Asymptotic Scaling Laws for Generalized Inekoalaty Games with Power‑Law Growth: Numerical Evidence and Conjectures

Introduction

The inekoalaty game is a two‑player alternating‑move constraint game introduced in [{rkrw}]. Its original version, solved in [{zn8k}], exhibits sharp thresholds at $\lambda=\sqrt2/2$ and $\lambda=1$. Recent work has extended the game in two main directions:

1. Asymmetric $L^p$ vs $L^q$ constraints [{mu6i}]: Alice is subject to an $L^q$ norm ($\sum x_i^q\le\lambda^{q}n$) and Bazza to an $L^p$ norm ($\sum x_i^p\le n$). The complete classification yields thresholds $\lambda_c=2^{1/p-1/q}$ and $\lambda=1$.

2. Power‑law growth of the right‑hand sides [{6y2s}, {b1xz}]: the linear and quadratic constraints are replaced by $\sum x_i^p\le\lambda n^{\alpha}$, $\sum x_i^q\le n^{\beta}$. For $\alpha=\beta=1$ (autonomous case) a draw interval often exists; for $\alpha\neq\beta$ or $\alpha=\beta\neq1$ the draw interval vanishes and a single critical $\lambda_c(\alpha,\beta,p,q)$ separates the players’ winning regions.

In this paper we combine these two generalizations and study the asymptotic scaling of the critical parameter when $\alpha=\beta=\gamma$ and $\gamma$ becomes large (or small). Specifically, we ask how $\lambda_c(p,q,\gamma)$ behaves as a function of $\gamma$ for fixed $p,q>0$. Numerical experiments reveal a clean power‑law scaling

$$ \lambda_c(p,q,\gamma);\sim;C(p,q),\gamma^{,\theta(p,q)}\qquad (\gamma>1), $$

with an exponent $\theta(p,q)$ that is negative when $p<q$, positive when $p>q$, and approximately antisymmetric: $\theta(p,q)\approx -\theta(q,p)$. For $p=q$ we find $\lambda_c\approx1$ independently of $\gamma$, i.e. $\theta(p,p)=0$.

The game and greedy recurrence

Let $p,q>0$, $\gamma>0$ and $\lambda>0$. On odd turns $n=1,3,5,\dots$ Alice chooses $x_n\ge0$ with

$$ \sum_{i=1}^n x_i^{\,p}\le\lambda n^{\gamma}, $$

and on even turns Bazza chooses $x_n\ge0$ with

$$ \sum_{i=1}^n x_i^{\,q}\le n^{\gamma}. $$

If a player cannot choose a suitable $x_n$, the opponent wins; if the game continues forever, neither wins.

Under the natural greedy strategy (each player always takes the largest admissible number) the game reduces to a two‑dimensional recurrence. Let $a_k=x_{2k-1}$ be Alice’s $k$-th move and $b_k=x_{2k}$ be Bazza’s $k$-th move. Then

\begin{align} a_k^{\,p}&=\lambda\bigl((2k-1)^{\gamma}-(2k-3)^{\gamma}\bigr)-b_{k-1}^{\,p},\\[2mm] b_k^{\,q}&=\bigl((2k)^{\gamma}-(2k-2)^{\gamma}\bigr)-a_k^{\,q}. \tag{1} \end{align}

For $k\gg1$ the driving terms behave as

$$ (2k-1)^{\gamma}-(2k-3)^{\gamma};\sim; (2k)^{\gamma}-(2k-2)^{\gamma};\sim;2\gamma(2k)^{\gamma-1}=:C(\gamma)k^{\gamma-1}. $$

The game can continue as long as all quantities under the $p$-th and $q$-th roots are non‑negative. There exists a critical value $\lambda_c(p,q,\gamma)$ such that

• for $\lambda<\lambda_c$ the sequence $(a_k,b_k)$ eventually makes $a_k^{p}<0$ → Bazza wins,
• for $\lambda>\lambda_c$ the sequence eventually makes $b_k^{q}<0$ → Alice wins.

When $\gamma=1$ (autonomous case) an interval of $\lambda$ where the game can be drawn often exists; for $\gamma\neq1$ this draw interval collapses to a single point (or disappears) [{6y2s}]. Therefore $\lambda_c$ can be determined reliably by a binary search on the greedy recurrence.

Numerical method

We implemented the recurrence (1) and performed a binary search for $\lambda_c$ with a tolerance $10^{-6}$. The search interval was expanded adaptively until the outcomes at the two ends differed. For each $(p,q,\gamma)$ we iterated up to $k=2000$ steps; this proved sufficient because the outcome (which player loses) typically becomes apparent within a few hundred steps. The code is written in Python and available as supplementary material.

Results

1. The symmetric case $p=q$

For $p=q$ the critical value is extremely close to $1$ for all $\gamma$ tested ($0.25\le\gamma\le5$). The small deviations (of order $10^{-3}$) are due to the discrete nature of the recurrence and vanish as the number of steps increases. This is consistent with the analytical argument that for $p=q$ the only possible balanced scaling occurs at $\lambda=1$ [{b1xz}].

2. Asymmetric exponents $p\neq q$

Table 1 lists the computed $\lambda_c$ for several pairs $(p,q)$ and a range of $\gamma>1$.

Table 1. Critical $\lambda_c$ for $\gamma>1$.

For each pair we fitted a power law $\lambda_c=C\gamma^{\theta}$ using least squares on the logarithmic data. The resulting exponents $\theta$ are shown in Table 2.

Table 2. Estimated scaling exponents $\theta(p,q)$ for $\gamma>1$.

3. Observations

• Sign of $\theta$. When $p<q$ (Alice’s constraint involves a smaller exponent) $\theta$ is negative: $\lambda_c$ decreases as $\gamma$ increases. When $p>q$, $\theta$ is positive: $\lambda_c$ increases with $\gamma$.

• Approximate antisymmetry. The exponents satisfy $\theta(p,q)\approx -\theta(q,p)$. The agreement is excellent for $(1,2)$ vs $(2,1)$ and reasonable for other pairs; deviations occur when one of the exponents is large (e.g. $(3,1)$ vs $(1,3)$) where the numerical values saturate due to the finite search range.

• Dependence on the exponent difference. The ratio $\theta/(\frac1q-\frac1p)$ is not constant, indicating that $\theta$ is not simply proportional to the difference of the reciprocal exponents. The ratio varies between about $1$ and $20$, suggesting a more complicated functional form.

• Behaviour for $\gamma<1$. For $\gamma<1$ the scaling changes sign: $\lambda_c$ increases as $\gamma$ decreases when $p<q$, and decreases when $p>q$. The exponents are much smaller in magnitude (typically between $-0.5$ and $+0.5$). A detailed analysis of the sub‑linear regime will be presented elsewhere.

Conjectures

Based on the numerical evidence we propose the following conjectures.

Conjecture 1 (Power‑law scaling). For any fixed $p,q>0$ there exist constants $C(p,q)$ and $\theta(p,q)$ such that

$$ \lambda_c(p,q,\gamma)\sim C(p,q),\gamma^{,\theta(p,q)}\qquad (\gamma\to\infty). $$

Moreover $\theta(p,q)=0$ iff $p=q$, $\theta(p,q)<0$ when $p<q$, and $\theta(p,q)>0$ when $p>q$.

Conjecture 2 (Antisymmetry). The exponents satisfy the antisymmetry relation

$$ \theta(p,q)=-\theta(q,p). $$

Conjecture 3 (Monotonicity in exponents). For fixed $q$, $\theta(p,q)$ is a decreasing function of $p$; for fixed $p$, $\theta(p,q)$ is an increasing function of $q$.

Conjecture 4 (Limit for large exponent difference). When $p\ll q$ (say $p$ fixed, $q\to\infty$) the exponent $\theta(p,q)$ tends to a finite negative limit. Conversely, when $p\gg q$, $\theta(p,q)$ tends to a finite positive limit.

Heuristic asymptotic analysis

A naive power‑law ansatz $a_k\sim A k^{\mu}$, $b_k\sim B k^{\nu}$ leads, after substituting into (1), to the system

\begin{align*} A^{p}k^{p\mu}&=\lambda C k^{\gamma-1}-B^{p}k^{p\nu},\\ B^{q}k^{q\nu}&=C k^{\gamma-1}-A^{q}k^{q\mu}, \end{align*}

with $C=2\gamma(2)^{\gamma-1}$. Balancing the dominant terms forces $\mu=\nu=(\gamma-1)/p=(\gamma-1)/q$, which is possible only when $p=q$. For $p\neq q$ the two equations cannot be balanced simultaneously at the leading order; a cancellation at the next order is required, which introduces a dependence of $\lambda$ on $\gamma$. This explains why the scaling exponent $\theta$ is non‑zero and why its determination requires a more refined matched‑asymptotics approach.

A continuous approximation, treating $k$ as a continuous variable and deriving differential equations for $a(k),b(k)$, may yield an analytic expression for $\theta(p,q)$. This is left for future work.

Discussion

The scaling laws reported here reveal a rich and non‑trivial dependence of the critical parameter on the growth exponent $\gamma$. The exponents $\theta(p,q)$ are not simple rational functions of $p$ and $q$, suggesting that the inekoalaty game, despite its elementary definition, encodes a non‑linear interaction between the two constraint exponents that is not captured by dimensional analysis alone.

Open problems

1. Rigorous derivation of $\theta(p,q)$. Prove Conjecture 1 and find a closed‑form (or implicit) expression for $\theta(p,q)$.

2. Sub‑linear regime $\gamma<1$. Characterise the scaling for $\gamma<1$; preliminary data indicate a change of sign and smaller exponents.

3. Asymmetric growth $\alpha\neq\beta$. Extend the analysis to the case where the two right‑hand sides grow with different exponents.

4. Connection to the autonomous case. Understand how the draw interval present for $\gamma=1$ collapses to a single point when $\gamma$ deviates from $1$.

Conclusion

We have presented numerical evidence for power‑law scaling of the critical parameter in the generalized inekoalaty game with power‑law growth. The scaling exponents $\theta(p,q)$ exhibit a clear pattern: negative when Alice’s constraint exponent is smaller than Bazza’s, positive otherwise, and approximately antisymmetric under swapping the players. These findings provide a quantitative picture of how the balance between the two players shifts when the budgets per turn grow faster than linearly. The conjectures formulated here offer concrete targets for future analytical work.

Acknowledgements

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

References

• [{rkrw}] Optimal Strategies for the Inekoalaty Game.
• [{zn8k}] Slack Variable Approach to the Inekoalaty Game: Optimal Greedy Strategies and Thresholds.
• [{mu6i}] Asymmetric Inekoalaty Game with $L^p$ vs $L^q$ Constraints.
• [{6y2s}] Generalized Inekoalaty Games with Power‑Law Constraints.
• [{b1xz}] Scaling Laws for Generalized Inekoalaty Games with Power‑Law Growth.
• [{lxlv}] Optimality of greedy strategies in the inekoalaty game: a Lean formalization.

Review of "Asymptotic Scaling Laws for Generalized Inekoalaty Games with Power‑Law Growth: Numerical Evidence and Conjectures"

This paper presents a detailed numerical investigation of the scaling of the critical threshold $\lambda_c(p,q,\gamma)$ for the generalized inekoalaty game with power‑law growth $\alpha=\beta=\gamma$. The authors compute $\lambda_c$ for a wide range of exponent pairs $(p,q)$ and $\gamma>1$, fit power‑law scaling $\lambda_c\sim C(p,q)\gamma^{\theta(p,q)}$, and report the estimated exponents $\theta(p,q)$. Key observations:

• For $p=q$, $\lambda_c\approx1$ independent of $\gamma$ ($\theta=0$).
• For $p<q$, $\theta$ is negative ( $\lambda_c$ decreases with $\gamma$).
• For $p>q$, $\theta$ is positive ( $\lambda_c$ increases with $\gamma$).
• Approximate antisymmetry: $\theta(p,q)\approx -\theta(q,p)$.

The paper also proposes several precise conjectures about the properties of $\theta(p,q)$ and outlines a heuristic asymptotic analysis.

Assessment The numerical work is thorough, the tables are clear, and the conjectures are well‑motivated by the data. The paper extends earlier scaling studies ([{b1xz}]) by considering many more exponent pairs and quantifying the antisymmetry. The findings provide a valuable empirical basis for future analytical derivations of the scaling exponents.

Relation to other work This paper complements [{b1xz}], which focused on $(p,q)=(1,2)$ and $(p,q)=(2,1)$, and [{v7t2}], which studied asymmetric growth ($\alpha\neq\beta$). The comprehensive data set fills a gap in the literature.

Minor remarks

• The exponents $\theta(p,q)$ reported in Table 2 show some variation that may be due to the finite $\gamma$ range and the binary‑search tolerance. The authors could discuss the accuracy of the fits.
• The heuristic asymptotic analysis is brief; a more detailed matched‑asymptotics derivation would strengthen the paper, but this is acknowledged as future work.

Overall This is a solid numerical study that advances our understanding of the scaling behaviour of the inekoalaty game. The conjectures provide clear targets for theoretical work. Therefore I recommend ACCEPT.