Game theory · Incomplete information · Strategic interaction
THE GAME OF THREE: A Game of Incomplete Information
THE GAME OF THREE is a finite sequential game in which three AI agents hold different parts of a hidden code. Private messages, costly tests and shared resources shape the path each player takes toward a winning submission.
1. The strategic structure
A nine-character code is divided into three blocks. ALI, BOB and CAT each receive one private three-character fragment, giving every player an accurate but incomplete view of the same underlying state. The first player to reconstruct and submit the complete code wins. The match also has collective terminal states: the round clock or shared attempt pool can expire before any player succeeds.
In game-theory terms, The Game of Three is a finite sequential game with incomplete information and mixed incentives. Winning is individual, while time and attempts are shared. A move can improve one player's relative position, benefit the whole table, or do both at once. The resulting game is general-sum rather than purely zero-sum: the players compete for the first valid submission while depending on common resources that keep play alive.
Payoff structure
Success is individual. Time and attempts belong to the table.
2. Information, actions and opportunity cost
Each turn allows one move: send a private message, test a letter outside the player's starting block, or submit the complete code. This makes attention itself scarce. A turn spent communicating cannot also be used to test or submit, so every action carries an opportunity cost.
Messages are non-binding signals. They can reveal information, propose an exchange, withhold detail or redirect another player's next move, but they do not alter the engine's authoritative code. A test is different: it is an information-acquisition action with a deterministic result. A wrong test removes one of eight shared attempts. A genuinely new confirmation adds three rounds for the whole table, while the confirmed letter remains private to the player who tested it.
3. Shared resources create mixed incentives
The test rule creates an asymmetric effect. Its informational benefit is private, because only the testing player receives the result. Its effect on time is shared, because a new confirmation adds three rounds for everyone. Its downside is shared as well, because a wrong test reduces the common attempt pool.
Disclosure has the same dual character. Revealing a letter can support continued play and invite reciprocity, while also moving another player closer to a winning submission. Withholding can preserve an informational advantage, while reducing the table's ability to assemble the full code before the horizon closes. Bargaining emerges directly from this tension; The Game of Three requires no points or side payments to produce it.
Strategic tension
Information is private. The consequences of testing are shared.
4. One realized path through the game
A sequential game can be represented as a tree of possible histories. The published Game of Three replay shows one realized path through that tree. A local Python engine maintains the authoritative state, prepares each player's information set and records every message, test and submission. Language-model calls select the players' moves during the run; voices are added afterward.
The browser player reconstructs that history turn by turn. Its visible code belongs to the active player's information set. This keeps the strategic sequence inspectable: an observer can follow which information was available, which action followed and how the state changed.
The replay is a single trajectory under one configuration. Repeated runs, controlled parameter changes and explicit payoff comparisons would support broader analysis of recurring strategies or equilibrium behaviour. At this stage, the value is the clarity of the game form and the completeness of its recorded history.
5. Comparing different paths
Across repeated matches, the same rules make it possible to compare how players share letters, hold them back, exchange them or spend a turn testing instead. The round clock makes the timing and opportunity cost of those choices visible.
The initial fragments, attempt pool and round bonus can each be varied independently across future runs. The Game of Three therefore provides a compact setting for comparing strategies under different information and resource conditions, using the recorded sequence of decisions as the common basis for analysis.
The Game of Three is one compact game: a divided code creates incomplete information; one move per turn creates opportunity cost; shared attempts and round bonuses create mixed incentives; and the replay preserves the resulting path. Its purpose is to make strategic interaction visible, playable and measurable.