The median game

We introduce a game which is played by two players on a connected graph G . The players I and I I alternatively choose vertices of the graph until all vertices are taken. The set of vertices chosen by player I is denoted by ? I , and by I I is denoted by ? I I . Let d ( x , π ) = ? y ? π d ( x , y ) and let M ( π ) = min { d ( x , π ) ? x ? π } be the median value of a profile π ? V ( G ) . The objective of player I is to maximize M ( π I I ) - M ( π I ) and the objective of player I I is to minimize M ( π I I ) - M ( π I ) . The winner of the game is the player with the smaller median value of her profile. We give a necessary condition for a tree so that player I (who begins the game) has a winning strategy for the game. We prove also that for hypercubes and some other symmetric graphs the player I I has a strategy to draw the game. Complete bipartite graphs are considered as well.