  A graph $G = (V,E)$ is said to be saturated with respect to a monotone
increasing graph property ${\mathcal P}$, if $G \notin {\mathcal P}$ but $G
\cup \{e\} \in {\mathcal P}$ for every $e \in \binom{V}{2} \setminus E$. The
saturation game $(n, {\mathcal P})$ is played as follows. Two players, called
Mini and Max, progressively build a graph $G \subseteq K_n$, which does not
satisfy ${\mathcal P}$. Starting with the empty graph on $n$ vertices, the two
players take turns adding edges $e \in \binom{V(K_n)}{2} \setminus E(G)$, for
which $G \cup \{e\} \notin {\mathcal P}$, until no such edge exists (i.e. until
$G$ becomes ${\mathcal P}$-saturated), at which point the game is over. Max's
goal is to maximize the length of the game, whereas Mini aims to minimize it.
The score of the game, denoted by $s(n, {\mathcal P})$, is the number of edges
in $G$ at the end of the game, assuming both players follow their optimal
strategies. We prove lower and upper bounds on the score of games in which the
property the players need to avoid is being $k$-connected, having chromatic
number at least $k$, and admitting a matching of a given size. In doing so we
demonstrate that the score of certain games can be as large as the Tur\'an
number or as low as the saturation number of the respective graph property. We
also demonstrate that the score might strongly depend on the identity of the
first player to move.
