A general position set of a graph G is a set of vertices S in G such that no three vertices from S lie on a common shortest path. In this paper we introduce and study the general position achievement game. The game is played on a graph G by players A and B who alternatively pick vertices of G. A selection of a vertex is legal if has not been selected before and the set of vertices selected so far forms a general position set of G. The player who selects the last vertex wins the game. Playable vertices at each step of the game are described, and sufficient conditions for each of the players to win is given. The game is studied on Cartesian and lexicographic products. Among other results it is proved that A wins the game on Kn□Km if and only if both n and m are odd, and that B wins the game on G∘Kn if and only if either B wins on G or n is even.

The general position number gp( G) of a graph G is the cardinality of a largest set of vertices S such that no element of S lies on a geodesic between two other elements of S. The complementary prism G[Formula: see text] of G is the graph formed from the disjoint union of G and its complement [Formula: see text] by adding the edges of a perfect matching between them. It is proved that gp( G[Formula: see text]) ≤ n( G) + 1 if G is connected and gp( G[Formula: see text]) ≤ n( G) if G is disconnected. Graphs G for which gp( G[Formula: see text]) = n( G) + 1 holds, provided that both G and [Formula: see text] are connected, are characterized. A sharp lower bound on gp( G[Formula: see text]) is proved. If G is a connected bipartite graph or a split graph then gp( G[Formula: see text]) ∈ { n( G), n( G)+1}. Connected bipartite graphs and block graphs for which gp( G[Formula: see text]) = n( G) + 1 holds are characterized. A family of block graphs is constructed in which the gp-number of their complementary prisms is arbitrary smaller than their order.