The diameter game

A large class of Positional Games are defined on the complete graph on $n$ vertices. The players, Maker and Breaker, take the edges of the graph in turns, and Maker wins iff his subgraph has a given -- usually monotone -- property. Here we introduce the $d$-diameter game, which means that Maker wins iff the diameter of his subgraph is at most $d$. We investigate the biased version of the game; i.e., when the players may take more than one, and not necessarily the same number of edges, in a turn. Our main result is that we proved that the $2$-diameter game has the following surprising property: Breaker wins the game in which each player chooses one edge per turn, but Maker wins as long as he is permitted to choose $2$ edges in each turn whereas Breaker can choose as many as $(1/9)n^{1/8}/(\ln n)^{3/8}$. In addition, we investigate $d$-diameter games for $d\ge 3$. The diameter games are strongly related to the degree games. Thus, we also provide a generalization of the fair degree game for the biased case.