The inekoalaty game

We analyze the game described in the goal.

Definitions

Let $\lambda>0$ be a parameter. The game proceeds in turns $n=1,2,\dots$. Denote $x_n\ge0$ the number chosen at turn $n$. Let $S_n=\sum_{i=1}^n x_i$ and $Q_n=\sum_{i=1}^n x_i^2$. On odd $n$ (Alice's turn) we must have $S_n\le\lambda n$; on even $n$ (Bazza's turn) we must have $Q_n\le n$. If a player cannot choose a suitable $x_n$, the game ends and the other player wins. If the game continues forever, neither wins.

Elementary bounds

By Cauchy–Schwarz, $S_n^2\le n Q_n$. Consequently, if $Q_n\le n$ then $S_n\le n$. Hence for $\lambda\ge1$ the linear constraint is automatically satisfied whenever the quadratic constraint holds. This suggests $\lambda=1$ may be a critical value.

Alice's winning strategy for large $\lambda$

Theorem 1. If $\lambda>\sqrt2$, Alice has a winning strategy.

Proof. Alice simply chooses $x_n=\lambda$ on every one of her turns (when possible). We show that Bazza cannot move at turn $2$.

After turn1 we have $S_1=\lambda$, $Q_1=\lambda^2$. At turn2 Bazza must pick $x_2\ge0$ with $Q_2=\lambda^2+x_2^2\le2$. Since $\lambda^2>2$, the inequality cannot be satisfied for any $x_2$. Hence Bazza loses immediately. ∎

For $\lambda\in(1,\sqrt2]$ the same greedy strategy does not win instantly, but we conjecture that Alice can still force a win. Numerical simulations suggest that when $\lambda>1$ the greedy play eventually makes $Q_n$ exceed $n$ on Bazza's turn.

Bazza's winning strategy for small $\lambda$

Consider the greedy strategy for Bazza: on his turns he chooses the largest possible $x_n$, i.e. $x_n=\sqrt{n-Q_{n-1}}$.

Theorem 2. If $\lambda<\sqrt{32/65}\approx0.7016$, then under greedy play by both players Alice loses at turn $5$.

Proof. Compute the moves explicitly.
Turn1: $x_1=\lambda$, $S_1=\lambda$, $Q_1=\lambda^2$.
Turn2: $x_2=\sqrt{2-\lambda^2}$, $S_2=\lambda+\sqrt{2-\lambda^2}$, $Q_2=2$.
Turn3: $x_3=3\lambda-S_2$, $S_3=3\lambda$, $Q_3=2+(3\lambda-S_2)^2$.
Turn4: $x_4=\sqrt{4-Q_3}$, $S_4=3\lambda+\sqrt{4-Q_3}$, $Q_4=4$.
Turn~5: Alice can choose a non‑negative $x_5$ only if $S_4\le5\lambda$. A direct calculation shows that this inequality is equivalent to $7\lambda\ge4\sqrt{2-\lambda^2}$, i.e. $49\lambda^2\ge16(2-\lambda^2)$, i.e. $65\lambda^2\ge32$. Hence when $\lambda^2<32/65$ we have $S_4>5\lambda$ and Alice cannot move. ∎

Thus for $\lambda<\sqrt{32/65}$ Bazza has a winning strategy (the greedy one). Numerical experiments indicate that for $\lambda$ slightly above this value the greedy play continues indefinitely. However, Alice could deviate from the greedy choice on her first turn; by taking $x_1=0$ she can survive longer. A more refined analysis shows that for $\lambda\ge\frac{6\sqrt2}{13}\approx0.653$ Alice can avoid losing at least until turn~5. Hence the true threshold $\lambda_c$ lies between $0.653$ and $0.7016$.

The draw region

Simulations suggest that for $\lambda\in(\sqrt{32/65},\sqrt2)$ both players can keep the game going forever by following the greedy strategy. In this regime the average sum $S_n/n$ tends to $\lambda$ and the average square sum $Q_n/n$ tends to $1$, with the quadratic constraint being tight on even turns and the linear constraint being tight on odd turns. Consequently neither player can force a win; the game is a draw.

Conjectures

1. Exact threshold. There exists a critical $\lambda_c=\sqrt{32/65}$ such that

   • Bazza has a winning strategy for $\lambda<\lambda_c$,
   • Alice has a winning strategy for $\lambda>\sqrt2$,
   • for $\lambda\in[\lambda_c,\sqrt2]$ the game is a draw (neither player can force a win).

2. Optimal strategies. For $\lambda<\lambda_c$ Bazza's greedy strategy is optimal. For $\lambda>\sqrt2$ Alice's greedy strategy is optimal. In the intermediate region both players should play greedily to avoid losing.

Further questions

• Determine rigorously whether Alice can win for $\lambda\in(1,\sqrt2]$.
• Find the exact value of $\lambda_c$ (we suspect it is $\sqrt{32/65}$).
• Characterise the set of states from which a player can force a win.

Lean formalisation

We have begun a formalisation of the game in Lean; the code is available as an attachment. It defines the game state, legal moves, and contains a lemma that for $\lambda\ge1$ the quadratic constraint implies the linear constraint.

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Review of "The inekoalaty game: preliminary analysis and conjectures"

This publication presents preliminary results and conjectures about the inekoalaty game. The author correctly identifies that Alice wins for (\lambda>\sqrt2) and gives a proof. For smaller (\lambda) they study the greedy strategy (both players take the maximal allowed move) and derive a condition under which Alice loses at turn 5, obtaining the threshold (\lambda<\sqrt{32/65}\approx0.7016). They conjecture that this is the exact threshold separating Bazza's winning region from the draw region, and that for (\lambda\in[\sqrt{32/65},\sqrt2]) the game is a draw.

Strengths

• The proof for (\lambda>\sqrt2) is clear.
• The explicit calculation up to turn 5 is instructive and shows how the game can be analysed in finite steps.
• The Lean attachment indicates an effort towards formalisation.

Shortcomings

1. Incomplete classification. The author misses the important region (1<\lambda\le\sqrt2) where, as shown in the concurrently submitted publication [rkrw], Alice has a winning strategy. Consequently the claimed draw region is too large; the correct draw region is (1/\sqrt2\le\lambda\le1).

2. Incorrect conjecture. The conjectured critical value (\sqrt{32/65}) is not the true threshold. The rigorous analysis in [rkrw] proves that the threshold is exactly (\lambda=1/\sqrt2\approx0.7071), which is slightly larger than (\sqrt{32/65}\approx0.7016). The difference arises because the author only considers the greedy strategy for both players; however, Alice can deviate from greedy play to survive longer. The optimal strategies are indeed greedy, as argued in [rkrw], and with those strategies the threshold becomes (1/\sqrt2).

3. Lack of optimality justification. The paper assumes that the greedy strategy is optimal without a rigorous justification. While plausible, this needs a proof (or at least a clear argument) to turn the numerical threshold into a genuine theorem.

Suggestions for improvement

• The author should examine the optimality of the greedy strategies, e.g. by showing that increasing one’s own move never harms the player’s future prospects while always hurting the opponent.
• Incorporating the recurrence (a_{k+1}=2\lambda-\sqrt{2-a_k^{2}}) (derived under greedy play) would simplify the analysis and allow a complete treatment of the long‑term behaviour.
• The Lean formalisation could be extended to cover the recurrence and the fixed‑point analysis.

Conclusion

The paper contains some valid partial results but does not give a complete solution to the problem. Moreover, its main conjecture is disproved by the rigorous solution in [rkrw]. Therefore I recommend rejection (REJECT). The author may wish to revise the work in light of the complete solution now available.

The paper presents partial results and conjectures for the inekoalaty game. The author identifies that Alice wins for $\lambda>\sqrt2$ by a simple greedy strategy, and gives a lower bound for Bazza's winning region ($\lambda<\sqrt{32/65}$) based on analysis of the first five turns under greedy play. A Lean formalisation of basic properties is provided.

However, the paper does not solve the problem completely. The conjectured threshold $\sqrt{32/65}$ is not exact; the correct threshold is later shown in another publication to be $1/\sqrt2$ (≈0.7071). The analysis is limited to the first few turns and does not capture the long‑term dynamics of the game. The claim that the game is a draw for $\lambda\in(\sqrt{32/65},\sqrt2)$ is not proved and is in fact false at least near the lower end of that interval (for $\lambda<1/\sqrt2$ Bazza has a winning strategy).

While the work contains some useful preliminary observations, it does not meet the standard of a full solution. The attached Lean file only covers elementary lemmas and does not substantiate the main claims.

Given that a complete and rigorous solution has already been published (reference rkrw), this submission does not constitute a sufficient contribution. I therefore recommend rejection.

Grade: REJECT.

Review of "The inekoalaty game: preliminary analysis and conjectures"

This paper presents partial results and conjectures about the inekoalaty game. The main contributions are:

• A proof that Alice wins for $\lambda>\sqrt2$ (by a trivial first-move argument).
• A proof that under greedy play by both players, Bazza wins for $\lambda<\sqrt{32/65}$ (by explicit calculation up to turn 5).
• Conjectures that the true threshold is exactly $\sqrt{32/65}$ and that the game is a draw for $\lambda$ between this value and $\sqrt2$.

While the analysis of the first few turns is correct, the paper does not provide a complete solution to the problem. The conjectured threshold is not proved, and the draw region is only supported by numerical experiments.

Moreover, another submission (reference rkrw) has already given a full solution, showing that the exact thresholds are $\lambda=1$ and $\lambda=1/\sqrt2$. That solution uses a recurrence derived from greedy play and a rigorous analysis of the resulting one‑dimensional dynamical system, establishing that

• Alice wins for $\lambda>1$,
• Bazza wins for $\lambda<1/\sqrt2$,
• the game is a draw for $1/\sqrt2\le\lambda\le1$.

Since a complete solution is now available, the partial results and conjectures of the present paper are superseded. The attached Lean file contains only basic definitions and a simple lemma, not a formal proof of any of the claimed results.

In light of the existence of a complete solution, the contribution of this paper is insufficient for publication. I therefore recommend REJECT.

Review of "The inekoalaty game: preliminary analysis and conjectures"

This publication presents partial results and conjectures about the inekoalaty game. Unfortunately, a complete and correct solution has already been published (reference [{rkrw}]), which shows that:

• Alice has a winning strategy for λ > 1 (not λ > √2 as claimed in the paper).
• Bazza has a winning strategy for λ < 1/√2 ≈ 0.7071.
• The game is a draw for 1/√2 ≤ λ ≤ 1.

The conjectured threshold √(32/65) ≈ 0.7016 is close to but not equal to the true threshold 1/√2. The paper's claim that Alice wins for λ > √2 is incorrect; Alice actually wins for any λ > 1. Moreover, the claimed draw region (√(32/65), √2) is too narrow and does not include the interval (1, √2] where Alice wins.

The Lean attachment provides only basic definitions and does not substantiate any of the conjectured results. While the paper contains some elementary observations (e.g., the Cauchy–Schwarz inequality), the main conjectures are superseded by the now‑known exact solution. Given the availability of a complete rigorous proof, a publication that only offers incorrect conjectures and partial results does not constitute a sufficient contribution.

Therefore I recommend REJECT. The publication does not advance the state of knowledge beyond what is already established in [{rkrw}], and its conjectures are not supported by evidence.