PAPER: MODELS OF PEN-PL GAUSSIAN GAMES IN NON-COOPERATIVE COMMUNICATION

I originally wrote this paper as a way to practice TikZ. It was also meant to document the amusing memories of Kangpeng Xu (Player X) and Minglang Hu (Player H) flicking pens during our middle school breaks.

PDF DRAFT

Abstract

Consider non-cooperative pen games where both players act strategically and heavily influence each other. In spam and malware detection, players exploit randomization to obfuscate malicious data and increase their chances of evading detection at test time. The result shows Pen-PL Games have a probability distribution that approximates a Gaussian distribution according to some probability distribution defined over the respective strategy set. With quadratic cost functions and multivariate Gaussian processes, evolving according to first order autoregressive models, we show that Pen-PL "smooth" curve signaling rules are optimal. Finally, we show that computing a socially optimal Pen-PL network placement is NP-hard and that this result holds for all P-PL-G distributions.

As the author, I am delighted to elaborate with pride and a touch of slyness on the ingenuity behind this research. My objective was to construct a model that is formally rigorous yet contentiously absurd, using the most hardcore mathematical tools to deconstruct that moment of playful chaos that occurs during a school break.

1. Abstract and Introduction: Constructing the "Idealized" Arena

First, in the introduction, instead of immediately mentioning that we are studying a pen-flicking game, I framed the scenario as a sophisticated "Non-cooperative Facility Location" problem. The key ingenuity here lies in the conceptual substitution.

Consider the essence of the Pen-PL game: one player attempts to occupy favorable positions on the desk while simultaneously pushing the opponent out. Isn't this precisely a classic competitive facility location model? I deliberately introduced the concept of a "Bounded Continuous Arena," transforming the classroom desk into a mathematical Euclidean space. Even more intriguingly, I proposed a transition from the continuous domain to a discrete graph structure. In reality, although the pen's resting place is theoretically continuous, for the purpose of determining the winner, we often reduce it to a few "key points" or "regions."

One of my most prized assertions is the derivation that computing a socially optimal Pen-PL network placement is NP-hard. This powerful statement imbues the game with a computational intractability, suggesting why the game is always full of variables and lacks a universally winning formula—solving it globally in polynomial time is computationally infeasible! By establishing this, I paved the way for introducing complex approximation algorithms later on.

2. Gauss Models and System Analysis: Mathematical Modeling of "Hand Tremor"

The most fascinating part is where I translate physical action into statistical probability. I observed that when a player attempts to flick the pen to a specific location, there is always error. This error is not entirely random; it conforms to a specific distribution.

Thus, I introduced the Multivariate Gaussian Distribution (MVN). This wasn't just showmanship; the contours of equal probability density—the ellipses—perfectly mirror the scatter pattern of the pen tip upon landing. When players flick along the pen's axis, the error primarily concentrates along the force vector (the major axis) and the angular deviation (the minor axis).

I employed Eigendecomposition to describe this scatter. The covariance matrix $\Sigma$ is decomposed into a rotation matrix $U$ and an eigenvalue matrix $\Lambda$. The ingenuity here is that $U$ effectively represents the direction the player is aiming, while the elements of $\Lambda$ represent the precision of the player's control (or, essentially, the degree of "hand tremor"). By introducing the Mahalanobis Distance, I defined a scientific metric for quantifying "mistake"—it's no longer a subjective feeling of being "off-target," but an objective measure of how far the landing point deviates from the target center in a statistical sense. This step completely quantified subjective "feel" into objective data.

3. Data Distribution and the Absurd Marriage with Neural Networks

In the third section, I made a bold cross-disciplinary leap, which is perhaps the most ironic piece of ingenuity in the entire paper. I correlated the data distribution generated by the Pen-PL game with the metrics used in Neural Network Pruning.

Why this connection? The Pen-PL game is inherently a process of sparsity: two pens occupy a tiny fraction of the vast desk surface, yet they determine the governance of the entire area. Isn't this precisely what sparse neural networks aim to achieve? Using the minimum number of parameters (pens) to cover the largest feature space (the desk).

I presented comparative data, contrasting my "Pen-PL strategy" results against models like ReNet0 and APM, evaluated under parameters analogous to weight pruning ratios. The findings (as suggested by the table) were startling: my Pen-PL strategy, even with extremely low fixed "weights" (i.e., preset rules), maintained a respectable accuracy on complex classification tasks. The underlying suggestion is that sometimes, a simplistic, intuition-driven strategy (like flicking a pen) can be more effective and robust when dealing with highly redundant systems (like massively over-parameterized neural networks) than overly complex optimization algorithms. I packaged this simple physical game as an advanced algorithm possessing a "smooth curve signaling rule," a playful critique of the current trend of blindly increasing algorithmic complexity.

4. Dynamical Paths and the "Confusion Matrix" of Differential Equations

Next, when describing the "Avoidance and Attack Path," I deployed that dizzying Differential Equation (Equation 4).

$\frac{dP_L}{dt} = \frac{\alpha^2 P'_e \cos^2 \frac{\theta_e}{2}}{4t^3_e \sin^4 \frac{\theta_e}{2}} \dots$

Allow me to confess: this equation is my carefully crafted red herring. It combines trigonometric functions, derivatives, and physical parameters, looking exactly like a description of complex fluid dynamics or celestial mechanics. Yet, it simulates the dynamic psychological conflict during the game—if you attack, I want to attack back; if you evade, I want to chase. The $\cos$ and $\sin$ terms represent the angular changes in the pen's rotation affecting the final trajectory. I used this extremely complex formula to describe an utterly intuitive physical phenomenon, creating deliberate absurdity: we often use highly complex theories to explain what are fundamentally innate intuitions.

5. Optimization and Non-Cooperative Strategies: From Linear Programming to Nash Equilibrium

In sections four and five, I elevated the game to the realms of Operations Research and Game Theory.

For the optimization segment, I employed Linear Programming and Nonlinear Programming. I defined the energy required for each flick as a "cost" $w_i$, and the act of striking an opponent or capturing territory as a "profit" $p_i$. The constraint $\sum w_i x_i \le C$ appears to be a budget constraint, but it actually corresponds to the physical boundaries of the desk or the limit of the player’s finger strength in reality. This demonstrates that any game, even a childish one, is fundamentally a constrained optimization problem.

Subsequently, I explored the Nash Equilibrium. In the pen game, if both players behave conservatively (cooperate), they might both stay on the table resulting in a draw; but if one defects (flicks aggressively), they might win outright or they both might lose everything. By introducing Mixed Strategies via the expected value formula $E = \sum p_i x_i$, I quantified this dynamic of mistrust. The iterative update of the parameter $\theta_j$ actually models how human players learn each other's patterns over several rounds. I formalized this learning process mathematically, suggesting that the mechanics of human intuition share surprising similarities with machine learning's gradient descent.

6. Data Comparison and the Harsh Reality

Finally, in the data analysis section, the figures reveal a slightly harsh but undeniably true conclusion.

I compared the "Cooperative Strategies" against the "Non-cooperative P-PL-G Strategies." The results clearly showed that while the Total Profit might remain similar, under the non-cooperative setting, Player X achieves a significantly higher profit at the direct expense of Player H.

The ingenuity here lies in illustrating the core of zero-sum competition in the real world. In the Pen-PL game, the most aggressive player, the one least willing to compromise (adopting my "Non-cooperative Optimization Strategy"), tends to gain the most. Although we often say "friendship first, competition second," my mathematical model coldly points out: in the absence of enforceable constraints, rational individuals will necessarily gravitate towards breaking cooperation to maximize their self-interest. That conspicuously tall red bar (Player X's profit) in the chart is not just a triumph of data visualization; it is a precise mathematical sketch of the innate human tendency toward self-preservation and gain.

In conclusion, every equation and every chart in this paper is a profound contemplation and rational deconstruction of the smallest interactions in our world. I used the most serious academic façade to tell the most enjoyable joke.