  The study of the number of collisions in a Poisson-Dirichlet coalescent leads
to the analysis of the following version of a stochastic leader-elec\-tion
algorithm. Consider an infinite family of persons, labeled by $1,2,3,\ldots$,
who generate iid random numbers from an arbitrary continuous distribution.
Those persons who have generated a record value, that is, a value larger than
the values of all previous persons, stay in the game, all others must leave.
The remaining persons are relabeled by $1,2,3,\ldots$ maintaining their order
in the first round, and the election procedure is repeated independently from
the past and indefinitely. We prove limit theorems for a number of relevant
functionals for this procedure, notably the number of rounds $T(M)$ until all
persons among $1,\ldots,M$, except the first one, have left (as $M\to\infty$).
For example, we show that the sequence $(T(M)-\log^{*}M)_{M\in\mathbb{N}}$,
where $\log^{*}$ denotes the iterated logarithm, is tight, and study its weak
subsequential limits. We further provide an appropriate and apparently new kind
of normalization (based on tetrations) such that the original labels of persons
who stay in the game until round $n$ converge (as $n\to\infty$) to some random
non-Poissonian point process and study its properties. The results are applied
to study subsequential distributional limits for the number of collisions in
the Poisson-Dirichlet coalescent.
