Stackelberg Game on the Boundary of Coexistence
Bibliographic record
Abstract
This paper combines Convex analysis and Game theory to investigate the problem of maximizing the sum rate of transceivers operating in the same frequency band. In our earlier work, we proved that for transceivers operating under a total power constraint, the power distribution that maximizes the sum rate lies on the boundary of the feasible set formed by the power constraint. In this paper, we first prove that for two users, the sum rate is convex on the boundary formed by the line segment representing the power constraint, and the maximum sum rate is achieved when all the power is allocated to one of the users. Obviously, such a power allocation is unfair to the other user. We consider a scenario in which the first user is willing to sell some of the power allocated to him to the second user. We use Stackelberg Game theory to analyze this scenario and prove the existence of a unique competitive equilibrium. We derive the best response functions, and determine the optimum price the first user must charge and the optimum amount of power the second user should buy at this price. We also use simulation to obtain the best response functions and the equilibrium point, and demonstrate their agreement with our analytical results.
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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 | 0.003 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.003 | 0.006 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 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".