  We study two types of two player, perfect information games with no chance
moves, played on the edge set of the binomial random graph ${\mathcal G}(n,p)$.
In each round of the $(1 : q)$ Waiter-Client Hamiltonicity game, the first
player, called Waiter, offers the second player, called Client, $q+1$ edges of
${\mathcal G}(n,p)$ which have not been offered previously. Client then chooses
one of these edges, which he claims, and the remaining $q$ edges go back to
Waiter. Waiter wins this game if by the time every edge of ${\mathcal G}(n,p)$
has been claimed by some player, the graph consisting of Client's edges is
Hamiltonian; otherwise Client is the winner. Client-Waiter games are defined
analogously, the main difference being that Client wins the game if his graph
is Hamiltonian and Waiter wins otherwise. In this paper we determine a sharp
threshold for both games. Namely, for every fixed positive integer $q$, we
prove that the smallest edge probability $p$ for which a.a.s. Waiter has a
winning strategy for the $(1 : q)$ Waiter-Client Hamiltonicity game is $(1 +
o(1)) \log n/n$, and the smallest $p$ for which a.a.s. Client has a winning
strategy for the $(1 : q)$ Client-Waiter Hamiltonicity game is $(q + 1 + o(1))
\log n/n$.
