  In this paper, we construct two hypergraphs which exhibit the following
properties. We first construct a hypergraph $G_{CP}$ and show that Breaker wins
the Maker-Breaker game on $G_{CP}$, but Chooser wins the Chooser-Picker game on
$G_{CP}$. This disproves an (informally stated) conjecture of Beck. Our second
construction relates to Beck's Neighbourhood Conjecture, which (in its weakest
form) states that there exists $c > 1$ such that Breaker wins the Maker-Breaker
game on any $n$-uniform hypergraph $G$ of maximum degree at most $c^n$. We
consider the case n=4 and construct a 4-graph $G_4$ with maximum vertex degree
3, such that Maker wins the Maker-Breaker game on $G_4$. This answers a
question of Leader.
