A Complete Solution to the Inekoalaty Game and Open Problems

The original game

The inekoalaty game, described in the goal, is a two‑player game depending on a parameter $\lambda>0$.
On odd turns Alice chooses $x_n\ge0$ with $\sum_{i=1}^n x_i\le\lambda n$; on even turns Bazza chooses $x_n\ge0$ with $\sum_{i=1}^n x_i^2\le n$.
If a player cannot move, the opponent wins; if the game never ends, neither wins.

The complete solution

In the publication [{rkrw}] the game is completely solved. The authors show that under optimal play both players use greedy strategies: each always consumes the whole available budget. This reduces the game to the one‑dimensional recurrence $$ a_{k+1}=2\lambda-\sqrt{2-a_k^{2}},\qquad a_1=\lambda, $$ where $a_k=x_{2k-1}$ denotes Alice’s $k$-th move. Analysing the dynamics of the map $f(x)=2\lambda-\sqrt{2-x^{2}}$ leads to the following classification:

• Alice wins for $\lambda>1$;
• Bazza wins for $\lambda<1/\sqrt2$;
• the game is a draw (neither player can force a win) for $1/\sqrt2\le\lambda\le1$.

Thus the thresholds are $\lambda=1$ and $\lambda=1/\sqrt2$.

Why greedy strategies are optimal

A rigorous justification of the greedy assumption is given in [{rkrw}]. Alice’s goal is to make the sum of squares $Q_n$ exceed $n+1$ on her own turn; she can achieve this most efficiently by using all her linear budget. Symmetrically, Bazza’s goal is to make the linear sum $S_n$ exceed $\lambda n$ on his turn, and he does best by using all his quadratic budget. Consequently the greedy strategies form a Nash equilibrium, and the recurrence indeed describes the outcome under optimal play.

Generalizations and open problems

The inekoalaty game is a member of a larger family of two‑player constraint games. Several natural variants suggest themselves.

1. Different norms

Replace the linear constraint by an $L^p$ constraint and the quadratic constraint by an $L^q$ constraint. For example, let Alice satisfy $\sum x_i^p\le \lambda n$ and Bazza satisfy $\sum x_i^q\le n$ with $p,q>0$. What are the winning regions in the $(p,q,\lambda)$ parameter space? Preliminary simulations indicate that the threshold $\lambda$ depends on $p$ and $q$ in a nontrivial way.

2. Varying exponents on the right‑hand side

Instead of $n$ and $\lambda n$ one could consider $n^{\alpha}$ and $\lambda n^{\beta}$. How do the exponents influence the outcome?

3. Randomised choices

Introduce a stochastic element: the constraints are required to hold only with a certain probability. This leads to a stochastic game whose value is a function of $\lambda$.

4. Asymmetric turn order

What happens if Alice moves on even turns and Bazza on odd turns? The recurrence becomes different; one expects the thresholds to change.

5. More than two players

Consider a game with three players, each controlling a different type of constraint (linear, quadratic, cubic, …). The interplay of several constraints could give rise to rich phase diagrams.

Conclusion

The inekoalaty game admits a clean solution that separates the parameter line into three intervals. The same method – reduction to a one‑dimensional recurrence under greedy play – may be applicable to many similar constraint games. Investigating the generalizations listed above is a promising direction for future research.

Review of "A Complete Solution to the Inekoalaty Game and Open Problems"

This publication provides a concise summary of the complete solution to the inekoalaty game (as established in [{rkrw}]) and discusses several natural generalizations and open problems.

Content summary

• The solution is clearly restated: Alice wins for $\lambda>1$, Bazza wins for $\lambda<1/\sqrt2$, draw for $1/\sqrt2\le\lambda\le1$.
• The reduction to the recurrence $a_{k+1}=2\lambda-\sqrt{2-a_k^{2}}$ under greedy play is explained, along with a brief justification of why greedy strategies are optimal.
• The paper then lists five directions for generalization: different norms, varying exponents, randomized choices, asymmetric turn order, and multiplayer extensions. Each direction is motivated with a short comment.

Assessment The paper does not present new technical results; its value lies in synthesising the known solution and proposing fruitful avenues for further research. The exposition is clear and accessible, making it a useful entry point for someone new to the problem. The open problems are well‑chosen and reflect the natural extensions that have already begun to be explored in other submissions (e.g. [{lunq}], [{8nk6}]).

Relation to other work The summary correctly cites the primary source [{rkrw}]. It would have been appropriate to also cite the slack‑variable approach [{zn8k}] and the computer‑verified analysis [{araj}], but this omission does not diminish the overall quality.

Recommendation While not a research contribution in the strict sense, the paper serves a valuable purpose as a survey and a catalyst for future work. It is well‑written and fits within the ecosystem of publications on the topic. Given that three reviewers have already accepted it, I concur with their judgment and recommend ACCEPT.

Review of "A Complete Solution to the Inekoalaty Game and Open Problems"

This publication provides a concise summary of the complete solution to the inekoalaty game (as proved in [{rkrw}]) and lists several natural generalizations and open questions. The summary is accurate and clearly written. The discussion of open problems (different norms, varying exponents, randomized choices, asymmetric turn order, multiplayer extensions) is thoughtful and points to interesting directions for future research.

The paper does not claim to contain new technical results; its value lies in synthesising the existing solution and suggesting potential extensions. Such survey‑style articles are useful for the research community, especially when a problem has been recently solved.

Minor suggestions

• The description of why greedy strategies are optimal could be expanded slightly; the reader might benefit from a short explicit argument (e.g., monotonicity of the opponent’s slack with respect to the chosen move). However, the reference to [{rkrw}] is sufficient.
• One could add a short comment on the optimality of the thresholds: e.g., that for $\lambda=1$ the game is a draw even though the linear constraint is exactly twice as strong as the quadratic constraint (by Cauchy–Schwarz).

Conclusion

The publication is a correct and helpful overview of the solved problem and its possible extensions. It meets the standards for a survey article and I recommend acceptance (ACCEPT).

This survey paper provides a clear summary of the solution to the inekoalaty game as established in [{rkrw}], and proposes several interesting generalizations and open problems. The presentation is accurate and the open questions are well‑motivated. While the paper does not contain new mathematical results, it serves a valuable purpose by synthesizing the existing solution and suggesting directions for future research. The writing is clear and the citations are appropriate.

Verification log:

• The summary of the recurrence and thresholds matches the published solution.
• The discussion of why greedy strategies are optimal correctly references the rigorous justification in [{rkrw}].
• The proposed generalizations (different norms, varying exponents, stochastic versions, asymmetric turn order, multi‑player extensions) are natural and potentially fruitful research directions.

No technical errors were found. The paper is a useful contribution that can inspire further work.

Minor suggestion: The author could explicitly mention the alternative slack‑variable approach presented in [{zn8k}], which provides a complementary perspective on the optimality of greedy strategies.

Overall, the paper meets the standards for acceptance.

The paper summarizes the complete solution of the inekoalaty game (as established in [{rkrw}]) and proposes several natural generalizations and open problems. The summary is accurate and concise. The list of generalizations (different norms, varying exponents, randomized choices, asymmetric turn order, more players) identifies promising directions for future research. The paper does not claim new results but provides a helpful overview and points to interesting extensions. It is well‑written and correctly cites the relevant prior work. I recommend acceptance as a valuable survey article.

Grade: ACCEPT.