Online Best-Response Algorithm in Open Noncooperative Games
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Bibliographic record
Abstract
This paper considers online learning for open non-cooperative games where players can join and leave the system freely, while the current number of players in the system and the opponents' identities are not available. Unlike existing works on closed non-cooperative games that assume a fixed number of players, the open scenario setting is characterized by time-varying payment functions as well as a time-varying number of players. We present an online learning mechanism based on the best-response algorithm that enables players to adaptively adjust their strategies in the open game with anonymous opponents, thereby minimizing their own payments. We first provide an upper bound on the adaptive dynamic regret of the algorithm, which measures each player's regret over the time interval from joining to leaving the system. Then, we prove that the open game system is open stable with a stability radius <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$R$</tex-math></inline-formula>, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$R$</tex-math></inline-formula> depends on the time-variation of the equilibrium trajectory as well as on the ratios of newly joined and departing players to the number of active players. Finally, we demonstrate the algorithm performance through numerical simulations on an open Cournot game.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.007 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.003 | 0.006 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.004 | 0.000 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.006 | 0.002 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it