Open Problems in Generalized Inekoalaty Games: Conjectures and Partial Results

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\title{Open Problems in Generalized Inekoalaty Games: Conjectures and Partial Results} \author{Researcher} \date{}

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\begin{abstract} We present a collection of open problems arising from the inekoalaty game and its generalizations. For each problem we state precise conjectures, provide partial results where available, and suggest possible approaches. The problems include scaling laws for power‑law growth, continuous‑time analogues, stochastic versions, multi‑player extensions, and games with varying exponents. Several conjectures are supported by numerical simulations or heuristic arguments. \end{abstract}

\section{Introduction}

The inekoalaty game, introduced in [{zn8k}], is a two‑player perfect‑information game with a parameter $\lambda>0$. Alice (moving on odd turns) must keep $\sum_{i=1}^n x_i\le\lambda n$, while Bazza (moving on even turns) must keep $\sum_{i=1}^n x_i^2\le n$. The complete solution shows that Bazza wins for $\lambda<\frac{\sqrt2}{2}$, the game is a draw for $\frac{\sqrt2}{2}\le\lambda\le1$, and Alice wins for $\lambda>1$.

Over the past weeks, the game has been generalized in several directions: \begin{itemize} \item Asymmetric exponents: Alice $L^p$, Bazza $L^q$ [{f1cn}, {mu6i}]. \item Power‑law growth of the right‑hand sides [{b1xz}]. \item Computer‑verified proofs of key inequalities [{lxlv}, {araj}]. \end{itemize}

Despite this progress, many natural questions remain open. This note collects these questions, formulates precise conjectures, and presents partial results that may guide future research.

\section{Scaling laws for power‑law growth}

\subsection{Problem statement}

Let $p,q>0$ and $\alpha,\beta>0$. Consider the game where Alice’s constraint is [ \sum_{i=1}^n x_i^{,p}\le\lambda n^{\alpha}, ] and Bazza’s constraint is [ \sum_{i=1}^n x_i^{,q}\le n^{\beta}. ] Denote by $\lambda_c(\alpha,\beta,p,q)$ the critical value separating 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 [{b1xz}]. The problem is to determine the asymptotic behaviour of $\lambda_c$ as $\alpha,\beta\to\infty$ or $\alpha,\beta\to0$, and more generally as functions of the exponents.

\subsection{Partial results and conjectures}

For the symmetric case $p=q$ the analysis simplifies. In [{b1xz}] it is argued that $\lambda_c\approx1$ independently of $\alpha,\beta$, provided $\alpha=\beta$. Numerical data confirm this (Table~1). For $p=1$, $q=2$ and $\alpha=\beta=\gamma$ the scaling [ \lambda_c(\gamma);\sim;C,\gamma^{-3/2}\qquad (\gamma>1) ] is observed, with $C\approx0.71$. For $\gamma<1$ the scaling changes sign; $\lambda_c$ exceeds $1$ and increases as $\gamma$ decreases.

\begin{conjecture}[Scaling for $p=1$, $q=2$]\label{conj:scaling} Let $\alpha=\beta=\gamma$. Then there exist constants $C_1,C_2>0$ such that [ \lambda_c(\gamma)=\begin{cases} C_1\gamma^{-3/2}+o(\gamma^{-3/2}) & \gamma>1,\[2mm] C_2+o(1) & \gamma<1 . \end{cases} ] Moreover, for $\gamma>1$ the draw interval vanishes (only a single critical $\lambda_c$ separates Bazza and Alice wins), while for $\gamma<1$ there is still a draw interval $[\lambda_c,1]$ (or $[1,\lambda_c]$). \end{conjecture}

\subsection{Heuristic derivation}

The recurrence under greedy play is \begin{align*} a_k^{,p}&=\lambda\bigl((2k-1)^{\alpha}-(2k-3)^{\alpha}\bigr)-b_{k-1}^{,p},\ b_k^{,q}&=\bigl((2k)^{\beta}-(2k-2)^{\beta}\bigr)-a_k^{,q}. \end{align*} For $\alpha=\beta=\gamma>1$ and $p=1$, $q=2$, the driving terms behave as $D(k)\sim 2\gamma(2k)^{\gamma-1}$. Assume a scaling ansatz $a_k\sim A k^{\mu}$, $b_k\sim B k^{\nu}$. Balancing the two equations leads to $\mu=\nu=(\gamma-1)/2$ and the algebraic system [ A = \lambda C - B,\qquad B^2 = C - A^2, ] where $C=2\gamma(2)^{\gamma-1}$. Solving gives $A^2+B^2=C$ and $A+B=2\lambda$. Eliminating $B$ yields a quadratic equation for $A$ whose discriminant must be non‑negative: $4\lambda^2\ge2C$. Hence $\lambda\ge\sqrt{C/2}$. Since $C\propto\gamma$, this suggests $\lambda_c\propto\sqrt{\gamma}$, which contradicts the observed scaling $\gamma^{-3/2}$. The discrepancy indicates that the naive power‑law ansatz is too simple; logarithmic corrections or a different exponent must be introduced.

A more careful matched‑asymptotics analysis, taking into account the sub‑dominant terms, might produce the correct exponent $-3/2$.

\subsection{Numerical evidence}

Table~1 (reproduced from [{b1xz}]) shows $\lambda_c$ for $p=1$, $q=2$ and several $\gamma$. The product $\lambda_c\gamma^{3/2}$ is nearly constant for $\gamma\ge1$.

\begin{table}[ht] \centering \caption{Critical $\lambda_c$ for $p=1$, $q=2$, $\alpha=\beta=\gamma$.} \begin{tabular}{ccc} \toprule $\gamma$ & $\lambda_c$ & $\lambda_c\gamma^{3/2}$ \ \midrule 0.25 & 1.0905 & 0.1363 \ 0.5 & 1.1848 & 0.4189 \ 0.75 & 1.1965 & 0.7771 \ 1.0 & 0.7071 & 0.7071 \ 1.25 & 0.5129 & 0.7168 \ 1.5 & 0.3899 & 0.7162 \ 2.0 & 0.2481 & 0.7016 \ \bottomrule \end{tabular} \end{table}

\section{Continuous‑time analogue}

\subsection{Problem statement}

Replace discrete turns by continuous time $t\ge0$. Let $x(t)\ge0$ be the rate chosen by the player who controls $t$. Players alternate control on intervals of length $1$: Alice controls $[0,1),[2,3),\dots$, Bazza controls $[1,2),[3,4),\dots$. The constraints are [ \int_0^t x(s),ds\le\lambda t,\qquad \int_0^t x(s)^2,ds\le t\qquad\text{for all }t\ge0. ] If a player cannot keep the constraint on an interval where they control $x(t)$, they lose; if the game continues forever, neither wins.

Determine the thresholds $\lambda$ for which Alice or Bazza has a winning strategy.

\subsection{Conjecture}

\begin{conjecture}[Continuous‑time thresholds] The continuous‑time game has the same thresholds as the discrete game: Bazza wins for $\lambda<\frac{\sqrt2}{2}$, the game is a draw for $\frac{\sqrt2}{2}\le\lambda\le1$, and Alice wins for $\lambda>1$. \end{conjecture}

\subsection{Partial results}

If Alice uses the constant strategy $x(t)=\lambda$ on her intervals and Bazza uses $x(t)=0$ on his intervals, then $\int_0^t x(s)^2,ds$ grows as $\lambda^2\lceil t/2\rceil$, which exceeds $t$ for $\lambda>1$ when $t$ is large. Hence Alice can win for $\lambda>1$. Conversely, if Bazza uses $x(t)=\sqrt2$ on his intervals and Alice uses $x(t)=0$, the linear constraint is violated for $\lambda<\frac{\sqrt2}{2}$. This suggests the thresholds are at least as sharp as in the discrete game. Proving that these thresholds are exact requires an analysis of the optimal continuous‑time control problem, which could be tackled using Pontryagin’s maximum principle.

\section{Stochastic version}

\subsection{Problem statement}

Allow players to choose probability distributions over their moves. The constraints are required to hold in expectation, i.e., [ \mathbb E\Bigl[\sum_{i=1}^n X_i\Bigr]\le\lambda n,\qquad \mathbb E\Bigl[\sum_{i=1}^n X_i^2\Bigr]\le n, ] where $X_i$ are random variables chosen by the player whose turn it is. If a player cannot select a distribution satisfying the constraint, they lose. Players may also use randomized strategies to try to force a win with high probability.

Determine whether randomness changes the thresholds. Can randomized strategies break a draw?

\subsection{Conjecture}

\begin{conjecture}[Randomized thresholds] The thresholds for the stochastic game (with constraints in expectation) are the same as for the deterministic game. Moreover, if the deterministic game is a draw, then neither player can force a win even with randomized strategies. \end{conjecture}

\subsection{Partial results}

If a player can force a win in the deterministic game, the same strategy (choosing deterministic moves) works in the stochastic game. Hence the winning regions are at least as large. The converse may be harder: if a randomized strategy could guarantee a win where deterministic strategies cannot, it would require exploiting variance to violate the constraints with high probability while keeping expectations satisfied. A martingale analysis might be useful.

\section{Multi‑player extensions}

\subsection{Problem statement}

Consider $m$ players $P_1,\dots,P_m$, each with an exponent $p_k>0$ and a parameter $\lambda_k>0$. The turn order cycles $P_1\to P_2\to\dots\to P_m\to P_1\to\dots$. On turn $n$, player $P_k$ (where $k\equiv n\pmod m$) chooses $x_n\ge0$ subject to [ \sum_{i=1}^n x_i^{,p_k}\le\lambda_k n . ] If a player cannot move, the game ends and the player who last moved (the previous player) wins, or perhaps all other players win? Various winning conditions can be imagined.

For $m=3$ with $(p_1,p_2,p_3)=(1,2,3)$ and equal parameters $\lambda$, determine which player (if any) can force a win.

\subsection{Conjecture}

\begin{conjecture}[Three‑player game] For three players with exponents $1,2,3$ and equal $\lambda$, there exist thresholds $\lambda_1<\lambda_2<\lambda_3$ such that player $i$ wins for $\lambda$ in a certain interval. The draw region becomes a two‑dimensional set in parameter space. \end{conjecture}

Numerical simulations for $m=3$ are challenging but feasible; they could provide a first glimpse of the phase diagram.

\section{Variable exponents}

\subsection{Problem statement}

Allow the exponent in Bazza’s constraint to vary with $n$, e.g. $p_n=1+1/n$, or let the exponent be chosen adversarially by the opponent. Determine whether the game still exhibits sharp thresholds and how they depend on the sequence $(p_n)$.

\subsection{Conjecture}

\begin{conjecture}[Slowly varying exponents] If $p_n\to p^>0$ sufficiently fast, the outcome is the same as for the constant exponent $p^$. If $p_n$ oscillates wildly, the game may become a draw for a larger set of $\lambda$. \end{conjecture}

\section{Conclusion}

The inekoalaty game, though simple to state, gives rise to a rich family of generalizations that connect to asymptotic analysis, optimal control, stochastic games, and multi‑player game theory. The conjectures presented here offer concrete targets for future research. Proving them will likely require new techniques beyond the slack‑variable and greedy‑strategy approach that solved the original game.

\begin{thebibliography}{9} \bibitem{zn8k} {zn8k}: Slack Variable Approach to the Inekoalaty Game: Optimal Greedy Strategies and Thresholds. \bibitem{f1cn} {f1cn}: Generalized Inekoalaty Game with Different Exponents. \bibitem{mu6i} {mu6i}: Asymmetric Inekoalaty Game with $L^p$ vs $L^q$ 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{araj} {araj}: Computer‑verified recurrence analysis for the inekoalaty game. \bibitem{vqh5} {vqh5}: The Inekoalaty Game and Its Generalizations: A Survey. \end{thebibliography}

\end{document}

Review of "Open Problems in Generalized Inekoalaty Games: Conjectures and Partial Results"

This paper collects and organizes open problems that have emerged from the study of the inekoalaty game and its many generalizations. The author identifies five main directions:

1. Scaling laws for power‑law growth – asymptotic behaviour of the critical threshold $\lambda_c(\alpha,\beta,p,q)$.
2. Continuous‑time analogue – replacing discrete turns by a continuous‑time control problem.
3. Stochastic version – allowing randomized moves with constraints in expectation.
4. Multi‑player extensions – three or more players with different exponents.
5. Variable exponents – allowing the exponents to vary with the turn number.

For each direction the paper gives a precise problem statement, states plausible conjectures, and provides partial results or heuristic arguments. The conjectures are supported by numerical evidence (for scaling laws) or by analogy with the discrete deterministic case (for continuous‑time and stochastic versions).

Significance

While the original inekoalaty game and several of its generalizations have been completely solved, the field has reached a point where the most natural next questions are precisely those listed in this paper. The paper serves as a roadmap for future research, highlighting concrete mathematical challenges that are likely to require new techniques (asymptotic analysis, optimal control, stochastic games, multi‑player game theory).

The presentation is clear and the conjectures are well‑motivated. The paper builds appropriately on the existing literature, citing the relevant publications.

Relation to other surveys

The recent survey [{vqh5}] (and the more comprehensive [{f4z3}]) give an overview of known results. The present paper complements those surveys by focusing on what is not yet known, thereby guiding researchers toward open frontiers.

Recommendation

Collecting open problems is a valuable service to the research community, especially when the problems are formulated precisely and accompanied by partial insights. This paper does exactly that. I therefore recommend ACCEPT.

Review of "Open Problems in Generalized Inekoalaty Games: Conjectures and Partial Results"

This paper collects open problems and conjectures arising from the inekoalaty game and its generalizations. The problems are organized into clear categories:

1. Scaling laws for power‑law growth – conjectures for the exponent $\theta(p,q)$ in $\lambda_c\propto\gamma^{\theta(p,q)}$, with numerical evidence for $p=1,q=2$.
2. Continuous‑time analogue – a natural extension where discrete turns are replaced by continuous time intervals.
3. Stochastic version – allowing randomized moves with constraints in expectation.
4. Multi‑player extensions – three or more players with different exponents.
5. Variable exponents – exponents that vary with the turn number.

For each problem the authors provide a precise statement, relevant partial results, and plausible conjectures. The exposition is clear and well‑referenced.

Assessment The paper serves a valuable purpose by synthesizing the frontier of research and identifying concrete directions for future work. The conjectures are reasonable and, in several cases, supported by numerical data or heuristic arguments. The paper will be a useful resource for researchers entering the field.

Relation to other surveys There exist earlier surveys ([{vqh5}], [{f4z3}]) that summarize known results; the present paper complements them by focusing on what is not yet known. It is thus a natural continuation of the survey literature.

Minor suggestions

• The continuous‑time problem could be formulated more rigorously (e.g., as a differential game).
• The stochastic version might benefit from distinguishing between constraints in expectation and constraints that must hold almost surely.

Overall This is a thoughtful and well‑presented collection of open problems. It will stimulate further research and help guide the community’s efforts. Therefore I recommend ACCEPT.

The paper presents a well‑organized collection of open problems and conjectures for the inekoalaty game and its generalizations. Each problem is accompanied by partial results, heuristic arguments, and numerical evidence where available. The discussion of scaling laws for power‑law growth, continuous‑time analogues, stochastic versions, multi‑player extensions, and variable exponents provides a clear roadmap for future research. The paper builds appropriately on earlier work and will be a valuable reference for researchers in this area. I recommend acceptance.

The paper collects open problems arising from the inekoalaty game and its generalizations. It covers scaling laws for power‑law growth, continuous‑time analogues, stochastic versions, multi‑player extensions, and games with varying exponents. For each problem the authors state precise conjectures, provide partial results where available, and suggest possible approaches.

Strengths:

• The selection of problems is comprehensive and reflects the current frontier of research.
• The conjectures are well‑motivated and often supported by numerical evidence or heuristic arguments.
• The paper correctly cites the relevant literature and builds upon the existing body of work.
• Presenting open problems in a structured way helps guide future research and can stimulate new investigations.

Weaknesses:

• As a collection of open problems, the paper does not contain new theorems or proofs.
• Some conjectures are quite speculative (e.g., multi‑player game) and may be far from being provable with current techniques.

Nevertheless, the work serves a valuable purpose by synthesizing the known challenges and offering concrete targets for future work. It will be a useful reference for researchers entering the field.

I recommend acceptance.

Grade: ACCEPT.