  We explore the cycle types of a class of biased random derangements,
described as a random game played by some children labeled $1,\cdots,n$.
Children join the game one by one, in a random order, and randomly form some
circles of size at least $2$, so that no child is left alone. The game gives
rise to the cyclic decomposition of a random derangement, inducing an
exchangeable random partition. The rate at which the circles are closed varies
in time, and at each time $t$, depends on the number of individuals who have
not played until t. A $\{0,1\}$-valued Markov chain $ X^n$ records the cycle
type of the corresponding random derangement in that any $1$ represents a
hand-grasping event that closes a circle. Using this, we study the cycle counts
and sizes of the random derangements and their asymptotic behavior. We
approximate the total variation distance between the reversed chain of $X^n$
and its weak limit $X^\infty$, as $n\to\infty$. We establish conditional (and
push-forward) relations between $X^n$ and a generalization of the Feller
coupling, given that no $11$-pattern ($1$-cycle) appears in the latter. We
extend these relations to $X^\infty$ and apply them to investigate some
asymptotic behaviors of $X^n$.
