  Positional games have been introduced by Hales and Jewett in 1963 and have
been extensively investigated in the literature since then. These games are
played on a hypergraph where two players alternately select an unclaimed vertex
of it. In the Maker-Breaker convention, if Maker manages to fully take a
hyperedge, she wins, otherwise, Breaker is the winner. In the Maker-Maker
convention, the first player to take a hyperedge wins. In both cases, the game
stops as soon as Maker has taken a hyperedge. By definition, this family of
games does not handle scores and cannot represent games in which players want
to maximize a quantity.
  In this work, we introduce scoring positional games, that consist in playing
on a hypergraph until all the vertices are claimed, and by defining the score
as the number of hyperedges a player has fully taken. We focus here on
Incidence, a scoring positional game played on a 2-uniform hypergraph, i.e. an
undirected graph. In this game, two players alternately claim the vertices of a
graph and score the number of edges for which they own both end vertices. In
the Maker-Breaker version, Maker aims at maximizing the number of edges she
owns, while Breaker aims at minimizing it. In the Maker-Maker version, both
players try to take more edges than their opponent.
  We first give some general results on scoring positional games such that
their membership in Milnor's universe and some general bounds on the score. We
prove that, surprisingly, computing the score in the Maker-Breaker version of
Incidence is PSPACE-complete whereas in the Maker-Maker convention, the
relative score can be obtained in polynomial time. In addition, for the
Maker-Breaker convention, we give a formula for the score on paths by using
some equivalences due to Milnor's universe. This result implies that the score
on cycles can also be computed in polynomial time.
