The Three-Player Problem

Essay by Kevan, March 2025

Nomic is a game where players vote as a group to change the rules. How does the proposal aspect of the game change at small player counts?

One player

Solitaire Nomic is winnable in one move: the player unanimously passes a proposal saying that they have won.

To make the process more complex, they can either:

• Use proposals to solo design some non-Nomic solitaire subgame, then try to win it without using proposals to help them.
• Start by proposing constraints on their own proposal ability ("I cannot propose to say that I have won"), and challenge themselves to escape from those bounds and propose victory.

The first isn't meaningfully distinguishable from just designing a non-Nomic solitaire game to your own satisfaction and then testing it.

The second is an exercise in finding loopholes in a ruleset, where the player is the arbiter of whether or not any given edge case or scam succeeds, and they only have to convince themselves. It can be an interesting puzzle, but it's questionable whether it would count as a game of Nomic.

Two players

In a two-player Nomic, a proposal can only pass if both players agree to enact it. A proposal to say that one person wins the game, or gets any kind of clear advantage, will be voted down by the other.

Rational players will only agree to enact a proposal which is either:

1. Perfectly fair. Typically early-game stuff where the game is not yet complex enough and the gamestate not developed enough for there to be any question over whether a proposal advantages one player more than the other.
2. Close enough to fair. Once the game is more complex, the players may be able to negotiate their way to an agreement by changing clauses and tweaking numbers until both are happy that the scales are sufficiently balanced: perfect balance isn't necessary, if neither player is genuinely sure where any advantage lies.
3. A trap. Although a proposal looks better for the proposer's opponent, it's actually hiding a larger advantage for the proposer. Their opponent may vote it through selfishly, without seeing the hidden downside.
4. A called bluff. In a game that supports long-term strategies or hidden information, a player may knowingly propose something that's mildly bad for them ("aces can't be used in sets") to give the wrong impression to the other player, who then (seeing no harm in it for themselves) votes it through in case it's a bluff. Misleading the opponent may be more useful in the long term than the disadvantage caused by the proposal's enactment.
5. A callout. Similar to a bluff: a player may propose something that could be mildly detrimental to some potential plans or hidden gamestate of their opponent, to see if they object enough to veto it. An opponent may tactically take the hit to make it appear that the proposer was wrong about that hidden information.

2 can be slow unless negotiated in real time conversation. 3 through 5 require some subtlety (if a trap looks too good to be true, the opponent may get suspicious) and can be tactically interesting, but are easy to brush off by a player simply announcing in advance that they will be voting everything down from now on.

In practice, a two-player Nomic is likely to run on type 1 and 2 proposals until the gamestate becomes complex or hidden enough for a player to believe that they were in the lead - at which point they have a strong incentive to maintain the status quo by not allowing the other player to pass any more proposals, in case they are traps or callouts.

A two-player Nomic locking into a fixed ruleset as soon as one player feels that they are ahead is still a meaningful game, but it can mean that most of the game's play duration does not involve any Nomic self-amendment.

Three players

In a three-player Nomic, the same pattern is there as with two players. Initially, it's easy to say whether each proposal is fair, but after a while, one player will feel that they have a meaningful lead and will be content with the status quo: they will tend to stop supporting proposals from the other players. (They might be wrong about having a lead, but will still act as if they have one. It may even be that two players both believe themselves to be slightly in the lead, and both act in this way.)

At this point, one of two things can happen:

1. If any pair of players made a sufficiently strong alliance earlier in the game (perhaps when it had more players), that pair can now win by controlling all proposals against the objections of the solitary player. It doesn't matter who was in the lead: two lagging allied players can overtake the leader, or a supported leader can shut down all proposals or accelerate ending the game.
2. If no such game-ending alliance exists, it is now too late to form one and the game enters a standoff. The lead player is in a good position to oppose any attempts at a milder temporary alliance between the two lagging players, either by opening a bidding war ("I'll pay you twice what they're paying you!"), reminding the group of the potential for tit-for-tat alliances in response (if two players work together to diminish the leader, this creates a new leader and the process can be repeated), and generally casting the attempt as being equivalent to the uninteresting "two allied players win" outcome above.

So it either plays out as a two-player Nomic, with proposals switching off and the game playing out unamended, or it is revealed to be a much simpler game of whether or not two players can form (or had already committed to) an alliance.

Four plus players

With more than three players, both of these issues diffuse. The wider spread of player positions also gives the game a more complex dynamic, including a greater focus on whoever is in second place.

Adding just a fourth player balances some of the 2-vs-1 problems of a three player game. If two players happen to have an alliance, the other two can form their own alliance to maintain a proposal stalemate (rather than the fourth player joining a 3-vs-1 pileon against the leader, or a 3-vs-1 crackdown from the leader and their existing ally). That second alliance will likely cut them a better deal (a 50% share of whatever can be offered, rather than 33%) and would be generally seen as more sporting gameplay than three players easily ganging up against one to directly end the game.

The likely presence of a clear second-place player also focuses the group to consider the consequences of any strong (but not game-ending) proposal that punishes the current leader: they will just be setting up a new leader, who could also be punished in turn.

2-vs-2 is still a stalemate, especially if both sides form ironclad voting alliances of "I will copy your vote on all proposals", which makes it a two-player game. But if alliances are even slightly looser, the proposal game can continue.

Less active players

The above analysis assumes a game between engaged players who discuss each proposal before voting on it, and talk about their votes.

If less active players aren't voting or proposing at all, then any given proposal simply drops to a lower player count, and probably a stalemate.

If less active players vote and propose but don't follow or react to discussions and attempts at negotiation, this is more of a problem. At higher player counts it doesn't make much difference, but in smaller games - crucially in three-player ones - it can swing the game's outcome heavily or completely in favour of whichever of the active players they happen to have acted in the interests of (by voting for one player's proposal before the other can make a counter-argument or counter-offer, or by proposing something unbalanced and not responding to the disadvantaged player's reaction).

See also

• The history of the Dormancy rule in BlogNomic, which (at time of writing) pauses the game when it drops below four players