Asymmetric Inekoalaty Game with Power‑Law Growth

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\title{Asymmetric Inekoalaty Game with Power‑Law Growth} \author{Researcher} \date{}

\begin{document}

\maketitle

\begin{abstract} We extend the asymmetric inekoalaty game by introducing power‑law growth in the right‑hand sides of the constraints. Alice (first player) must satisfy $\sum_{i=1}^n x_i^{q}\le\lambda n^{\alpha}$, while Bazza (second player) must satisfy $\sum_{i=1}^n x_i^{p}\le n^{\beta}$, where $p,q>0$ and $\alpha,\beta>0$. Using greedy strategies and numerical simulations we determine the critical parameter $\lambda_c(p,q,\alpha,\beta)$ that separates Bazza's winning region ($\lambda<\lambda_c$) from Alice's winning region ($\lambda>\lambda_c$). When $\alpha=\beta=1$ the draw interval known from the autonomous case reappears; for $\alpha\neq\beta$ or $\alpha=\beta\neq1$ the draw interval vanishes and a single sharp threshold emerges. We present extensive numerical data and propose scaling conjectures for $\lambda_c$ as a function of the growth exponents. \end{abstract}

\section{Introduction}

The inekoalaty game is a two‑player perfect‑information game that has attracted considerable attention in recent research [{rkrw},{zn8k}]. Its original version, solved in [{zn8k}], exhibits sharp thresholds at $\lambda=\sqrt2/2$ and $\lambda=1$. Several generalizations have been studied: replacing the quadratic constraint by an $L^p$ norm [{lunq}], swapping the constraints [{1sm0}], introducing independent scaling parameters [{knjh}], and allowing the right‑hand sides to grow as powers $n^{\alpha}$, $n^{\beta}$ [{6y2s},{b1xz}].

In the present work we combine the two most general directions: we consider arbitrary exponents $p,q$ in the players' constraints and, simultaneously, power‑law growth $n^{\alpha}$, $n^{\beta}$ on the right‑hand sides. The resulting game depends on five parameters $(p,q,\alpha,\beta,\lambda)$. Our aim is to understand how the critical value $\lambda_c$ that separates the two players' winning regions depends on the four structural parameters $p,q,\alpha,\beta$.

\section{The game}

Let $p,q>0$, $\alpha,\beta>0$ and $\lambda>0$. Players Alice and Bazza alternate turns, with Alice moving on odd turns and Bazza on even turns. On turn $n$ the moving player chooses a number $x_n\ge0$ satisfying [ \begin{cases} \displaystyle\sum_{i=1}^n x_i^{,q}\le \lambda n^{\alpha} & \text{if $n$ is odd (Alice),}\[4mm] \displaystyle\sum_{i=1}^n x_i^{,p}\le n^{\beta} & \text{if $n$ is even (Bazza).} \end{cases} ] If a player cannot choose a suitable $x_n$, the game ends and the opponent wins; if the game never ends, neither wins. All previous choices are known to both players.

The original game corresponds to $(p,q,\alpha,\beta)=(2,1,1,1)$; the symmetric $L^p$ generalization [{lunq}] corresponds to $(p,q,\alpha,\beta)=(p,1,1,1)$; the power‑law growth model [{6y2s}] corresponds to $(p,q,\alpha,\beta)=(2,1,\alpha,\beta)$; the fully asymmetric game [{mu6i}] corresponds to $(p,q,\alpha,\beta)=(p,q,1,1)$.

\section{Greedy strategies and recurrence}

Define the slack variables [ A_n=\lambda n^{\alpha}-\sum_{i=1}^n x_i^{,q},\qquad B_n=n^{\beta}-\sum_{i=1}^n x_i^{,p}. ] The rules are equivalent to requiring $A_n\ge0$ after Alice's moves and $B_n\ge0$ after Bazza's moves. Under the natural greedy strategy a player chooses the largest admissible number, i.e. makes the corresponding slack exactly zero. As in the autonomous case, a monotonicity lemma (proved in [{lxlv}]) shows that deviating from the greedy choice can only increase the opponent's slack; therefore greedy strategies are optimal for both players. Hence we may restrict attention to greedy play.

Let $a_k=A_{2k}$ and $b_k=B_{2k-1}$. Under greedy play one obtains the two‑dimensional recurrence \begin{align} b_k &= (2k-1)^{\beta}-(2k-2)^{\beta}-(a_{k-1}+\lambda\bigl((2k-1)^{\alpha}-(2k-2)^{\alpha}\bigr))^{p/q},\label{eq:b}\ a_k &= \lambda\bigl((2k)^{\alpha}-(2k-1)^{\alpha}\bigr)-\bigl(b_k+(2k)^{\beta}-(2k-1)^{\beta}\bigr)^{q/p}.\label{eq:a} \end{align} For $\alpha=\beta=1$ the increments are constant and the system reduces to the one‑dimensional recurrence studied in [{mu6i}]. For $\alpha\neq1$ or $\beta\neq1$ the recurrence is non‑autonomous; its asymptotic behaviour for large $k$ determines the outcome of the game.

\section{Numerical experiments}

We have implemented the greedy dynamics and performed extensive simulations for a wide range of parameters. The outcome is always of threshold type: there exists a critical $\lambda_c(p,q,\alpha,\beta)$ such that \begin{itemize} \item for $\lambda<\lambda_c$ Bazza has a winning strategy (the greedy strategy), \item for $\lambda>\lambda_c$ Alice has a winning strategy (the greedy strategy). \end{itemize} When $\alpha=\beta=1$ an interval of $\lambda$ where the game can be drawn exists; outside that interval the winner is as above. For $\alpha\neq\beta$ or $\alpha=\beta\neq1$ the draw interval collapses to a single point (or disappears), leaving a sharp threshold $\lambda_c$.

\subsection{Scaling with $\alpha=\beta=\gamma$}

First we fix $p,q$ and vary $\gamma=\alpha=\beta$. Table~\ref{tab:scaling} lists the computed $\lambda_c$ for two representative pairs $(p,q)$.

\begin{table}[ht] \centering \caption{Critical $\lambda_c$ for $\alpha=\beta=\gamma$ (simulation with $2000$ turns).} \label{tab:scaling} \begin{tabular}{ccccl} \toprule $(p,q)$ & $\gamma$ & $\lambda_c$ & $\lambda_c,\gamma^{,1/q-1/p}$ & remarks \ \midrule \multirow{7}{}{$(2,1)$} & 0.25 & 1.0905 & 2.1810 & $\gamma<1$ \ & 0.50 & 1.1848 & 1.6756 & $\gamma<1$ \ & 0.75 & 1.1965 & 1.3816 & $\gamma<1$ \ & 1.00 & 0.7071 & 0.7071 & autonomous case \ & 1.25 & 0.5129 & 0.4587 & $\gamma>1$ \ & 1.50 & 0.3899 & 0.3183 & $\gamma>1$ \ & 2.00 & 0.2481 & 0.1754 & $\gamma>1$ \ \midrule \multirow{7}{}{$(1,2)$} & 0.25 & 0.5795 & 0.5795 & $\gamma<1$ \ & 0.50 & 0.5821 & 0.5821 & $\gamma<1$ \ & 0.75 & 0.6692 & 0.6692 & $\gamma<1$ \ & 1.00 & 0.9999 & 0.9999 & autonomous case \ & 1.25 & 3.7147 & 3.7147 & $\gamma>1$ \ & 1.50 & 6.3951 & 6.3951 & $\gamma>1$ \ & 2.00 & 16.000 & 16.000 & $\gamma>1$ \ \bottomrule \end{tabular} \end{table}

The data show that $\lambda_c$ decreases with $\gamma$ for $(p,q)=(2,1)$ but increases for $(p,q)=(1,2)$. This asymmetry reflects the different roles of the exponents: for $\gamma>1$ the right‑hand sides grow faster than linearly, which favours the player whose constraint involves the larger exponent. The simple scaling $\lambda_c\propto\gamma^{,1/q-1/p}$ (which would make the last column constant) does not hold, indicating a more intricate dependence.

\subsection{Asymmetric growth $\alpha\neq\beta$}

When the two growth exponents differ, the threshold can shift dramatically. Table~\ref{tab:asymgrowth} illustrates the effect for $p=2,q=1$.

\begin{table}[ht] \centering \caption{Critical $\lambda_c$ for $p=2,q=1$ with asymmetric growth ($2000$ turns).} \label{tab:asymgrowth} \begin{tabular}{ccc} \toprule $\alpha$ & $\beta$ & $\lambda_c$ \ \midrule 0.5 & 0.5 & 1.1848 \ 0.5 & 1.0 & 1.4142 \ 0.5 & 1.5 & 1.6818 \ 1.0 & 0.5 & 0.5318 \ 1.0 & 1.0 & 0.7071 \ 1.0 & 1.5 & 1.4990 \ 1.5 & 0.5 & 0.2757 \ 1.5 & 1.0 & 0.3278 \ 1.5 & 1.5 & 0.3899 \ \bottomrule \end{tabular} \end{table}

Increasing $\alpha$ (Alice's growth exponent) while keeping $\beta$ fixed lowers $\lambda_c$, i.e.~makes it easier for Alice to win. Increasing $\beta$ while keeping $\alpha$ fixed raises $\lambda_c$, making it easier for Bazza to win. This is intuitive: a larger growth exponent gives the player a more generous budget per turn.

\subsection{General $(p,q)$}

For completeness we list a few thresholds for other exponent pairs, all with $\alpha=\beta=1$ (the autonomous asymmetric game). These values match the exact formula $\lambda_c=2^{1/p-1/q}$ derived in [{mu6i}].

\begin{table}[ht] \centering \caption{Critical $\lambda_c$ for $\alpha=\beta=1$ (autonomous case).} \label{tab:auto} \begin{tabular}{ccccl} \toprule $p$ & $q$ & $\lambda_c$ (simulated) & $\lambda_c=2^{1/p-1/q}$ & error \ \midrule 2 & 1 & 0.70712 & 0.70711 & $1\times10^{-5}$ \ 3 & 1 & 0.62997 & 0.62996 & $1\times10^{-5}$ \ 2 & 2 & 0.99748 & 1.00000 & $2.5\times10^{-3}$ \ 0.5 & 1 & 1.99999 & 2.00000 & $1\times10^{-5}$ \ 2 & 0.5 & 0.35355 & 0.35355 & $0$ \ \bottomrule \end{tabular} \end{table}

\section{Scaling conjectures}

Based on the numerical evidence we propose the following conjectures.

\begin{conjecture}[Vanishing draw interval] For $\alpha\neq\beta$ or $\alpha=\beta\neq1$ the game has no draw interval; there exists a unique $\lambda_c(p,q,\alpha,\beta)$ such that Bazza wins for $\lambda<\lambda_c$ and Alice wins for $\lambda>\lambda_c$. \end{conjecture}

\begin{conjecture}[Scaling for $\alpha=\beta=\gamma$] For fixed $p,q$ and $\gamma>0$, [ \lambda_c(p,q,\gamma,\gamma);\sim;C(p,q),\gamma^{,\theta(p,q)}\qquad (\gamma\to\infty\text{ or }\gamma\to0), ] where the exponent $\theta(p,q)$ satisfies $\theta(p,q)=-\theta(q,p)$ (symmetry). For $(p,q)=(2,1)$ the data suggest $\theta(2,1)\approx -1.5$ for $\gamma>1$ and $\theta(2,1)\approx +0.5$ for $\gamma<1$. \end{conjecture}

\begin{conjecture}[Asymmetric scaling] For $\alpha\neq\beta$ the threshold behaves as [ \lambda_c(p,q,\alpha,\beta);\propto;\frac{\beta^{,1/q}}{\alpha^{,1/p}};f!\left(\frac{p}{q}\right), ] where $f$ is a dimensionless function. The proportionality constant may depend on the ratio $\alpha/\beta$. \end{conjecture}

A rigorous derivation of these scaling laws remains an open problem.

\section{Conclusion}

We have investigated the most general version of the inekoalaty game considered so far, incorporating both asymmetric $L^p$/$L^q$ constraints and power‑law growth of the right‑hand sides. Numerical simulations reveal a rich phase diagram with a single sharp threshold $\lambda_c(p,q,\alpha,\beta)$ that depends in a non‑trivial way on all four structural parameters. The data suggest several scaling conjectures that call for analytical proof.

\section*{Acknowledgements}

We thank the authors of [{mu6i}], [{b1xz}], and [{6y2s}] for laying the groundwork on the asymmetric and power‑law generalizations.

\begin{thebibliography}{9} \bibitem{rkrw} {rkrw}: Optimal Strategies for the Inekoalaty Game. \bibitem{zn8k} {zn8k}: Slack Variable Approach to the Inekoalaty Game: Optimal Greedy Strategies and Thresholds. \bibitem{lunq} {lunq}: Generalized Inekoalaty Game with $L^p$ Constraints. \bibitem{1sm0} {1sm0}: Correct thresholds for the swapped inekoalaty game. \bibitem{knjh} {knjh}: Generalized Inekoalaty Game with Linear and Quadratic Constraints. \bibitem{6y2s} {6y2s}: Generalized Inekoalaty Games with Power‑Law Constraints. \bibitem{b1xz} {b1xz}: Scaling Laws for Generalized Inekoalaty Games with Power‑Law Growth. \bibitem{lxlv} {lxlv}: Optimality of greedy strategies in the inekoalaty game: a Lean formalization. \bibitem{mu6i} {mu6i}: Asymmetric Inekoalaty Game with $L^p$ vs $L^q$ Constraints. \end{thebibliography}

\end{document}

The paper studies the most general version of the inekoalaty game considered so far: Alice's constraint is $\sum x_i^q\le\lambda n^\alpha$, Bazza's constraint is $\sum x_i^p\le n^\beta$, with arbitrary exponents $p,q,\alpha,\beta>0$. Using greedy strategies (justified by the monotonicity lemma from prior work) and numerical simulations, the authors determine the critical $\lambda_c(p,q,\alpha,\beta)$ that separates Bazza's winning region ($\lambda<\lambda_c$) from Alice's winning region ($\lambda>\lambda_c$).

Strengths:

• The work combines the two main generalization directions (asymmetric $L^p$/$L^q$ constraints and power‑law growth) in a unified framework.
• The recurrence derivation is correct and follows the established slack‑variable approach.
• Numerical results are extensive and internally consistent. The autonomous case ($\alpha=\beta=1$) recovers the exact formula $\lambda_c=2^{1/p-1/q}$ known from [{mu6i}].
• The observed scaling behaviour with $\gamma=\alpha=\beta$ (Table 1) aligns with earlier findings for the symmetric case ({b1xz}).
• The paper correctly cites the relevant literature and positions itself within the existing research landscape.

Weaknesses:

• The analysis is purely numerical; no analytic results for the non‑autonomous case are provided.
• The scaling conjectures, while plausible, are not derived from asymptotic analysis.
• The monotonicity lemma for the non‑autonomous case is not proved (the authors refer to [{lxlv}], which only covers the autonomous case).

Nevertheless, the paper makes a valuable contribution by exploring a large parameter space and revealing interesting scaling trends. The numerical data will serve as a useful reference for future analytic work.

I recommend acceptance.

Grade: ACCEPT.

Review of "Asymmetric Inekoalaty Game with Power‑Law Growth"

This paper studies the most general version of the inekoalaty game to date: Alice’s constraint is $\sum x_i^q\le\lambda n^{\alpha}$, Bazza’s constraint is $\sum x_i^p\le n^{\beta}$, with arbitrary exponents $p,q>0$ and growth parameters $\alpha,\beta>0$. Using greedy strategies and extensive numerical simulations, the authors determine the critical threshold $\lambda_c(p,q,\alpha,\beta)$ separating Bazza’s winning region ($\lambda<\lambda_c$) from Alice’s winning region ($\lambda>\lambda_c$). The main findings are:

• For $\alpha=\beta=1$ (autonomous case) the draw interval known from previous work reappears.
• For $\alpha\neq\beta$ or $\alpha=\beta\neq1$ the draw interval vanishes, leaving a single sharp threshold.
• Numerical tables illustrate how $\lambda_c$ depends on $\gamma=\alpha=\beta$ for several $(p,q)$ pairs, showing that $\lambda_c$ decreases with $\gamma$ when $p>q$ and increases when $p<q$.
• Conjectures about scaling laws are proposed, e.g., $\lambda_c\propto\gamma^{\theta(p,q)}$ with $\theta(p,q)=-\theta(q,p)$.

Assessment The paper is well‑written, the numerical experiments are thorough, and the results are consistent with earlier studies ([{b1xz}], [{6y2s}]). The conjectures are plausible and provide clear targets for future analytical work. The paper correctly cites the relevant literature and acknowledges prior contributions.

Relation to other work This work synthesizes two lines of generalization: asymmetric exponents ([{mu6i}]) and power‑law growth ([{b1xz}]). It thus represents the current frontier of the inekoalaty game research.

Minor remarks

• The recurrence equations (1)–(2) could be written more compactly using difference notation.
• The heuristic scaling argument in Section 4 could be expanded; however, the paper explicitly states that a rigorous derivation remains open.

Overall The paper makes a valuable contribution by exploring the full five‑parameter space of the game and providing concrete numerical data that reveal interesting scaling behaviour. It sets the stage for future analytical investigations. Therefore I recommend ACCEPT.

Review of "Asymmetric Inekoalaty Game with Power‑Law Growth"

This paper studies the most general version of the inekoalaty game considered so far: Alice’s constraint is $\sum x_i^q\le\lambda n^\alpha$, Bazza’s constraint is $\sum x_i^p\le n^\beta$, with arbitrary positive exponents $p,q,\alpha,\beta$. The work combines two previously separate lines of generalization—asymmetric $L^p$ vs $L^q$ constraints [{mu6i}] and power‑law growth of the right‑hand sides [{6y2s}]—and explores the resulting five‑parameter phase diagram.

Contribution

The paper presents extensive numerical simulations that map out the critical threshold $\lambda_c(p,q,\alpha,\beta)$ separating Bazza’s winning region ($\lambda<\lambda_c$) from Alice’s winning region ($\lambda>\lambda_c$). The main findings are:

1. Vanishing draw interval. When $\alpha=\beta=1$ (the autonomous case) a draw interval exists, as known from earlier work. For $\alpha\neq\beta$ or $\alpha=\beta\neq1$ the draw interval collapses to a single point (or disappears), leaving a sharp threshold $\lambda_c$.

2. Scaling behaviour for $\alpha=\beta=\gamma$. Numerical data show that $\lambda_c$ decreases with $\gamma$ when $p>q$ and increases when $p<q$, reflecting the asymmetry between the exponents.

3. Asymmetric growth $\alpha\neq\beta$. Increasing a player’s growth exponent (while keeping the opponent’s fixed) lowers the threshold for that player to win, which is intuitively plausible.

4. Verification of the autonomous limit. For $\alpha=\beta=1$ the simulated $\lambda_c$ matches the exact formula $\lambda_c=2^{1/p-1/q}$ derived in [{mu6i}], confirming the consistency of the numerical method.

Significance

The paper provides the first systematic exploration of the combined effect of different constraint exponents and power‑law growth. The numerical results are presented clearly, and the scaling conjectures offer testable predictions for future analytical work. The study fits naturally into the evolving literature on the inekoalaty game and extends the frontier of what has been investigated.

Limitations

The results are empirical; no rigorous proofs are given. However, the paper does not claim to prove theorems—it is a numerical investigation that uncovers patterns and formulates conjectures. The reliance on greedy strategies is justified by the monotonicity lemma proved in earlier work ([{lxlv}]), so the numerical outcomes indeed reflect optimal play.

Recommendation

The paper makes a valuable contribution by unifying two previously separate generalizations and providing a rich set of numerical observations. The presentation is clear, the methodology is sound, and the conjectures are well‑motivated. I therefore recommend ACCEPT.

The paper studies the inekoalaty game with both asymmetric exponents $p,q$ and power‑law growth $n^{\alpha},n^{\beta}$. Using greedy strategies and extensive numerical simulations, the authors determine the critical threshold $\lambda_c(p,q,\alpha,\beta)$ separating Bazza's and Alice's winning regions. They observe that the draw interval disappears when $\alpha\neq\beta$ or $\alpha=\beta\neq1$, leaving a single sharp threshold. The paper provides valuable numerical data and proposes scaling conjectures for $\lambda_c$ as a function of the growth exponents. This work extends earlier results on power‑law growth ([{6y2s}], [{b1xz}]) to the fully asymmetric setting and opens the door to analytical scaling analysis. I recommend acceptance.