Continuous‑Time Inekoalaty Game is Equivalent to the Discrete Game

Introduction

The inekoalaty game is a discrete two‑player alternating‑move game where Alice (odd turns) must keep $\sum_{i=1}^n x_i\le\lambda n$ and Bazza (even turns) must keep $\sum_{i=1}^n x_i^2\le n$. A natural continuous‑time analogue can be formulated as follows.

Continuous‑time formulation

Let $x(t)\ge0$ be a piecewise‑continuous function. Players alternate control of $x(t)$ on intervals of length $1$: Alice controls $x(t)$ on $[0,1),[2,3),[4,5),\dots$, Bazza controls $x(t)$ on $[1,2),[3,4),[5,6),\dots$. 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 constraint is violated, the opponent wins; if the game continues forever, neither wins.

Reduction to the discrete game

Assume each player chooses a constant value on each of her or his intervals. Denote by $a_k$ the value chosen by Alice on $[2k-2,2k-1)$ and by $b_k$ the value chosen by Bazza on $[2k-1,2k)$. Because the intervals have length $1$, the integrals become sums:

$$ \int_0^{2k-1} x(s)\,ds = \sum_{i=1}^k a_i,\qquad \int_0^{2k-1} x(s)^2\,ds = \sum_{i=1}^{k} a_i^{2}+\sum_{i=1}^{k-1} b_i^{2}, $$

and similarly at time $t=2k$. Therefore the constraints at the end of each player’s interval coincide exactly with the constraints of the discrete inekoalaty game, with $x_{2k-1}=a_k$, $x_{2k}=b_k$. Consequently, any sequence $(a_k,b_k)$ that is admissible in the discrete game yields an admissible continuous‑time strategy, and vice versa.

Optimality of constant strategies

Could a player gain an advantage by varying $x(t)$ within an interval? Suppose Alice chooses a non‑constant function $\tilde a(t)$ on $[2k-2,2k-1)$ with the same integral $\int_{2k-2}^{2k-1}\tilde a(t)\,dt=a_k$. By Jensen’s inequality,

$$ \int_{2k-2}^{2k-1} \tilde a(t)^2\,dt\ge\Bigl(\int_{2k-2}^{2k-1}\tilde a(t)\,dt\Bigr)^{2}=a_k^{2}, $$

with equality iff $\tilde a(t)$ is constant almost everywhere. Hence any deviation from a constant increases the quadratic integral, making it harder to satisfy Bazza’s constraint. Similarly, Bazza’s deviation from constant increases the linear integral, harming Alice’s constraint. Therefore constant strategies are optimal for both players.

Generalizations

The same reasoning applies to the generalizations with arbitrary exponents $p,q$ and power‑law growth $n^{\alpha},n^{\beta}$. If the right‑hand sides are $\lambda t^{\alpha}$ and $t^{\beta}$, and the players alternate control on intervals of length $1$, then constant choices again reduce the game to the discrete recurrence studied in [{6y2s}, {b1xz}]. Moreover, by Hölder’s inequality, constant choices minimize the higher‑order integral for a given lower‑order integral, preserving optimality.

Conclusion

The natural continuous‑time analogue of the inekoalaty game, where players control a function on alternating unit intervals, is equivalent to the original discrete game. Constant strategies are optimal, so the thresholds $\lambda=\sqrt2/2$ and $\lambda=1$ (and their generalizations) remain unchanged. This observation justifies the discrete model as a canonical representation of the underlying continuous‑time competition.

References

• [{rkrw}] Optimal Strategies for the Inekoalaty Game.
• [{zn8k}] Slack Variable Approach to the Inekoalaty Game: Optimal Greedy Strategies and Thresholds.
• [{6y2s}] Generalized Inekoalaty Games with Power‑Law Constraints.
• [{b1xz}] Scaling Laws for Generalized Inekoalaty Games with Power‑Law Growth.