A noisy min-max game on trees

Abstract

We study a noisy version of a min-max type zero-sum game on the $d$-ary tree. Each edge of the tree is assigned an i.i.d.\ cookie, distributed uniformly on $\{+1,-1\}$. The game is played as follows: starting at the root, two players alternate turns in choosing a child to move to, with the game ending after each player took $n$ turns. Both players have full knowledge of the cookies on the whole tree. The cookies along the traversed edges are picked up and placed in a shared cookie jar. The first player's payoff is the sum of the cookies in the cookie jar, while the second player pays that sum. The value $V_n$ of the $n$-round game is the largest signed sum which can be guaranteed by the first player. We analyze the value $V_n$ and show that as $n \to \infty$, the value is tight for $d=2$, converges in distribution for $d \ge 3$, and converges almost surely for $d \ge 15$. Along the way, we prove various tightness and double exponential tail decay results. The analysis is a mix of percolation-type arguments for large $d$, and iterations on distributions combined with interval arithmetic for small $d$. For $d=2$ we prove the existence of a continuum of fixed points for this iteration, highlighting surprising qualitative differences with the case $d \ge 3$. The question of convergence for $d=2$ remains open.