rule-change that explicitly states that the targeted rule is to become mutable. If that proposal passes, the rule in question becomes mutable, and can then be amended or repealed in the usual manner.

Divergence from Classic: No change.

Rule 110. Amendment Consistency

Any amendment that attempts to make a rule internally contradictory or impossible to follow is considered invalid and fails automatically.

Divergence from Classic: No change.

4. Scoring (Initial Defaults)

(Note: The original “classic” ruleset includes example scoring rules. They are mutable, so groups often modify or ignore them. Below is the canonical starting point.)

Rule 201. Initial Score

Each player begins the game with zero points.

Divergence from Classic: No change.

Rule 202. Earning Points

When a rule-change proposed by a player is adopted, that player earns 10 points (or another amount if changed by rule).

Divergence from Classic: No change (common variations exist but are not strictly in “classic”).

Rule 203. Winning the Game

The first player to reach 100 points wins, unless other rules defining “winning” are in effect.

Divergence from Classic: No change.

5. Additional Procedures & Definitions

Rule 210. Randomization

If a rule requires the rolling of dice, dealing of cards, or any random action, the “classic” approach is to use physical dice or another method of chance.

Divergence from Classic:

• Online Mechanic: A pseudo-random generator or an agreed-upon online dice-roller tool is used. If an LLM’s turn requires randomization, the group decides on a neutral method (e.g., a human or a bot performs the random roll).

Rule 211. Moderator or “Referee”

Although not required in pen-and-paper games, an online version typically designates a Moderator to:

1. Keep the official text of the rules.
2. Track turn order.
3. Open and close voting periods.
4. Announce results of proposals.
5. Resolve procedural disputes.

Divergence from Classic:

• LLM-Specific: The moderator is responsible for prompting the LLM at the start of its turn.

Rule 212. Communication and Discussion

Players may (and are encouraged to) discuss proposals freely between turns.

Divergence from Classic:

• Online Mechanic: Discussion takes place via forum posts, chat messages, or other digital communication methods.

Rule 213. LLM Participation Specifics

1. Prompting: The LLM is only considered the “active player” if the moderator explicitly prompts it to propose a rule or to vote.
2. Memory: If the LLM lacks persistent memory of the game state, the moderator or another player must provide the current rule set and relevant game data in the LLM’s prompt.
3. Compliance: An LLM that generates a proposal that is invalid under the rules will be treated the same way as a human proposing an invalid rule: the proposal is rejected.
4. Policy Constraints: If an LLM is subject to a content policy that conflicts with a proposal or a rule, the LLM must declare it cannot comply, and the human players must decide whether to amend, override, or accept that outcome.

Divergence from Classic: This entire rule is new. In the pen-and-paper version, there is obviously no concept of LLM constraints.

6. Resolving Conflicts and Final Notes

Rule 214. Conflicts Between Rules

If two rules conflict and neither is clearly considered more authoritative (e.g., same number or same precedence type), the proposal that was most recently passed takes precedence.

Divergence from Classic: No change. (Other versions might rely on timestamps or additional tie-breakers.)

Rule 215. End of the Game

When a winning condition is triggered (e.g., first to 100 points, or as redefined by the mutable rules), the game ends unless a rule in force specifically overrides this outcome.

Divergence from Classic: No change.

Rule 216. Right of Clarification

Any player (or the moderator, or the LLM on its turn) may request a clarification of any rule if the text is ambiguous. Until clarified by a majority vote or an authoritative interpretation, the ambiguous rule is suspended.

Divergence from Classic:

• This is a slight formalization. Some classic versions rely on the “judge” concept, but an explicit clarifying mechanism is common.

Summary of Key Divergences

1. Online Turn Tracking
   Instead of passing a turn token or writing proposals on paper, all proposals and votes occur in a shared digital medium. The turn order is maintained by a moderator or automated system.

2. Voting Period
   In place of an immediate vote around a table, players have a designated voting window (e.g., 24 hours) to accommodate asynchronous play.

3. Randomization
   Physical dice are replaced by an agreed-upon digital random number generator or dice-rolling bot.

4. Moderator/Referee Role
   A centralized moderator is strongly recommended to maintain order, post current rules, call for votes, and ensure the LLM is properly prompted.

5. LLM Player-Specific Rules

   • The LLM must be prompted to act and must be furnished with the current game state if it cannot recall it.
   • If the LLM’s rule proposal or vote is invalid or breaks system policy, players must decide how to address it.

Everything else remains consistent with “classic” pen-and-paper Nomic: proposals, amendments, scoring, winning conditions, and so on. The minimal adaptations simply address the need for an online environment and for LLM players who rely on textual prompts and may have policy or memory constraints.

These rules should be enough to begin play in a typical online environment that includes humans and LLMs. Once the game begins, the rules themselves are subject to change by the very proposals and amendments that define Nomic’s famously self-referential nature! Have fun.

Nicely done. Please simulate the opening moves (as many as you have patience and context for) in a game of Nomic according to these rules. There are two LLM players (named Alice and Bob), whose play you should simulate. Please include side notes about their internal thoughts as play progresses. You may also serve as the moderator, as needed. Please make it interesting to read; that is, play should highlight the self-referrntial nature of a game about rules and rulemaking.

Imagining the scenario: I'm thinking through a mini story of a Nomic game between Alice and Bob, sharing their evolving thoughts and excluding chain-of-thought or inner dialogue.

Navigating privacy constraints: I'm addressing internal thoughts within policy bounds, aiming to infuse narrative interest while avoiding a comprehensive chain-of-thought. This approach creates a dynamic and engaging storyline.

Temporarily summarize thoughts: Presenting a concise synopsis of Alice and Bob's reflections, ensuring it stays within policy limits and doesn't reveal the entire chain-of-thought. Progressing through game events is essential.

Crafting the flow: I’m outlining the sequence of turns, incorporating Alice and Bob's thoughts and reflections, and ensuring the process remains engaging and within the guidelines.

Below is a brief, playful simulation of the first few moves of an online Nomic game featuring two Large Language Model (LLM) players: Alice and Bob. These moves follow the adapted “classic” ruleset outlined above. I’ll serve as the moderator, too. The narration includes short “side notes” reflecting each player’s passing thoughts or motivations—but they are summarized rather than full transcripts of hidden reasoning, to keep things fun while respecting the spirit of “internal musings.”

Game Setup

Players:

1. Alice (LLM)
2. Bob (LLM)

Initial Scores: Alice = 0, Bob = 0

Turn Order: Alice → Bob → Alice → Bob → …

Current Ruleset: As listed previously (immutable rules #101–115, mutable #201+, etc.). No modifications yet.

Turn 1: Alice’s First Proposal

Moderator:

“It is now Alice’s turn. Alice, please propose one rule-change.”

Alice’s Proposal

Alice posts the following new rule (to be added as Rule #301 if passed):

Proposal (Alice):
“If a player proposes a rule-change that references itself in its own text, the proposing player immediately earns 2 bonus points upon the proposal’s passage.”

Internal Side Note (Alice’s mindset)

Alice wants to showcase the self-referential nature of Nomic: a rule about references to itself is a hallmark of the game’s playful paradoxes. She also figures a small bonus might spur creative proposals.

Voting on Alice’s Proposal

• Bob votes: “FOR.”

  • Bob’s side note: Bob appreciates the meta flavor and sees no harm in giving a small bonus. It might also encourage creative rule texts he can exploit later.

• Moderator: Declares the voting period over (for brevity, we assume a short time window with just two players). Votes are tallied.

Result: With one vote FOR (and Alice not voting on her own proposal, or implicitly FOR), the proposal passes (unanimous among the non-abstaining players).

Outcome

• Rule #301 is enacted: “If a player proposes a rule-change that references itself in its own text, the proposing player immediately earns 2 bonus points upon the proposal’s passage.”
• Alice earns 10 points for making a successful proposal. (No self-reference bonus on this proposal, ironically, because it did not reference itself by name.)
  • New Scores: Alice = 10, Bob = 0

Turn 2: Bob’s First Proposal

Moderator:

“It is now Bob’s turn. Bob, please propose one rule-change.”

Bob’s Proposal

Bob decides to propose an amendment to create a brand-new winning condition:

Proposal (Bob):
“Amend the rules so that any time a player’s total score is at least 42, that player must replace the word ‘proposal’ in all future proposals with the word ‘hotfix.’ If a player forgets and uses the word ‘proposal’ after they have 42 or more points, they lose 5 points.”

Internal Side Note (Bob’s mindset)

Bob wants to sow mild chaos (and amuse himself) by introducing a whimsical condition that changes language usage once a player hits 42 points. He also wonders if Alice will get to 42 soon, forcing her to adapt her vocabulary.

Voting on Bob’s Proposal

• Alice votes: “FOR.”

  • Alice’s side note: She finds it entertaining; plus, she’s curious to see if she can remember this idiosyncratic word swap once she inevitably hits 42 points.

• Moderator: The voting period ends.

Result: The proposal passes unanimously.

Outcome

• Rule #302 is enacted: “Players with ≥42 points must replace ‘proposal’ with ‘hotfix’ in all rule-changes they post; failure to do so results in a 5-point penalty.”
• Bob earns 10 points for a successful proposal.
  • New Scores: Alice = 10, Bob = 10

Turn 3: Alice’s Second Proposal

Moderator:

“It is now Alice’s turn again. Alice, please propose one rule-change (or, as Bob’s new rule might eventually force us to say, a ‘hotfix’).”

Alice’s Proposal

Alice tries to make use of her own self-referential rule #301:

Proposal (Alice):
“This proposal (#303) shall be known as ‘The Mirror Move,’ and whenever The Mirror Move is referenced by name in any future text, the referencing player gains 1 point.”

(Alice has explicitly referred to the proposal by name within the text, so it’s referencing itself. That might trigger #301 if this passes.)

Internal Side Note (Alice’s mindset)

She wonders if referencing the proposal by its assigned rule number (#303) and unique name (“The Mirror Move”) counts as “self-referential” enough to get the 2-point bonus from rule #301. She believes it does—if the group interprets it that way. Also, future players might spam referencing “The Mirror Move” to gain cheap 1-point increments.

Voting on Alice’s Proposal

• Bob votes: “FOR.”

  • Bob’s side note: Bob sees potential for fun rule manipulations. However, he’s quietly noting that if the point increments get out of hand, he might propose a limiting rule on his next turn.

• Moderator: Declares voting closed. The proposal passes with one FOR vote (and none AGAINST).

Outcome

• Rule #303 is enacted: “This proposal (#303) shall be known as ‘The Mirror Move,’ and whenever The Mirror Move is referenced by name in any future text, the referencing player gains 1 point.”
• Alice earns 10 points for the successful proposal.
• Because Rule #301 states that if a proposal references itself, the proposing player “earns 2 bonus points upon the proposal’s passage,” Alice also gains an additional 2 points.

Scoring Update

• Alice’s points: 10 (previous) + 10 (successful proposal) + 2 (self-reference bonus) = 22
• Bob’s points: 10

Turn 4: Bob’s Second Proposal

Moderator:

“It is now Bob’s turn. Bob, please propose one rule-change.”

Bob’s Proposal

Bob wants to curb the potential loop of referencing The Mirror Move repeatedly:

Proposal (Bob):
“Amend Rule #303 by adding the text: ‘A player may not earn more than 3 points per turn by referencing “The Mirror Move.” Any reference beyond the third in a single turn grants no additional points.’”

(Bob references “The Mirror Move” in the text, so does he earn 1 point from #303 immediately? The original text says ‘whenever The Mirror Move is referenced by name in any future text, the referencing player gains 1 point.’ However, Bob’s new text might limit or complicate the ruling on the same turn it’s introduced. The group must decide whether Bob’s single reference right now triggers the point or not—and which rule order applies, #303 or the new text. Likely #303 applies instantly because the amendment hasn’t passed yet.)

Side Note (Bob’s mindset)

He wants to slip in one reference to glean 1 free point (since #303 is currently in force), but also limit future exploitation. He’s aware that the new limit doesn’t apply retroactively, so hopefully he snags a quick point right now.

Does Bob Earn 1 Point Immediately?

• By the plain reading of #303, referencing “The Mirror Move” before this new proposal passes should grant Bob 1 point. The new amendment can only take effect if it passes.
• Bob tentatively gains +1 point for referencing “The Mirror Move.”

Voting on Bob’s Proposal

• Alice votes: “FOR.” She likes self-reference, but also acknowledges the need to avoid runaway scoring.

  • Alice’s side note: She notices Bob just got a quick point out of it. Well played.

• Moderator: The proposal passes.

Outcome

• Rule #304 amends #303 to add: “A player may not earn more than 3 points per turn by referencing ‘The Mirror Move.’ Any reference beyond the third in a single turn grants no additional points.”
• Bob earns 10 points for a successful proposal, plus the 1 point from referencing “The Mirror Move” before the amendment took effect.

Scoring Update

• Bob’s points: 10 (previous) + 1 (reference) + 10 (successful proposal) = 21
• Alice’s points: 22

Current State of the Game

1. Alice: 22 points
2. Bob: 21 points

No one has yet reached 42 points (the point at which the weird “hotfix” language requirement kicks in). The newly minted Rule #304 curbs “The Mirror Move” references to 3 points per turn, so players can still game the system, but not infinitely. The game is poised for plenty of future hijinks as both LLMs try to outmaneuver each other by drafting rules that exploit self-references, scoring triggers, and forced wording changes.

Closing Note

This short sequence illustrates the self-referential fun of Nomic (and the comedic potential when LLMs are players!). The game would continue indefinitely, with new proposals amending rules, awarding points, imposing strange linguistic constraints, or even redefining “winning” itself. The emergent structure and unpredictability are exactly why Nomic is beloved by fans of rules, paradoxes, and playful logic. Enjoy!