Signaling Games for Log-Concave Distributions: Number of Bins and\n Properties of Equilibria
Bibliographic record
Abstract
We investigate the equilibrium behavior for the decentralized cheap talk\nproblem for real random variables and quadratic cost criteria in which an\nencoder and a decoder have misaligned objective functions. In prior work, it\nhas been shown that the number of bins in any equilibrium has to be countable,\ngeneralizing a classical result due to Crawford and Sobel who considered\nsources with density supported on $[0,1]$. In this paper, we first refine this\nresult in the context of log-concave sources. For sources with two-sided\nunbounded support, we prove that, for any finite number of bins, there exists a\nunique equilibrium. In contrast, for sources with semi-unbounded support, there\nmay be a finite upper bound on the number of bins in equilibrium depending on\ncertain conditions stated explicitly. Moreover, we prove that for log-concave\nsources, the expected costs of the encoder and the decoder in equilibrium\ndecrease as the number of bins increases. Furthermore, for strictly log-concave\nsources with two-sided unbounded support, we prove convergence to the unique\nequilibrium under best response dynamics which starts with a given number of\nbins, making a connection with the classical theory of optimal quantization and\nconvergence results of Lloyd's method. In addition, we consider more general\nsources which satisfy certain assumptions on the tail(s) of the distribution\nand we show that there exist equilibria with infinitely many bins for sources\nwith two-sided unbounded support. Further explicit characterizations are\nprovided for sources with exponential, Gaussian, and compactly-supported\nprobability distributions.\n
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How this classification was reachedexpand
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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 teacher head, 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".