Consider the following word game, which we'll call Geography. You have a set of names of places, like the capital cities of all the countries in the world. The first player begins the game by naming the capital city c of the country the players are in; the second player must then choose a city c′ that starts with the letter on which c ends; and the game continues in this way, with each player alternately choosing a city that starts with the letter on which the previous one ended. The player who loses is the first one who cannot choose a city that hasn't been named earlier in the game.
For example, a game played in Hungary would start with "Budapest," and then it could continue (for example), "Tokyo, Ottawa, Ankara, Amsterdam, Moscow, Washington, Nairobi."
This game is a good test of geographical knowledge, of course, but even with a list of the world's capitals sitting in front of you, it's also a major strategic challenge. Which word should you pick next, to try forcing your opponent into a situation where they'll be the one who's ultimately stuck without a move?
To highlight the strategic aspect, we define the following abstract version of the game, which we call Geography on a Graph. Here, we have a directed graph G = (V, E), and a designated start node s e V. Players alternate turns starting from s; each player must, if possible, follow an edge out of the current node to a node that hasn't been visited before. The player who loses is the first one who cannot move to a node that hasn't been visited earlier in the game. (There is a direct analogy to Geography, with nodes corresponding to words.) In other words, a player loses if the game is currently at node v, and for edges of the form (v, w), the node w has already been visited.
Prove that it is PSPACE-complete to decide whether the first player can force a win in Geography on a Graph.
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