MathLabs

Problem 5

Alice and Bazza are playing the inekoalaty game, a two-player game whose rules depend on a positive real number λ\lambda which is known to both players. On the nnth turn of the game (starting with n=1n=1) the following happens: if nn is odd, Alice chooses a nonnegative real number xnx_n such that x1+x2+⋯+xn≤λnx_1+x_2+\cdots+x_n\le\lambda n; if nn is even, Bazza chooses a nonnegative real number xnx_n such that x12+x22+⋯+xn2≤nx_1^2+x_2^2+\cdots+x_n^2\le n. If a player cannot choose a suitable number xnx_n, the game ends and the other player wins. If the game goes on forever, neither player wins. All chosen numbers are known to both players. Determine all values of λ\lambda for which Alice has a winning strategy and all those for which Bazza has a winning strategy.
Step 2 of 4: Bazza's greedy reply makes every pair total at least √2
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t+2−t2≥2for all t∈[0,2]t+\sqrt{2-t^2}\ge\sqrt2\quad\text{for all }t\in[0,\sqrt2]
Detailed analysis

Bazza's strategy is to always play the maximum legal value; by induction his running sum of squares equals 2i2i exactly after turn 2i2i, so on turn 2i2i he plays x2i=2−x2i−12x_{2i}=\sqrt{2-x_{2i-1}^2} (since the budget added since turn 2i−22i-2 is exactly 22). The function t↦t+2−t2t\mapsto t+\sqrt{2-t^2} on [0,2][0,\sqrt2] has derivative zero only at t=1t=1 (a maximum, value 22) and is smallest at the endpoints t=0,2t=0,\sqrt2, where it equals 2\sqrt2; hence t+2−t2≥2t+\sqrt{2-t^2}\ge\sqrt2 always. So regardless of Alice's choice x2i−1x_{2i-1}, the pair contributes x2i−1+x2i≥2x_{2i-1}+x_{2i}\ge\sqrt2 to the running total.