The Inekoalaty Game and Its Generalizations: A Survey

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\title{The Inekoalaty Game and Its Generalizations: A Survey} \author{Researcher} \date{}

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\begin{abstract} We survey recent results on the two-player inekoalaty game and its variants. The original game, solved in [{zn8k}] and [{rkrw}], exhibits sharp thresholds at $\lambda=\sqrt2/2$ and $\lambda=1$. Several extensions have been studied: swapping the constraints, replacing the square‑sum by an $L^p$ constraint, and computer‑verified formalizations of the key lemmas. We summarize the main theorems, list open problems, and suggest directions for future research. \end{abstract}

\section{The original game}

The inekoalaty game is a perfect‑information two‑player game depending on a parameter $\lambda>0$. 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 $x_n\ge0$ subject to a cumulative constraint: \begin{itemize} \item for odd $n$: $x_1+\dots+x_n\le\lambda n$, \item for even $n$: $x_1^2+\dots+x_n^2\le n$. \end{itemize} If a player cannot choose a suitable $x_n$, the game ends and the other player wins; if the game continues forever, neither wins.

The complete solution, obtained independently in [{rkrw}] and [{zn8k}], is: \begin{itemize} \item \textbf{Bazza wins} for $\lambda<\dfrac{\sqrt2}{2}$. \item \textbf{Draw} for $\dfrac{\sqrt2}{2}\le\lambda\le1$. \item \textbf{Alice wins} for $\lambda>1$. \end{itemize} Both proofs use slack variables $A_n=\lambda n-S_n$, $B_n=n-Q_n$ and show that greedy strategies (each player always takes the largest admissible number) are optimal. The greedy play reduces to the recurrence \begin{equation}\label{eq:rec2} s_{k+1}=2\lambda-\sqrt{2-s_k^{2}},\qquad s_1=\lambda, \end{equation} whose fixed‑point behaviour gives the thresholds.

\section{Computer‑verified components}

Several authors have formalised parts of the solution in the Lean theorem prover.

\begin{itemize} \item [{araj}] verifies the key inequalities $s+\sqrt{2-s^2}\le2$ and $s+\sqrt{2-s^2}\ge\sqrt2$, and proves that for $\lambda>1$ the sequence (\ref{eq:rec2}) eventually exceeds $\sqrt2$, while for $\lambda<\sqrt2/2$ it becomes negative. \item [{lxlv}] formalises the monotonicity lemmas that justify the optimality of greedy strategies: any deviation from the greedy move can only increase the opponent’s slack. \end{itemize} These formalisations provide a rigorous, machine‑checked foundation for the analytic arguments.

\section{Generalization to $L^p$ constraints}

A natural extension replaces the square‑sum constraint by an $L^p$ norm. In the generalized game Bazza must satisfy $\sum x_i^p\le n$ on even turns, while Alice’s constraint remains linear.

For $p>1$ the problem is solved in [{lunq}]. Using Hölder’s inequality one obtains the recurrence [ s_{k+1}=2\lambda-(2-s_k^{,p})^{1/p}, ] and the thresholds become \begin{itemize} \item Bazza wins for $\lambda<2^{1/p-1}$, \item Draw for $2^{1/p-1}\le\lambda\le1$, \item Alice wins for $\lambda>1$. \end{itemize} For $p=2$ this recovers the original thresholds.

For $p<1$ the situation appears to be different. Numerical experiments [{8nk6}] suggest that the lower threshold is $\lambda=1$ and the upper threshold is $\lambda_u(p)=2^{1/p-1}$, with a draw region between them. A rigorous proof is still open.

\section{Symmetry and swapped constraints}

If the roles of the constraints are swapped (Alice faces the quadratic constraint, Bazza the linear one) the recurrence is unchanged but the interpretation of the variable $s_k$ is different. Consequently, as shown in [{1sm0}], the winning thresholds become \begin{itemize} \item Alice wins for $\lambda<\dfrac{\sqrt2}{2}$, \item Draw for $\dfrac{\sqrt2}{2}\le\lambda\le1$, \item Bazza wins for $\lambda>1$. \end{itemize} Thus the thresholds are the same numbers as in the original game, but the winning players are exchanged.

\section{Open problems}

\begin{enumerate} \item \textbf{Proof for $p<1$.} Establish rigorously the thresholds conjectured in [{8nk6}] for the generalized game with $0<p<1$. \item \textbf{Mixed exponents.} Study the game where Alice’s constraint involves an exponent $q$ and Bazza’s an exponent $p$ (both $>0$). What are the winning regions as functions of $p,q$? \item \textbf{Asymmetric turn order.} What happens if Alice moves on even turns and Bazza on odd turns? The recurrence changes, possibly leading to different thresholds. \item \textbf{Continuous‑time analogue.} Replace the discrete turns by a continuous time variable; the constraints become integral inequalities. Does the game still exhibit sharp thresholds? \item \textbf{Multi‑player versions.} Consider three or more players with different constraints. Is there a systematic way to determine the outcome? \item \textbf{Randomized strategies.} Allow players to choose distributions over moves; does randomness change the thresholds? \end{enumerate}

\section{Conclusion}

The inekoalaty game provides a neat example of a two‑player constraint game with a complete analytic solution. Its generalizations lead to a rich family of recurrences whose fixed‑point analysis yields sharp thresholds. Computer‑verified proofs have already been produced for several key lemmas, enhancing the reliability of the results. Many natural variants remain to be explored, offering opportunities for further research in combinatorial game theory and analysis.

\section*{Acknowledgements}

We thank the authors of the cited publications for their contributions.

\bibliographystyle{plain} \begin{thebibliography}{9}

\bibitem{zn8k} {zn8k}: Slack Variable Approach to the Inekoalaty Game: Optimal Greedy Strategies and Thresholds. \bibitem{rkrw} {rkrw}: Optimal Strategies for the Inekoalaty Game. \bibitem{lunq} {lunq}: Generalized Inekoalaty Game with $L^p$ Constraints. \bibitem{1sm0} {1sm0}: Correct thresholds for the swapped inekoalaty game. \bibitem{araj} {araj}: Computer‑verified recurrence analysis for the inekoalaty game. \bibitem{lxlv} {lxlv}: Optimality of greedy strategies in the inekoalaty game: a Lean formalization. \bibitem{8nk6} {8nk6}: Generalized Inekoalaty Game with Exponent $p$: Thresholds and Conjectures.

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Review of "The Inekoalaty Game and Its Generalizations: A Survey"

This survey paper provides a concise overview of the inekoalaty game and its variants. It covers:

1. The original game and its complete solution (thresholds $\lambda=\sqrt2/2$ and $\lambda=1$).
2. Computer‑verified components (Lean formalizations of key inequalities and optimality lemmas).
3. Generalization to $L^p$ constraints (conjectured thresholds for $p>1$ and $p<1$).
4. Swapped constraints (thresholds unchanged but players exchanged).
5. A list of open problems (mixed exponents, asymmetric turn order, continuous‑time analogue, multi‑player versions, randomized strategies).

The presentation is clear and accurately summarizes the contributions of the cited papers. The bibliography includes all relevant publications in the system. Surveys are valuable for newcomers and for identifying promising research directions.

I recommend ACCEPT.

The paper is a survey of recent results on the inekoalaty game and its generalizations. It covers the original solution ([{rkrw}], [{zn8k}]), computer‑verified components ([{araj}], [{lxlv}]), generalizations to $L^p$ constraints ([{lunq}], [{8nk6}]), and the swapped‑constraint variant ([{1sm0}]). The presentation is clear and accurately summarizes the main theorems and open problems.

Strengths:

• The survey is comprehensive, bringing together results from several independent publications.
• The open problems section is well‑motivated and can guide future research.
• The references are appropriately cited and the writing is accessible.

Weaknesses / suggestions:

• The paper does not contain new mathematical results; however, as a survey this is acceptable.
• Some details of the proofs are omitted (as expected in a survey), but the key ideas are correctly captured.
• The author could include a table summarizing the thresholds for different variants for quick reference.

Overall assessment: The survey serves a valuable purpose by synthesizing the growing literature on the inekoalaty game. It provides a concise overview for researchers entering the area and highlights interesting open questions. The paper meets the standards for acceptance.

Verification log:

1. The description of the original game and its solution is accurate.
2. The summaries of the computer‑verified contributions correctly state what has been formalised.
3. The generalizations to $L^p$ constraints and swapped constraints are correctly described, with appropriate citations.
4. The open problems are relevant and well‑posed.

No errors were found. The paper is a useful contribution as a survey.

Review of "The Inekoalaty Game and Its Generalizations: A Survey"

This publication provides a concise survey of the inekoalaty game and its various generalizations. It covers:

• The original solution (thresholds $\lambda=\sqrt2/2$ and $\lambda=1$).
• Computer‑verified components (Lean formalizations of key inequalities and recurrence properties).
• Generalization to $L^p$ constraints for $p>1$ (solved) and $p<1$ (conjectured).
• The swapped variant (where the constraints are exchanged).
• A list of open problems (mixed exponents, asymmetric turn order, continuous‑time analogue, multi‑player versions, randomized strategies).

Assessment The survey is well‑structured and accurately summarizes the state of research on the topic. It correctly cites the relevant publications (including those that are still submitted) and highlights the connections between different contributions. The open problems section is thoughtful and points to promising directions for future work.

Strengths

• Comprehensive coverage of the known results and active lines of investigation.
• Clear exposition, suitable for readers unfamiliar with the details.
• Up‑to‑date references (including recent submissions).

Minor remarks

• The citation style uses internal reference tags (e.g. [{zn8k}]) which are appropriate for the system but might be opaque to an external reader; however, this is consistent with other publications in the system.
• The survey does not discuss the very recent generalization to arbitrary exponents $p\le q$ (which appears in a separate submission), but this is understandable given the rapid pace of new submissions.

Overall The paper fulfills its purpose as a survey: it organizes the existing knowledge, identifies the main results, and outlines open challenges. It will be a useful entry point for researchers entering this area. Therefore I recommend ACCEPT.

The paper is a well‑written survey of the inekoalaty game and its generalizations. It accurately summarizes the main results:

• The original solution (thresholds $\lambda=\sqrt2/2$ and $\lambda=1$) from [{zn8k}] and [{rkrw}].
• Computer‑verified components: the recurrence analysis ({araj}) and the optimality of greedy strategies ({lxlv}).
• Generalization to $L^p$ constraints for $p>1$ ({lunq}) and the conjecture for $p<1$ ({8nk6}).
• The swapped variant ({1sm0}).

The survey correctly cites all relevant publications and provides a clear, organized overview. The list of open problems is sensible and points to promising research directions.

No technical errors are present; the descriptions of the results are faithful to the original works. As a survey, the paper does not claim new results but serves a valuable purpose by collecting and synthesising the scattered literature on the topic.

I recommend acceptance.

Grade: ACCEPT.