The other day I was playing a game called Trans Europa (or Trans America) which is quite graph theoretic in flavour. The game takes place on a triangular lattice graph with certain distinguished nodes (called cities). Each player is given five cities at random. The players then take it in turns to 'claim' an edge by laying a 'rail' on it (ie. by assigning an edge as being a distinguished edge). However, a player can only set down a rail by moving their player piece from one node to another, with the starting node being assigned at random.
If there is an unbroken path of rails connecting two of a player's cities, then the two cities are said to be connected, and the first person to create a railed path which connects all five of their cities wins.
The information is not perfect, since the other player does not know which cities the other player is trying to connect. Does this game already have a known name in the mathematics literature, or is there something similar? What would the optimal strategy be? The game is essentially who can be first to move along a path which connects five random nodes on a triangular lattice, with at least one other player attempting to do the same and being able to use the other player's path. An optional variation allows a player to lay a few rails which cannot be used by the other player.
Edit: I did a literature search and it looks as if might be possible to understand the game in the framework of this article of Erdös and Selfridge.
Best Answer
This is a non-answer that's growing a little large for a comment.
First, a mathematical statement of a slightly generalised version of the game. The board is a graph. Each player has a starting vertex and a set of goal vertices. At any stage there is a subgraph of marked edges. A legal move for a player is to mark any edge incident on the connected component containing their starting vertex. Players take it in turns to move, and a player wins as soon at their goal vertices are in one connected component of the marked subgraph. (This does mean that draws are possible.)
Trans Europa/America are played on a graph which is a subgraph of the triangular lattice. Goal vertices are assigned secretly and randomly from a smaller subset split into 5 colours, with each player receiving one of each colour (with no repeats). The starting vertex is chosen by each player in turn order. Players make two moves each round, but some edges are double-weighted so that they take up a whole turn, which sets up some situations in which you have to be careful about parity.
I don't think Erdős-Selfridge type positional games are the way to think about this problem, as that framework typically supposes that claimed objects belong to the claiming player. Most of the results I know about building structures in graphs are in that framework or the closely-related world of maker-breaker games. I'm pleasantly surprised by how much people take to Trans Europa and have wondered idly myself about the maths behind it to no particular end.