Question $5$: Geography Game.

Introduction

Basic Geography Game: The Geography game is a simple word game

where players take turns naming a geographical location $($such as a city,

country, river, etc.$)$ that starts with the last letter of the previously mentioned location. The game continues until a player cannot come up with a

valid location or repeats a previously mentioned location.

The Geography game can be represented using a simple graph structure.

We define a graph where the vertices represent geographical locations. There

is a directed edge from vertex x to vertex y if and only if the first letter of y

is the same as the last letter of x$.$ During the game, two players alternately

choose vertices, creating a path in this graph. A player loses if they cannot

move to a new vertex $($i$.$e$.,$ the current vertex has no outgoing edges$)$ or if

they attempt to revisit a vertex that has already been visited.

This idea can be generalized and induce the following game played on

general directed graphs.

Generalized Geography Game $($GG$)$:

Generalized Geography Game $($GG$)$: The Generalized Geography

game is played on a directed graph G $=($V$,$ E$)$ starting from a designated

vertex v$0.$ Two players, Player $$ and Player $,$ take turns beginning with

Player $.$

On each turn, a player moves from the current vertex to an adjacent

vertex $($following the direction of an edge$)$ that has not been visited before.

The turn then passes to the other player.

A player loses if they cannot move to an unvisited vertex. A player wins

if they have a strategy to always force the other player to lose, regardless of

the other player$$s strategy.

E$.$g$.,$ when the game is played on a tree starting from the root, where

the tree has exactly two branches, one with an odd number of vertices and

the other with an even number of vertices, Player $$ can ensure a win by

moving first towards the branch with an even number of vertices.

The GG decision problem:

Consider the following Generalized Geography decision problem.

GG $=\{$G$,$ v$0$ : G $=($V$,$ E$)$ is a directed graph, and Player $$ wins from v$0$ in V $\}$Prove that GG in PSPACE $($In fact, GG is PSPACE $$ Complete$).$

Hint: Use a space efficient recursion as in the proof that TQBF in

PSPACE.