Nash Equilibria for Exchangeable Team-Against-Team Games, Their Mean-Field Limit, and the Role of Common Randomness
Bibliographic record
Abstract
Abstract. We study stochastic exchangeable games among a finite number of teams consisting of a large but finite number of decision makers as well as their mean-field limit with infinite number of decision makers in each team. For this class of games within static and dynamic settings, we introduce sets of randomized policies under various decentralized information structures with privately independent or common randomness for decision makers within each team. (i) For a general class of exchangeable stochastic games with a finite number of decision makers, we first establish the existence of a Nash equilibrium under randomized policies (with common randomness) within each team that are exchangeable (but not necessarily symmetric, i.e., identical) among decision makers within each team. (ii) As the number of decision makers within each team goes to infinity (that is, for the mean-field limit game among teams), we show that a Nash equilibrium exists under randomized policies within each team that are independently randomized and symmetric among decision makers within each team (that is, there is no common randomness). (iii) Finally, we establish that a Nash equilibrium for a class of mean-field games among teams under independently randomized symmetric policies constitutes an approximate Nash equilibrium for the corresponding prelimit (exchangeable) game among teams with finite but large numbers of decision makers. (iv) We thus establish a rigorous connection between agent-based-modeling and team-against-team games, via the representative agents defining the game played in equilibrium, and we furthermore show that common randomness is not necessary for large team-against-team games, unlike the case with small-sized ones.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.004 | 0.018 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.005 |
| Scholarly communication | 0.004 | 0.007 |
| Open science | 0.003 | 0.003 |
| Research integrity | 0.004 | 0.003 |
| Insufficient payload (model declined to judge) | 0.007 | 0.001 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".