The tree search game for two players

TL;DR

This paper analyzes a two-player game on trees where players alternately guess vertices to find a hidden target, revealing subtrees after each guess, and determines winning probabilities under various strategies.

Contribution

It introduces a new two-player search game on trees and characterizes winning probabilities under optimal and random strategies.

Findings

Optimal play yields approximately 50% win probability for each player.

Mixed strategies result in win probabilities between 9/16 and 2/3.

Random play leads to win probabilities between 13/30 and 17/30.

Abstract

We consider a two-player search game on a tree $T$. One vertex (unknown to the players) is randomly selected as the target. The players alternately guess vertices. If a guess $v$ is not the target, then both players are informed in which subtree of $T \smallsetminus v$ the target lies. The winner is the player who guesses the target. When both players play optimally, we show that each of them wins with probability approximately $1/2$. When one player plays optimally and the other plays randomly, we show that the player with the optimal strategy wins with probability between $9/16$ and $2/3$ (asymptotically). When both players play randomly, we show that each wins with probability between $13/30$ and $17/30$ (asymptotically).