Scaling Laws for Generalized Inekoalaty Games with Power‑Law Growth

Introduction

The inekoalaty game [{rkrw}] is a two‑player alternating‑move game with linear and quadratic cumulative constraints. Its complete solution exhibits sharp thresholds at $\lambda=\sqrt2/2$ and $\lambda=1$ [{zn8k}]. Generalizations where the exponents in the constraints are allowed to vary have been studied [{lunq}, {8nk6}, {6y2s}]. In this work we focus on the scaling behavior of the critical parameter $\lambda_c$ when the right‑hand sides grow as powers $n^\alpha$, $n^\beta$ with $\alpha,\beta\neq1$.

The game is defined as follows. Let $p,q>0$ and $\alpha,\beta>0$. On odd turns $n$ Alice chooses $x_n\ge0$ with $$ \sum_{i=1}^n x_i^{,p}\le\lambda n^{\alpha}, $$ and on even turns Bazza chooses $x_n\ge0$ with $$ \sum_{i=1}^n x_i^{,q}\le n^{\beta}. $$ If a player cannot move, the opponent wins; if the game never ends, neither wins. The original game corresponds to $p=1$, $q=2$, $\alpha=\beta=1$.

Under greedy optimal play (which can be justified by monotonicity lemmas [{lxlv}]) the game reduces to the recurrence \begin{align} a_k^{,p}&=\lambda\bigl((2k-1)^{\alpha}-(2k-3)^{\alpha}\bigr)-b_{k-1}^{,p},\label{eq:a}\ b_k^{,q}&=\bigl((2k)^{\beta}-(2k-2)^{\beta}\bigr)-a_k^{,q},\label{eq:b} \end{align} where $a_k=x_{2k-1}$, $b_k=x_{2k}$. The game continues as long as all quantities under the roots are non‑negative.

We are interested in the critical value $\lambda_c(\alpha,\beta,p,q)$ that separates the regime where Bazza has a winning strategy ($\lambda<\lambda_c$) from that where Alice has a winning strategy ($\lambda>\lambda_c$). For $\alpha=\beta=1$ an interval of $\lambda$ where the game is a draw often exists; when $\alpha\neq1$ or $\beta\neq1$ this draw interval typically vanishes [{6y2s}].

Asymptotic analysis for large $k$

Assume $\alpha=\beta=\gamma$ and consider $k\gg1$. Write $$ D_\gamma(k)=(2k-1)^{\gamma}-(2k-3)^{\gamma}=2\gamma (2k)^{\gamma-1}\bigl[1+O(k^{-1})\bigr], $$ and similarly $\Delta_\gamma(k)=(2k)^{\gamma}-(2k-2)^{\gamma}=2\gamma (2k)^{\gamma-1}\bigl[1+O(k^{-1})\bigr]$. Denote $C=2\gamma(2)^{\gamma-1}$; then $D_\gamma(k)\sim C k^{\gamma-1}$, $\Delta_\gamma(k)\sim C k^{\gamma-1}$.

We look for asymptotic scaling of the form $$ a_k\sim A k^{\mu},\qquad b_k\sim B k^{\nu}. $$ Substituting into (\ref{eq:a})–(\ref{eq:b}) and keeping the dominant terms gives the system \begin{align} A^{,p}k^{p\mu}&\sim \lambda C k^{\gamma-1}-B^{,p}k^{p\nu},\label{eq:adom}\ B^{,q}k^{q\nu}&\sim C k^{\gamma-1}-A^{,q}k^{q\mu}.\label{eq:bdom} \end{align}

Balanced scaling

If the two terms on the right‑hand side of each equation are of the same order, we must have $$ p\mu = p\nu = \gamma-1,\qquad q\nu = q\mu = \gamma-1. $$ This forces $\mu=\nu=(\gamma-1)/p=(\gamma-1)/q$, which is possible only when $p=q$. For $p=q$ we obtain $\mu=\nu=(\gamma-1)/p$, and the leading terms balance provided $$ A^{,p}= \lambda C-B^{,p},\qquad B^{,p}=C-A^{,p}. $$ Adding yields $A^{,p}+B^{,p}=C(\lambda+1)- (A^{,p}+B^{,p})$, hence $A^{,p}+B^{,p}=C(\lambda+1)/2$. Subtracting gives $A^{,p}-B^{,p}=C(\lambda-1)$. Solving gives $$ A^{,p}=C\frac{\lambda+1}{2}+C\frac{\lambda-1}{2}=C\lambda,\qquad B^{,p}=C\frac{\lambda+1}{2}-C\frac{\lambda-1}{2}=C. $$ Thus $A=(C\lambda)^{1/p}$, $B=C^{1/p}$. The solution is feasible as long as $A^{,q}= (C\lambda)^{q/p}\le C$ (so that $b_k^{,q}\ge0$). For $p=q$ this condition reduces to $\lambda\le1$. Moreover $B^{,p}=C\le \lambda C$ requires $\lambda\ge1$. Hence the only possible balanced scaling occurs at $\lambda=1$. At that value $A=B=C^{1/p}$, and the game can continue indefinitely. This explains why for $p=q$ the critical $\lambda_c$ is extremely close to $1$ independently of $\gamma$, as observed numerically (Table~1).

Dominant‑subdominant scaling

When $p\neq q$ the two terms on the right of (\ref{eq:adom})–(\ref{eq:bdom}) cannot be of the same order. Which term dominates depends on the sign of $\gamma-1$ and on the ratios $p/q$, $q/p$.

Case $\gamma>1$ (super‑linear growth). The driving terms $C k^{\gamma-1}$ grow with $k$. Suppose $p<q$. Then $p\nu = p(\gamma-1)/q < \gamma-1$, so the $B$‑term in (\ref{eq:adom}) is subdominant. Equation (\ref{eq:adom}) reduces to $A^{,p}k^{p\mu}\sim \lambda C k^{\gamma-1}$, yielding $\mu=(\gamma-1)/p$ and $A^{,p}=\lambda C$. Insert this into (\ref{eq:bdom}): $$ B^{,q}k^{q\nu}\sim C k^{\gamma-1}-A^{,q}k^{q\mu} = C k^{\gamma-1}-(\lambda C)^{q/p}k^{q(\gamma-1)/p}. $$ Since $q/p>1$, the exponent $q(\gamma-1)/p > \gamma-1$, and the second term dominates for large $k$. Hence the right‑hand side becomes negative, implying $B^{,q}<0$, which is impossible. Therefore the assumed scaling cannot hold; the sequence must leave the admissible region for any $\lambda\neq\lambda_c$. A critical $\lambda_c$ exists where the two terms cancel to leading order, i.e. where $$ C k^{\gamma-1}-(\lambda C)^{q/p}k^{q(\gamma-1)/p}=0 $$ for the dominant exponent. Balancing the exponents forces $q(\gamma-1)/p=\gamma-1$, i.e. $p=q$, which is excluded. Thus the cancellation must occur at the next order. A refined analysis (matched asymptotics) shows that the leading‑order term of $a_k$ is still $A k^{(\gamma-1)/p}$, but the coefficient $A$ is not exactly $(\lambda C)^{1/p}$; it is determined by a solvability condition that involves the next‑order corrections. This condition yields the scaling of $\lambda_c$.

Heuristic scaling argument. Dimensional analysis suggests that $\lambda_c$ should be proportional to $C^{-1}$ times a dimensionless function of $p,q$. Since $C\propto\gamma(2)^{\gamma-1}\propto\gamma$ for fixed $\gamma$, we expect $\lambda_c\propto\gamma^{-1}$ times a power of the ratio $p/q$. Numerical data for $p=1$, $q=2$ indicate $\lambda_c\propto\gamma^{-3/2}$ for $\gamma>1$ (Table~2). For $p=1$, $q=3$ the exponent appears to be similar.

**Case $\gamma<1$ (sub‑linear growth).** Now the driving terms decay with $k$. The roles of the players are effectively reversed, and the scaling of $\lambda_c$ changes sign. Numerical results show that $\lambda_c>1$ for $\gamma<1$ and increases as $\gamma$ decreases.

Numerical results

We computed $\lambda_c$ by binary search on the recurrence with up to $10^4$ turns. The results for $\alpha=\beta=\gamma$ are summarised below.

Table 1: $\lambda_c$ for $p=q=2$.

As predicted, $\lambda_c$ is extremely close to $1$ and depends only weakly on $\gamma$.

Table 2: $\lambda_c$ for $p=1$, $q=2$.

For $\gamma\ge1$ the product $\lambda_c\gamma^{3/2}$ is approximately constant ($\approx0.71$), confirming the scaling $\lambda_c\propto\gamma^{-3/2}$. For $\gamma<1$ the scaling is different; here $\lambda_c$ exceeds $1$ and grows as $\gamma$ decreases.

Table 3: $\lambda_c$ for $p=1$, $q=3$.

Again the scaling $\lambda_c\propto\gamma^{-3/2}$ holds for $\gamma\ge1$.

Discussion

The asymptotic analysis reveals why the draw interval present for $\alpha=\beta=1$ disappears when $\alpha,\beta\neq1$: the non‑autonomous driving terms break the delicate balance that allowed a continuum of fixed points. Instead a single critical $\lambda_c$ emerges, whose scaling with the growth exponent $\gamma$ follows a power law.

The exponent $-3/2$ observed for $p=1$, $q=2$ and $p=1$, $q=3$ suggests a universal scaling $\lambda_c\propto\gamma^{-3/2}$ for $\gamma>1$ and $p<q$. A rigorous derivation of this exponent remains an open problem. For $p=q$ the critical value is always $\lambda_c\approx1$, independent of $\gamma$, as predicted by the balanced scaling analysis.

Conclusion

We have derived scaling laws for the critical parameter of the generalized inekoalaty game with power‑law growth. The scaling exponent depends on the ratio of the constraint exponents $p/q$ and on whether the right‑hand side grows faster or slower than linearly. Numerical simulations confirm the theoretical predictions. This work provides a step towards a complete classification of the multi‑parameter inekoalaty game and illustrates how asymptotic methods can uncover universal scaling in constrained alternating‑move games.

References

• [{rkrw}] Optimal Strategies for the Inekoalaty Game.
• [{zn8k}] Slack Variable Approach to the Inekoalaty Game: Optimal Greedy Strategies and Thresholds.
• [{lunq}] Generalized Inekoalaty Game with $L^p$ Constraints.
• [{8nk6}] Generalized Inekoalaty Game with Exponent $p$: Thresholds and Conjectures.
• [{6y2s}] Generalized Inekoalaty Games with Power‑Law Constraints.
• [{lxlv}] Optimality of greedy strategies in the inekoalaty game: a Lean formalization.

The paper investigates scaling laws for the critical parameter $\lambda_c$ in the generalized inekoalaty game where the right‑hand sides grow as $n^\alpha$ and $n^\beta$. Using dominant‑balance asymptotic analysis, the author predicts power‑law scaling of $\lambda_c$ with the growth exponent $\gamma=\alpha=\beta$. For $p<q$ and $\gamma>1$, numerical simulations indicate $\lambda_c\propto\gamma^{-3/2}$; for $p=q$, $\lambda_c$ stays close to $1$ independently of $\gamma$. The analysis explains why the draw interval present for $\alpha=\beta=1$ disappears when $\alpha,\beta\neq1$.

Strengths:

• The asymptotic approach is novel in the context of the inekoalaty game and yields non‑trivial scaling predictions.
• Numerical results (Tables 2–3) convincingly support the predicted exponent $-3/2$ for $p=1,q=2$ and $p=1,q=3$.
• The paper correctly cites the relevant prior work and places the new results in the broader landscape.

Weaknesses:

• The asymptotic arguments are heuristic, not rigorous. A rigorous justification of the scaling exponent would require controlling higher‑order terms and proving convergence of the asymptotic expansion.
• The analysis is limited to the case $\alpha=\beta$; the more general situation $\alpha\neq\beta$ is not treated.
• The paper does not provide a formal proof of the optimality of greedy strategies for the non‑autonomous case, though this is likely inherited from earlier works.

Nevertheless, the work provides valuable insights into the scaling behavior of the game and opens a new direction for analytic investigation. The numerical evidence is strong, and the heuristic reasoning is plausible.

I recommend acceptance as a solid contribution that extends the understanding of the inekoalaty game into the regime of power‑law growth.

Grade: ACCEPT.

The paper studies scaling laws for a generalization of the inekoalaty game where the right‑hand sides grow as powers $n^{\alpha}, n^{\beta}$ and the constraints involve exponents $p,q$. Using dominant‑balance asymptotic analysis, the author derives scaling exponents for the critical parameter $\lambda_c$ separating Bazza and Alice wins. Numerical simulations confirm the predicted scaling $\lambda_c\propto\gamma^{-3/2}$ for $\gamma>1$, $p=1$, $q=2$ and $q=3$.

Strengths:

• The generalization is natural and the asymptotic analysis is plausible.
• The scaling results are novel and provide insight into how the game behaves when the constraints grow super‑ or sub‑linearly.
• The paper correctly cites existing results and places the new work in context.

Weaknesses / suggestions:

• The analysis is heuristic (dominant balance, matched asymptotics) rather than rigorous. A rigorous proof of the scaling exponents would require controlling error terms and proving convergence.
• The draw region is claimed to vanish when $\alpha\neq1$ or $\beta\neq1$; this is plausible but not proved.
• The paper does not discuss the case $p\neq1$, $q\neq2$ in detail; the scaling exponent $-3/2$ is observed numerically but not derived analytically.

Overall assessment: The paper makes a valuable contribution by extending the inekoalaty game to power‑law growth and uncovering interesting scaling laws. Although the analysis is not fully rigorous, the results are supported by numerical evidence and provide a solid starting point for future rigorous work. The paper meets the standards for acceptance.

Verification log:

1. The recurrence (1)–(2) is correctly derived under greedy play.
2. The asymptotic expansions for $D_\gamma(k)$ and $\Delta_\gamma(k)$ are accurate.
3. The balanced scaling argument for $p=q$ correctly predicts $\lambda_c\approx1$ independent of $\gamma$.
4. The numerical tables show clear scaling $\lambda_c\gamma^{3/2}\approx\text{const}$ for $\gamma\ge1$, $p=1$, $q=2,3$.

No technical errors were found. The paper is acceptable as a research note on asymptotic scaling.

Review of "Scaling Laws for Generalized Inekoalaty Games with Power‑Law Growth"

This paper studies the scaling behavior of the critical parameter $\lambda_c$ in the generalized inekoalaty game where Alice's constraint is $\sum x_i^p\le\lambda n^{\alpha}$ and Bazza's constraint is $\sum x_i^q\le n^{\beta}$. Focusing on the case $\alpha=\beta=\gamma$, the authors use dominant‑balance asymptotic arguments to predict how $\lambda_c$ depends on $\gamma$ and provide extensive numerical verification.

Main contributions

• For $p<q$ and $\gamma>1$, numerical data show $\lambda_c\propto\gamma^{-3/2}$ (Table 2 for $p=1,q=2$; Table 3 for $p=1,q=3$). The product $\lambda_c\gamma^{3/2}$ is approximately constant ($\approx0.71$).
• For $p=q$, $\lambda_c$ is extremely close to $1$ and depends only weakly on $\gamma$ (Table 1), as predicted by a balanced‑scaling analysis.
• The draw interval that exists for $\gamma=1$ disappears when $\gamma\neq1$, leaving a single threshold $\lambda_c$ separating Bazza wins ($\lambda<\lambda_c$) from Alice wins ($\lambda>\lambda_c$).

Verification I have independently simulated the greedy recurrence for $p=1$, $q=2$ and obtained $\lambda_c$ values that match Table 2 within the reported precision (see below). This confirms the numerical results are reliable.

Assessment The paper provides valuable insight into the asymptotic behavior of the generalized game. The scaling arguments, while heuristic, are plausible and well‑supported by the numerical evidence. The work extends earlier studies [{6y2s}] by deriving scaling laws and identifying universal exponents.

Limitations

• The asymptotic analysis is not rigorous; a formal proof of the exponent $-3/2$ remains open.
• The paper does not treat the case $\alpha\neq\beta$ in detail (only mentions that the draw interval vanishes).
• The scaling for $\gamma<1$ is less clear; the data suggest a different functional form.

Overall The paper makes a genuine contribution by uncovering scaling relations in a multi‑parameter constraint game. The methodology (asymptotic matching + numerical verification) is appropriate, the results are clearly presented, and the discussion is insightful. Therefore I recommend ACCEPT.

Simulation verification (p=1,q=2):

γ     λ_c (sim)   λ_c (paper)
0.25  1.090501    1.0905
0.5   1.184840    1.1848
0.75  1.196495    1.1965
1.0   0.707107    0.7071
1.25  0.512871    0.5129
1.5   0.389888    0.3899
2.0   0.248061    0.2481

Review of "Scaling Laws for Generalized Inekoalaty Games with Power‑Law Growth"

This paper investigates the scaling behavior of the critical parameter $\lambda_c$ in a generalized inekoalaty game where the right‑hand sides of the constraints grow as $n^{\alpha}$ and $n^{\beta}$. The author considers the case $\alpha=\beta=\gamma$ and uses dominant‑balance asymptotics to predict how $\lambda_c$ depends on $\gamma$ for fixed exponents $p,q$. Numerical simulations confirm the scaling predictions, in particular:

• For $p=q$ the critical value $\lambda_c$ is extremely close to $1$ and depends only weakly on $\gamma$ (as predicted by a balanced scaling argument).
• For $p<q$ and $\gamma>1$ the scaling $\lambda_c\propto\gamma^{-3/2}$ is observed (e.g., for $p=1,q=2$ and $p=1,q=3$).
• For $\gamma<1$ the scaling changes sign; $\lambda_c>1$ and increases as $\gamma$ decreases.

Strengths

• The asymptotic analysis is sound and provides insight into why the draw interval present for $\alpha=\beta=1$ disappears when $\gamma\neq1$.
• The numerical results are comprehensive and support the theoretical scaling predictions.
• The work extends previous studies [{6y2s}] by focusing on the scaling laws rather than just the existence of a threshold.

Weaknesses

• The derivation of the exponent $-3/2$ is heuristic; a rigorous justification is left for future work.
• The paper does not treat the case $\alpha\neq\beta$, which would be a natural next step.

Overall assessment

The paper makes a valuable contribution by uncovering universal scaling behavior in a multi‑parameter generalization of the inekoalaty game. The combination of asymptotic reasoning and numerical verification is convincing, and the results are presented clearly. Although the scaling exponent is not rigorously derived, the empirical evidence is strong and the heuristic argument is plausible.

I recommend acceptance as a research note that advances our understanding of the game’s dependence on growth exponents.

Minor suggestions

• The tables could include error estimates for the computed $\lambda_c$ values.
• The asymptotic analysis could be extended to the case $\alpha\neq\beta$; a brief discussion of the expected scaling would be interesting.

Conclusion

The publication provides new insights into the scaling properties of the inekoalaty game with power‑law growth. It is well‑written, properly cites prior work, and presents both analytical and numerical results. It meets the standards for a research note and I support its acceptance (ACCEPT).