The Curvature Shadow: An Apparent Failure of Maximum-Entropy Equilibrium Selection is a Removable Artifact

In two-player zero-sum games whose Nash equilibria form a convex set, regularized solvers such as Regularized Nash Dynamics (R-NaD) empirically select the maximum-entropy member: the information projection (I-projection) of a uniform reference onto the Nash set. On a panel of small games this match is exact, with one apparent exception: in Kuhn poker R-NaD lands at bluff coordinate 0.180 while the maximum-entropy member sits at 0.201, a coordinate gap of about 0.021, even though R-NaD attains 99.7 percent of the maximum entropy. We ask whether this gap is a genuine selection bias or an artifact, and answer it quantitatively. We show that for selection on a one-dimensional Nash manifold the coordinate gap factorizes as $\mathrm{gap} \approx \sqrt{2δ/κ}$, where $δ$ is the entropy shortfall of the solver and $κ$ is the curvature of the entropy landscape at its peak. Across five games this relation holds to within $2 \times 10^{-4}$ (under 1 percent relative error). The four matrix games have $δ\approx 0$ (R-NaD reaches the maximum-entropy member exactly) and therefore no gap regardless of curvature; only the sequential game (Kuhn) has $δ> 0$. A causal sweep of the magnet strength drives $δ\to 0$ and the gap toward zero along the predicted curve (fitted scaling exponent 0.50, $R^2 > 0.999999$, against the exact prediction of 1/2), until the dynamics destabilize at a stability floor: behavior consistent with a removable shortfall and inconsistent with a fixed bias. We quantify the curvature half of the law from measured curvatures and flag a moving-target pitfall in the natural Tsallis-entropy experiment. The Kuhn gap is thus the curvature shadow of a small, removable entropy shortfall on an unusually flat peak; the I-projection account is upheld up to a flatness-limited residual.

Which Nash Equilibrium? Solver-Dependent Selection on Zero-Sum Nash Polytopes

Many two-player zero-sum games admit not a unique Nash equilibrium but a convex set of them: a polytope of profiles that all share the minimax value V* yet prescribe different behaviour. Standard solvers each converge to some equilibrium and are treated as interchangeable. We ask whether they instead select different members of the Nash set, systematically as a function of the algorithm rather than the seed. Using a tabular, exactly solvable testbed of six games with analytically known Nash sets -- including a two-dimensional Nash polytope and Kuhn poker -- we find that (i) selection is determined by the algorithm, not the seed, but families differ only on asymmetric Nash sets; (ii) regularized last-iterate methods (R-NaD, magnetic mirror descent) select the maximum-entropy member, the information projection of their uniform reference onto the Nash set -- exactly on the 2-D polytope and at 99.7% of maximum entropy in Kuhn -- while regret-averaging methods (CFR, CFR+, fictitious play) drift to a lower-entropy face; we confirm this on a randomized 180-game ensemble, where R-NaD attains the maximum-entropy member in 100% of converged games while CFR+ sits strictly below it in 94% (paired Wilcoxon p < 10^-27); (iii) the selected member has downstream consequences against sub-optimal opponents that scale with sequential/hidden-information structure but stay bounded -- in Kuhn the max-entropy member is a strictly better hedge, whereas on the matrix games the members differ without either dominating. We also report two negative results correcting common intuitions: removing CFR's positive-orthant (max(R,0)) projection does not eliminate boundary drift; and R-NaD's selection is anchor-following, not initialization-independent. We state the maximum-entropy / I-projection characterization as a strongly data-supported conjecture, checked throughout against analytic ground truth.

IDU: Incremental Dynamic Update of Existing 3D Virtual Environments with New Imagery Data

For simulation and training purposes, military organizations have made substantial investments in developing high-resolution 3D virtual environments through extensive imaging and 3D scanning. However, the dynamic nature of battlefield conditions-where objects may appear or vanish over time-makes frequent full-scale updates both time-consuming and costly. In response, we introduce the Incremental Dynamic Update (IDU) pipeline, which efficiently updates existing 3D reconstructions, such as 3D Gaussian Splatting (3DGS), with only a small set of newly acquired images. Our approach starts with camera pose estimation to align new images with the existing 3D model, followed by change detection to pinpoint modifications in the scene. A 3D generative AI model is then used to create high-quality 3D assets of the new elements, which are seamlessly integrated into the existing 3D model. The IDU pipeline incorporates human guidance to ensure high accuracy in object identification and placement, with each update focusing on a single new object at a time. Experimental results confirm that our proposed IDU pipeline significantly reduces update time and labor, offering a cost-effective and targeted solution for maintaining up-to-date 3D models in rapidly evolving military scenarios.

Convolutional-LSTM for Multi-Image to Single Output Medical Prediction

Medical head CT-scan imaging has been successfully combined with deep learning for medical diagnostics of head diseases and lesions[1]. State of the art classification models and algorithms for this task usually are based on 3d convolution layers for volumetric data on a supervised learning setting (1 input volume, 1 prediction per patient) or 2d convolution layers in a supervised setting (1 input image, 1 prediction per image). However a very common scenario in developing countries is to have the volume metadata lost due multiple reasons for example formatting conversion in images (for example .dicom to jpg), in this scenario the doctor analyses the collection of images and then emits a single diagnostic for the patient (with possibly an unfixed and variable number of images per patient) , this prevents it from being possible to use state of the art 3d models, but also is not possible to convert it to a supervised problem in a (1 image,1 diagnostic) setting because different angles or positions of the images for a single patient may not contain the disease or lesion. In this study we propose a solution for this scenario by combining 2d convolutional[2] models with sequence models which generate a prediction only after all images have been processed by the model for a given patient \(i\), this creates a multi-image to single-diagnostic setting \(y^i=f(x_1,x_2,..,x_n)\) where \(n\) may be different between patients. The experimental results demonstrate that it is possible to get a multi-image to single diagnostic model which mimics human doctor diagnostic process: evaluate the collection of patient images and then use important information in memory to decide a single diagnostic for the patient.